Underweysung der Messung

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Albrecht Dürer

De Symmetria Partium in Rectis Formis Humanorum Corporum Nuremberg, 1532

Underweysung der Messung Nuremberg, 1538

Y The Warnock Library

O

Table of Contents Commentary by David Price Binding and Collation Provenance *** English Translations De Symmetria Partium in Rectis Formis Humanorum Corporum Underweysung der Messung

*** Book Images De Symmetria / Underweysung der Messung

*** Octavo About This Edition

©2003 Octavo. All rights reserved. Click here to see End User License Agreement.

albrecht dürer De Symmetria Partium in Rectis Formis Humanorum Corporum / Underweysung der Messung nuremberg, 1532 and 1538

Dürer’s De Symmetria and Underweysung der Messung David Price

Albrecht Dürer (1471–1528), German painter, engraver, and draftsman, was the most celebrated artist of the Northern Renaissance.

His reputation spread throughout Europe during his lifetime,

beyond his native city of Nuremberg, a thriving center of art and commerce. Like Leonardo da Vinci, Dürer became deeply involved

in scientific and mathematical studies. Dürer’s application of scientific principles to the creation of art, especially as recorded in his Underweysung der Messung (Instruction in measurement), marks

the beginning of art theory in Northern Europe.

This Octavo Edition presents a sammelband (or volume in

which multiple works are bound together) of the first two books of

Dürer’s Von menschlicher Proportion (Four books of human proportions) in the Latin translation of 1532 (De Symmetria), along with

the revised and expanded 1538 second edition of Underweysung der Messung. Other copies of this assemblage are recorded. The arms

of Rudolf II (1552–1612), Emperor of the Holy Roman Empire, and his first-cousin Charles of Burgau, Archduke of Austria, are tooled in

gold on the vellum binding of this copy. Rudolf II was a passionate

collector of Dürer’s art; among the masterpieces he had acquired were Virgin of the Rose Garlands (1506), Adam and Eve (two panels, 1507), and Martyrdom of the Ten Thousand (1508) as well as the famous watercolor Hare (1502).

Line-drawing instrument, from Underweysung der Messung (view 103). 1

earnest on the preparation of books for artists. Manuscript frag-

ments survive from as early as 1513 of a work entitled Speiss für Malerknaben (Nourishment for apprentice painters). In one frag-

ment, he claimed his intention to replace ancient Greek treatises that had been irretrievably lost.

underweysung der messung Underweysung der Messung (1525; revised and expanded, 1538)

was the first scientific book Dürer published. It presents a wide range of geometric subjects, as indicated by the original German

title: Underweysung der Messung, mit dem Zirckel und Richtscheyt,

in Linien, Ebnen und gantzen Corporen (Instruction in measure-

ment, with a compass and rule, in lines, planes, and solids). The basics of linear, plane, and solid geometry often lay a foundation for descriptions of practical applications for architecture and art,

including construction of columns, design of alphabets, and even

a model for a bishop’s crosier. The book’s most notable achieve-

ment, a topic that found crucial enlargement in the 1538 second edition, is the analysis of perspective.

That edition, reproduced here, is the first complete printing of

Dürer’s final version of the study. To the first edition of 1525, it adds several corrections, lengthy expansions, and twenty-two new illustrations designed by Dürer. The second edition, for instance,


After his second trip to Italy (1505–07), Dürer began working in

Artist drawing a nude, from Underweysung der Messung (view 181).

ironic, undercutting the instructional nature of the handbook, if not the attitude toward art as technique in general—and it may have looked that way to Renaissance eyes as well.

Like many posthumously printed books, the 1538 edition has a

dash of mystery. A complete manuscript for the augmentations

does not survive, although drafts for a few passages are extant. It is often thought that the artist entered much of the new material

onto an exemplar of the first edition. As indicated by the colophon, we owe the publication of this book to the artist’s widow, Agnes

Frey Dürer (1475–1539), who served as publisher of his posthumous

works in the original German as well as in the important Latin

translations. It is likely that Thomas Venatorius (ca. 1488–1551) served as editor. An acquaintance of the artist who memorialized him in several epicedial poems, Venatorius was one of the

preachers instrumental in the propagation of the Reformation

in Nuremberg. In addition to poetry and theology, he was well

versed in mathematics and had the distinction of editing the editio princeps of Archimedes (Basel, 1544). The presswork for Dürer’s

offers an improved calculation of π as 3B/h. Also among the new

books was done by Hieronymus Formschneider (d. 1556), called

material is the well-known woodcut of an artist making a picture

Andreae in Latin, who almost certainly also transferred Dürer’s

of a reclining nude with drastic foreshortening in order to demon-

designs to blocks and cut them for the illustrations. The German

strate the operation of a perspective-grid. To modern eyes, this fig-

name Formschneider means “one who cuts woodblocks.”

ure of an artist mechanically drawing a voluptuous nude seems

2

ning of scientific prose in German, even though Dürer drew on several native sources for the work. Hans Rupprich, the editor of the manuscripts of Dürer’s writings, cited several works that

were known to Dürer, including Matthäus Roriczer’s architectural study Püchlen der fialen gerechtigkeit (Regensburg, 1486) and the anonymous Geometria deutsch (Regensburg, ca. 1500), the latter a source for a few of Dürer’s figures. Dürer certainly profited from

the research and publications of Johannes Werner, a Nuremberg mathematician, in particular from his study of cones, Super vigen-

tiduobus elementis conicis (Nuremberg, 1522). He was also familiar

with a wealth of material from classical sources, the most important of which was by Euclid, whose works he owned in the Latin

translation of Bartolommeo Zamberti (Venice, 1505). There are

also traces of other classical influences, including the works of Ptolemy, Archimedes, and Eutocius.

The pioneering work of the Italian humanists Leon Baptista

Alberti, Leonardo da Vinci, Piero della Francesca, and Luca Pacioli also had a profound influence on Dürer. The precise nature of

influence in each of these cases is difficult to ascertain. With the

exception of Pacioli, the published works of these authors postdate Dürer’s book. Piero della Francesca’s De prospectiva pigendi,

for example, was not published until 1899, although it was written between 1470 and 1490. Dürer demonstrably knew that work in manuscript since a passage from it, translated into German,


Underweysung der Messung has been heralded as the begin-

secret art of perspective.” We do not know who his instructor was, although it is generally thought to have been Pacioli.

organization of the underweysung Dürer organized the material of Underweysung der Messung in four books. Book One, which begins with Euclid’s definitions of point,

line, surface, and solid, concerns linear geometry. Discussions focus on the properties of curves and spirals, with extensive analysis

of lines derived from conic sections (ellipse, parabola, and hyperbola). This book is the context for Dürer’s infamous blunder, the assertion—not corrected in the second edition—that an ellipse is

egg-shaped (he called the ellipse an “Eierlinie”), i.e., that it is symmetrical only along one axis. This is a bafflingly elementary error

for someone who thought so deeply and extensively about geometry. Dürer describes triangles, rectangles, and regular polygons in Book Two. This segment contains such well-known principles as

survives among his papers. It is, furthermore, obvious that Dürer

learned much from personal instructions of people versed in the

ideas of the masters. Dürer wrote to Pirckheimer on October 13, 1506, saying that he planned to journey to Bologna to learn “the

Three basic types of line, from Underweysung der Messung (view 84).

3

the age-old problem of squaring a circle (the context of a poor

calculation of π as 3B/i), although Dürer is quite aware that his techniques yield mere approximation. Several of the polygons are quite complex, and it is in this context that the reader is treated

to a richly illustrated excursus on intricate geometric patterns for tiles.

Book Three, which begins the treatment of solids, is devoted

predominantly to practical applications. It takes up an account of

cones and cylinders, but devolves into a fascinating dissertation

on architectural styles of columns (along with their bases and capitals) and techniques for constructing different types of sun-

dials. To the lengthy description of columns, Dürer appended a

discussion of three commemorative columns: the first for a military victory over a powerful adversary, the second for the victory over the insurgents in the great Peasants’ War of 1525, and the

third for a deceased drunk. The plans appear to be capricious, and

Dürer may have been influenced in this jocularity by a fantastic design by Leonardo da Vinci.

The book ends with the famous analysis of lettering, one of the

earliest attempts to rationalize principles of typographic design. Dürer also describes a technique for optically sizing inscriptions

on the face of tall monuments in order to make the words at different heights appear the same size to a viewer on the ground. His example is Martin Luther’s translation of the phrase from 1 Peter 1:25


the Pythagorean theorem (not so-named by Dürer) and also treats

“Das Wort Gottes bleibt ewig” (The Word of God will last forever).

Detail of Dürer’s study of letterforms (view 148).

Colloquy, the city council officially instituted Lutheran doctrine

for churches in its territory, claiming that the Bible provided legitimacy for its seizure of the Catholic Church’s property. Without

mentioning this momentous event, Dürer’s monument implicitly endorses it.

Book Four contains analysis of the properties of polyhedra, and

it concludes with Dürer’s description of perspective. It explains

the properties of five regular polyhedra as well as seven Archimedean solids; the 1538 edition adds descriptions of a truncated icosahedron and an icosidodecahedron. The five so-called Platonic

solids (based on a passage in Timaeus) were significant among Renaissance Neoplatonists, who considered them representative of the elements of the universe (the cosmos, earth, fire, water,

Like the column commemorating the Peasants’ War, this paper

and air are represented by the dodecahedron, cube, pyramid,

monument has tremendous cultural relevance. This Bible phrase

icosahedron, and octahedron, respectively). Dürer also explains

had become a slogan of the Lutheran evangelicals. And, in March

the flattening of polyhedra into two-dimensional schemes (often

1525, in the immediate aftermath of the Nuremberg Religious

called “nets”) as well as the approximate flattening of a sphere 4

segments. Furthermore,

he offers three solutions to the “Delic Problem”

of doubling the size of a cube.

Of note in his analy-

sis of perspective is the

inclusion of a system for making an exact drawing of the intersection

of the visual pyramid at the picture plane of

an object. Dürer also

proposes a less demand-

ing system, which he calls “der nähere Weg”

(the shorter way), for Scheme for doubling a cube (view 162).

determining the correct location of transversals,

although his scheme has defects (see Kemp 1989, 59). It is possible that he derived this technique from the writings of Jean Péllerin,

as Walter L. Strauss argues (Dürer 1977, 28–31), although the system is similar to Alberti’s method as well. Dürer was always concerned with practical applications despite the demanding nature


divided into meridian

of his methods. Even in the context of defining geometric perspec-

de symmetria

The other book bound in this volume is the first edition, lavishly

illustrated with the original woodcuts, of the first two books of

Von menschlicher Proportion (Four books of human proportions)

in the Latin translation De Symmetria by Joachim Camerarius (1500–1574). This imprint appeared in 1532 (as a separately printed

book) and was followed in 1534 by the translation of Books Three

and Four. Camerarius was one of the most prolific humanist schol-

ars of his generation. He served as rector of the new gymnasium in Nuremberg, where he associated with Dürer, before becoming professor at Tübingen and, then, rector at Leipzig. A modern reader

may find it difficult to see how Dürer’s decision to publish his books in German was a problem for Renaissance humanists. Yet

even Erasmus qualifies his praise of Underweysung der Messung in his De recta pronunciatione with the observation that the otherwise excellent book was written in the vernacular. Camerarius’

Latin translation, the publication of which used the original 85 woodblocks, was the foundation of the international reception of

Dürer’s writings. All told, Dürer’s Von menschlicher Proportion appeared in German (1528 and 1603), Latin (1532/34, 1537, 1557), French

(1557 and 1614), Italian (1591 and 1594), Spanish (ca. 1599), and Dutch (1622 and 1662).

As the title indicates, De Symmetria … humanorum corporum is

an extensive study of the human body. Book One has five types

(for both male and female) based on the system of Vitruvius, who described lengths of units as fractions of the whole. For example,

tival techniques he pays particular attention to mechanical aids

in the Vitruvian canon, a normal body has a head B/h of the total

for drawing, including four devices for achieving perspectival

height. Book Two adds eight new types, suggesting Dürer’s doubt

projection on the picture plane. His grid system and glass-pane

that there is a finite number of types for human proportions. In-

apparatus have antecedents in da Vinci.

stead of fractions, the second book uses units of measure based 5


on Alberti’s system, but with refinements (one of which is that Dürer’s ruler has 600 unit divisions).

Dürer’s books, noble works of art themselves, enjoyed wide

distribution in the Renaissance. They were familiar to subsequent

artists, scientists, and art theorists. Johannes Kepler, the great astronomer at the court of Rudolf II, knew Dürer’s Underweysung der Messung; he noted the faulty ellipse in Book One, but otherwise

made use of Dürer’s work in his presentation of Archimedean

solids. It is, however, questionable if Dürer’s studies were of much

practical use to painters. In fact, it has become a humorous commonplace in Dürer scholarship to cite the reaction of an unnamed

painter at the convent of Bergen. Eufemia Pirckheimer wrote to her brother Willibald (to whom the book was dedicated) that

the sisters at Bergen had enjoyed Dürer’s book, even though the convent’s painter claimed that “she doesn’t need it because her art is successful without it” (Dürer 1956–66, 1:278).

David Price has written extensively on the culture of the Renaissance and Reformation, including books and articles on poetry, drama, the Bible, and the visual arts, especially on the writings and art of Albrecht Dürer. He holds a Ph.D. from Yale University

and is presently Associate Professor of History and Associate Professor of Church History at Southern Methodist University. He is

the author of Albrecht Dürer’s Renaissance: Humanism, Reformation and the Art of Faith (Ann Arbor: University of Michigan Press,

2003).

Female figure from De Symmetria (view 59). 6

Dürer, Albrecht. Dürer: Schriftlicher Nachlaß. Edited by Hans Rupp-

rich. 3 volumes. Berlin: Deutscher Verein für Kunstwissenschaft, 1956–66.

——. Dürer’s Record of Journeys to Venice and the Low Countries. Translated by Rudolph Tombo and edited by Roger Fry. New York: Dover, 1995.

——. The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler. Translated and with

a Commentary by Walter L. Strauss. New York: Abaris, 1977.

Gombrich, Ernst. “The Evidence of Images.” In Interpretation:

Theory and Practice, edited by Charles S. Singleton, 35–104.

Baltimore: Johns Hopkins University Press, 1969.

Hutchison, Jane Campbell. Albrecht Dürer. Princeton: Princeton University Press, 1990.

Kaufmann, Thomas DaCosta. The Mastery of Nature: Aspects of Art,

Science and Humanism in the Renaissance. Princeton: Prince-

ton University Press, 1993.

——. “Rudolf II.” In The Dictionary of Art 13: 912–15. Edited by Jane Turner. New York: Grove, 1996.

Kemp, Martin. The Science of Art. New Haven: Yale University Press, 1989.

Panofsky, Erwin. “Albrecht Dürer and Classical Antiquity.” In

Erwin Panofsky. Meaning in the Visual Arts, 236–85. Chicago:


selected bibliography

Schröder, Eberhard. Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner “Underweysung.” Basel:

Birkhäuser, 1980.

Strieder, Peter. Albrecht Dürer. Translated by Nancy M. Gordon and Walter L. Strauss. New York: Abaris, 1982.

——. “Albrecht Dürer.” In The Dictionary of Art 9: 427–45. Edited by Jane Turner. New York: Grove, 1996.

Detail from view 177 of Underweysung der Messung, showing one method for drawing objects in perspective.

University of Chicago Press, 1955.

——. The Life and Art of Albrecht Dürer. Princeton: Princeton University Press, 1955.

Price, David. Albrecht Dürer’s Renaissance: Humanism, Reformation

and the Art of Faith. Ann Arbor: University of Michigan Press,

2003.

7

De Symmetria partium in rectis formis humanorum corporum, Libri in latinum conversi…. Nuremberg, [H. Andreae] in aedibus viduae

Durerianae, 1532. Underweysung der Messung, mit dem Zirckel und richtscheyt, in Linien Ebnen und gantzen Corporen…. Nuremberg,

durch Hieronymum Formschneyder, 1538.

binding These copies are bound together as a sammelband in

a Bohemian gilt-stamped limp vellum binding of about 1600, covers with an outer border of interlaced fleurs-de-lys, front cover

with arms of Emperor Rudolf II, rear cover with arms of Rudolf

II’s cousin Charles, Margrave of Burgau and Archduke of Austria, each centered in a panel with top and bottom sections containing

two quatrefoils and a central composite design of two rosettes flanked by reposing fleurons, the whole enclosed by corkscrew

borders and with large fleurons emerging from the corners, all gilt, flat gilt-tooled spine divided into six compartments, five with

a quatrefoil, top compartment with title in sepia: ‘symmetria /

geometria’, head of spine with Dürer’s monogram in black in a large unknown hand, all edges gilt, simply gauffered.

The bookplate of the archaeologist and numismatist Jean-

Baptiste Seroux d’Agincourt (1730–1814) is on the front pastedown

and the last leaf has a small eighteenth-century red seal bearing a coat-of-arms attributed to the Agges family of Amsterdam.


Binding and Collation

O4 blank], 173 unnumbered leaves (two with foldout extensions as issued), woodcut illustrations in text include 85 full-length

anatomical figures, mostly full-page pairs; 146 mostly numbered

geometric diagrams or groups of diagrams (two folding, two with small extensions pasted on as issued), numerous letterforms, eight full-page alphabets, and four depictions of artists using perspective machines.

contents A1 a title to De Symmetria; A1b dedication to Emperor

Charles V; A2a –A4a note to the reader; A4b –A5b Joachim Camerarius’ dedication of the Latin translation to Dürer’s friend Christophorus Colerus together with a discussion of Dürer’s illustrations and a

list of anatomical parts contained in them; A6a/b two laudatory poems on Dürer by Eobanus Hessus; B1 a –G3b Book I; G4a –O3b Book

measurements The cover measures 12 * 8D/i inches (305 * 212 mm)

II, ending with colophon; O4a/b blank; 2A1 a title to Underweysung

and the textblock 11H/i * 7H/i inches (301 * 200 mm).

der Messung; 2A1b dedication to Willibald Pirckheimer; 2A2a – 2D6b

Book I; 2E1 a– 2G1b Book II; 2G2a – 2M4a Book III; 2M4b – 2Q4a Book IV,

collation 2o : A–E6 F 4 G–N6 O4; 2A–O6 P 4 (2P3 + ‘P3’) Q4 [$4 (—F4, O3,

ending with colophon; 2Q4b blank.

O4, 2H4, 2K4, 2P3, 2P4, 2Q4) signed; E4, F4, G4 incompletely signed ‘4’;

8

The two works by Albrecht Dürer reproduced here were bound together in limp vellum some fifty years after publication. Books that are so bound are commonly described using the German

term sammelband—the plural is sammelbände. There are many reasons for binding things together. Several dozen papal bulls, each of only a few pages, might well be preserved for reference by assembly in a sammelband, as might two related texts by different authors—for instance, an edition of Sophocles and a Sophoclean lexicon. In the case of this Dürer sammelband, the four

books of the second (but first complete) edition of Dürer’s book on

solids have been bound with the first Latin translation of the first

two books of Von menschlicher Proportion (1528)—this Latin edi-

tion was the form in which the work achieved its European and not simply its German reputation. The last two books, published

in Latin in 1534, are not present, but this is a common state of affairs and would not have disturbed the sixteenth-century mind.

Camerarius, after all, might have issued his Latin translation of all four books together, just as they had been published together

in German. The separate books have their own significance and integrity when considered (as intended) as a practical manual for the artist, for they are clearly addressed to different audiences.

Dürer on solids (the Underweysung der Messung) might well be all


Provenance

The insignia of Rudolf II (left) and Charles of Burgau (right) from the binding of this copy of De Symmetria/Underweysung der Messung.

combination plainly reflecting a particular ecological niche of the artistic market.

As the gilt arms testify, the binding is associated with the Holy

Roman Emperor Rudolf II (1552–1612) and his first cousin Karl, Margrave of Burgau and Archduke of Austria (1560–1618). The front cover bears the Emperor’s large gilt arms, the back cover those of

the Archduke. Both were great-great-grandsons of the Emperor Maximilian I (1459–1519), himself a patron of Dürer. The precise signification of these two coats of arms, however, is a more vexing question. The volume might easily have been a gift from one to the

other, in a presentation binding. Alternatively, since Rudolf even-

that a sixteenth-century still-life painter required as a handbook.

tually bought the collections at Schloss Ambras that his cousin had

a portrait painter’s needs. And Books Three and Four would be

from Karl von Burgau’s library that Rudolf had had rebound, add-

copies are recorded that are similarly constituted, the distinctive

ous provenance. No similar bindings, however, have been located.

Books One and Two of Von menschlicher Proportion would satisfy

largely inherited from his father Ferdinand II, it might be a book

consulted only by a caricaturist or a history painter. Many other

ing the two coats of arms to indicate current ownership and previ-

9

four years before his birth, but he lived in the afterglow of his fame, and served as patron of several generations of German and

Austrian artists for whom Dürer was a continuing, almost overwhelming inspiration. Rudolf was the great royal collector of his

age, with purchasing agents throughout Europe. He owned works by many disciples and imitators of Dürer, including the goldsmith

Paulus van Vianen, the painter Hans Hoffmann, and the miniaturist Daniel Fröschl. His Imperial engraver Aegidius Sadeler was

yet another artist inspired by Dürer. Hoffmann (ca. 1530–ca. 1591)

was hired by Rudolf in 1585 both as artist and connoisseur, to serve

as copyist, expert valuer, and detector of forgeries. His expertise proved invaluable when Rudolf was offered the Imhoff collection of Dürer’s works in 1588. Other paintings and drawings had been

acquired from the Granvell collections, and from Nuremberg,

Augsburg, and Venice. Rudolf collected Dürer’s landscapes, as well as botanical and zoological studies: the familiar Hare now

at Vienna was once a part of Rudolf’s museum. Dürer’s Virgin of the Rose Garlands, however, remains in Prague. It was one of the

few religious paintings in Rudolf’s collection: the Emperor’s tastes generally ran to the mythological and the erotic.

The binding of this sammelband is almost certainly Bohemian,

representative of an efflorescence of the art inspired by Imperial patronage, for in 1583 Rudolf had moved his court to Prague from

Vienna, which was uncomfortably near the Turks. After his death


Rudolf II was too young to have known Dürer, who died twenty-

in 1612, the Habsburg court returned to Vienna. This brief interval

sixteenth century, Prague acted as a magnet to artists, philosophers, and scientists, on the order of Queen Christina’s Stockholm,

Periclean Athens, or the Florence of the Medici. In addition to the disciples of Dürer, Rudolf patronized such artists as Arcimboldo, Bartholomäus Spranger, Adriaan De Vries, and Joris Hoefnagel. Rudolf’s esoteric, alchemical, and astronomical tastes brought Johannes Kepler, Tycho Brahe, and John Dee to his court.

The fame of such men has made the name of Rudolf live in

memory as the creator of a Golden Age in Central Europe long after

other more worthy monarchs have been forgotten. To misapply an

adage, life is short but art is long. Rudolf was an intolerant, incompetent, moody, eccentric, depressive, mentally unstable, reclusive

collector who neglected the chief duty of a monarch: to be fruitful and multiply. His Habsburg relations eventually put his empire into receivership. Rudolf was compelled to surrender portions of his territory to his younger brother, the Archduke Matthias.

The relics of this inspired collector and patron are now found

scattered around the world. The collections were immediately

redistributed and reduced on the return of the court to Vienna after Rudolf’s death 1612. Between 1619 and 1635, further items were

dispersed by sale or gift of the Bohemian Estates. Much of what remained in Prague was then carried off as war booty by the Swedish armies in 1648, at the close of the Thirty Years War. It passed

into the collections of Queen Christina, who had acquired the title

of the Hyperborean Minerva by assembling at her court many of the leading scholars of France and Holland. These included René

is known in the history of the city as the Rudolfine era. It is the

Descartes, who advised her on philosophical books, Nicolaus Hein-

only period in Czech history when the country could claim to

sius, her Latin adviser, the French librarian Gabriel Naudé, Pierre-

be the site of a “New Athens” or a “Third Rome,” boasting an as-

Daniel Huet, Samuel Bochart, consultant on oriental languages,

semblage of “all the talents.” For a few decades at the end of the

Ismaël Boulliau, and Claude Saumaise, advisor on philology. 10

ian and authority on Greek texts for most of her reign. He bought

collections for her abroad, including the magnificent humanist library that he had inherited from his father Gerardus Joannes

Vossius in 1649. He had his

own books sent to Stockholm too, but in his absence

and in the confusion of excessive acquisition, they

were inextricably mingled

with the Queen’s collection. When she abdicated in 1654 at the age of 27, Vossius had

ten large cases of books, containing a selectissima

bibliotheca, sent to Amster-

dam in compensation. He and several of his assistants The Hyperborean Minerva: Christina of Sweden, bronze bust by Bernini.

were later paid their long arrears in salary in natura,

i.e., in books of their choice. Vossius was pleased with the selection

he had been allowed to make, referring to it as spolia optima, or “rich booty.”

On her slow progress to Rome, Christina unpacked her books in

Antwerp, where Vossius catalogued them and was able to remove


The Dutch scholar Isaac Vossius (1618–89) served as her librar-

retaining chiefly the classical literature (of which he published numerous editions) and selling the rest at auction in 2,452 lots at

Leiden in 1656, in the guise of a sale of his father’s books: Catalogus

variorum & exquisitissimorum librorum Gerardi Ioannis Vossii. The

books were in fact largely from Stockholm, and included several illustrated with woodcuts by Dürer, as well as portions of the libraries of Hugo Grotius and Rudolf II.

The library that Vossius kept eventually became one of the

most famous in Europe and was sold en bloc to the University of Leiden after his death. These facts suggest a plausible itinerary for

the Dürer sammelband: Prague—Stockholm—the Netherlands. It

was certainly in the Netherlands by the early eighteenth century,

for the last leaf is adorned with a small seal, with Fortune on the coat of arms, apparently that of the Agges family of Amsterdam. There was in fact a copy of De Symmetria in the 1656 Vossius sale

catalogue (page 43, lot 21 of the “Libri mathematici”) but problems of identification remain. An auction catalogue was originally (and largely remained until well into the twentieth century) a

mere auxiliary to the sale, a menu rather than the recipe book that

it has become. Inspection of the goods by prospective purchasers (or their agents) was essential. So Vossius’ copy is summarily identified by title and date, without mention of condition, edition, binding, or provenance:

Albertus Durerus de Symmetria partium in rectis formis humanorum corporum. Norinbergae. 1543.

some of the many duplicates for himself. He also received books

Unfortunately, 1543 is almost certainly a misprint: no edition of

out these various acquisitions, which included a remarkable series

mon in book catalogues; errors in transcription of place names,

from her library when it was in Brussels and in Paris. After sorting

that date is now known to exist. Errors in dating are not uncom-

of alchemical manuscripts from Rudolf II, the Codices chymici now

however, are rare. Although a couple of sixteenth-century Pa-

at the University of Leiden, Vossius reconstituted his own library,

risian editions are recorded, only one Nuremberg edition of De 11

sius catalogue entry does indeed refer to our edition, if not necessarily to our copy. The fact that the Underweysung der Messung is not mentioned is less troublesome: a hurried cataloguer might

easily transcribe only the first title page and fail to note that a sec-

ond book was bound in at the back—this is the sort of mistake that is still common in the book trade.

The evidence that this copy was once owned by Vossius is thus

inconclusive. It may equally well have reached the Netherlands as

part of the “spoils” of one of his assistants—or it may never have been in Stockholm to begin with. The sceptic, after all, will rightly

note that no crow would dream of taking the polar route to get from Bohemia to the Netherlands. The believer might reply that great religions have been founded on considerably less evidence. Let us leave it as an intriguing possibility.

By the next century, at the latest, the book had continued on

its journey, travelling this time to France. The front pastedown

bears the later amateur bookplate of the French art historian Jean-Baptiste

Louis

Georges

Seroux d’Agincourt (1730–1814), author of the six-volume Histoire

de l’art par les monumens, depuis sa décadence au IVe siècle jusqu’à


Symmetria is known—1532—so it seems very likely that the Vos-

son renouvellement au XVIe (Paris, Bookplate of Jean-Baptiste Seroux d’Agincourt from front pastedown of this sammelband (view 2).

1810–23). In this book he describes

Dürer as “a great master” who is “rightly regarded as the founder

12

of the German school” and as a “lofty genius capable of perceiv-

ing Nature’s highest beauties and of conveying them with a truth worthy of their subject.”

The Editors

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english translation

Albrecht Dürer

De Symmetria Partium in Rectis Formis Humanorum Corporum Nuremberg, 1532 Translation by Silvio Levy for Octavo.

Using This Octavo Translation note Octavo Editions are imaged as uncropped page spreads, as if the actual open book were being viewed. Each spread view is sequentially numbered in the Acrobat pdf file. This translation contains divisions corresponding to the page breaks of the original text; in most instances, each page of the translation file corresponds to one page of the original book. All illustrations are included in the translation file at actual size. Each page of the translation has the Acrobat view number and the original page number indicated in the upper right corner, thus: view 3: page A1r

“ view 3 ” refers to the Acrobat image number; “page A1r” refers to the specific page number in the book itself, with r denoting a recto or righthand page (lefthand pages are denoted v, for verso). Recto and verso designations are necessary in the case of the present work, since it was paginated by leaf rather than by individual page (i.e., leaf numbers appear on the rectos only, and are in the form of paired letters and numerals 1–6). This translation begins with the title page and continues consecutively until the end of the text. Illustrations are included and caption material within them is translated, but blank pages are not indicated in the pagination references. Further information about this book’s structure and condition can be found in the Binding and Collation statement.

© 2003 Octavo. All rights reserved.

Translation Note The Books on the normal proportions of the parts of the human form were written by Dürer in German, and were later translated into Latin so that they could reach a wide readership. The translator, Joachim Camerarius, brought to the task both earnestness and the necessary competence, but his style is far less direct than Dürer’s: this can be seen especially in Camerarius’ address “To the Fair-Minded Reader” (views 4–6), but also in the translation of the text itself. On several occasions he departs from the original. In this edition we have not detailed such textual variations from the German, except where they render the Latin text obscure or contradictory, in which cases we have restored the intended meaning by reference to the German. Significant corrections and editorial comments appear in brackets and the errata given at the end of the book have been incorporated where appropriate. Almost any Latin text of the period was apt to contain a few Greek words. Greek was known to many scholars, though not as widely as Latin. In the case of our book, the very title incorporates the word συµµετρία, meaning not what we understand by symmetry, but the notion of combined or relative measurements, likewise implied by the etymology of the term. This word occurs in Greek characters in the title of the Preface dedicated to Christopher Coler (view 7), and in Latin characters elsewhere. Later in the Preface, Camerarius mentions that this Greek word is said to have no real equivalent in Latin, but proceeds to give two good ones himself (commensus, commensuratio). Certainly a reader of the title, where Symmetria occurs in Latin characters, would have been expected to understand its meaning. Other Greek words are interspersed in the front matter; they were only marked in our translation when their use in Latin seemed justified, including all occurrences in the list of parts of the body (views 7–8). This list calls for special comment. In Hellenistic times, Herophilus, Erasistratus, and their respective schools had named and described a great many anatomical features of the human body, down to recondite ones such as the fallopian tubes, the cardiac valves, and the torcular Herophili (the confluence of the four cranial venous sinuses). Most of these features, at least the internal ones, would not be known and discussed again until the seventeenth and eighteenth centuries. There were no “scientific” names for the parts of the body in Dürer’s time, even in Latin. For a work of detailed measurements such as Dürer’s, precision is desirable, and Camerarius supplies the Latin equivalents in his list with many circumlocutions and some euphemisms (such as natura), generally offering more than one alternative. He also humbly remarks that these choices may not necessarily be optimal. In fact he does not adhere to them with perfect consistency, and Dürer’s labeled drawings are therefore an invaluable aid to understanding.

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Books on the normal proportions of the parts of the human form, by Albrecht Dürer, famous painter and geometer, translated into Latin. [1532]

To the reader. Should you perchance peruse these bold writings of a German hand recast into Italic sounds, forgive the mistakes which, we fear, are bound to exist. Let the novelty of the work recommend it to you: for the undertaking lacked a model, and there were no earlier footsteps that we might follow. But if somehow the writings please you, may your favor grow and be as a crown to them. Jo[achim Camerarius].

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Albrecht Dürer De Symmetria Partium in Rectis Formis Humanorum Corporum Nuremberg, 1532 the warnock library

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Warning: By order of Charles V, Holy Roman Emperor, no unauthorized printing of the writings of Albrecht Dürer—whether of this work or of those on the elements of geometry or on fortifications— shall be made or sold with impunity. Infractors will be subject to the many penalties listed in the certificate of privilege.

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To the Fair-Minded Reader, Interested in the Fine Arts: If I were to say something about painting here, I would speak not so much in praise as in weeping. For who among you can be ignorant of the encomia heaped on this most praised of arts? Yet who would be so perverse as not to agree that the loss of what is blooming and the neglect of what is thriving should rather be a cause for grief and tears? However, I did not set out to talk about art, but to say some words about the artist and author of this book, and about the work itself. And I trust that his excellence and merit have brought him renown not only in his country but abroad. I am fully aware that he does not need our commendation, particularly since his distinguished works themselves bestow on him eternal glory. But, both because we would be publishing his creation and because we would have the occasion of committing to writing the life and character of a famous friend, we thought it convenient to weave together what we know about him, partly from hearsay and partly firsthand, including a eulogy of the skill and singular capacity of the artist and man, hoping also that this would bring some pleasure to readers. We have learned that our Albrecht originates from Pannonia, but that his forebears moved to Germany. There is no point in dwelling on his origins and birth; if we are to speak frankly, they are more ennobled by him than he is by them. Nature gave him a striking body in proportion and height, a match for the splendid mind it enshrined; mindful of her fairness, so often praised by Hippocrates, by which she usually gives suitable bodies to exceptional minds, just as she made for the absurdly fidgety monkey an equally absurd body. He had an expressive head, darting eyes, an honorable nose of the type the Greeks call τετράγωνον [four-sided], a long neck, a wide chest, a flat stomach, sinewy thighs, sturdy legs. As for his fingers, one would say nothing more elegant was ever seen. His speech was of such wit and charm that the listener could not bear its coming to an end. He had little in the way of formal study, but what study brings—particularly a knowledge of the natural sciences and mathematics—he managed to acquire. He could equally well grasp what was important, develop a subject in its outline, and explain it to others. This is attested by his geometric writings, in which I see nothing that can be faulted regarding that science, within the limits of what he undertook to discuss. He passionately embraced uprightness in his habits and life, and behaved so as to be justly regarded as an excellent man. Nor was he forbiddingly severe or grave, the kind who never indulges in humor or pleasantry and finds them incompatible with honor and virtue; on the contrary, he cultivated the lighter side of life even into his old age—for example with physical exercise and music: but above all when he molded nature into a painting, and for this reason he devoted all his might to the pursuit of art, and surrounded himself with the works of painters and admirers of all sorts, in order to know their techniques and to imitate what he approved of. He experienced the benevolence and obtained the favor of kings and princes, and especially of the emperors Maximilian himself and his grandson Charles, by whom he was paid a not negligible allowance. And when his hand had acquired maturity, so to speak, one could discern behind it all the more the mind sublime and loving of excellence, so great were all his works and so praiseworthy their subjects. There can be found eulogies from Maximilian and also immortal works of astronomical history from which one can 5


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gauge the truth of what I have just said, not to mention other testimonials, any of which would have made a painter of any land and age glad to have the same said of him. There is certainly no more accurate and truer indicator of someone’s nature than what can be gleaned from the works made in the exercise of one’s craft. And two types of crafts leave the others far behind, proceeding as it were from the workshop of Nature herself; they are particularly powerful indicators. I refer to what are called the Μιµητικά [imitative arts] and Ποιητικά [creative arts]: those that depict the likeness of things, either in words or through manual skill. The verbal type is properly called poetic, whereas the manual type does not to my knowledge have a general name, but is generally understood to be fourfold: it may involve statuary, painting, sculpture, or molding. Let us take painting, since the occasion arises and it is the most celebrated among the four, having even been equated to poetry by one of the Greeks [Simonides]. Clearly, then, one will judge painters wholly according to their works, just as with poets. It is said that Polygnotus painted everything greater, and Pauso everything less [Aristotle, Poetica 1448a]. Who does not deduce that the former had the greater mind, the latter the humbler one? Martial said [in 8.55] that he would be not a Virgil but a Marsus [an epigrammatist and Virgil’s contemporary], were he to find a Maecenas. And of course no one who has read Martial would expect him to become a Virgil; his verse, admirable though it is in the epigram line, lacks the qualities needed for greater works—it is in fact utterly inappropriate for them, being often coarse and disgusting. So I cannot help laughing when I think of Catullus’ maxim that a poet comme il faut should be chaste, but his verse need not be [Catullus 16]; and of how Martial wanted to show that his character was very different from his poems. But now let us put poets aside. How many painters are there, or rather, are there any painters at all, whose nature is not declared by their works? I will not even mention examples from antiquity, satisfying myself with those from our own times. We all know that many court the praise and admiration of the populace by depicting what cannot be done honorably except in private, and, what is more, what even when privately done is evil and shameful. No one considers chaste those whose minds and hands are occupied with such things. We have also seen many artists’ quite nicely colored miniature paintings, attesting to their authors’ skill and industry but lacking in art. Here therefore I will give Albrecht his due. He was a diligent guardian of virtue and piety, and by the greatness of his paintings he showed he was conscious of his own strengths. Even among his lesser works there is nothing to look down upon: you will not find in them a single line drawn at random or at cross purposes, nor one superfluous dot. And let me say something about the constancy and firmness of his hand. Looking at his lines you would swear they were drawn with straightedge or compass, whereas in fact he used no other instrument than his brush or pen when working, to the great admiration of onlookers. And with what coordination of hand and mind he would, in a trice, make a pen sketch on paper of any figure that he fancied! And this too will seem a wonder to the reader, that he would sometimes draw separately parts not only of a scene but also of bodies, and when brought together they would match so perfectly that nothing could seem more apt. Undoubtedly the mind of this exceptional artist, educated in all knowledge and truth and in the agreement of the parts among themselves, governed his hand and told it “Trust me,” needing no props. Similar was his deftness holding a brush, with which he would draw minute images on canvas or on wood, having made no sketch beforehand, and the result would not only be devoid of defects but 6


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elicit the highest praise. This was most admirable to accomplished painters, who know how difficult such a feat is, having plenty of experience in the matter. Here I cannot fail to mention what took place between him and Giovanni Bellini, the Venetian whose art was famous all over Italy. When Albrecht came to Venice, he easily struck up a friendship with Bellini, and, as is usual in such cases, they took to showing each other their works, Albrecht admiring and praising all the other had done, and Bellini approving everything about Albrecht’s work, but particularly the subtlety of the hair in portraits. And it so happened that after they finished talking about their art, Bellini asked, “Would you do me a kindness to please your friend?” “Anything that is within my power.” “I would like you to give me one of the brushes you use to paint hair.” Albrecht immediately brought forth several brushes like any others, like those that Bellini himself used, and said, “Choose any one you like, or all of them.” Bellini, thinking he had been misunderstood, elaborated, “I did not mean these, but the ones with which you paint several hairs in a single stroke. They must be fairly sparse, with some separation between the brush hairs. For if you painted such long threads individually you could not possibly make their distance and curvature so consistent.” Albrecht replied, “I use none other than these,” and to prove it he took one of the brushes he had proffered and painted very long, wavy hairs such as might be seen in a woman’s head, keeping them utterly regular. The amazed Bellini would later confess to many people that he would not have believed from anybody what he himself had seen that day. A similar tribute was paid by Andrea Mantegna, who lived and worked in Mantua, earning praise for restoring to painting a certain rigor and discipline and for being the first to dig up scattered and broken classical statues and put them forth as models for contemporary art. His works were hard and rigid, made by hands not accustomed to following the intelligence and fluidity of the mind; nevertheless they were widely held up as models of excellence and perfection. Now Mantegna, being sick in Mantua, heard that Albrecht was in Italy, and immediately sent word that he wanted to see him, hoping to impart to him some of his own theoretical knowledge; for Mantegna would often, within his circle, bemoan the fact that he did not have Albrecht’s sure hand and facility with a brush, saying at the same time that the latter lacked his knowledge. Without delay, putting aside everything else, Albrecht undertook the trip; but before he could reach Mantua Mantegna died, and he would later say that no event in his life had been so sad. For, accomplished as he was, he nonetheless always sought to improve his splendid mind. We have been astonished to see the bearded image of a man he made with the brush alone on canvas, without the usual pencil sketch, as I have already mentioned. The hairs of the beard are almost two feet long, and drawn so exquisitely and cunningly, with such regularity, that the more one knows about painting, the more one admires them, thinking it incredible that one could draw them freehand, without the help of any instruments. Moreover one cannot find in his work any filth, anything indecorous: indeed the thoughts of his pure soul were averse to all such things. O artist worthy of such success! How true to life, how free of all blemish were the portraits you painted! All this you accomplished by bringing your experience to your art and method, which were unknown until now and unheard of, at least among painters in our country. For who else, among even the most famous of our artists, could explain his method in such a way as to 7


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demonstrate that his fame owes more to knowledge than to chance? But Albrecht had everything prepared, thought out and ready, having brought painting into the realm of precepts and method and planning, without which, as Cicero wrote, even if you do something well with the help of Nature, you may not be in a position to do it again, since your success was fortuitous. This method he perfected first for himself, but soon, given his liberal and open temperament, he set out to explain to the honorable Willibald Pirkheimer in writing, dedicating the books to him with a most elegant letter, which we have not translated into Latin lest we fail to do justice to its pristine beauty. But before he could complete everything and publish the book, as he wished, death took him—a peaceful and desirable death to be sure, but in my opinion premature. If there was anything in that man approaching a vice, it was his infinite earnestness, which was often too stern toward him, pushing him too hard. Although death did not allow him to see through the publication of his work, his friends took over, following his directions; and this leads me to say a few words about the book and this translation. The work itself was started by him in a geometric vein, without great polish or literary aspirations, so it may seem a bit rough; but the quality of the content will easily compensate for that. A few days before passing away he asked me if I would translate it into Latin, and I willingly agreed. He was then still polishing it, but death took this worry and this ability away from him. Later, when his friends published the work, they prevailed upon me, by their authority no less than by entreaty, to undertake the translation, and to carry out with zeal after his death this project which I accepted while he was alive. Of course I was aware of the tremendous undertaking this would be, because of my incomplete familiarity with the subject and because there was no Latin model that I might imitate in this endeavor; nor did it escape me how much harder the task would be than my other, ordinary occupations. I realized I could not devote to the translation as much time and application as I would have liked and as the work deserved. But what was I to do? Both out of respect to my great friend, now deceased, and because of the will of his friends and as it were their lien upon me, I set foot against all difficulties on this path, which was none too smooth or easy for me. Therefore I have put into these two books of the whole work as much effort as I could spare from my unavoidable business, and I have tried to convey somehow in Latin what he taught so beautifully in German. In this I know I will easily meet with your indulgence, good readers devoted to the arts, for I neither would nor could defend them from the charge of foolhardiness. Yet I think it would be well that my boldness be not only forgiven but supported: if I know how well received the effort spent on this part of Albrecht’s writings has been, I will be encouraged to undertake to translate into Latin, no less carefully and diligently, the remainder of his writings on painting. And do not expect his writings on this subject only, but also those on geometry and those on the science of fortifications, where he describes the defense of contemporary cities. Those, moreover, are the only subjects on which he wrote books. When I hear certain people write or claim verbally that they will put out a booklet by Albrecht on the proportions of the horse’s body, I very much wonder where they will get after his death something he did not complete in life. It is not unknown to me that he had started to make investigations in this area, nor that more remained to be done than he did while alive, 8


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nor (which most contradicts the claims made) that he had not started taking measurements. Indeed I know that whatever he had put on paper was lost through the malice of certain individuals, who caused his notes to be stolen, so that never again did he feel motivated to restart that work. Not that he lacked a suspicion—indeed a certainty—about who had sent these thieving drones; only, given his generosity, he chose to keep it secret, to his own loss and pain, rather than shed his meekness and forgiveness in pursuit of these men. So whatever there may exist along those lines, we will not allow it to appear under Albrecht’s name, since it will not be worthy of this great artist. A few years ago there also came out some little papers in German containing rules on that subject, beggarish productions and quite unsuitable. I will say no more about them here, other than that, unless I am wholly mistaken, their “author” rued them more than once. As for the material I mentioned a bit earlier, it can easily be shown to be Albrecht’s: the author prepared it for publication himself, and its quality is obvious. Moreover I think you will find it much better here than anywhere else: not because it is more polished or eruditely presented here, but because interpreting it in the author’s own workshop, using his annotated and corrected manuscripts and the wealth of examples he left, it basks, so to speak, in the open light, while it must seem obscure to others, however learned. This fact is shown by Albrecht’s book on geometry, now in the hands of a most learned gentleman, who will soon have it published in a much more elaborate and more fully explained form than can now be found, since it will include no less than twenty-six new drawings and numerous corrections and improvements added by the author on rereading the published edition, and since he, as if guessing he would not be publishing again, left instructions about the type and the figures to those who were to publish the work. We will also see that the book comes out as soon as possible, in German, as written by the author. Surely these will be well received by you, and you will bestow your gratitude both on the studies of the author, an exceptional man with a passion for sharing something useful with the public, and on our efforts and desire to make available to all nations what appeared to be written for just one.

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Preface to the Books on Proportion by Albrecht Dürer, To the Most Renowned and Learned Christopher Coler, V. C. When, thanks to your great encouragement, I undertook to translate these books of Albrecht’s (your most beloved friend) into Latin, I thought it good to render to you an account of the project, so your judgment would recommend the book to the reader and so I would be emboldened by having you as the guide, so to speak, for this publication. Therefore when I heard that our work did not displease you, I decided to include at this point the account I prepared, to show that it meets with your approval and thus to ennoble the work. I thought also that the reader might derive some profit from this imparting of our reasoning and thoughts. Therefore if the readers have reason to be thankful for anything in this part of the work, it is all due to you, Excellency, who stood not only as my supporter but also as my instigator.

To Christopher Coler, Mayor. When, at your urging, I took upon myself the task of translating Dürer’s books on painting, it seemed good to show you before publication the nomenclature I followed for the most commonly named things in the whole work, with a view to either retaining or changing it according to your opinion. That the author researched, in apt and tasteful drawings of bodies, how far the parts of the body should be from one another, and what elegant and tasteful combination of features constitutes beauty—by considering this and the title of this work we can predict that Proportion is its subject. As Pliny says, proportion was first applied to painting by Parrhasius; it was later used most of all by Euphranor, who also wrote a book about it. Although it is said that the Greek word symmetria [used in the Latin title of this work] has no real equivalent in Latin, we shall interpret it as a measuring together or relative measurement [commensus, commensuratio]. Thus by Art, which follows Nature, a method is investigated by means of which anything may be made in its graphical expression just as it is in reality. Since, as you know, there are two ways to measure the parts of a body, in height and in width, we get a twofold method. When lengths are marked in profile views, I think it will be understood, keeping in mind the three-dimensional character of the body, that they refer to thickness. This is related to what is called “depth,” when one moves back, so to speak, from the feature in question: for example, going from the eyes to the ears in a view of the head. The foot is a special case, because its longest dimension is horizontal. Therefore in the profile view is given a designation of the height perpendicularly. Notice that whenever parts of the body are individually named, their beginning or uppermost point should be understood. This is commonly spelled out in the text, unnecessarily in my opinion. In some places it is downright superfluous—but I felt that in someone else’s work it is best not to change too much. (But who would need the addition of “top of” when hearing the word “crown” in reference to the top of the head?) Moreover, although the author carefully and diligently sought the best terms for the parts of the body, even introducing new terms for certain parts so the measurements would be more exact, anyone familiar with translation will appreciate the difficulties that we faced in this area, not least because no 10


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ancient text exists whose terminology we could imitate. Nonetheless we tried our best. The reader will judge how appropriate our Latin equivalents for Dürer’s terms may be: it is not for us to predict this, though we expended much care and time on the choice. I hope you will consider these terms carefully; if they seem satisfactory to you, it will be a good sign that others will find them so. Corrections and suggestions of better terms not only will not offend us—they will be a source of pleasure and help. But let me hide nothing: here is a list of the parts of the human body in Latin, together with Dürer’s German terms, so the reader can see all that I did, and how I did it.

Names of the Parts of the Human Body Mentioned in These Books. The crown [sinciput]; Greek, Βρέγµα. By this should be understood the top of the head, where the ancients placed the cowlick. Die schaytel [Scheitel]. The cowlick [vertex] in the middle of the scalp around which the hairs turn. Der hinderst wirbel. The occiput or back of the head [occipitium]. Hinden der kopff ob [ober] dem genick. Hairline [radices capilli]. Harwachs. Only above the forehead and temples. The forehead [frons]. Die stiern [Stirn]. The eyebrows [supercilia]. Augpraen [Augenbraue]. The eyelids [ palpebrae]. Augschedel. The nose [nasus]. Die nase. The nostrils [nares]. Naslöcher [Nasenlöcher]. The mouth [bucca]; munde [Mund]. The word is used for the feature on the face consisting of the meeting of the lips [coniunctio labiorum]; Greek, Συναρµογήν or Συµβολήν. The cheeks [genae, malae, buccae]. Die packen [Backen]. The temples [tempora]. Neben das haubt [Haupt]. The neck [collum, gula, guttur, ceruix]. Der hals. The shoulder blades [scapulae]. Schulterflaysch. The shoulders proper [humeri]. Achsel. From here comes out the humerus [humeri os]; its end fits into the socket of the shoulder blade or ὠµοκοτύλη, acetabulum. Die achselbain [Achselbein]. Throat [iugula, iuguli, iugulum]. Halsgrüblein. Refers to the σφαγή, slight cavity at the base of the neck. The chest [ pectus]. Die pruste [Brust]. The armpits [alae, axillae]. üchsen. They appear in front and back views. The breasts [mamae]. prüste [Brüste]. The nipples [papillae]. die tütlein. The lower chest [infra mamas]. undern prüsten. Also called in some places the lower thorax [imus thorax]. The loins or waist [lumbi, latera]. At the lower back, where we gird ourselves. die waychen [Weiche]. The curve of the hip [sinus coxarum], or protruding portion of the hip bone above the socket where the femur attaches. The latter is the cotyloid cavity κοτύλαι, acetabulum. The Greeks call it simply ἰσχία [hip joint]. der huftart. Top of thigh or hip joint [summum femur, coxa]. der huftende. Sometimes we call it coxendix, but mostly we use this 11


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term for the hip bone [caput coxae], die peyn in hüften. Belly [aluus]. End des pauchs [Bauches]. The pubic area [verenda, pudenda, pubes]. die Scham. This includes the glans, the scrotum, the penis [virgula] and the testes. But in the woman “top of the pubis” [initium pubis] refers to the top of the cleft [natura], σχίσµατος, and “bottom of the pubis” [ima pubes] to where the cleft ends, or where the legs fork. Den spaldt [Spalt] des weybs. Bottom of the buttocks [extremae nates]. End des hindern. Thigh [ femur]. das ober peyn [Bein]. Narrowing of the thigh [ feminis sulcus]. Einpeisung des peins. Upper knee [supra genu]. Ob dem knie. Inner upper knee [supra genu interius]. poplites. Mid knee [medium genu]. In der mittel. Lower knee [infra genu]. Under dem knie. The calf [sura]. waden [Wade]. Again, we mark it as middle or lower, inner or outer. The shin [tibia]. Schinpein [Schienbein]. The ankles [tali]. Die knorren. On both sides and called inner and outer. Top of the foot [mons pedis or conuexus pes]. Des fues ritz. The sole [ planta]. Die solen [Sohle]. The foot [calx]. den fues [Füss]. The heel [calcaneum]. die fersen [Ferse]. The arm [bracchium]. Arm. The shoulder [humerus]. achsel. The armpit [ala]. üchsen. Muscles [musculi, tori]. die mause. The elbow joint [nexus cubiti], Greek, ἀγκών, where the arm bends. der elnpogen [Ellenbogen]. The wrist joint [junctura manus], Greek, καρπος [i.e., καρπός]. Das gelengk der hand. The hand [manus]. hand. The open hand [ palma]. die offen hand. Fingers and toes [digiti]. finger und zehen. The joints [articuli]. Gelengk. These are the things we wished to bring to your attention, and which you, Excellency, will consider, and thus keep your good will toward us and our efforts. Jo[achim]. Ca[merarius]. Qu[aestor].

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H. Eobanus Hessus Wrote These Verses for Dürer. Now, reader, you have in a small volume much new material on proportion, with noble illustrations, and when you read it you will perhaps notice that the author had not only the hands of an artist, but also immense knowledge, good judgment, and a sharp mind; nothing escaped him. In matters involving the mathematical arts, no one surpassed him in judgment; when it came to surrounding cities with walls, Dürer could draw them with sure hand. He not only painted, but also taught, warlike things that even famous generals could not. But this is not the place to repeat the praise that has been bestowed on him in another poem. But reader, you know Dürer’s immortal accomplishments; consider him worthy of a lofty prize. Be thankful to him and to his Nordic homeland, by means of which this monument is passed on to you.

And These Too. What great proofs of his art Apelles [the famous Greek painter] left are attested by many writers; but nothing has survived that can be attributed to his brush. Could the same fate await Dürer? I think not: the cause of the loss of one will work toward the preservation of the other. The strokes of Apelles were recorded with the fleeting pen, whereas we safeguard Dürer’s with shining bronze, so no misfortunes can wipe them out.

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Apelles left few copies of his works, and a single ruthless moment could destroy them: we, in a short time, print more thousands than a single hand could have written in thrice-four years. Things being so, my faith is confirmed: Dürer will live a thousand years more. By his art monuments have long been made ready which—books aside— were unworthy to die. For who better than he has mastered the art of painting? Who cleverer in mind or hand? He has left all others behind, predecessors and contemporaries alike. The threnodies I have written on him show that the works of such a man are more fitting than praise.

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The First Book on Proportions by the Famous Painter and Geometer Albrecht Dürer To draw the human form, follow this procedure: Take a ruler longer than the height of the desired image, and mark on it a straight line of the same length as that height, so that one end of the line will represent the crown of the head, and the other end the foot. And, as the figures you have in mind to draw are of different sizes, some short, some tall, so should you arrange lines of different lengths, and subdivide them independently. Therefore, whenever you encounter a fraction number from now on, regard it as a fraction of the line representing the total height of the figure you are drawing. This one line should be carefully distinguished from all others, and this distinction should be carried through from the division into two to the division into fifty, one hundred, or however many parts may be used. Place all these parts with their numbers next to the long line drawn on the ruler, so that all the initial points agree and the endpoints are staggered as needed. In this way longer parts get lower numbers and shorter parts higher numbers. Thus you should make a segment half the total length and label it 2, one of a third the total length and label it 3, and so on. And later, the parts thus numbered should be subdivided more finely into some number of parts, even or odd, if the occasion so requires according to your judgment. For anyone who wants to measure something accurately, so long as everything cannot be encompassed by one fixed subdivision, must move to smaller subdivisions. You will see many occasions when I have done this; they are indicated by the presence of double or even triple numbers, large or small, even or odd, by means of which I seem to have been able to express most nearly the true measure of things. Moreover the composition of the numbers that I have indicated can be changed and used differently according to the designs of others. For the better understanding of what I have said so far and will say later, I have drawn on the side a rule with its parts and labels, made according to the instructions above. And when you set out you should make sure that your diagram of parts is well-made and exact; otherwise much labor and time will be spent whenever it is necessary to construct a segment representing a desired fraction of the height from top to foot.

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Now I will explain how you can measure out the image to be drawn. Suppose, e.g., that you want to draw a man, the stout rustic type, having a head one-seventh the length of the body. Draw a horizontal straight line, and three lines perpendicular to it, as tall as the image you want to draw. The first vertical line, or axis, will be for a profile, the next for a front view, and the last for a back view. Having drawn these parallel axes as explained, mark out the height of each of the important features of the body, from the crown to the sole. Thus you will notice in the examples that we have marked the vertical position of each feature with a horizontal line crossing the vertical axes, each crossline being set off by its numbers, and that we have added everywhere the fraction number indicating the width of the feature. The correctness of the whole method will be apparent to anyone who examines it carefully. Although sometimes fewer and sometimes more lines will be marked, the main horizontal lines you will encounter are the following, from top to bottom: first the crown; then the forehead; then the eyebrows; then the chin; and so down to top of shoulders, throat, chest, shoulder blades, nipples, lower chest, loins, navel, curve of hip, hip joint, belly, pubis, tip of penis, lower buttocks, narrow part of thigh, upper knee, upper of ham, mid knee, lower ham, lower outer calf, lower inner calf, top of feet, lower outer ankle, sole. Now we will add these names next to the horizontal lines that indicate the height of the parts of the body; this will make clear the method that I have followed throughout the book. After arranging things in this way you will measure the lengths of the parts. From the crown to the throat let there be one tenth part and one eleventh. To the top of shoulders, two eleventh parts. To the bottom of the chin, B/h. The cowlick is halfway between the crown and the forehead. From the chin to the base of the hair, B/ba. If this tenth part is divided equally into three intervals, the first will mark the forehead, the second eyes and nose, the third mouth and chin. From the throat to the top of the chest, B/da. To the armpits, B/bd. To the nipples, B/ba. Bottom of pectorals, B/i. Loins, C/bb. From the loins to the navel, B/ea. To the curve of the hip, B/da. To the hip joint, B/ba. To the crotch, B/i. To the thip of the penis, B/g. To the lower buttocks, B/ba plus B/bb. From the lower buttocks to the narrow part of the thigh, halfway down the thigh, B/bi. From the sole to the to the bottom of the ankle, B/ci. From the sole to the top of the foot, B/ca. Having measured out the body down to the bottom [i.e., top] of the thigh, there remains the placement of the knees. Once this is done, the height of the body will be divided into three unequal portions: the first and longest runs from the throat to the top of the thigh; the middle one, from there to mid knee, is shorter; the last and shortest one runs from the knee to the bottom of the shin. For the upper limbs [sic: the intended meaning, present in the German original, is the hind limbs of animals] are usually longer and stronger, as is easily observed in man. The word “trunk” will be used for the long portion, which, although made of many parts, is considered as a whole, and is strong enough to move the other parts. Returning to the three lengths, you should know that they ought to keep a certain proportion among themselves, so that whatever fraction of the length of the trunk is represented by the length of the thigh, measured from hip joint to mid knee, that same fraction of the thigh should represent the shin. 16


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This you can arrange with a construction that I shall not repeat in later drawings. Draw a triangle abc, with side ab horizontal and bc vertical, so b is a right angle. Divide bc into three equal parts at points d and e. From a draw lines to d and e, from which you will derive the proportion I mentioned as follows. Mark on a ruler two points whose distance equals the length of the trunk from the throat to the hip joint. Likewise mark the base of the shin on the ruler. Call the throat point f, the hip joint point g, and the base of the shin h. Then place the ruler so that the point g touches the line ad of the triangle mentioned earlier, and do not let this point move away from that line. Next turn and slide the ruler so that f is made to lie on the edge ac of the triangle and the lowest point h is made to lie on the horizontal edge ab. This done, the line a will intersect the rule somewhere between g and h; call this point I, and you will have found the three proportional lengths: for the ratio of the length fg to that of gI is the same as that of gI to Ih.* Point f is to be understood as marking the throat, point g the hip joint, point I the middle of the knee, and point h the bottom of the shin. Note that the ruler applied to the triangle should be turned so that the point f moves toward the vertical edge eb [i.e., cb]. This triangle can be called an “equator,” and you will also be able to use this idea backwards when you want to invert something. Indeed, to those who use it correctly, it will be useful in accomplishing many things. * [Note: Dürer was probably aware that this proportionality is only approximate: the ratio fg/gI equals gI/Ih only when the ruler is vertical. The more the ruler is tilted, the greater the error: at 30 degrees from the vertical it is already 17% (if ab = bc, as in the diagram). But the author implicitly assumes that the three segments of interest are of nearly equal lengths, so the ruler is almost vertical and the error is small.—Translator]

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Now we come to the measurement of the arm, as follows. From the shoulder joint at the height of the throat to the elbow, C/bb. But from the shoulder to the end of the deltoids, B/ba. From the elbow to the fingertips, B/e. From the fingertips back to the hand joint, B/ba. And if you like you can extend that to B/j with no loss of elegance.

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After correctly placing the knee in the image, measure out the remaining features thus: From mid knee to upper knee B/cb To the lower knee B/ea. And so ends the knee: middle, upper, and lower. To the lower outer calf C/bj Inner B/i Before continuing I will show a diagram with the triangle discussed above.

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[Note: The text on this page is incorporated from the errata section (view 82).—Editor]

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So we have covered the vertical measurements of all the features. Now we will indicate the widths to be marked on the horizonal lines drawn across the features of the image in profile. The head, at the height of the cowlick, shall be made to have width B/j At the top [i.e., bottom] of the forehead B/be plus B/bf At the eyelids B/h At the nose B/i At the chin and lower [i.e., upper] neck likewise B/i At the neck itself B/bc At the throat B/j At the upper chest C/bd At the armpits B/g At the nipples also B/g At the lower chest B/bc plus B/bd At the loins where we gird ourselves likewise B/bc plus B/bd At the hips B/g At the upper femur or hip joint and at the belly and buttocks B/bb plus B/bc At the pubis also B/bb plus B/bc Where the thigh meets the buttocks B/h At mid thigh B/be plus B/bf At the upper knee B/ba At mid knee B/bc At the lower knee B/bc At mid calf at the widest point B/ca plus B/cb At the lower calf B/bd At the ankle were the leg connects with the foot B/bd The foot itself or heel shall be extended to B/g. Next, the features of the arm in profile shall have the following widths: Shoulder B/cb Under the armpit B/bd Elbow B/bi Muscles of the forearm B/bi Joint of the hand B/dc The hand itself hanging as shown B/da In the examples the measurements of the arms will be indicated separately from the body

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to make things clearer and less involved. We now start with the width across of each feature, one at a time. Let the width of the head at the cowlick be B/ba At the forehead B/i At the eyelids B/j At the ears B/i At the nose B/ba Let the width of the neck under the chin be B/bc At the throat B/f At the chest D/ba Let the distance between the armpits be B/f Between the nipples C/bf At the loins B/f At the top of the hips B/j plus C/bj At the thigh B/e There you should also mark the hip joints, which are B/g apart At the pubis B/e At the thighs immediately under the buttocks C/bh At mid thigh B/ba Here notice the already mentioned narrowing of the thigh on the inside. Above the knee B/bc At mid knee B/be Under the knee C/cg At mid calf B/cc plus B/ce At the lower inner calf B/be At the bottom of the calf above the ankle B/ch At the ankle B/cc The foot itself should be drawn B/bf wide at the toes Next you should make the width of the arm as follows: At the armpits B/bi Above the elbow joint B/cb Below B/bg At the forearm above the hand joint


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B/cf. The palm should be B/bf wide. Lastly, the back view, built around the third vertical axis, should be B/e wide at the shoulder blades, between the armpits.

Make the length of the cleft between the buttocks B/i. Make the heel B/ce wide.

And now that the height and width of all the parts of the body have been triply described according to this method, you should sketch the contour and other features of the figure as best you can, according to your abilities. It is quite useful to have as a model a man’s image of approximately the same height, and I often work that way; the result is better and more elegant than if you fill in the lines casually. I have laid out everything as exactly as possible in the example figures. And here more than anywhere one can admire how artfully nature has built man as if from two parts, attaching the trunk onto the legs at the hip joint, as if onto a base. In the example figures we carefully signaled this feature with a curve in the area of the belly and the pelvis. We give this first set of figures a label, say the letter A, so as to make things as clear as possible by referring back to it.

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Heights of the features of the male body. Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulders Throat Upper chest Armpits Nipples Lower chest Loins where we gird ourselves Navel Top of hip Hip joint Pubis Tip of penis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee Lower outer calf Lower inner calf

Top of foot Lower outer ankle Sole

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After this we will describe the image of a woman, a bit stocky and rustic, whose shape is as much as possible like that of the man discussed earlier; let her height likewise be seven of her heads. You should follow the same method as before, measuring out the parts of her body. In the examples I have omitted the horizontal lines to begin with, writing only the numbers indicating the width; however I have marked the height with little points. In this way we avoid an unpleasant excess of lines. You should form the parts of the body of such a woman using the measurements given here.

From the crown to the top of the shoulders let there be B/ba plus C/cd From the crown to the throat, C/bb Still from the crown to the lower chin, B/h From the chin to the cowlick, B/i From the chin up to the top of the forehead, B/ba Divide this B/ba into three equal parts; the topmost will contain the forehead, the middle one the nose, eyes and ears, the bottom one the mouth and chin Under the chin draw a fleshy circle, like a dewlap From the throat to the back B/f From the throat to the bottom of the breasts B/h From the throat to the nipples B/j From the throat to the armpits B/bf Likewise from the throat to the upper breast B/ci Here observe that the armpits are lower in back than in front. From the loins to the navel B/cc From the loins to the hip joint B/j From the loins to the lower belly B/i From the loins to the lower pubis C/bb From the loins to the lower buttocks B/f From the soles to the convexity of the foot B/ca From the soles to the outer ankle B/ci From the inner ankle to the mid knee C/j You can also place the knee using the equator diagram, as shown earlier. Then it will come out a bit lower. You can choose whichever version you prefer. From mid knee to the bottom of the thigh B/i Moreover make the length of the foot B/g

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Then make the length of the arm from the top of the shoulder to the elbow equal to C/bb. From the elbow to the fingertips B/e. And from there back to the wrist joint B/ba. Now, the height of the features having been marked, you should mark their width. Work first with the profile view, as follows: The profile of the head at the height of the cowlick should have width B/j When I mention the cowlick or any other feature, make sure you look back at the example figures, so things will be clear. At the forehead B/be plus B/bf At the eyebrows likewise B/be plus B/bf At the nose B/i At the chin and neck B/bc At the throat let the width be B/ba At the chest, above the breasts, B/h At the armpits C/bd At the nipples B/bc plus B/bd At the bottom of the breasts B/h At the loins or lower back, where we gird ourselves, also B/h At the navel B/ba plus B/bb At the hip joint B/i plus B/ba The thigh immediately below the buttocks should have width B/h Above the knee let the width be C/bj At mid knee B/bc Below the knee likewise B/bc At mid calf B/ca plus B/cc At the lower calf B/bc Above the foot B/bi Then mark the width of the arm thus: At the shoulder let it be B/bb At the biceps B/bc At the elbow B/ca


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At the muscle of the forearm B/h At the wrist joint B/de At the hand itself, drawn hanging, B/ch After that, mark the widths in the front view thus: The head at the cowlick should have width B/bi plus B/bj At the forehead B/bf plus B/bg At the eyebrows B/j At the ears B/i At the nose B/ba At the neck below the chin B/bc At the throat B/f The distance between the shoulder joints should be C/bb At the chest E/bf Above the breasts, between the armpits B/g Below the nipples B/i Below the breasts B/ba plus B/bb At the loins or lower back, where we gird ourselves, B/f At the navel B/e At the upper thigh B/h plus B/i Also there make the distance between the hip joints C/bb At the thigh just next to the lower pubis B/bf plus B/bg Above the knee C/cb

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At mid knee C/cf At the upper calf B/cc plus B/ce At the lower calf B/be Above the foot B/cg At the ankle B/cf The foot itself at the toes should have width B/bg And the heel B/ci In the back view the width of the back between the armpits should be B/f. The length of the cleft between the buttocks should be B/g. The width of the arm in the front view at the biceps should be B/bg At the elbow B/bj At the muscle of the forearm B/bf At the wrist joint B/ch The palm itself B/bg And thus, having measured out all the parts of the body, you should fill in the outline and other features as required—just as you can see in the example provided, with all the curves and all the appropriate strokes carefully laid out, each in its place. This drawing too shall be given a name for future reference, say A I.


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Heights of the features of the female body. Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulders Throat Upper chest Armpits Nipples Lower breasts Loins Navel Hip joint Lower belly

Lower pubis Lower buttocks

Upper knee

Mid knee Lower knee Lower outer calf

Top of foot Lower ankle Sole

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We now turn to the proportions of a male body whose head corresponds to one eighth of the height of the body. You should proceed in the same way, establishing the height of the features thus: From the crown to the throat there should be a distance of B/g From the crown to the lower chin B/i From the lower chin to the upper forehead B/ba This tenth part can be divided into three parts, as shown earlier. From the crown to the loins where we gird ourselves B/d From the throat to the armpits B/be. Seen from the back, the armpits appear lower From the throat to the chest B/ba From the crown to the point where the legs diverge B/c From the loins to the navel B/cj From the loins to the curve of the hip B/bi From the loins to the hip joint B/ca plus B/bi From the loins to the pubis C/bd From the loins to the lower buttocks B/ba plus B/bb From the lower buttocks to the lower scrotum B/ea From the lower buttocks to mid thigh B/bf From the sole to the outer ankle B/ch From the ankle to where the foot starts to arch B/cb From the ankle to mid knee B/e In placing the knee you can also follow the method of the equator, shown earlier From mid knee to the upper knee B/da From mid knee to the lower knee B/da, and the knee is contained in these parts From mid knee to lower outer calf B/j From mid knee to lower inner calf B/bf plus B/bg And so you will move on to the arms From the top of the shoulder to the elbow B/f From the elbow to the fingertips B/e From the fingertips to the wrist B/ba

Now you will approach the widths of the profile thus: At the cowlick make the head B/ba wide At the eyebrows B/i At the nose B/j At the chin and neck B/ba The neck should be B/bg wide just below the chin and B/be a bit further down Let the body be B/bc wide at the throat At the chest B/h At the nipples B/h At the lower chest B/be plus B/bf At the loins B/bg plus B/bh At the navel B/bh plus B/bi At the curve of the hip B/i At the hip joint B/h At the pubis or buttocks and thigh C/bf; the thigh itself just under the buttocks should be B/j wide At mid thigh B/bj plus B/ca Above the knee B/be At mid knee B/bf Below the knee B/bg At mid calf B/bd At the lower calf B/bh Above the foot B/ce Make the foot itself have length B/g Then mark the width of the arm thus: At the shoulder B/bd At the armpits B/bh At the elbow B/ce At the muscle of the forearm B/cc At the wrist joint B/ba At the hand itself, drawn hanging, B/de After that, mark the widths in the front view thus: The head at the forehead has width B/j At the eyebrows B/ba At the ears C/bh

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At the nose B/bc The width of the neck itself below the chin should be B/bg At the height of the throat make the trunk have width B/g, and make the distance between the shoulder blades B/bb plus B/bc Make the chest and shoulders B/e wide The distance between armpits will be B/g Between the nipples B/j At the loins C/bd At the curve of the hip B/g At the top of the thigh B/ba plus B/bb. At the same height the distance between hip joints will be B/be plus B/bf. Make the thigh itself below the buttocks have width B/bb; at mid thigh B/bd Above the knee it will be B/bg At mid knee B/bi Below the knee B/ca At mid calf B/bf

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At the lower calf B/ca At the bottom of the shin B/de At the ankles let the width be B/ch The foot at the tip of the toes B/bg Now turning to the arms: at the biceps, below the armpit, make the width B/ce Above the elbow B/cg At the muscle of the forearm B/bj At the wrist joint B/bg Let the width of the palm be B/bg Make the back view at the shoulder blades have width B/f The cleft of the buttocks will have length B/ba The heel will extend to B/ci And having observed these proportions you should fill in the figure carefully, as we strove to do in the example. Just as we called the first figure A, we shall denote this one B.


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Heights of the features of the male body. Crown Forehead Eyelids Nose Chin Throat Armpits Nipples

Loins where we gird ourselves. Navel Top of hip Hip joint Pudenda Cleft of the buttocks Lower buttocks Bottom of scrotum Narrowing of thigh

Upper knee Mid knee Lower knee

Outer calf Inner calf

Top of foot Ankle Sole

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To this male shape we will add, as before, a corresponding female form, whose head represents one eighth of the total height. This is how you should draw her: From the crown to the throat the height will be B/g From the crown to the lower chin B/i From the chin back up to the upper forehead B/ba; and as before you will divide this tenth into three equal parts, corresponding to forehead, nose, chin. From the throat to the loins B/f From the throat to the armpits B/bg From the throat to the breasts B/bb From the throat to the lower breasts B/j From loins to lower buttocks B/bb plus B/bc From loins to lower pubis B/bd plus B/be From the loins to the hip or top of thigh B/ba From the loins to the navel B/ea From the sole back up to the ankle B/da From the sole to the top of the foot B/cb From the sole to mid knee B/i plus B/j Or you can certainly place the knee using the equator, as explained earlier. From mid knee to upper knee B/cg; that is where the knee starts From mid knee to lower knee B/i Then you will move on to the arm. From the shoulder to the elbow the length should be B/f From the elbow to the fingertips B/e From the fingertips to the wrist joint B/ba And this is how you will measure out the width of the woman’s features in profile. At the forehead it will be B/j At the eyebrows B/bg plus B/bh At the nose C/bj At the chin and neck B/bb The neck itself below the chin should have width B/bh At the throat B/be At the chest, above the breasts, B/j At the nipples B/i Below the breasts B/j At the loins or lower back, were they are girt, B/j At the navel B/i At the top of the thigh or hip joint B/g At the thigh just below the buttocks C/bh 32

Upper knee B/be At mid knee B/bg At mid calf B/be At the lower calf B/bi At the bottom of the shin B/cg At the ankle B/bh The width of the foot itself will be B/h And the arm at the shoulder should have width B/be Below the armpits B/bi. Above the elbow B/cg At the muscle of the forearm B/ce At the wrist joint B/ea The hand itself, drawn hanging, B/dg Next you will ascertain the width of the features in the front view, as follows: At the forehead it will be C/bj At the eyebrows B/ba. At the nose B/bc Let the neck under the chin have width B/bh At the throat B/h. A bit below mark the shoulder joints, separated by B/h. At the upper chest B/j plus B/ba Between the armpits B/h. At the nipples B/ba At the loins where they are girt B/h At the navel C/bb At the upper thigh B/j plus B/ba; also place the hip joints there, separated by a distance of B/h At the thigh below the pubis B/ba Above the knee B/bf. At mid knee B/bi At mid calf B/bg At the lower calf B/bj At the bottom of the shin B/de At the ankle B/cj Below the ankle, that is, at the top of foot B/db The width of the foot itself at the tip of the toes will be B/ca Then make the arm thus: At the biceps let the width be B/ce. Above the elbow B/ci At the muscle of the forearm B/cc At the wrist joint B/de Make the palm itself B/ca wide Finally make the back view with width C/bd at the shoulder blades, that is, between the armpits Let the cleft of the buttocks have length B/j The heel will extend to B/da


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So now you will complete the image by drawing all the features nicely, just as you can see we tried to do in the example figure. We will call this figure B I.

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Heights of the features of the female body. Crown Forehead Eyelids Nose Chin Throat Armpits Breast Lower breast Loins where they are girt Navel

Top of thigh Lower pubis Lower buttocks


Lower calf


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One can also modify the preceding figures in various places, as follows. First make the height of the man down to where the legs split off equal to B/c From the crown to the throat C/bb From crown to top of shoulders B/h From crown to lower chin B/i From the lower chin to the upper forehead B/ba Then divide this distance equally into three, for the forehead, nose, and chin, as shown several times before. From the throat to the loins C/bb From the throat to the the lower chest B/j From the throat to the nipples B/bb Also from the throat to the armpits, that is, the upper chest, B/bg From loins to lower buttocks B/bc plus B/bd From loins to pubis B/bh plus B/bi From loins to top of thigh B/ba From loins to curve of hip B/ce Still from loins to navel B/df From the lower buttocks B/ea down, the thigh has a marked furrow. Also from the lower buttocks B/be down is the narrow part of the thigh, frequently mentioned before. From the sole to the top of the foot B/cc From the sole to the lower ankle B/dc From lower ankle to mid knee B/e From mid knee to upper knee, on the outside, B/ca But on the inside, B/da Likewise from mid knee to lower knee, on the outside, B/ia But on the inside, B/ea From mid knee to lower outer calf B/ba From mid knee to lower inner calf B/j The length of the foot will be B/g And this is the length of the arm: From the shoulder joint to the elbow B/ba plus B/bb From the elbow joint to the wrist joint

B/h From the wrist joint to the fingertips B/ba After that you will mark the widths of the features, much as before, starting with the profile view. At the forehead B/ba At the eyebrows B/bh plus B/bi At the nose and temples B/bj At the chin and neck B/ba The width of the neck below the chin is B/bh At the shoulders B/bf At the throat B/bc At the upper chest B/i At the armpits, above the breasts, C/bf At the nipples also C/bf At the lower chest B/bf plus B/bg At the loins C/bj At the navel B/j At the curve of the hip B/bg plus B/bh At the top of the thigh C/bf Let the width of the thigh itself below the buttocks be B/bj plus B/ca At mid thigh, that is, at the narrowing of the thigh, B/ba At the upper knee, outside, B/ch [i.e., C/ch] At the upper knee, inside, B/be At mid knee B/bg At the lower knee, outside, B/bh At the lower knee, inside, B/bg At mid calf B/be At the lower calf, outside, B/bg At the lower calf, inside, B/bg At the bottom of the shin, just above the top of the foot, B/cg At the top of the foot B/ca Now turn to the arms. Make the width at the shoulders B/be Under the armpits B/bj At the elbow joint B/cg At the muscles of the forearm B/cf At the wrist joint B/ee The hand itself, drawn hanging, has width B/de And so you get to the front view.

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At the forehead make it C/bj wide At the ears B/j At the nose B/bc At the neck under the chin B/bh Above the throat B/be At the throat B/g A bit further down, the distance between the shoulder blades is B/g Below the throat, at the chest and shoulders, B/i plus C/h Between the armpits B/bc plus B/bd At the nipples B/j At the loins where we gird ourselves C/bd At the curve of the hip B/bb plus B/bc At the thighs D/ca plus B/cb There too lie the hip joints, B/h apart from each other At the thigh under the buttocks B/bb At the narrowing of the thigh B/bd At the upper knee, outside, B/bg At the upper knee, inside, B/bh At mid knee B/bj At the lower knee, outside, B/ca At the lower knee, inside, also B/ca

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At mid calf B/bg At the lower outer calf B/bi At the lower inner calf B/ca At the bottom of the shin B/dh At the ankles B/cj Below the ankles B/da The width of the foot at the toes B/bh Modify the arm in the front view thus: At the biceps let its width be B/cf Above the elbow B/ch At the widest point of the forearm B/cb At the wrist joint B/de The palm should be made with width B/bi Modify the back view so it has width C/bb between the armpits, and so the cleft of the buttocks is B/bb long Make the foot B/dd wide After this has been done fill in the outline carefully as I have taught earlier. And as an example of all of this I offer the figures that follow.


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Crown Forehead Eyebrows Nose Chin Top of shoulders Throat Armpits Nipples Lower chest

Loins where we gird ourselves Navel Curve of hip Hip joint Pudenda Lower buttocks Narrowing of thigh Upper knee, outer Upper knee, inner

Mid knee Lower knee, outer and inner Lower outer calf Lower inner calf

Top of foot Ankle Sole

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Modify the image of the woman thus: From the crown where the hairs part to the throat the height will be C/bb To the shoulder blades B/h To the lower chin B/i From the lower chin back to the cowlick B/j From the lower chin to the forehead B/ba And this tenth should be divided as shown earlier to place the features of the face. From the throat to the armpits B/bh To the nipples B/bb To the lower breast B/j To the loins were they are girt C/bb From there to the navel B/ea To the thigh B/ba To the lower belly C/bh To the top of the pubis B/h To the lower pubis C/bd To the lower buttocks B/bb plus B/bc From the sole up to the top of the foot the height will be B/cd To the ankle B/df To mid knee B/e Or you can choose to place the knee by using the equator. From mid knee to upper knee B/cf Lower knee B/dg To the lower outer ankle B/bb To the lower inner ankle B/j Let the length of the foot be B/bc plus B/bd The arm comes next; let its length from top of shoulder to elbow be C/bb From the elbow to the fingertips B/e From there back to the wrist joint B/bb Next come the width measurements of the modified figure. First sideways: at the forehead B/bi plus B/bj At the eyebrows B/j At the cheeks and nose B/ba

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At the neck and throat B/bb Let the neck below the chin have width B/bj At the top of the shoulder blades let the width be B/bg At the throat B/be At the breasts B/bi plus B/bj At the nipples B/bg plus B/bh Below the breasts B/bi plus B/bj At the loins where they are girt B/j At the navel B/i At the thighs B/g At the belly C/bd The thigh below the buttocks B/j Above the knee B/bd At mid knee B/bg Below the knee B/bh At mid calf B/bg At the lower inner calf B/bh Make the shin above the foot B/cg wide At the top of the foot B/ce The profile of the arm too can be given proportions different from the ones above. Under the armpit let the width be B/bi At the elbow B/cg At the forearm near the elbow B/ce At the wrist joint B/ce The palm should have width B/dg Now here are the modifications for the front view. Let the head, at the height of the forehead, have width B/bj plus B/ca At the ears B/j At the nose B/bc Make the neck under the chin B/bi wide At the top of the shoulders B/bj The shoulder joints, next to the throat, should be B/h apart At the chest and shoulders let the width be C/g [i.e., C/j] Between the armpits C/bf Between the nipples B/ba At the loins where they are girt B/h At the navel C/bb At the hip or top of thigh B/f


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At the hip or top of thigh B/f At the same height, make the distance between the hip joints B/h Let the thigh below the buttocks be B/ca plus B/cb wide Above the knee the width should be B/bf At mid knee B/bh Below the knee also B/bh At the lower outer calf B/bg At the lower inner calf B/bi At the bottom of the shin B/df At the top of the foot and at the ankle B/da The tip of the foot, that is, the toes, should occupy a width of B/bi

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Now the arm, according to this scheme, should have width B/cc under the armpit Above the elbow B/cj Below the elbow at the forearm B/cc At the wrist B/dg But let the hand itself have width B/ca Finally the back view in this case should have width B/g between the armpits Let the cleft of the buttocks have length B/j The heel should have width B/de And so you will flesh out the image thus constructed by drawing the appropriate lines everywhere; and as usual we have appended an example.


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Crown where the hair is parted Cowlick Forehead Eyebrows Nose Chin Top of shoulders Throat Armpits Breast Lower breast

Loins where they are girt Navel Top of thigh, or hip Lower belly Upper pubis Lower pubis Lower buttocks

Upper knee Mid knee Lower knee Lower outer calf Lower inner calf

Top of foot Bottom of shin Sole

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Next come the proportions of a male body whose length equals nine times the head height. The method is similar to the one we followed above: you should measure out the image one feature at a time, starting with the vertical distances. From the crown to the lower chin let the height be B/j From the chin back to the forehead B/ba, and the cowlick should be at the same height. This B/ba should be apportioned to the various parts of the face as explained previously. From the crown to the top of the shoulders let the height be B/bf plus B/bg From the crown to the throat B/g From the throat to the top of the chest B/ci Under the armpits B/be To the nipples B/bc To the lower chest C/bj To the loins where we gird ourselves B/g From there to the navel B/cg To the curve of the hip B/cc To the top of the thigh or hip joint B/j To the lower belly B/i To the pubis B/h To the lower buttocks B/g From the lower buttocks to the narrowing of the thigh B/bb From the sole to the top of the foot B/cd From the sole to the ankle B/df From the lower ankle to mid knee B/e From mid knee to upper outer knee B/ca From mid knee to upper inner knee B/da From mid knee to lower outer knee B/ia From mid knee to lower inner knee B/ea Likewise from mid knee to lower outer ankle B/ba Likewise from mid knee to lower inner ankle B/j Let the length of the foot be C/bd After this you should mark the length of the arm: measured from the shoulder to the elbow,

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C/bb From the elbow to the fingertips B/e From the fingertips back to the wrist joint B/ba Having dealt with the height of the features you should now mark their widths. First the profile, as follows: Let the head at the forehead have width B/bc At the eyebrows B/j At the nose B/ba At the chin it should be made C/cd At the neck under the chin B/bi At the top of the shoulders B/bh At the throat B/bc At the upper chest B/i By the armpits B/bf plus B/bg At the nipples B/i At the lower chest B/bg plus B/bh At the loins where we gird ourselves B/bi plus B/bj At the navel B/bi plus B/ca At the curve of the hip B/bi plus B/bj At the hips or top of thighs B/bf plus B/bg At the lower belly and buttocks B/i At the pubis B/bg plus B/bh The thigh itself below the buttocks should have width B/ba And further down at the narrow part B/bb Above the outer knee let the width be B/bf Above the inner knee C/db At mid knee B/bi Below the outer knee B/bj Below the inner knee B/bi At mid calf B/bf At the lower outer calf B/bg At the lower inner calf B/bi At the bottom of the shin B/ci At the top of the foot B/ce There follows a description of the arm in profile At the shoulder let the arm have width B/bf Below the armpit B/ca At the elbow B/cg Above [i.e., below] the elbow B/cf At the wrist joint B/ei


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The palm itself should have width B/di Mark the widths of the front view as follows: At the forehead B/bb A bit further down B/ba At the eyebrows B/bb At the ears B/bi plus B/bj At the nose B/bc At the neck below the chin B/bi At the top of the shoulders B/bg At the throat B/g And just below that, the shoulder joints will be C/bd apart At the chest and shoulders let the width be C/j Between the armpits B/h Between the nipples B/j At the loins where we gird ourselves B/h At the navel B/bc plus C/cf At the curve of the hip B/bc plus B/bd At the top of the thigh or hip B/ba plus B/bc Also mark there the hip joints, the distance between them being B/bf plus B/bg The thigh below the buttocks will have width B/bc Likewise at the narrowing of the thigh B/be Above the outer knee the width will be B/bi Inner knee B/bj Mid knee B/cb Below the outer knee likewise B/cb Inner knee B/ca At mid calf B/bh At the lower outer calf B/bj Inner calf B/cb

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At the bottom of the shin where it is narrowest B/ec At the ankle B/dd At the toes, that is, at the extremity of the foot B/bj To this should be added the width of the arm in the front view Below the armpits it will be B/cg Above the elbow B/db Below the elbow at the forearm B/cc At the wrist joint B/di At the palm let it be B/bj Finally make the back view with width B/g between the armpits, and B/df at the heel. The cleft of the buttocks will be B/bb long. After these distances, points and numbers have been marked as shown, you will draw the shape of the desired image by making the appropriate contours. We have provided an example, labeled D. Should you perhaps wish to make the head of this man slightly larger and taller, you can take an eighth part of the whole body length from cowlick to sole, or to be more precise in this argument from crown to sole, and use this dimension for the head, thus making it a little larger without changing anything else. Then the cowlick will take the position previously occupied by the crown of the head. It will also be advisable in the profile view to make the forehead above the eyebrows a bit more prominent. In this way in the full length of this body will be almost eight times the height of the head. The following pages illustrate this variation as well.


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Crown Forehead Eyebrows Nose Chin Top of shoulders Throat Upper chest Armpits Nipples Lower chest Loins where we gird ourselves Navel Curve of hip Top of thigh or hip joint Lower belly Pubis Lower buttocks Narrowing of thigh

Upper knee, outer

Upper knee, inner Mid knee

Lower knee, outer

Lower knee, inner Lower outer calf

Lower inner calf


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This male body is complemented by a matching female one, whose height also equals nine of its own heads. Measure it out feature by feature, thus: From the crown to the lower chin let the distance be B/j From the lower chin back to the top of the forehead B/ba From this B/ba mark out three equal portions for the features of the face, as often mentioned before. From crown to top of shoulders let the height be B/bf plus B/bg From the crown to the throat B/g From the throat to the upper chest B/da To just below the armpits B/bg To the nipples B/bb To the lower breasts B/j To the loins where they are girt C/bb From there to the navel B/cg To the top of the thigh or hip joint B/ba To the lower belly C/bh To the top of the pubis B/h To the lower pubis B/bc plus B/bd To the lower buttocks B/bb plus B/bc And from there to mid thigh B/bc From the sole to the top of the foot B/ec From sole to mid knee B/e From mid knee to upper knee B/cg From mid knee to lower knee B/da To the lower outer calf B/bb To the lower inner calf B/j Let the length of the foot be B/h The length of the arm from the top of the shoulder to the elbow will be C/bb From the elbow to the fingertips B/e From there back to the wrist joint B/bb Now you mark out the widths of the profile view, as follows: Let the head at the forehead have width B/bc

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At the eyebrows B/ba At the nose B/bb At the chin B/bd At the neck below the chin B/cb Top of shoulders B/bj Throat B/be Upper chest B/ba By the armpits let the width be C/bj At the nipples B/bi plus B/bj At the lower breast C/bj At the loins where they are girt C/cb At the navel B/j At the top of the thigh or hip B/h At the lower belly and buttocks B/be plus B/bg At the thigh below the buttocks B/ba At mid thigh C/cb Above the knee B/bf At mid knee B/bi Below the knee B/bj At mid calf B/bg At the lower inner calf B/bj At the bottom of the shin B/ci At the top of the foot and ankle B/cg To this add also the widths of the arm seen in profile. Make the width at the shoulder B/bi Below the armpits B/ca At the elbow B/dj Below the elbow at the forearm B/cg At the wrist joint B/ej The thickness of the open hand is B/ea Now the front view is done as follows: Make the top of the forehead B/bb wide Mid forehead B/ba At the eyebrows B/bb At the ears B/ba At the nose B/bd At the neck below the chin B/cb At the top of the shoulders B/bj At the throat C/bd


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Below this point mark the shoulder joints, separated by a distance of B/h At the upper chest B/f Between the armpits B/i Between the nipples B/bb At the lower breast B/bf plus B/bg At the loins where they are girt B/i At the navel B/bb plus B/bc At the top of the thigh or hip B/ba plus C/cb And there too mark the hip joints, separated by B/bf plus B/bg At the lower belly B/ba plus C/cb At the thigh below the buttocks C/cb At mid thigh B/cf Upper knee B/bh At mid knee B/bj Lower knee B/ca At mid calf B/bh At the lower inner calf B/ca At the bottom of the shin B/dj

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At the top of the foot B/db Let the foot at the tip of the toes have width B/ca To this should be added a proportional arm: Below the armpits B/cg wide Above the elbow B/dd Below the elbow at the forearm B/cd At the wrist joint B/ea The width of the palm will be B/cc Finally in the back view make the width between the armpits equal to B/h The cleft of the buttocks B/ba The heel will have width B/dh And so, having marked all the features, fill out the shape as we have often explained and as we have done in the example figures below, which we have labeled D I. Should you want to make the head a little larger on the same body, follow the procedure discussed in connection with the preceding figure.


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Crown where the hair is parted Forehead Eyebrows Nose Chin Top of shoulders Throat Upper chest Armpits Nipples Lower breast

Loins where they are girt Navel

Top of thigh, or hip Lower belly Upper pubis Lower pubis Lower buttocks Mid thigh


Lower outer calf Lower inner calf


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I will put forth one more shape, slender and taller, whose height is ten times that of the head. I will first teach how to make the male body. The heights of its features will be these: From the crown to the lower chin let it be B/ba From the lower chin back to the upper forehead B/bb In this space the shape of the face will be drawn. From the crown to the top of the shoulders C/bh To the throat B/bd plus B/be To the shoulders B/g From the shoulders to the top of the chest B/cf To the armpits B/bh To the nipples B/bd To the lower chest C/cb To the loins were we gird ourselves C/bd From there to the navel B/da To the curve of the hip B/cb To the top of the thigh or hip B/i To the private parts B/be plus B/bf To the bottom of the scrotum C/bd To the lower buttocks C/bb From there to the narrowing of the thigh B/bb Place the middle of the knee between the lower buttocks and the sole. From the sole back up to the lower ankle let the height be B/df From the sole to the top of the foot B/cg From mid knee to upper knee B/da To lower knee B/ea From mid knee to lower outer calf B/ba To lower inner calf B/j The length of the foot itself will be B/h Then measure out the lengths of the arm, as follows: From the shoulder joint to the elbow make the lengths C/bb There are two possibilities for the lengths from here down. Either make the distance from elbow to fingertips B/e and go

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back B/bb to the wrist joint; or make the distance from elbow to fingertips D/bb and go back C/cb to the wrist joint. Then come the widths. First mark them in the profile view, next to the points indicating height, as follows: At the forehead let the width be B/bd At the eyebrows B/bb At the nose and temples B/bb At the mouth B/bd At the chin B/be At the neck below the chin B/cc At the top of the shoulder blades B/ca At the throat B/bd At the shoulder joints B/bb At the upper chest B/bh By the armpits B/i At the nipples likewise B/i At the lower chest C/bh At the loins where we gird ourselves B/bb At the navel also B/bb At the curve of the hip B/ba At the top of the thigh B/bh plus B/bi At the private parts B/j The thigh itself below the buttocks should be given width B/bb The middle or narrowing of the thigh B/bc Above the knee let the width be B/bh At mid knee B/bj Below the knee B/ca At mid calf B/dc plus B/de At the lower outer calf B/bh At the lower inner calf B/bj Above the top of the foot B/dc At the top of the foot B/cj At mid foot, that is, below the ankle B/cd The profile view of the arm shall have these measurements: At the shoulder B/bh Under the armpits B/cb At the elbow B/da Above [i.e., below] the elbow B/ci At the wrist joint B/fa


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The palm itself B/ce There follow the widths in the front view. Make them as follows: At the forehead B/be At the upper temples B/bc At the eyebrows B/bd At the ears B/bc At the nose B/bf At the neck below the chin B/cc At the top of the shoulder blades B/ca At the shoulders B/bc plus B/bd Just below that, mark the shoulder joints separated by B/bc plus B/bd At the chest and shoulders D/bi plus B/bj Between the armpits B/h Between the nipples B/ba At the lower chest C/bd At the loins where we gird ourselves C/bf At the navel B/bd plus C/ch At the curve of the thigh B/bd and B/be At top of the thigh B/g Also here make the distance between the hip joints C/bf Let the thigh below the buttocks have width B/bd At the narrowing of the thigh B/bg Above the knee let the width be B/ca At mid knee B/cc Below the knee B/cd At mid calf B/bj At the lower inner calf B/cd At the shin B/ef

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At the ankles B/df Below the ankles B/eg The width of the foot at the toes B/cb Now the arm in the front view should have width B/ci below the armpits Above the elbow B/de Below the elbow at the forearm B/ce At the wrist joint B/ec Let the width of the palm be B/cc Next let the width of the back view at the armpits be B/bc plus B/bd Let the cleft of the buttocks be B/bb long Make the heel B/dh wide And having laid down all these features, complete the image by drawing the appropriate contours, as you can see we have striven to do in the example figures. Should you wish to make the head slightly larger, take a B/j length segment, make its lower end the chin and its upper end crown. Thus the height the head has grown slightly, and the width should also be increased proportionally; the rest of the body remains as stated earlier. We have included an example of this in the following figures. The proportion between head and body is now no longer 1 in 10 but almost 1 in 9. Let the label of this image be E.


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Crown Forehead Eyebrows Nose Lower chin Top of shoulder blades Throat Shoulders Upper chest Armpits Nipples Lower chest Loins where we gird ourselves Navel Curve of hip Top of thigh Pudenda Bottom of scrotum Lower buttocks Narrowing of thigh



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Next I will show the proportions of a female body matching the male figure just described. Its heights will be as follows: From the crown where the hairs part to the lower chin B/ba From there back to the top of the forehead B/bc And in this B/bc will be distributed the features of the face, as often shown before. From the crown to the flesh below the chin—the dewlap as it were—B/bi plus B/bj To the top of the shoulder blades B/i To the throat C/bd To the shoulders B/g To [i.e., From] shoulders to upper chest B/cc Below the armpits B/bg To the nipples B/bb To the lower breasts B/j To the loins where they are girt C/bb From there to the navel B/ea To the top of the thigh B/ba To the lower belly let the distance be C/bh To the top of the pubis B/bd plus B/be To the lower pubis B/bc plus B/bd To the lower buttocks B/bb plus B/bc From there to mid thigh B/bc From the sole back to mid knee let the distance be C/h To the lower ankle B/di From mid knee to the top of the foot B/e To the lower inner calf B/j Lower outer calf B/ba To the lower knee B/da Again from mid knee to upper knee B/da And the length of the foot B/be plus B/bf The lengths of the arm shall be as follows: From the shoulder to the elbow let it be B/bb From the elbow joint to the fingertips B/e From the fingertips back to the [wrist] joint B/bb After the height of the features has been set down as above and marked everywhere, you will mark the widths, starting with

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the profile view: At the forehead let the width be B/bd At the eyebrows B/bb At the nose and eyebrows B/bc At the chin B/be Make the neck below the chin B/ce wide At the top of the shoulder blades B/cc At the throat B/bh At the shoulder joints B/bd At the chest B/ba At the nipples B/bj plus B/cb At the lower breasts B/ca plus B/cb At the loins where they are girt B/bb At the navel B/ba At the top of the thigh, or hip, B/be plus B/bf At the lower belly B/bf plus B/bg At the top of the pubis and the buttocks B/bg plus B/bh The thigh below the buttocks B/ba Further down at the narrowing of the thigh C/cd Above the knee B/bh At mid knee B/bj Below the knee B/ca At mid calf B/bh At the lower outer calf B/bi At the lower inner calf B/bj At the bottom of the shin B/dc At the top of the foot and the ankle B/ch And the arm will be as follows. At the shoulder it has width B/bj Below the armpits B/cd At the elbow B/de Below the elbow B/da At the wrist joint B/ga The open hand has thickness B/ef Now come the widths in the front view. At the forehead let it be B/bd At the upper temples B/bd At the eyebrows B/bd At the ears B/bc At the nose B/bg Let the neck below the chin have width B/cf


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At the top of the shoulder blades take the width as B/cc At the throat C/bh At the top of the shoulders B/h. Almost at the same height are to be placed the shoulder joints, separated by C/bf At the upper chest let the width be C/bb Between the armpits B/j Between the nipples B/bc At the lower breasts B/i At the loins where they are girt likewise B/i At the navel C/bd At the top of the thigh B/bb. Also nearby are the hip joints, separated by B/i At the lower belly the width will be C/bb At the thigh below the buttocks B/bc At the narrowing of the thigh C/ch Above the knee B/bj At mid knee B/cc Below the knee the same At mid thigh B/bj At the lower outer thigh B/cb Lower inner thigh B/cd At the bottom of the shin B/ei

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At the top of the foot and ankle B/ed Further down B/eg The width of the foot itself at the toes shall be B/cc The arm belonging to this drawing will have width B/ci below the armpit Above the elbow B/de Below the elbow B/cg At the wrist joint B/eg Let the palm have width B/ce Next, the width of the back view between the armpits shall be C/bf The cleft of the buttocks B/ba The heel B/ea Then, as often mentioned, you must fill in the right shape around and inside the points marked, completing each view: profile, front, and back. And if you wish to make the head slightly larger, do so as we showed for the male body, and as exemplified in the following figures. Let the label for these figures be E I.


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Crown where the hair is parted Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Armpits Nipples Lower breast Loins where they are girt Navel

Top of thigh, or hip Lower belly Upper pubis Lower pubis Lower buttocks




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It is likely that the reader should know how the images drawn according to the method shown are to be bent at the spine and joints. Nonetheless, in order for our explanations to be understood more quickly, the preceding profile drawings use little triangles to mark the length and curvature of the spine, which everywhere lies about a quarter of the way from back to front. Likewise in the front view you will see little circles marking several joints. And although the preceding images have all been made of the same height, using the same length ruler to save effort, if they were to be used together in a single drawing it should be kept in mind that the thinner ones should be made slightly taller. If you make a man and a woman using the same, or rather matching, proportions, the length of the ruler giving the height of the woman should be B/bi less than that of the man. If this precaution is not taken the woman appears to be bigger than the man, since the nude female body should be drawn somewhat fleshier than the male. The male body is more compact and taut, but less fleshy. We have covered in the images above a range of body types, from the stockiest to the slenderest, so that any reader would have the ability to handle fully and variously whatever work he decided to do based on these models. We will discuss this in more detail in the sequel. As you recall, we described above the scheme of distances of four human body types. We will next investigate in detail and more exactly the features and composition of one part of the body, the head, starting with the male. We will continue to follow the same method, except that the ruler (fundamental length) will be longer than the one used for the earlier figures, since we will deal with smaller parts, and a short ruler would not allow a clear picture. Having before you a profile view of a male head, whose height is B/i of the total body height, you will draw around it a quadrilateral with equal sides, that is, a square. Call a the side that touches the nose, b the back side, c the top, and d the bottom. After this you will measure out the vertical position of all the parts of the head, and draw parallel horizontal lines at those heights, labeling them with appropriate letters. Thus you will easily collect the measurements of all the parts of the head, as we have done in the examples offered below. The procedure is this: Draw a horizontal line e at a height B/ba above d, as done earlier in the discussion of the male body eight heads tall. This line will divide hair from forehead. Place the cowlick a third of the way up from e to c. Draw two lines dividing cd into three equal parts: the upper line, f, will touch the eyebrows and the top of the ears, and the lower one, g, the bottom of the nose and ears; also here the occiput meets the neck. With two points divide ef into equal thirds; through the lower point draw a horizontal line h, marking where the forehead begins to project beyond the eyebrows; above h the forehead curves back to the hairline. With two points divide fg into equal thirds; through the upper point draw a horizontal line i. Lines f and i delimit the eyes, whose inner and outer corners lie halfway between the two. Likewise, a quarter of the way up from g to f, draw a horizontal line k, marking the upper limit of the flare of the nostrils and of the earlobe. Divide gd in half with a horizontal line l, marking the top of the chin, whereas d marks the bottom. Divide gl in half with a line m, which will cross the mouth at the middle. After you divide gm into three 59


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equal parts, the lowest will be taken by the upper lip, and the other two by the little cleft below the nose. After dividing lm into two equal parts, the upper one will be taken by the lower lip, and the other by the depression between mouth and chin. And so you will have all the main features of the face delimited above and below by horizontal lines. There remains now to indicate the horizontal position of the features: their width or thickness, or as some say their depth. This will be specified by means of vertical lines. The word depth should be understood to mean the distance to the front plane, containing the [tip of] the nose. Divide sides a and b into seven equal parts by means of six vertical lines nopqrs. Line n will touch the pupil of the eye and the corners of the mouth. The corner of the eye near the temples will be midway between n and o. These instructions are to be understood to apply whether the profile faces right or left, the ordering of the lines n to s being chosen accordingly. Draw a slanted line from where c meets n to where a meets f, and label it t. Draw another slanted line from the same intersection cn to where a meets g, and call it u. The forehead will curve along line nt [i.e., t], and the nose should be drawn along line nu. The crossing of the horizontal line e with nt marks the hairline, or top of the forehead. Next draw a vertical line [halfway] between n and a, delimited by the horizontal lines d and g; this new line x is tangent to the upper lip and to the chin. Divide the space between n and x into three, by means of two vertical lines: the foremost will touch the lower lip and the other the already mentioned depression between lower lip and chin. Draw the eyebrows almost to o, sticking a bit above f at the middle. The whole ear is enclosed between the verticals q, r and the horizontals f, g; the earlobe lies in the front half of the space between q and r, and between k and g. Put the cowlick a third of the way back from s to b, and a third of the way up from e to c. Place the connection of the head with the neck on line g, a quarter of the way from s to b. After having done this, start drawing the outline of the head inside the grid: the top of the head should touch line c between p and q, and the back should touch b between h and f, continuing down still in this quadrilateral to meet the neck, which then slopes down to the bottom of the square. Next draw the top of the head up to the forehead, and from there down to the nose and mouth. The tip of the nose will touch a between k and g. In this way the head will be touching all sides of the square in which it is contained. Then draw the jaw up and back past the ears, and put the ears and eyelids in their place. Delimit the whole forehead by drawing the hairline along e and down between p and q, roughly to mid ear. The chin curves down to meet the neck on line d, a quarter of the way back from o to p. The width of the neck can be made equal to B/be or C/ch. One should fill in these outlines with great care, otherwise the figure will appear somehow deformed even if no great error has been made in marking down the features and making the grid. Having thoroughly discussed the shape of the head in profile, we now deal with the front view in the same way, inscribing it in a rectangle. The height will be the same, but the width less. Inside the rectangle you will draw horizontal lines in order, following the same instructions as for the profile, and label them as before, from c, the topmost, to d, the bottommost. 60


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These lines will bound the features of the face above and below. Next you will cut them with vertical lines, to delimit the widths. Draw two vertical lines a and b, B/ba apart, joining the top and bottom of the rectangle and so completing its border. Divide the space between a and b equally into five, with vertical lines cdef. Halfway between d and e place the top of the head, between the horizontals c and e. Draw the nose so the flares of the nostrils touch d and e respectively, and stay between the horizontals k and g. Place the corners of the mouth where m crosses the verticals d and e; you will draw the chin between the same two lines. Place the four corners of the eyes halfway between the horizontals f and I, on the verticals cdef. Divide the vertical strips ac and fb in half lengthwise with new lines g and h; the crossing of these new lines with k will mark the upper cheeks. The eyebrows end between g and c and between f and h, just below the horizontal f. Divide the strips ag and hb in half, again lengthwise, with lines i and k; they mark the outer limit of the hair line, and between them is the expanse of the forehead, limited above and below by e and f. The temples go outside the lines ik, so that one half of each ear is covered and disappears from sight between the horizontals f and g. The ears themselves should be formed between ai on one side and kb on the other, delimited above and below by f and g; the top goes beyond the borders a and b, but the bottom approaches the verticals i and k. Having done all of this, draw the contours so that the top of the head touches the horizontal line c between the verticals d and e, then curves down on either side toward the verticals a and b, touching them near the horizontal h; from there to the ears at the horizontal f, between a and i on one side, k and b on the other. Draw the expanse of the forehead, as well as the jaw, along the verticals g and h, down to the horizontal m, and concluding with the chin. Also draw the neck from the horizontal m down to d, making it B/bf wide or a bit more. Complete the face by drawing in appropriately the eyes, nose, mouth, chin, and other parts. Finally you will turn to the third quadrilateral, also a rectangle, in which you will draw the back view of the head. Do it according to the description of the contour of the front view, and place the cowlick in its place. And below, where the head curves in, form the neck, nicely and harmoniously. Now suppose you want to project the head made according to the method we have explained, so that the circumference of all its parts, as well as its boundaries and surfaces, appears at a given horizontal level, as if the head were made of wax and were cut by one of the horizontal lines indicated earlier—one would then know the shape of the plane figure determined by the cut, called a cross section. If you wish to approach this question, make yourself a triangle as I will now show. I call this triangle a transferent, because it allows one to transfer or modify one thing into another maintaining the correct proportions. It is prepared as follows. Take a triangle having one right angle and label its vertices 1 2 3. There will be exactly one right angle, as I said, but otherwise the triangle can be arbitrary, with sides equal or different, long or short. One either side of this triangle—it does not matter which—mark certain points, such as 4 5 6 in the following figure. From these points draw parallel lines to the other side of the triangle, which will thus become divided in the same proportion; from its division by the new points draw other parallels to the hitherto undivided side, and so it too will be divided in the same proportion.

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From the diagram just described you will be able to determine the distances of many things. We will begin by showing how it can be applied to the task of finding the cross sections.

Transferent diagram

This is how you apply this transferent to the cross sections, as they are called. Have before you both the profile and the front view of the face, prepared as previously said. From the square that contains the profile view extend all the vertical lines anopqrs as far down as necessary. Likewise from the rectangle containing the front view extend the vertical lines aigcdefgkb. Beneath the two views, crossing all the vertical lines that you have extended in this way, draw now a horizontal line, and mark the points where it intersects the vertical lines a and b of the front view with the labels 1 and 2, respectively. From the point labeled 1 measure down a distance equal to the width of the front view, and label 3 the point at the end of this interval; then connect 2 and 3 with a slanted line or hypotenuse, and you will have completed the triangle that I call the transferent. Next notice where the slanted line 23 intersects the vertical lines extended from the front view, and through those intersection points draw horizontal lines so they meet the vertical lines extended from the profile view. At the points where the horizontal lines 21 and 3 cross the vertical lines extending the sides ab of the profile view, write labels 4 and 5 for the points in front and 6 and 7 for those in the back. This is the rectangle inside which will be contained the cross sections of the head.

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This rectangle having been made and marked with 4 5 6 7, and the lines of the front view and profile having been drawn with the help of the triangle you prepared, the rectangles that you see inside will indicate where a given section of the face should be placed. For these little rectangles will hold the location of the corresponding parts, such as eyes, nose, mouth, chin, ears, neck—in short, of all the lines of the whole head. Below are examples of all we have discussed: profile, front view, back view, and cross sections. It is extremely useful to have these things investigated and the plane sections of the parts known. Moreover, should one prefer to make the head so its length is B/h that of the body, as that of the earlier figure was B/i, he should first enlarge the profile view of the head, while preserving all the other parts. The neck should be C/ch wide. Likewise also the front view of the head can be suitably enlarged so that the width at the forehead is C/bh. But it should narrow again near the ears; they too will protrude more. Likewise the line of the hair and the expanse of the forehead will be wider, all of which the interested reader can see in the examples below. Keep in mind that the correct proportioning of a body equal to seven of its own heads is in no way to be scorned.

Using the same method just explained for drawing cross sections of the head, one can make cross sections of the whole body, whether erect or in any position.

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I will now describe the hand of a strong man more exactly, part by part; as mentioned before I will make its length B/ba. Draw two horizonal lines c and d separated by a vertical line of height B/ba, called ab. Likewise make another vertical line ef at a fair distance from the first. Using little points or ticks, divide ab into 19 equal parts. Place ef so it will go through the middle of the hand, extending from the tip of the middle finger where it will touch c down to d, which marks the joint of hand and forearm, that is, the wrist. Draw through point 1 a horizontal line g; it will touch the tip of the next to last finger, the one the Greeks call paramesos. Through the midpoint between 1 and 2 draw a horizontal line h; it will touch the tip of the index finger. Through point 4 draw a horizontal i; it will touch the tip of the little finger. Finally, through point 7 draw a horizontal k; it will touch the tip of the thumb. After you have marked the tips of the fingers, you will set up their lower parts, the roots as it were, where they come out of the hands. From point 11 draw a horizontal line l; it will cross the lowermost knuckles of the index and middle fingers, both of which can be seen to lie at the same level in the hand. Thus the middle finger is longer than the rest of the hand below it. Write m where the line l crosses the vertical ef, and place the center of a compass at f, the other tip at m. From there draw an arc through the hand toward the little finger, as far as necessary; on this arc you will place the lowermost knuckle of the little finger and of the next, which are lower than those of the other fingers. Then draw from tick 14 a horizontal line n, which will go through the thumb’s lower knuckle. Having measured out the heights of the fingers you should now locate all the joints, in the following way. From point 10 draw a horizontal line o, which will go through the upper knuckle of the thumb. Divide the index finger with a horizontal line halfway between h and l; it will go through its middle knuckle. Divide the part of the finger above this line into 7 equal parts by means of 6 points; through point 3 draw a horizontal line, which will go through the upper knuckle. A horizontal line p drawn from point 6 will cross the middle knuckle of the middle finger. Likewise divide the space between the lines c and p into 7 equal parts by means of 6 points, and draw a horizontal line through point 3; it will cross the top knuckle of the middle finger. The horizontal line k can be seen to cross the middle knuckle of the next to last finger. Divide the portion of the finger above this line into 9 equal parts by means of 8 points, and draw a horizontal line through point 4; it will cross the top knuckle of the same finger. The length of the little finger can be left as determined from the preceding procedure, or it can be made equal to the distance between the middle to the lower knuckles of the middle finger. Divide this length into 11 equal parts by means of 10 points, and draw a horizontal line through point 5: it will cross the middle knuckle of the little finger. Again use 10 points to divide the upper portion of this interval into 11 parts, and draw a horizontal line through point 6: it will cross the top knuckle of the little finger. Make sure not to confuse the numbers and measurements on the line ab with those on the other, shorter, vertical segments. Make the length of the nail of each finger equal to half the distance between the finger’s extremity and its first, or uppermost, knuckle. Note also that the gap between the middle finger and either of its neighbors is located at midpoint of the length of the hand, but the gap between the little finger and its neighbor is somewhat lower, and between the thumb and the index finger still lower. 64


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Now that you have fixed the lengths of the hand and of its parts, you will next investigate their widths. Moreover we must depart from the procedure used earlier for drawing the body and face, where we would first investigate of the profile and then the other views. In the first place you will set down the widths of the open hand (which does not belong to the profile view), part by part, as follows: Make the width of the hand along line d, where it meets the arm, equal to H/bj of the length, which equals the distance ab. Along the horizontal line m [i.e., n] make the width BB/bj of the length: the hand should be widest at the thumb knuckle. Next, the width from the outer edge of the index finger, at the level of the horizontal line l, going through the lower knuckles and along the arc of circle mentioned earlier, to the outer edge of the little finger, should be made equal to the length of the index finger. The middle finger, the longest, has width at the base equal to B/f of its length; but near the top the width tapers down by B/e. The index finger is as wide at the base as the middle finger, but is a bit narrower near the top. The little finger and its neighbor should have width B/f of their own height at the base, tapering down by B/e near the top. Make the width of the thumb at the middle equal to B/d of its height. The fingers are inserted in the hand in such a way that the root of each one is placed at a depth equal to the width of the finger; this is indicated by means of little circles in the example figures. It should be noticed here that no two fingers have the same shape, as can easily be seen. Notice also that the edge of the hand is not straight but bulges somewhat just before the little finger. Having done this you should now turn to the width or thickness of the hand in profile. At the wrist joint, where the hand meets the arm, let its width be F/bj of the length ab. At the lower knuckle of the thumb, at the level of line n, make the thickness half the width prescribed above for the open hand. Make it the same at the fleshy part of the palm. But since the hand curves in, so to speak, at that point, where the base of the thumb and of the little finger get closer together, the actual width of the hand is not as much as it appears from the profile or as would be indicated by the outer contours. At the first knuckle make the thickness of the thumb equal to the length of the little finger above the upper knuckle; but at the top decrease the thickness by B/e. Again, at the bottom the thumb should be made fairly thick. The fingers are almost as thick in this direction as they are wide in the earlier view of the open hand; only at the knuckles is the thickness significantly less. And so, having marked down the lengths, widths, and thicknesses of the hand, complete its form by drawing the appropriate lines, as you can see in the example figure given below. And if one seeks to imitate this model, he should work with great diligence in order not to deform anything or go astray in drawing the lines, since indeed the proper accomplishment of this drawing is not for the common talent or the lazy hand. (continued)

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Now then I will describe the foot of a strong man more exactly and in detail, as I did for the hand. We continue to work with a ratio of head to body length of B/i. The foot will have length B/g and width B/bh. Make these the dimensions of a rectangle, drawn with the longer sides horizontal. Label its top side a and the bottom side b. Of the shorter sides call c the one on the right, where the toes will be, and d the one on the left, touching the heel. In this rectangle the foot will be drawn, seen from above, in such a way that its contours touch all four sides of the rectangle. First divide the length of the rectangle between cd into three equal parts by means of two vertical lines e and f, with e toward the front. Note that although letter labels will be assigned in mixed order, any new group of letters should be assumed to label elements front to back. Having made these subdivisions, in the first, between c and e, you will place the toes; in the middle one, between e and f, the portion of the foot between toes and ankle, containing bones and nerves; and in the last the heel and the meeting of shin and foot. Divide line d into seven equal parts using six points ghiklm, and from g draw a [horizontal] line to c. Call n the point where this line meets c, and o the point where it meets f. Then divide lm in half at p, and from p draw a horizontal line to the vertical line f, labeling q the point where the lines meet. From there draw a slanted line to k. Next place a point r on side c halfway between a and b, and divide rb into three equal parts with two points S and T. Then divide be into three equal parts with two points u, x, divide ef into three equal parts with two lines y, z, and divide fd into three equal parts with two points st and sz. Where g intersects e place the label tz. Then from point h draw a horizontal line toward line f, and label their intersection a1. Divide both rs and xe into three equal parts by means of two points, in each case labeling them 1 and 2, and divide sz.d into five equal parts with four points, labeling them 1, 2, 3, 4. From point 3 draw a vertical line down to the horizontal ha1 and there mark the point b1. (You should know that according to our procedure the labels for all the short lines are to be found written above the horizontal line a.) Now draw some slanted lines: one diagonally from e to n; one from r to tz; one from point 2 on rs to point y; one from t to z; another diagonal from b to o; and one from u to a1. Draw from point st a [vertical] line down to slanted line kg, calling the intersection c1. Next place the center of the compass at tz and the pencil tip at n, and draw an arc down to the slanted line joining points r and tz. Still with same center and radius draw an arc between the slanted lines tz and bo. Still with the same center, close the compass a bit more so it reaches point 2 on line b between x and e; from there draw an arc up to the slanted line bo. Still with same center, open the compass again so it reaches point r, and draw an arc down to the oblique line joining t and z. These short arcs will mark the tips of the toes. And the innermost [proximal] toe joints, where the toe connects with the foot, shall be placed as follows. Mark a point on line e halfway between the lines bo and ua1, and place there the inner joint of the next to last toe. Then open the compass with center at z, placing the other tip at the point just made on line e, and draw an arc from this point up to the horizontal line gn.

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The inner joint of the long toe will be at the intersection point of the arc with the line gn. Mark two more points on the same arc, the first halfway between the slanted lines sy and tz and the other halfway between tz and bo. At those points place the inner joints of the middle toe and the one next to it; you will see these points marked by elongated circles in the drawing below. The inner joint of the little toe is on line e, halfway between the horizontal line b and the slanted line ua1. The big toe has two segments; the one before the joint is a bit longer than the outer one. The three middle toes are divided so that the first segment has the same length as the outer two together. The last segment of these toes should exceed the middle one in width by B/e, because the flesh below the nails extends to both sides. Next to the inner joint of the little toe make a fairly sizable bulge, and place the toe’s end at line x. Make this toe’s inner segment a bit longer than the others. Each toenail should be half as long as the toe segment to which it belongs. Then, centering the compass at y and placing the tip on line e just above the inner joint of the little toe, draw short arcs indicating where the foot branches into toes. Next extend the clefts between the toes further in slightly, and make little gulfs, as it were. Note that line f touches the top part of the foot, where it meets the leg. A horizontal cross section of the foot taken a bit further down (nearer the sole) will extend to C/d of the way between lines f and z. Finally divide sz d into three [i.e., five] equal parts by means of four points, and from point 3 draw a vertical line, which will touch the back of the shin next to the foot, at the Achilles tendon, called in Greek tenon. Having so arranged all horizontals, verticals, diagonals, and curves, by means of which the top view of the foot is to be described, we now turn to the actual drawing of this view. For the big toe start from e and stay slightly above the slanted line en, as the toe’s first part sticks out. Soon you will cross line en and stay slightly below it, and again back up to line en, as the other part of the toe sticks out on both sides. From there the contour curves so it is tangent to the line nr, and then folds back in, following above the line rtz to the meeting of the two toes, where the foot branches. Likewise, enclose each of the other toes within a curved boundary drawn between the prescribed slanted lines. Note that the big toe does not stick forward straight in a well-formed foot, as it does in some inelegant bodies. In order to make this clear I have marked the place of the toes in the figure using slanted lines, by means of which the toes are compressed more or less, as required by the elegance and the plan of the joints. So note that between the big toe and its neighbor there is some space, since all others are turned toward the little toe, whereas the little toe is turned in toward the others, along the slanted line ua1. Make the length of the little toe a third of the length of line b, measuring from point [i.e., line] e where its innermost joint is located. I would like you to know well and consider attentively the appearance of each toe (they are all different), the bulge of the joints, and the shapes of other parts that do not even have names. For this reason I refer you to the example figure, where these details can and should be observed. Then, moving back from the little toe, draw a bulge as it were tangent to line b at the corner with y. Make the inner contour of the foot, starting at e on edge a, convex up to line y, staying between 68


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the lines a and ng; then concave till it reaches line f within the segment fo, then convex again all the way to the heel. There start to curve down until it touch line d at point h. Continue the asymmetric curve of the heel on to the slanted line kq. Near line st there is again a concavity. From point q continue on until the already mentioned bulge near the little toe, stopping at the intersection of lines y and b. In our drawing the contour of the sole or footprint is shown by a dotted line. And the boundary, so to speak, where the foot meets the leg has width B/cg between the opposite sides of the ankle, along line st, but B/db below the ankle; the forward limit of this boundary lies C/d of the way from line f toward z. Make the inner ankle more rounded and a bit further forward from the heel in comparison with the more angular outer ankle; also the concavity between the inner ankle and the Achilles tendon at the back is less curved and longer than the one between the latter and the inner [i.e., outer] ankle. Then draw the slight mound next to the leg on the outer side going from line f to line z, to make the foot more elegant; make likewise the mound near the little toe, starting halfway beween lines y and z. Take special care here so the figure’s appearance is not distorted by erroneous lines. I have also supplied two separate figures showing vertical cross sections of the foot through lines e and f, to make our method all the easier to perceive. In the same way as in these two figures, one can investigate the cross section of any part of the body by whatever plane, vertical or oblique. Next you will draw the foot lying on the ground, seen from the side, as if part of a profile view. Mark out a vertical segment of height B/ce, as I have done in the example figure. Using the transferent, extend from the previously made top view lines c, e, y, z, f, d, and others, preserving their order and distances. After drawing these vertical lines, close them off with two horizontals. Thus you will have made a rectangle that is longer than it is tall: it has the same length as the top view of the foot, but its height is B/ce. This is where you will draw the side view. Then divide the space between a and b into three equal parts by means of two horizontal lines e and f. Thus the vertical lines will indicate the lengthwise placement of all features, such as the tips of the toes, then the toe joints and separations, the positions of the nails, and finally the whole view of the foot as it stands, and the boundary where the shin connects with the foot. The horizontal lines will show you the heights (which in this case mean thickness), as follows: Place the sole or footprint flat on line b. Then divide side c between e and f into three equal parts by means of points 1 and 2, and divide the vertical line e between the horizontals e and f into four equal parts by means of points 1, 2, 3. Draw another horizontal line between the horizontals a and e, going from z to d. Guided by these labels, draw the lines of the foot: starting from the intersection of a and f, make the top of the foot down to point 3 of the vertical line e. From there level off along the horizontal line f at the inner joint of the big toe, and continue with this toe’s profile out to line c, which it touches a bit further down at 1. But the inner joint of the little toe and its corresponding mound reach back up D/e of the way from b to f. Make sure you draw the remaining toes correctly: in order, they are each a bit longer and thicker than the next, and the tip of the little toe tapers a lot. The mound behind the little toe should be extended back beyond line y. The other remarkable protuberance of the foot appears further up: its top is level with the horizontal 69


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line e, the bottom reaches B/d of the way from the horizontal line f to b, in the direction of the toes it reaches the crossing of the horizontal line z with the vertical f, and toward the heel it goes past the vertical line st between the horizontals e and f. Round off the heel so it is tangent to the vertical line d at the crossing with f. Continue up to the crossing of the horizontal e with the vertical line at point 3, and on to cross line a between sz and d. From there on continue with the Achilles tendon. Draw the outer ankle, the one that is visible, from point sz to slightly below the horizontal that sits halfway between a and e, and above the top of the foot; the width or thickness of the ankle is B/ce. The contour of the shin is picked up from the crossing of c with the vertical line f, going up and toward the back. The shape of the heel below the ankle is drawn by a line extending forward and down from the horizontal line e. Next we set forth a description of the back view of the foot. Enclose it in a rectangle having the foot’s height or thickness, and likewise the same width as the foot. Using the transferent, bring over the lines and points from the earlier drawings so that all parts of the foot are located. Keep the two sides of the ankle at different heights: the inner ankle is higher, the outer lower. The width at either ankle bulge will be B/cg; at the line halfway between a and e make it B/db. Make the shin above the ankle B/de wide. Using a similar method one can also draw the front view of the foot. And just as we encompassed all these things by means of rules, so we have strived to offer complete drawings to the eyes of the interested reader, thus shedding some light onto the rules, for it is well known how helpful examples and illustrations are in understanding.

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Having finished the foot, I will now describe a female head in more detail than before, taking the same road as for the earlier study of the male head. Just as in the [initial] drawings you were told to make the height of the woman’s head B/i, so it will be here, and this stipulation will make the proportions agree with the male head studied earlier. The whole thing is done as follows. Set up three rectangles, with the help of the transferent: each will have height B/i, and the first will have width B/i as well, so it is a square, but the other two should have width B/ba. The square will be devoted to the profile; the other two rectangles are for the front and back views, respectively. Label the two vertical sides [of the square]: the right edge, faced by the nose and mouth, is a, and the left edge is b. Using vertical lines between a and b you will locate the back-to-front position, or depth as we say, of all the features of the head, starting from the face and moving back. You will mark the lines with letter labels. The measurements will be in fractions of the height of the whole body. The first line c is B/ia back from a; it is touched by the part of the forehead where the hair starts, by the tips of the eyebrows (which should not join above the nose), and by the upper lip. The next line is d, B/ge back from a; it is touched by the eyelids, the lower lips, and the tip of the chin. Next comes line e, B/fg back from a; it is touched by the concavity above the eyelids, by the pupil, the flare of the nostril, the corner of the mouth, and the depression between mouth and chin. Next is line f, B/ea back from a; it is touched by the corner of the eye, near the temples. Next draw line g, B/dc back from a; it is touched by the ends of the eyebrows, near the temples, and by the fold under the chin. Then comes line h, B/cc back from a; it marks the edge of the neck. Then line i, B/be back from a; it marks the beginning of the ear. Then line k, B/bb back from a, which is touched by the rear contour of the ear; but the earlobe only goes to halfway between i and k. Then line l, B/j back from a, containing the cowlick and the junction of occiput and neck; the latter continues down. Finally draw line m at a distance of C/bh from a; it marks the middle of the back of the neck. Having drawn all these vertical lines and marked all the parts that touch them, you will now determine the heights, that is, the vertical position, and this is done using parallel horizontal lines. The first such line, the top of the square, is called n and is touched by the crown of the head. The last is the bottom, called y. Then make a horizontal o B/ia below n; it contains the cowlick. Then another, p, B/da below n; the hairline it tangent to it. Then q, B/bg below n; it is touched by the top of the eyelids and the top of the ears. The eyebrows also lie on this line, surpassing it slightly at the middle of each. Then make line t halfway between q and y; it marks the bottom of the nose, and also the meeting between neck and occiput. Next draw a line r a third of the way down from q to t; the lower eyelids are tangent to it. The corners of the eyes and the pupils are placed halfway between q and r. Then draw a line s a fourth

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of the way up from t to q. This line touches the top of the nostrils’ flare and the bottom of the earlobe. Next draw x halfway between t and y; it touches the top of the chin. Draw u halfway between t and x; the mouth is divided in half by u. Divide the space tu into three equal parts, and place the lower lip in the lowermost; the upper two contain the little cleft between nose and mouth, which I find is called philtrum by the Greeks. Likewise divide the space between u and x in half: in the upper part put the lower lip, and in the lower part the depression between mouth and chin. Then draw a diagonal line from the crossing of lines c and p to that of lines a and y: the forehead goes slightly beyond this line, but the nose follows down along it. After doing these things, take up the next rectangle, marking it with horizontal lines made with the help of the transferent. Label one of its vertical sides a and the other b, and insert nine vertical lines, dividing it into ten equal parts. Label these lines defghiklm, and further divide in half the strips between a and d and between m and b with vertical lines c and n, respectively. Now that all the parts of the face have been located and marked, you will outline the images in the rectangles you prepared, but with lighter lines and as it were softer and less stark than in the male face. First start the head in profile so it touches the top line n at the crossing with i and so the occiput meets side b between p and q. Then draw in the crown, forehead, nose, mouth, chin, cheeks, ears, and eyes one at a time, in their previously determined places. Make the width of the neck B/be. Now draw the front view thus. Let the top touch line n in the middle, and let the temples go down to sides a, b, which they touch between p and q. The curves of the jaw start at c and n, below line r, and come together at the chin. The ears are drawn in the strips ac and nb, between the horizontals q and s; but the earlobes slope in a bit, going beyond line c on one side and n on the other. The neck goes down from the ears to the lower edge y, where its width is B/bf. Make the temples between the verticals a and c on one side and n and b on the other, and between the horizontals p and q. The hairline starts at the crossing hp and goes up and to either side, halfway between p and o. Place the eyes so the corners are determined by the lines e,g and i,l. The eyebrows jut above the line q at f and k and slant down, stopping between d,e on one side and l,m on the other. Next draw the nose, delimiting the flares of the nostrils on either side by the vertical lines g and i; the corners of the mouth touch these same lines. Then, in the remaining rectangle, you will place the image and shape of the head seen from the back, similar to what you did with the front view. And finally you will come to the cross sections, or level sets of constant height, which you should approach in the same way we prescribed earlier, in dealing with the male head. I have tried to diligently and carefully display all these things in the example figure, so that even without an written explanation I can trust that the matter would be understandable. And if you wish to make the woman’s head taller, follow the method given earlier for the male head, but use moderation, since all the features of the female sex should appear smaller.

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To these images of adult bodies of either sex I will add the description of an infant. The method will be similar to the earlier ones. For the three images, as before, here too you should draw three vertical lines, one of which will serve as the guide for the profile view, and the others for the front and rear views. Then draw horizontal lines at the top and bottom, labeling the upper one a and the lower one z. The heights of the little body’s features are to be determined by means of horizontal lines, as follows. The whole length of the line will be occupied by the body, from the top of the head at a to the sole of the feet at z. Therefore measure down B/e from the top of the head or crown, and there draw a horizontal line b, which marks the throat or lower limit of the neck at front and back. Use this B/e as the height of a square that will frame the head. (We will describe the head part by part in detail before turning to the other measurements.) Let the front side of this square, the one near the nose, be called a, and the back side b. Start by fixing the heights of the features using horizontal lines, as we taught in connection with earlier drawings. Make one such line B/ce below a: it will touch the top of the forehead at the front. Make another B/bg below a, and call it c: it will go through the cowlick. Another horizontal line d, B/g down from c, will touch the bottom of the chin and the meeting of occiput and neck; also the shoulders jut up to this line, for the neck is to be drawn with curved lines. 73


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Draw a horizontal line e two thirds of the way down from d to b; it marks the bottom of the double chin, and so you see how very short the infant’s neck is. Then draw a line f halfway between a and b, which will contain the ends of the eyebrows, whereas their middle juts slightly above f. The top of the ear also grazes f, and at this same height the back of the head rests on side b. Then draw line g halfway between f and d, marking the lower limit of the nostrils and earlobes. Now divide the strip fg into three equal parts using horizontal lines h and i; the upper one, h, goes through the pupils and the corners of the eyes. The whole eye is included between f and i, and i marks the top of the flare of the nostril. The earlobe lies between i and g. Divide the strip gd into five equal parts using four horizontal lines k,l,m,n: the first touch the upper lip and the second goes through the mouth. Then divide lm in two: in the upper half place the lower lip, in the other half the depression between lip and mouth. At line l the occiput is particularly curved, as you can see more precisely in the example figure. Line n goes through mid chin, where there is a dimple; also through the lower cheeks and upper neck. Then draw vertical lines crossing the horizontal lines; they will indicate the horizontal placement, or depth, of all the features, as follows. Dividing sides a and b halfway, draw a perpendicular c: it marks where the ear arises, and the middle of the trunk front-to-back. Draw another vertical line d [halfway] between a and c; it touches the tip of the eyebrows and the temples next to the forehead, continuing down to where the neck starts. Next divide strip ad into five equal parts with four lines e,f,g,h. Line e will touch the upper lip and the top of the nose where it is concave. Line f will touch the tip of the eyebrows and of the nose, the curve of the cheek next to the mouth, and the depression above the chin. Line g grazes the hairline at the forehead, the depression of the eyelids, the pupils, and the cleft of the double chin. Line h touches the outer corner of the eyes; the inner corner lies between g and f. Then draw line i halfway between a and e; the forehead is tangent to it, and the tip of the nose goes slightly beyond it. Draw another vertical line halfway between e and f, between the horizontals l and d; it will touch the lower lip and the chin. At a distance of D/ca from a set down a line k, which delimits the back edge of the ear. The earlobe will be drawn in the other half of the space ck, next to c. Line l, at a distance of B/f from a, will go through the place where the back of the head meets the neck; the latter ends halfway between k and l. Finally divide lb into four spaces using three points: the third point will mark the cowlick. And so, having set down all the parts of the head in the square by height and width, I will now give the measurements of the other parts of the little body in the various views, before presenting the example figures. (continued)

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Start by placing the top of the shoulders B/ga below b. Make the distance to the top of the chest B/bg To the armpits B/ba To the nipples B/j Below the nipples B/h To the loins where we gird ourselves B/f From there to the navel B/cb To the hip joint B/be

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To the top of the thigh B/bf To the lower belly B/g To the penis B/ba plus B/bb To the lower scrotum B/j To the lower buttocks B/e Down B/fa from here you start making folds in the thigh, for at this age children are unusually plump. Make another fold above the knee, B/bi from the buttocks.


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Place the mid knee [halfway] between the top of the thigh or hip bone and the sole. From mid knee to lower inner ankle B/dg From the same place to the bottom of the calf B/j From the sole back up to the top of the foot let the distance be B/ca. And the arm is described thus: From the shoulder to the elbow joint make the distance C/bb From the elbow joint to the fingertips B/e From there back to the wrist joint B/j There follow the data about the width of the profile view, which we also call thickness. At the shoulders let it be C/bf At the upper chest B/g At the nipples C/bb Below the nipples B/bb plus B/bc At the loins where we gird ourselves B/g At the navel C/bb At the hip joint B/ba plus B/bb At the upper thigh B/j plus B/ba At the lower belly and buttocks B/f At the lower scrotum B/g The thigh itself immediately below the buttocks should have width B/h, and a bit further down B/be plus B/bf Above the knee let the width be B/i At mid knee B/ba Below the knee B/bb At mid calf B/ba At the lower calf B/bc Above the top of the foot B/bg The foot’s length will be B/bd plus B/be But the parts of the arm in profile shall have the following widths: At the shoulders B/ba At the armpits B/bc At the elbows B/bg At the forearm near the elbow B/bf A bit further down B/bi

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Just above the hand B/cd And the open hand shall have width B/cb Having set down all the features in the profile view, go now to the area of the front view, and use the transferent to copy over the locations of the horizontal lines. Next you need to examine the widths as well. First you will do the head in detail, making here too a rectangle—no longer a square since its width will be C/j and so a bit less than the height. Label the vertical sides of this rectangle p [i.e., a] and b. Divide it halfway with a vertical line c, which will cross the crown, nose, mouth, and chin. Then divide the strips ac and cb in half with vertical lines d and e, which will mark the ends of the eyebrows next to the temples, as well as the neck. Then divide the strip de into three equal parts with vertical lines f and g; they will mark the corners of the eyes near the nose, the edge of the flare of the nostrils, the corners of the mouth, and also the limits of the chin. Divide the strip ad into three equal parts with two lines h and i; likewise divide eb with two lines k and l. Lines h and l mark the width of the face along the horizontal line h. The ears reach all the way out to the sides a and b; but the earlobes, because of the slope, reach in past h and l. The two vertical lines i and k are tangent to the chubby parts of the cheek. Draw a vertical line m one-fifth of the way from a to h, and likewise n one-fifth of the way from b to l; these new lines delimit the head between the horizontals c and f. The forehead goes just beyond h and l on the sides, again between the horizontals c and f. The eyebrows stop between i and d on one side and between e and k on the other, below the horizontal line f.


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Having finished marking the features in this rectangle too, you will now move on to the widths of the rest of the little body. Make the width at the shoulders B/e; place the shoulder joints there as well, C/j apart. At the upper chest the width will be C/h Between armpits B/f Between nipples B/h At the loins were we gird ourselves B/f At the navel E/bh At the curve of the hips the same. At the hip joints B/bf plus B/bg; the distance between them will be C/bb At the lower belly E/bf The width of the thigh next to the lower scrotum will be B/bf plus B/bg At the lower buttocks B/i At the narrowing of the thigh B/bh Above the knee B/bj plus B/ca At mid knee B/bc Below the knee B/bd At mid calf C/cd At the lower calf B/bg

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At the shin above the top of the foot B/bj The width of the foot at the toes will be C/ch The arm is specified as follows: Width below the armpits B/bh At the elbow joint B/bg Below the elbow B/be At the wrist joint B/ca Let the width of the open hand be B/bd Make the length of the infant’s body a third of the mother’s. After having measured out everything as taught here, fill in the forms in the profile, front and rear views, as we show in the following figures, which help comprehension as much or more than my words. The drawings also show the bending of the spine and the placement of the joints. The rear view has the same silhouette as the front view, but the distance between the armpits is a bit less than B/f. The length of the cleft of the buttocks is B/j; the width of the foot is B/cc.


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Crown Forehead

Eyebrows Nose Chin Double chin Throat Top of shoulder blades Top of chest Armpits Nipples Lower chest Loins where we gird ourselves Navel Curve of hip Top of thigh, or hip joint Lower belly Penis Lower scrotum Lower buttocks Upper knee Mid knee Lower knee Lower calf Top of foot Sole

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Second Book on Proportions by Albrecht Dürer. This second book of the proportions of the parts of the human body will treat the same subject as the first, but will show a different and more exact method for achieving the right dimensions of the body’s parts, using a ruler and its subdivisions, as follows. Prepare a ruler or measuring rod whose length is one third the total height of the body image you wish to draw. We will use this symbol for the rod’s length: ——— Divide the ruler into ten equal segments, each of which will be called a numerus and will be denoted by this sign: ——————————————————— Then divide the numerus likewise into ten equal segments. We will call each of them a portio and denote it thus: ——————————————————— Finally divide the hundredth into three equal parts, each of which will be called a frustulum and will be denoted thus: ————————————————— This is how you shall proceed. Before starting, take a piece of paper and make five vertical lines, thus forming four columns, and label each column: the first, for rods, with the symbol ; the second, for numeri, with ; the third, for portiones, with ; and the last, for frustula, with . When you start investigating the dimensions of each part of the body, add below the column heads the appropriate figures, indicating the number of rods, numeri, portiones, and frustula. If it so happens that one of the units is not represented, write 0 in that column. For the measurements, follow the order set down earlier. Draw three vertical lines whose length is the full height of the picture to be drawn: one will be for the profile view, another for the front view, and the last for the rear view. Across all three draw one long horizontal line at the top and one at the bottom. Then set down the vertical subdivisions by means of parallel horizontal lines, first across the profile, then extended to the front view, as needed; for the heights are the same for all the views. We will make all of this clear with example figures, as before and as always. You will see that for all the profile views we have written the measurements of length or height in the margin, using as guides the extensions of the individual horizontal lines. Each number is given with the symbol for its unit: rod, numerus, portio, frustulum. Similarly you will see the distances indicating width or thickness. And having explained all of this and prepared the ground I now offer the list of measurements, in the same sequence as before, with the heights first.

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From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the throat ———————————— To the top of the shoulders —————— Divide the space between the upper forehead and the lower chin into three equal parts using two lines. Place the forehead in the top part, the nose in the middle part, and the mouth and chin in the lowest. From top of shoulders to chest ———— To the armpit as seen from the front —— To the nipples ———————————— To the lower chest —————————— To the loins where we gird ourselves — From there to the navel ——————— To the curve of the hips ——————— To the top of the thigh, or hip ————— To the private parts ————————— To the lowest point of the scrotum —— To the lower buttock ————————— From the lower buttocks to the narrowing of the thigh —————————— Above the knee ——————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer calf ———————— Lower inner calf ——————————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— The length of the foot will be ————— Measure out the arm as follows: From the shoulder to the elbow ———— From the elbow to the wrist —————

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0 0 0 1 1

0 1 8 0 1

5 6 5 6 6

0 0 0 0 0

0 0 0 0 1 0 0 0 0 1 1

2 4 5 6 5 1 3 6 8 0 1

3 0 4 9 0 5 0 7 0 6 0

0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 1 1 1

4 8 1 1 6 7 2 4 5 0

5 0 0 6 0 2 6 0 9 3

0 0 0 0 0 0 0 0 0 0

1 0 6 0 0 7 6 0

rod numerus portio frustulum


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From there to the fingertips ————— The widths or thicknesses of the body in profile should be as follows. At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the lip —————————————— At the chin and neck ————————— Also there let the width of the neck be — At the throat ———————————— At the chest ————————————— By the armpits ——————————— At the nipples ———————————— At the lower chest —————————— At the loins where we gird ourselves — At the navel ————————————— At the curve of the hip ———————— At the top of the thigh, or hip ————— At the private parts ————————— Next to the lower buttocks let the width of the thigh be —————————— Nearby, below the buttocks —————— Next to the narrowing of the thigh —— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the bottom of the shin at its narrowest At the top of the foot ————————— At the lower outer ankle ——————— Next come the widths of the arm in profile: At the shoulder let it be ———————— At the biceps ———————————— At the elbow ———————————— At the forearm ———————————


0 6 5 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0

5 7 8 7 7 7 5 6 9 0 0 9 8 8 9 0 9

0 0 2 5 0 0 0 0 7 4 3 7 6 4 5 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

7 7 7 5 5 4 5 5 4 3 3 5

6 3 5 7 3 5 4 3 5 2 5 3

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

5 4 3 3

5 7 5 6

0 0 0 0

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At the wrist joint —————————— Let the width of the open hand be ——— Moving to the next vertical line, which as mentioned is for the front view, you should now mark the widths as follows: Let the width of the head at the cowlick be At the forehead ——————————— At the temples ———————————— At the eyebrows ——————————— At the ears ————————————— At the nose ————————————— Let the width of the neck be ————— At the bottom of the chin ——————— At the throat ———————————— At the shoulders ——————————— Likewise make the distance between the shoulder blades —————————— At the chest ————————————— Between the armpits ———————— Between the nipples ————————— At the loins where we gird ourselves — At the navel ————————————— At the curve of the hip ———————— At the top of the thigh, or hip ————— Also located there are the hip joints, the distance between them being ——— At the pubis ————————————— The thigh below the buttocks ————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ———————————

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0 2 1 0 0 2 3 0

0 0 0 0 0 0 0 0 1 1

5 7 7 7 8 6 4 6 4 5

5 2 8 3 0 0 3 6 3 8

0 0 0 0 0 0 0 0 0 0

1 1 1 0 1 1 1 1

2 8 2 8 1 2 2 4

4 0 3 2 4 7 8 6

0 0 0 0 0 0 0 0

1 1 0 0 0 0 0

0 4 6 6 4 4 4

5 5 7 0 6 4 0

0 0 0 0 0 0 0



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At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot and ankles ——— Below the ankles —————————— The toes at the tips have width ———— Set the width of the arm in the front view as follows: Below the armpit —————————— Above the elbow ——————————— Below the elbow at the forearm ———— At the wrist ————————————— Let the width of the open hand be ———

0 0 0 0 0 0 0

5 4 3 2 2 2 4

0 6 9 2 7 4 2

0 0 0 0 0 0 0

0 0 0 0 0

3 3 4 2 4

5 4 0 4 3

0 0 0 0 0

Having measured out all the parts, draw appropriately the shape of both images. Then move on to the third vertical line, meant for the rear view, and draw the same silhouette as for the rear [i.e., front] view. But make the distance between the armpits ——————————————— 1 3 7 0 In the rear view the armpits appear lower than in the rear [i.e., front] view, so make the distance from the top of the shoulders to the armpits ——————— 0 5 0 0 Let the length of the cleft of the buttocks be 0 6 7 0 Let the width of the heel be —————— 0 3 0 0 All this is illustrated below.


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Crown Cowlick Forehead Eyebrows Nose Chin Throat Top of shoulders Chest Armpits seen from the back [i.e., front] Nipples Lower chest Loins where we gird ourselves Navel Curve of hip Top of thigh, or hip joint Private parts Bottom of scrotum Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee Lower outer calf Lower inner calf


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To this male body we add a similar female one, which you should measure out using the same method. First take down the vertical distances. From the crown to the cowlick ——— To the forehead ——————————— To the lower chin —————————— To the shoulder blades ———————— To the throat ———————————— Divide the distance from the hairline to the lower chin into three equal parts using two lines. The first third is for the forehead, the next for the nose, eyes, and ears, and the last for the mouth and chin. From the upper throat to the shoulders * To the upper chest —————————— To the armpits seen from the front —— Seen from the back ————————— To the nipples ———————————— To the lower chest —————————— To the loins where they are girt ———— From the loins to the navel —————— To the top of the thigh, or hip joint —— To the lower belly —————————— To the beginning of the pubis, where the body forks out —————————— To the lower pubis —————————— To the lower buttocks ———————— From the lower buttocks to the narrowing of the thigh —————————— From there to the upper knee ————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer shin ———————— Inner shin ————————————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— The length of the foot shall be ———— Make the length of the arm as follows:

0 0 0 0 1

0 1 8 9 1

8 6 3 0 4

0 0 0 0 0

0 0 0 0 0 0 1 0 0 0

0 2 4 4 6 8 2 1 5 7

6 7 4 8 7 2 5 6 6 2

0 0 0 0 1 0 0 0 0 0

0 9 0 0 0 9 5 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1

3 7 9 2 6 8 2 3 5 0

5 2 5 0 4 2 6 6 5 0

0 0 0 0 0 0 0 0 0 0



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From the shoulder to the elbow ———— From there to the wrist ———————— Let the length of the hand be ————— To this we should add the measurements of width in profile, or thickness. They are as follows. At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the lip —————————————— At the chin ————————————— Likewise the neck has width ————— At the top of the shoulders —————— At the throat ———————————— At the shoulders ——————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the top of the thigh, or hip joint —— At the lower belly —————————— At the top of the pubis, thigh, and buttocks At the lower pubis —————————— The width of the thigh next to the lower buttocks will be —————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot ————————— At the lower outer heel ———————— Make the foot of the woman less width than the man’s. Set the widths of the arm in profile as follows:

1 0 2 0 0 7 6 0 0 6 8 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

5 6 8 7 6 6 4 4 5 6 8 9 9 8 8 9 2 1 4 0

3 9 1 0 5 0 0 4 5 6 3 2 0 5 0 3 0 0 0 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

8 7 6 5 5 5 5 4 3 3 5

5 8 4 0 0 2 0 2 1 1 7

0 0 0 0 0 0 0 0 0 0 0

* [Note: Measurement figures in this column are printed offset one row downward in the Latin edition beginning at this point; their alignment is corrected in this translation, per the German original.—Translator] 85


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At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below the elbow ——————————— A little further down ————————— At the wrist ————————————— The width of the open hand will be —— Then record also the widths of the body in front view: At the cowlick let it be ———————— At the hairline ——————————— At the forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose ————————————— The neck below the chin will have width At the top of the shoulder blades ——— At the throat ———————————— At the shoulders ——————————— The distance between the shoulder joints should be ————————————— Across the upper chest and arms ——— Between the armpits ———————— Between the nipples ————————— At the loins where they are girt ———— At the navel ————————————— At the top of the thigh, or hip joint —— The distance between the hip joints should be ————————————— At the lower belly —————————— At the top of the pubis ———————— At the lower pubis —————————— At the lower buttocks likewise ———— At the narrowing of the thigh ————

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0 0 0 0 0 0 0

5 4 3 3 2 2 2

3 5 3 7 7 0 4

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1

5 6 7 7 8 5 4 4 1 2

2 9 5 2 0 6 0 2 0 2

0 0 0 0 0 0 0 0 0 0

1 1 0 0 1 1 1

0 4 9 7 0 3 5

1 8 6 2 2 7 2

0 0 0 0 0 0 0

1 1 1 0 0 0

1 5 5 7 7 6

1 4 6 6 6 9

0 0 0 0 0 0



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Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the top of the foot ————————— Likewise below the ankle —————— And the width of the foot at the toes will be The widths of the arm are these: Below the armpits at the biceps ———— Above the elbow ——————————— Below the elbow at the forearm ———— At the wrist ————————————— The open hand will have width ————

0 0 0 0 0 0 0 0 0

5 4 4 5 4 3 2 2 4

3 5 5 0 5 8 5 5 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

3 3 3 2 3

4 0 8 4 9

0 0 0 0 0

Having finished this you should draw the shape of the body, in profile and in front view, using appropriate strokes. The silhouette of the rear view is the same as that of the front view; but since the armpit appear lower in the rear view, the distance from them to the shoulders is indicated above. Between the armpits the width will be 1 2 0 0 The cleft of the buttocks ——————— 0 7 5 0 The heel will have width ——————— 0 2 5 0 We have illustrated all these things in the example figures with great care.


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Throat Shoulders Chest Armpits seen from the front Nipples Below the breasts Loins where they are girt Navel Top of thigh, or hip joint Lower belly Top of pubis Lower pubis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee



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We now present another set of proportions for the human body, using the same method. First measure out the lengths. From the crown to the cowlick make it To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the throat ———————————— Divide the space from the hairline to the lower chin into three equal parts: one for the forehead, another for the nose, eyes, and ears, and the third for the mouth and chin. Then continue with the measurements. From the upper throat to the shoulders To the chest ————————————— To the armpits as seen from the front — To the armpits as seen from the back — To the nipples ———————————— To the lower chest —————————— To the loins where we gird ourselves — From there to the navel ——————— To the top of the thigh, or hip joint —— To the private parts ————————— To the lower buttocks ———————— From there to the narrowing of the thigh Above the knee ——————————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer calf ———————— Inner ————————————————

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0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

1 2 8 9 1

0 2 4 5 5 7 0 2 6 7 9 5 9 2 1 5 6

0 2 6 2 4

6 2 0 0 8 0 7 2 0 7 8 0 7 1 9 9 6

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



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To the top of the foot ————————— To the lower ankle —————————— To the sole ————————————— And the length of the foot will be ——— The arm should be measured out as follows: From the shoulder to the elbow ———— From the elbow to the wrist ————— From the wrist to the fingertips ———— Then the widths should be recorded, as follows: At the cowlick let it be ———————— At the hairline ——————————— At the eyebrows ——————————— At the nose ————————————— At the lips —————————————— At the neck and chin ————————— Here the neck should have width ——— At the top of the shoulder blades ——— At the throat ———————————— At the shoulders ——————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— At the lower chest —————————— At the loins where we gird ourselves — At the navel ————————————— At the hip —————————————— At the private parts ————————— The width of the thigh below the buttocks will be ——————————— At the narrowing of the thigh ———— Above the knee ———————————


1 1 1 1

3 4 6 0

3 3 0 0

0 0 0 0

1 0 0 0 0 8 6 0 0 6 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 7 7 7 6 6 4 4 5 6 8 8 8 8 7 7 8 7

3 1 8 0 4 4 2 5 5 6 0 7 8 1 4 5 6 7

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 6 6 0 0 6 0 0 0 5 5 0

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At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot ————————— At the lower outer ankle ——————— The widths of the arm are these: At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below the elbow ——————————— At the wrist ————————————— The width of the open hand is ———— Now move on to the next vertical line, for the front view. Mark its widths as follows: At the cowlick let it be ———————— At the hairline ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— The neck, next to the chin, has width — At the top of the shoulder blades ——— At the throat ———————————— The distance between the shoulder joints is At the chest ————————————— Between the armpits ———————— Between the nipples ————————— At the lower chest —————————— At the loins where we gird ourselves —

0 0 0 0 0 0 0 0

4 4 4 4 3 2 2 4

0 0 4 1 9 5 8 4

0 0 0 0 0 0 0 0

0 0 0 0 0 0

4 4 2 2 1 2

8 0 6 7 7 0

0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 0 1 0

5 6 6 6 5 4 5 1 0 4 3 7 1 9

5 8 3 9 0 1 0 2 0 7 0 2 1 9

0 1 0 0 0 0 0 0 0 0 0 0 1 0



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At the navel ————————————— At the hip —————————————— Let the distance between the hip joints be At the private parts ————————— Let the width of the thigh below the buttocks be ————————————— At the narrowing of the thigh ———— Above the knee let it be ——————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot and ankle ———— Below the ankles —————————— The width of the foot at the toes will be And here are the widths of the arm: At the biceps let it be ————————— Below the elbow ——————————— At the wrist ————————————— The width of the open hand will be —— After marking out the measurements, make a suitable drawing of the profile and front views. The rear view has the same silhouette as the front view. Its width will be: Between the armpits ———————— The cleft of the buttocks has length —— The heel has width —————————

1 1 0 1

3 1 8 1

0 9 8 8

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

5 4 3 3 3 4 3 3 1 2 2 3

6 4 5 0 3 0 6 1 6 2 0 5

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

2 3 2 3

6 0 0 2

0 0 0 0

1 1 5 0 0 6 3 0 0 2 0 0

If you enclose this body into a square, the feet will touch the base and the outstretched hands will each touch a side. Thus the body can be drawn according to the square, as you will in fact see in the example figures.

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Crown Cowlick Forehead Eyebrows Nose Chin Throat Shoulders Chest

Armpits seen from the front Nipples Lower chest Loins where we gird ourselves Navel Curve of hip Top of thigh, or hip joint Private parts Lower buttocks

Narrowing of thigh



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To this male body I will now add a corresponding female one. For it too, as before, start by measuring the lengths along the first of the three vertical lines, as follows. From the crown to the cowlick ———— To the hairline ———————————— To the lower chin —————————— To the top of the elbows ——————— To the throat ———————————— Divide the space from the hairline to the lower chin into three equal parts; the top one will enclose the forehead, the middle one the nose and eyes, the bottom one the mouth and chin. Then continue with the measurements. From the throat to the upper chest —— To the armpits as seen from the front — To the armpits as seen from the back — To the nipples ———————————— To below the breasts ————————— To the loins where they are girt ———— From there to the navel ——————— To the top of the thigh, or hip joint —— To the beginning of the pubis, where the body forks out —————————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To the upper knee —————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer calf ———————— Inner ———————————————— To the top of the foot ————————— To the lower ankle —————————— To the sole ————————————— And the length of the foot will be ———

94

0 0 0 1 1

1 2 8 1 2

0 4 6 0 0

0 0 0 0 0

0 0 0 0 0 1 0 0

1 3 4 6 7 1 1 6

5 4 0 0 4 6 9 3

0 0 0 0 0 0 0 0

0 1 1 0 0 1 0 0 0 1 1 1 0

9 0 1 2 7 0 2 6 7 2 3 5 9

1 0 0 6 7 1 0 3 4 7 5 3 8

0 0 0 2 0 0 0 0 0 0 0 0 0



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Measure out the arm as follows: From the shoulder to the elbow ———— From the elbow to the wrist ————— From the wrist to the fingertips ———— And thus, after marking all the heights by means of horizontal lines across the first vertical line, which is meant for the profile view, and after transferring these lengths to the other two vertical lines, you will now mark the widths, starting with the profile view, where they are thicknesses. And these are the widths: At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose and nape ————————— At the upper lip and nape ——————— Let the neck at the chin have width —— At the top of the shoulders —————— At the throat ———————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the hip —————————————— At the top of the pubis ———————— At the lower pubis and buttocks ——— Let the thigh below the buttocks have width —————————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ———


1 0 0 0 0 7 6 0 0 6 2 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

4 7 7 6 6 4 5 5 7 8 8 7 6 8 0 9 8

6 1 2 5 0 0 0 5 0 0 2 4 4 0 4 0 5

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

7 7 5 4 4 4 4 3 2

2 1 1 1 0 4 0 5 5

0 0 0 0 0 0 0 0 0

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At the top of the foot ————————— At the lower outer ankle ——————— And the widths of the arm will be these: At the shoulder ——————————— Below the armpits at the biceps ———— At the elbow ———————————— Below the elbow at the forearm ———— At the wrist ————————————— The open hand will have width ———— Then mark the widths against the second vertical line, for the front view, as follows: At the cowlick make the width ———— At the forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— Let the neck by the lower chin have width —————————————— At the top of the shoulders —————— At the throat ———————————— The distance between the shoulder joints shall be ————————————— At the chest and [lower] shoulders —— Between the armpits ———————— Between the nipples ————————— Below the breasts —————————— At the loins where they are girt ———— At the top of the thigh, or hip ————— Also there the distance between the hip joints, situated directly above the outer ankle, is ———————————— At the upper pubis where the body forks

95

0 2 9 0 0 4 4 0 0 0 0 0 0 0

4 3 2 2 1 1

4 8 7 6 5 8

0 0 0 0 0 0

0 0 0 0 0

4 6 6 6 4

8 5 0 5 9

0 0 0 0 0

0 3 8 0 1 0 2 0 1 2 2 0 0 1 0 0 0 0 1

9 3 9 7 9 9 4

3 9 0 4 9 3 0

0 0 0 0 0 0 0

1 0 0 0 1 3 8 0



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The thigh next to the lower pubis has width —————————————— Next to the lower buttocks —————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the ankles ———————————— Below them ————————————— The width of the foot at the toes will be To this we add the widths of the arm in front view: At the biceps let it be ————————— Above the elbow ——————————— Below the elbow at the forearm ———— At the wrist ————————————— The width of the open hand will be ——

0 0 0 0 0 0 0 0 0 0 0 0 0

7 6 6 4 4 3 4 3 3 1 2 1 3

0 7 2 4 0 8 0 5 0 7 0 8 4

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

2 1 2 1 3

9 5 9 9 0

0 0 0 0 0

After this make drawings of the body in profile and from the front, agreeing with the marks made. The silhouette seen from the back is the same as seen from the back [i.e., front]. Also we have indicated above the vertical distance between the throat and the armpits seen from the back. And the width between the armpits will be ————————— 1 0 8 0 The cleft of the buttocks has length —— 0 6 6 0 The heel has width ————————— 0 1 8 0 We have made all these things clear in the example figures below.


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Crown Cowlick Forehead Eyebrows Nose Chin

Shoulders Throat Chest Armpits Nipples Lower breasts Loins where they are girt Navel Curve of hip Top of pubis Lower pubis Lower buttocks Narrowing of thigh

Upper knee Mid knee Lower knee Lower outer calf Lower inner calf Top of foot Lower outer ankle Sole

96


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97


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We will now describe another male body, different from the previous ones, using the same technique. First I will give the length, or height, of its parts, which you should mark next to the first vertical line, used for the profile view, and then transfer to the other lines. From the crown to the cowlick let the length be ————————————— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— From there to the upper chest ————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To the lower chest —————————— To the lower thorax ————————— To the loins where we gird ourselves — From there to the navel ——————— To the top of the thigh, or hip ————— To the private parts ————————— To the lower buttocks ———————— From the lower buttocks to the narrowing of the thigh —————————— To above the knee ——————————

98

0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1

0 1 7 8 0 0 2 3 4 5 6 7 1 1 5 7 0

7 4 8 4 0 6 7 5 3 0 2 5 0 9 9 3 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 4 5 0 0 9 0 0



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To mid knee ————————————— From mid knee to lower knee ————— To the lower outer calf ———————— To the lower inner calf ———————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— And the length of the foot will be ——— Make the arm have the following lengths: From the shoulder to the elbow ———— From the elbow to the wrist ————— From the wrist to the fingertips ———— Having marked the height, turn now to the widths and thicknesses, as follows: Let the profile head at the cowlick have width ————————————— At the forehead ——————————— At the eyebrows ——————————— At the nose and temples ——————— At the upper lip and nape ——————— Let the neck next to the chin have width At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— At the chest ————————————— At the armpits ————————————


1 0 0 0 1 1 1 0

1 1 6 7 4 5 6 9

6 7 2 0 3 0 8 3

0 0 1 0 0 0 0 0

1 1 2 0 0 8 0 0 0 6 5 0

0 0 0 0 0 0 0 0 0 0 0

4 6 7 6 5 3 4 4 4 7 7

7 0 2 1 7 8 0 5 9 0 7

0 0 0 0 0 0 0 0 0 0 0

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At the nipples ———————————— At the lower chest —————————— At the bottom of the thorax —————— At the loins where we gird ourselves — At the navel ————————————— At the hip —————————————— At the private parts ————————— Let the thigh below the buttocks have width —————————————— A little further down ————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot ————————— At the lower outer ankle ——————— Let the widths of the arms be as follows: At the shoulder ——————————— Below the armpits at the biceps ———— At the elbow ———————————— Below the elbow at the forearm also —— At the wrist ————————————— The open hand will have width ———— Then we describe the widths of the front view. At the cowlick let it be ———————— At the forehead ———————————

99

0 0 0 0 0 0 0

7 7 7 6 6 7 7

9 7 8 4 4 6 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

5 5 5 4 3 3 4 3 3 2 2 3

7 5 5 0 6 6 0 6 3 2 5 6

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

4 3 2 2 1 1

1 6 5 5 4 7

0 0 0 0 0 0

0 4 5 0 0 5 8 0



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At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— Let the neck at the chin have width —— At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— Let the distance between the shoulder joints be ————————————— At the upper chest let the width be —— Between the armpits ———————— Between the nipples ————————— At the lower thorax ————————— At the loins where we gird ourselves — At the navel ————————————— Moreover between the loins and the navel the flesh bulges slightly; the width there will be ———————————— At the hip —————————————— The distance between hip joints will be At the private parts let the width be —— At the thigh below the buttocks ———— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ————————————————


0 0 0 0 0 0 0 1

6 6 6 4 3 4 9 1

2 0 4 8 7 2 5 3

0 0 0 0 0 0 0 1

0 1 0 0 1 0 0

9 4 9 6 0 9 9

6 0 7 5 2 2 7

0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

9 8 7 0 4 4 3 3 3 3 3 2

9 0 9 7 8 0 2 0 0 7 2 8

0 1 0 0 0 0 0 0 0 0 0 0

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At the lower shin at its narrowest ——— At the top of the foot ————————— At the ankles ———————————— Below them ————————————— The width of the foot at the toes will be There follow the widths of the arm in front view: At the biceps let it be ————————— Above the elbow ——————————— At the widest point of the forearm —— A bit further down —————————— At the wrist ————————————— The open hand will have width ————

100

0 0 0 0 0

1 1 2 1 3

5 8 0 9 4

0 0 0 0 0

0 0 0 0 0 0

2 2 3 2 1 3

5 2 0 2 8 4

0 0 0 0 0 0



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And so having marked the heights and widths of the features, you will now draw the profile and front views. The rear view, as we have often said, has the same silhouette as the front view. The distance between armpits in the rear view is —— 1 1 5 0 The height difference between armpits and throat was given earlier. Let the cleft of the buttocks have length 0 6 0 0 Make the width of the heel —————— 0 1 8 0 We show below example figures made according to these measurements.


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Nipples Lower chest Bottom of thorax Loins where we gird ourselves Navel Top of thigh, or hip Private parts Lower buttocks Narrowing of thigh

Upper knee Mid knee Lower knee Lower outer calf Inner


101


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102


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To this man we join a woman of similar proportions. As always, start by measuring out the lengths of her features, as follows: From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— From there to the chest ———————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To the lower chest —————————— To the loins where they are girt ———— From the loins to the navel —————— To the hip —————————————— To the lower belly —————————— To the beginning of the pubis, where the body forks out —————————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To the upper knee —————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer shin ———————— Inner shin ————————————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— Let the length of the foot be ————— The arm will be as follows: From the shoulder to the

103

0 0 0 0 1 1 0 0 0 0 0 1 0 0 0

0 2 7 8 0 1 1 3 4 6 7 1 2 6 7

6 0 7 3 4 1 7 2 0 1 6 3 1 1 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 1 0 0 1 0 0 0 1 1 1 0

8 9 0 4 9 1 2 6 7 3 4 6 9

5 6 3 5 0 2 0 4 2 4 4 0 0

0 0 1 0 0 1 0 0 0 0 0 0 0



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elbow —————————————— From the elbow to the wrist ————— From the wrist to the fingertips ———— To these measurements we add the widths of the profile view. At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the lip and nape ————————— At the neck and chin ————————— Also there the neck will take ————— At the shoulders ——————————— At the throat ———————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the hip —————————————— At the lower belly —————————— At the top of the pubis ———————— At the lower pubis —————————— At the thigh next to the lower buttocks At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot ————————— Below the ankle ———————————


1 0 6 0 0 8 8 0 0 5 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 7 6 5 5 3 4 5 6 7 7 7 6 8 0 9 9 7 6 6 4 3 3 4 3 3 2 2 4

5 7 0 0 6 5 5 4 0 6 3 6 0 1 0 0 3 3 6 9 0 5 9 8 1 4 2 3 6 1

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

O

The profile view of the arm has these widths: At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below the elbow ——————————— At the wrist ————————————— The open hand itself has thickness —— The widths of the front view as as follows: At the cowlick let the width be ———— At the forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— The neck at the lower chin has width — At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— Also there the distance between the shoulder joints will be ——————— At the upper chest —————————— Between the armpits ———————— Between the nipples ————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the hip —————————————— The distance between the hip joints is — At the lower belly —————————— At the top of the pubis ———————— Let the thigh next to the lower pubis have width —————————————— Next to the lower buttocks —————— At the narrowing of the thigh ———— Above the knee let the width be ———— At mid knee —————————————

104

0 0 0 0 0 0

4 3 2 2 1 1

0 5 4 4 3 6

0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

4 6 5 6 4 3 3 8 0

5 2 7 4 8 3 6 7 2

0 0 0 0 0 0 0 0 2

0 1 0 0 0 0 1 1 0 1 1

8 2 8 5 9 8 1 2 8 2 2

3 6 6 6 0 5 4 5 7 4 3

0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0

6 6 5 4 3

1 0 5 2 5

0 1 0 0 0



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Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ——— At the top of the foot ————————— At the ankle ————————————— Below them ————————————— The width of the foot at the toes will be Make the front view of the arm as follows: At the biceps let the width be ————— Above the elbow ——————————— Below the elbow at the forearm ———— At the wrist ————————————— The open hand shall have width ———

0 0 0 0 0 0 0 0 0

3 3 3 3 1 1 2 1 3

4 9 4 0 5 8 0 9 2

0 0 0 0 0 0 0 0 0

0 0 0 0 0

2 2 2 1 3

7 3 9 7 0

0 0 0 0 0

After setting all this down you will draw the shape of the body as required, for the profile and front views. The lines should be soft and elegant, as befits a woman’s body. The rear view has the same silhouette as the front view, and the distance between the armpits is —— 1 0 0 0 As we have said, the armpits appear lower when seen from the rear; their distance from the throat was indicated above. The cleft of the buttocks will have length 0 6 5 0 The heel will have width ——————— 0 1 7 0 All of this can be seen in the example figures presented below.


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Armpits seen from the front Nipples Lower breasts Loins where they are girt Navel Top of thigh, or hip Lower belly Top of pubis Lower pubis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee

Lower outer calf Inner Top of foot Lower outer ankle Sole

105


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106


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Still using the same procedure, I will now describe another female body, also complementary to the last male figure described. The lengths are these: From the crown to the cowlick let it be To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— From there to the chest ———————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To below the breasts ————————— To the loins where they are girt ———— From there to the navel ——————— To the top of the thigh ———————— To the beginning of the pubis, where the body forks out —————————— To the lower pubis —————————— To the lower buttocks ———————— From the lower buttocks to the narrowing of the thigh —————————— To the upper knee —————————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer shin ———————— Inner ———————————————— From mid knee to the top of the foot ——

107

0 0 0 0 1 1 0 0 0 0 0 1 0 0

0 1 7 8 0 1 2 3 4 6 7 1 1 6

5 9 6 6 2 0 0 3 0 2 3 4 9 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 8 6 0 0 9 7 0 1 0 4 0 0 0 1 0 0 0 1

4 8 1 2 6 7 3

4 9 1 2 3 1 5

0 0 0 0 0 0 0



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To the lower outer ankle ——————— To the sole ————————————— Let the length of the foot be ————— The arm is done as follows: From the shoulder to the elbow let the length be ————————————— From the elbow to the wrist ————— From the wrist to the fingertips ———— Then mark down the widths on the horizontal lines. Start with the profile view, thus: At the cowlick let the width be ———— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip and nape ——————— Let the neck at the chin have width —— At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— At the chest ————————————— At the armpits ———————————— Between the nipples ————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the top of the thigh ———————— At the top of the pubis ———————— At the lower pubis —————————— The thigh below the buttocks has width At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ———————————


1 4 3 2 1 6 1 0 0 9 2 0 1 0 8 0 0 8 7 0 0 5 7 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 7 7 6 6 3 3 4 4 7 7 7 7 6 8 9 9 8 7 6 4 3 3

5 0 3 5 1 6 7 3 9 0 5 6 1 3 0 7 0 4 1 2 4 8 7

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Lower ——————————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin —————————— At the top of the foot ————————— At the lower ankle —————————— Give the arm in profile these widths: At the shoulder ——————————— At the biceps below the armpit ———— At the elbow ———————————— Below the elbow ——————————— At the wrist ————————————— Let the open hand have thickness ——— Next come the widths in the front view, which (as we have often said) you should mark across the second of the three lines. They are: At the cowlick ———————————— At the forehead ——————————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— Let the neck at the chin have width —— At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— The distance between the shoulder joints at that point should be ——————— At the chest ————————————— Between the armpits ———————— Between the armpits seen from the back

108

0 0 0 0 0 0 0

3 4 3 3 2 2 4

8 1 7 4 4 7 1

0 0 0 0 0 0 0

0 0 0 0 0 0

4 4 2 2 1 1

0 0 4 5 4 6

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1

4 6 6 5 6 4 3 4 9 0

0 2 4 8 3 9 5 2 4 8

0 0 0 0 0 0 0 0 0 0

0 1 0 1

8 2 8 0

5 8 5 0

0 0 0 0



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Between the nipples ————————— Below the breasts —————————— A bit further down —————————— At the loins or waist ————————— At the navel ————————————— At the top of the thigh ———————— At that point the distance between the hip joints will be ————————— At the top of the pubis ———————— The thigh at the lower pubis will have width —————————————— At the lower buttocks ———————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin —————————— At the top of the foot ————————— At the ankle ————————————— Below them ————————————— The width of the foot at the toes will be The heel has width ————————— After these preparations the shape of the body should be drawn, each next to the its axis, as we have often said. Again, use the same silhouette for the rear view as the front view, and draw the cleft of the buttocks with a length of ————— The example figures submitted below illustrate all of this.


0 0 0 0 1 1

5 8 8 9 1 3

7 9 7 2 7 5

0 0 0 0 0 0

0 9 0 0 1 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 6 5 4 3 3 4 3 3 1 1 2 1 3 1

8 7 7 1 6 4 0 5 1 6 8 0 9 3 7

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 5 7 0

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Crown Cowlick Forehead Eyebrows

Nose Chin Top of shoulder blades Shoulders Throat Chest Nipples Lower breasts Loins or waist Navel Top of thigh Upper pubis, where the body forks Lower pubis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee Lower outer calf Inner


109


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110


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Now I will present a male body of different proportions, still keeping the same order, giving the heights first. From the crown to the cowlick the vertical distance is ——————————————— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the throat ———————————— Then mark three equal spaces between the upper forehead and the lower chin, dividing them with two lines. In the topmost you will place the forehead, in the middle one the nose, eyes, and ears, and in the third the mouth and chin. Then continue with the measurements. From the throat to the shoulders ——— To the armpits as seen from the front — To the armpits as seen from the back — To the nipples ———————————— To the lower chest —————————— To the loins or waist ————————— From there to the navel ——————— To the upper thigh —————————— To the lower belly —————————— To the private parts ————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer ankle ——————— To the lower inner ankle ——————— To the top of the foot ————————— From there to the sole ———————— From the sole back to the outer ankle —

111

0 0 0 0 1

0 1 7 8 0

5 5 6 3 6

0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0

0 3 4 5 6 0 2 6 7 8 0 5 9 2 1 7 7 4 2 1

6 6 2 4 3 3 2 5 2 2 0 0 8 3 7 0 6 3 5 7

0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0



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The length of the foot will be ————— The lengths of the arm are these: From the shoulders to the elbow ——— From the elbow to the wrist ————— From the wrist to the fingertips ———— After this come the widths, starting with those of the profile view, that is to say, the thicknesses. At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip and nape ——————— Let the neck at the chin have width —— At the top of the shoulder blades ——— At the throat ———————————— At the shoulders ——————————— At the armpits ———————————— At the nipples ———————————— At the lower chest —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— At the lower belly —————————— At the buttocks and private parts ——— The thigh next to the lower buttocks will have width ———————————— A bit further down —————————— At the narrowing of the thigh ———— Above the knee let it be ——————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at its narrowest ———


0 9 4 0 1 0 6 0 0 9 3 1 0 6 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 6 6 5 3 4 4 5 7 7 7 6 6 7 7 6

5 2 8 3 8 7 0 6 7 7 6 5 0 2 5 0 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

5 5 5 3 3 3 3 3 2 2

6 4 4 8 3 2 8 2 9 1

0 0 0 0 0 0 0 0 0 0

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At the top of the foot ————————— At the lower outer ankle ——————— To this add also the thickness of the arm. At the shoulder let it be ———————— At the height of the armpits seen from the back ————————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— Give the open hand a thickness of ——— There follow the widths in the front view. At the cowlick let it be ———————— At the top of the forehead —————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— The width of neck at the lower chin will be —————————————— At the top of the shoulder blades ——— At the throat ———————————— Let the distance between the shoulder joints be ————————————— At the chest ————————————— Between the armpits ———————— Between the nipples ————————— At the lower chest —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— Also there the distance between the hip joints will be ——————————— At the lower belly —————————— At the private parts ————————— The thigh at the lower buttocks has width At the narrowing of the thigh ————

112

0 2 4 0 0 3 7 0 0 4 1 0 0 0 0 0 0

3 2 2 1 1

6 1 5 3 6

0 0 0 0 0

0 0 0 0 0 0

4 5 6 5 6 4

1 5 0 5 4 4

0 0 0 0 0 0

0 3 4 0 0 4 0 0 1 1 4 0 0 1 0 0 0 0 0 1

9 3 8 6 9 8 9 0

5 50 8 3 9 2 0 2

0

0 1 1 0 0

7 0 0 4 4

3 1 1 8 0

0 2 0 0 0

0 0 0 0 0 0



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Above the knee let the width be ———— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the lower shin at its narrowest ——— At the top of the foot and ankles ——— Below the ankles —————————— The width of the foot at the toes will be Set the widths of the arm as follows: At the biceps ———————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— Make the width of the open hand ——— Then draw accurately the profile and front views of the man, each in its place. The back view has the same silhouette as the front, and its armpits are separated by ————————————————— It was already said earlier how far the armpits are from the throat. The cleft of the buttocks has length —— The heel has width —————————

0 0 0 0 0 0 0 0 0 0

3 2 2 3 2 2 1 1 1 3

1 8 8 3 6 4 4 8 8 2

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

2 2 2 1 3

5 0 7 7 2

0 0 0 0 0

1 0 2 0 0 5 5 0 0 1 6 0

If you take a compass and place its center at the navel, opening it so the pencil tip touches the tips of the toes, the arc drawn will also touch the fingertips when the arms are spread out and raised slightly above the horizontal. All these things can be seen in the figures given below.


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Throat Shoulders Armpits seen from the front Nipples Lower chest Waist Navel Top of thigh Lower belly Private parts Lower buttocks Narrowing of thigh



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To this man we add a corresponding woman, whose body we will describe in three sets of measurements as above. The lengths therefore are these: From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the throat ———————————— Divide the space from the top of the forehead to the lower chin into three equal parts, the first of which will encompass the forehead, the second the nose, eyes, and ears, and the third the chin and mouth. Then continue with the measurements. From the throat to the shoulders ——— To the upper chest —————————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To below the breasts ————————— To the waist ————————————— From there to the navel ——————— To the top of the thigh ———————— To the top of the pubis ———————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh —————————————— To above the knee —————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer calf ———————— Inner ———————————————— To the top of the foot —————————

116

0 0 0 0 1

0 1 7 8 4

6 6 7 7 0

0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0 1

1 2 3 4 5 6 1 2 6 8 9 0

0 4 7 4 0 6 4 2 3 2 2 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 1

4 9 2 1 6 7 3

3 7 2 4 2 0 6

1 0 1 0 0 0 1



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From the top of the foot to the sole —— From the sole back to the outer ankle — The length of the foot will be ————— Make the arm as follows: From the shoulders to the elbow ——— From the elbow to the wrist ————— From the wrist to the fingertips ———— Having set this down proceed to the widths, making first the profile view. At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip ——————————— At the neck and chin ————————— The neck itself at the same height ——— At the top of the shoulder blades ——— At the throat ———————————— At the shoulders ——————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh and buttocks —— At the top of the pubis ———————— At the lower pubis —————————— Let the thigh next to the lower buttocks have width ————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ————————————————


0 2 3 0 0 1 4 0 0 8 4 0 0 9 6 0 0 9 1 0 0 5 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 6 6 5 5 3 3 4 6 7 7 7 7 5 7 9 8 7

4 2 7 0 5 3 3 8 7 0 0 3 5 0 6 8 4 0 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0

6 5 4 3 3 3 3 2

5 8 0 5 3 7 2 9

0 0 0 0 0 0 0 0

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At the lower shin where it is narrowest At the top of the foot ————————— Below the ankles —————————— Then make the widths of the arm in profile as follows. At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— The thickness of the open hand is ——— Next come the widths of the front view. At the cowlick let it be ———————— At the top of the forehead —————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and cheeks ———————— The neck at the lower chin will have width At the top of the shoulder blades ——— At the shoulders ——————————— The distance between the hip [i.e., shoulder] joints will be ————————— At the chest and shoulders —————— At the armpits seen from the front —— At the armpits seen from the rear ——— Between the nipples ————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— Also there the distance between the hip joints will be ——————————— At the top of the pubis ———————— At the lower pubis ——————————

117

0 2 0 0 0 2 2 0 0 4 0 0 0 0 0 0 0 0

3 3 2 2 1 1

9 5 5 6 3 6

0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

4 5 6 5 6 4 3 3 1

2 4 0 8 2 8 3 7 2

0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 1 1

8 3 8 9 6 85 7 1 1

5 0 5 4 5 0 8 0 9

0 0 0 0 0 0 0 0



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At the lower buttocks ———————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the shin ————————————— At the top of the foot ————————— Below the ankles —————————— The foot at the toes will have width —— The arm should be done as follows: At the biceps let the width be ————— Above the elbow ——————————— Below it ——————————————— At the wrist ————————————— The open hand will have width ———— You should now make drawings agreeing with the values given for the three dimensions of the various parts of the woman’s body. The back view, as often said, will have the same silhouette as the back [i.e., front] view. The cleft of the buttocks will have length ————— The heel has width —————————

0 0 0 0 0 0 0 0 0 0 0 0

6 5 3 3 3 3 3 2 1 1 1 2

1 6 7 2 1 7 1 9 2 7 8 9

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0

2 2 2 1 2

5 0 6 6 9

0 0 0 0 0

0 5 7 0 0 1 5 0

If you draw this body with feet out and arms stretched and raised slightly above the vertical, and if you center a compass at the navel, you can graze with a single circle the tips of the fingers and toes. The figure illustrates this.

0 8 6 0 1 2 5 0 0 6 2 0


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Throat Shoulders Chest Armpits seen from the front Nipples Lower breasts Waist Navel Top of thigh Upper pubis Lower pubis Lower buttocks Narrowing of thigh



118


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I will now describe two bodies, one male and one female, doing first the heights as always. In the man let the vertical distance from the crown to the lower chin be —— From there back to the top of the forehead From the top of the forehead to the eyebrows —————————————— From the eyebrows to the lower nose — From the nose to the lower chin ———— From the crown to the upper shoulder blades —————————————— To the upper shoulders ———————— To the throat ———————————— From there to the upper chest ————— Below the armpits —————————— To the nipples ———————————— To the lower chest —————————— To the waist ————————————— From the waist to the navel —————— To the curve of the hip ———————— To the hip joint ——————————— To the lower belly —————————— To the private parts ————————— To the lower buttocks ———————— From there down a bit ———————— And still further down ———————— To the narrowing of the thigh ———— To mid knee ————————————— From mid knee to upper knee on the outside —————————————— Inside ——————————————— From mid knee to lower knee on the outside —————————————— Inside ——————————————— To the lower inner calf ———————— To the lower outer calf ————————

121

0 7 0 0 0 6 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

7 9 0 2 3 5 6 9 2 2 6 7 8 9 0 2 5 2

7 5 4 0 6 0 0 6 0 7 5 6 6 7 6 5 6 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 2 8 0 0 1 9 0 0 0 0 0

0 1 6 6

7 5 2 9

0 0 0 0



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To the top of the foot ————————— From there to the sole ———————— From the sole back to the outer ankle — The length of the foot will be ————— To the arm give the following lengths. Vertically from the upper throat down to the elbow ———————————— From the elbow to the wrist ————— Length of the hand ————————— To this we add the widths of the man’s body, in profile first. At the forehead let it be ——————— At the eyebrows ——————————— At the nose and nape ————————— At the lip and nape ————————— At the chin and neck likewise ————— The neck at the same height has width At the upper shoulder blades ————— At the shoulders ——————————— At the throat ———————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— At the lower chest —————————— At the waist ————————————— At the navel ————————————— At the curve of the hip ———————— At the hip joint ——————————— At the lower belly —————————— At the private parts and buttocks ——— The thigh’s width next to the lower buttocks will be ——————————— A bit further down —————————— And still further down ———————— At the narrowing of the thigh ———— Above the outer knee ————————


1 0 0 0

4 2 1 9

7 8 8 0

0 0 0 0

1 1 4 0 0 8 5 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 6 6 5 5 3 3 4 5 7 8 8 7 6 6 6 8 7 6

1 6 0 4 4 4 5 4 3 4 2 3 6 5 3 5 0 2 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

5 5 5 5 4

9 9 8 4 0

0 0 0 0 0

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Inner ———————————————— At mid knee ————————————— Lower outer knee —————————— Inner likewise ———————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the bottom of the shin ——————— At the top of the foot ————————— Below the ankles —————————— There follows the width of the arm in profile, or thickness. At the shoulders let it be ——————— Below the armpits at the biceps ———— At the elbow ———————————— Below it ——————————————— A bit further down —————————— At the wrist ————————————— The open hand has thickness ————— The widths in the front view should be marked thus: At the top of the forehead —————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the lower nose —————————— The next at the lower chin has width — At the upper shoulder blades ————— At the shoulders ——————————— At the throat ———————————— The distance between the shoulder joints is ———————————————— At the chest and shoulders —————— At the armpits ———————————— Between the nipples ————————— At the waist ————————————— At the navel ————————————— At the curve of the hip ————————

122

0 0 0 0 0 0 0 0 0 0

3 3 3 3 4 3 3 2 2 4

7 5 4 4 0 7 4 1 4 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

4 3 2 2 1 1 1

5 3 4 4 7 4 5

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

5 5 5 6 4 3 4 9 2

1 8 5 0 6 3 0 2 4

0 0 0 0 0 0 0 0 0

0 1 1 0 0 1 1

9 4 0 7 9 0 0

7 4 2 0 0 2 0

0 0 0 0 0 0 0



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At the hip joint ——————————— The distance between the hip joints will be —————————————— At the lower belly —————————— At the private parts ————————— At the thigh near the lower buttocks — A bit further down —————————— And still further down also —————— At the narrowing of the thigh ———— Above the outer knee ———————— Above the inner knee likewise ———— At the upper knee —————————— At mid knee ————————————— The two widths just given are the same. But when the outline of the body is drawn, it should be made slightly curved at this point. At mid calf let the width be —————— Near the [lower] outer calf —————— Near the [lower] inner calf —————— At the bottom of the shin ——————— At the ankle and the top of the foot —— Below the ankle likewise ——————— The foot at the toes will have width —— The arm should be done as follows. In the front view, below the armpit at the biceps the width will be —————— Above the elbow ——————————— Below it ——————————————— A bit further down —————————— At the wrist ————————————— The width of the hand will be ———— Then turn to the measurements of the female body, starting with the heights,


1 1 3 0 0 1 1 0 0 0 0 0 0 0 0

8 1 1 5 5 5 4 3 3 2 2

5 4 3 5 1 1 2 2 2 9 9

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

3 3 3 1 1 1 3

5 0 2 4 9 9 0

0 0 8 0 0 0 0

0 0 0 0 0 0

2 2 3 2 1 3

5 1 0 2 6 1

0 0 0 0 0 0

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as follows. From the crown to the upper neck ——— From there back up to the upper forehead From upper forehead to eyebrows ——— From eyebrows to the lower nose ——— From there to the neck likewise ———— From the top of the neck the chin should be made to curve up slightly. From the crown to the top of the shoulder blades —————————————— To the throat ———————————— From the throat to the top of the chest — To below the armpits ————————— To the nipples ———————————— To below the breasts ————————— To the loins where they are girt ———— From there to the navel ——————— To the top of the thigh ———————— To the lower belly —————————— To the beginning of the pubis, where the body forks ———————————— To the lower pubis —————————— To the lower buttocks ———————— From the lower buttocks to a bit further down —————————————— To the narrowing of the thigh ———— To above the knee —————————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer calf ———————— Inner ———————————————— To the top of the foot ————————— From there to the sole ———————— From the sole back up to the outer ankle The length of the foot will be —————

123

0 0 0 0 0

7 6 2 2 2

0 0 0 0 0

0 0 0 0 0

0 1 0 0 0 0 1 0 0 0

7 0 2 3 5 7 1 1 6 7

9 0 1 7 5 0 2 5 0 0

0 0 0 0 0 0 0 0 0 0

0 8 8 0 0 9 6 0 1 0 4 0 0 0 0 1 0 0 0 1 0 0 0

2 4 9 1 2 6 7 4 2 1 8

3 3 2 3 0 0 2 5 6 8 5

0 0 0 0 0 0 0 0 0 0 0



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The lengths of the arm are these: From the shoulder at the height of the throat down to the elbow —————— From the elbow to the wrist ————— The length of the hand will be ———— After marking the heights do the widths. For the profile view they are as follows: At the forehead ——————————— At the eyebrows ——————————— At the nape and nose ————————— At the lip and nape ————————— At the chin and neck likewise ————— The neck itself at the same height ——— At the top of the shoulder blades ——— At the throat ———————————— At the chest ————————————— By the armpits ——————————— At the nipples ———————————— Below the breasts —————————— At the loins where they are girt ———— At the navel ————————————— At the upper thigh —————————— At the lower belly —————————— At the top of the pubis ———————— At the lower pubis —————————— At the thigh by the lower buttocks —— A bit further down —————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ————————


1 1 5 0 0 8 5 0 0 5 8 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 6 5 5 5 3 3 4 6 6 7 6 5 6 9 8 8 7 6 6 5 4 3 3 3 3

1 3 5 2 2 2 3 0 2 8 0 6 5 8 0 6 0 5 5 0 4 0 5 5 9 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Inner ———————————————— At the bottom of the calf ——————— At the top of the foot ————————— Below the ankle ——————————— The widths of the arm are as follows: At the shoulder ——————————— Below the armpit —————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— The hand itself ——————————— Then mark on the next vertical line the widths in front view, as follows: At the top of the forehead let it be ——— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— The neck by the chin will have width — At the top of the shoulder blades ——— At the throat ———————————— Below this, the distance between the shoulder joints is ————————— At the chest ————————————— Between the armpits ———————— Between the nipples ————————— The width of the lower chest is ———— At the loins where they are girt ———— At the navel ————————————— At the top of the thigh ———————— Also here the distance between the hip joints is ————————————— At the lower belly —————————— At the top of the pubis ————————

124

0 0 0 0

3 2 2 3

2 0 2 5

0 0 0 0

0 0 0 0 0 0

4 3 2 2 1 1

4 3 4 4 3 4

0 0 0 0 0 0

0 0 0 0 0 0 0 0

5 5 5 5 4 3 3 8

1 7 3 8 5 0 5 7

0 0 0 0 0 0 0 0

0 1 0 0 0 0 1 1

8 2 8 6 8 7 4 2

5 2 2 0 4 7 0 0

0 0 0 0 0 0 0 0

0 8 4 0 1 1 3 0 1 2 4 0



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The thigh at the lower pubis has width At the lower buttocks ———————— A bit further down —————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the bottom of the shin ——————— At the top of the foot ————————— Below the ankle likewise ——————— And the foot at the toes will have width Then prepare the arm in the following way: Make its width below the armpits —— Above the shoulder ————————— Below it ——————————————— At the wrist ————————————— Make the width of the open hand ——— Now draw the male and female bodies based on the distances marked, using appropriate curves, as you can see we strove to do in the examples. If you wish to make the head slightly bigger, increase its height by ————————————— Then apply corresponding increases to the widths and thicknesses, in both the profile and the front view, as I have explained earlier and as I have demonstrated in the examples.


0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 6 6 5 3 3 3 3 3 3 1 2 2 2

2 1 0 3 8 5 1 5 4 0 4 6 6 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

2 2 2 1 2

4 1 9 5 8

0 0 0 0 0

0 0 8 0

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Crown Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Armpits Nipples Below these Loins where we gird ourselves Navel Curve of hip Top of thigh Lower belly Private parts Lower buttocks A bit further down Yet further down Narrowing of thigh Upper outer knee Inner Mid knee Lower outer knee Inner Lower outer calf Inner


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Crown Forehead Eyebrows Nose Chin Top of shoulder blades Throat Chest Armpits Nipples Lower breasts Loins where they are girt Navel Top of thigh Lower belly Top of pubis where the body forks Lower pubis Lower buttocks A bit further down Narrowing of thigh Upper knee Mid knee Lower knee Lower outer calf Inner


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There follow the measurements for another male body. The main components of its width [i.e., height] should be marked next to the first vertical line, meant for the profile view, as follows. From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the top of the shoulders —————— To the shoulder joints ———————— To the throat ———————————— From there to below the armpits ——— To the nipples ———————————— Below them ————————————— To the waist ————————————— From there to the navel ——————— To the top of the thigh ———————— To the private parts ————————— To the lower buttocks ———————— From the lower buttocks to the narrowing of the thigh —————————— To above the knee —————————— To mid knee ————————————— From mid knee to lower knee ————— From mid knee again to lower outer calf Inner ———————————————— To the top of the foot ————————— To the bottom of the shin ——————— To the sole ————————————— The foot should be given a length of —— The arm will be as follows: From shoulder to elbow let the height be From elbow to wrist ————————— From wrist to fingertips ——————— Having thus marked the heights, you should now deal with the thicknesses of the profile view

127

0 0 0 1 1 1 0 0 0 1 0 0 0 1

1 2 8 0 1 1 2 4 5 0 2 7 8 1

0 2 4 1 0 5 1 0 3 0 3 5 5 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 1 1 1

3 7 0 1 6 7 4 5 7 0

2 4 5 6 8 7 6 4 0 0

1 0 0 0 0 0 0 0 0 0

1 0 0 0 0 9 0 0 0 6 0 0



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and the widths of the front view. The thicknesses are: At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip and nape ——————— The neck at the chin has width ———— At the upper shoulders ———————— At the shoulder joints ———————— At the throat ———————————— At the armpit ———————————— At the nipples ———————————— At the lower chest —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— At the private parts ————————— At the thigh next to the lower buttocks At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot ————————— Below the ankle ——————————— The arm should be made thus: Thickness at the shoulder —————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— Yet further down —————————— At the wrist ————————————— The thickness of the hand will be ——— The widths in the front view will be these: At the cowlick ———————————— At the forehead ———————————


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 7 7 7 6 4 4 5 6 7 8 8 7 6 8 7 6 6 4 3 3 4 4 3 2 3 4

2 0 5 0 2 2 6 0 7 9 5 3 3 7 4 6 1 1 7 8 7 4 0 7 5 0 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

5 3 2 2 2 1 1

0 7 6 8 0 6 9

0 0 0 0 0 0 0

0 5 5 0 0 6 6 0

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At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— Let the neck at the chin have width —— At the top of the shoulders —————— Let the distance between the shoulder joints be ————————————— At the chest and shoulders next to the throat —————————————— At the armpits ———————————— At the armpits seen from the back ——— Between the nipples ————————— At the lower chest —————————— At the waist ————————————— At the navel ————————————— At the curve of the hip ———————— The distance between the hip joints will be At the private parts ————————— At the thigh next to the lower buttocks At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot ————————— Below the ankle ——————————— And the foot at the toes will have width

128

0 0 0 0 1

5 6 4 4 0

8 5 8 0 0

0 0 0 0 0

1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0

5 0 1 7 1 9 0 1 8 1 5 4 3 3 3 3 3 3 1 2 2 3

0 0 8 6 6 6 0 7 5 9 4 5 5 1 0 8 4 0 6 0 2 9

0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0



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Given next are the widths of the arm. At the biceps let it be ————————— Above the elbow ——————————— Below it ——————————————— Still further down —————————— At the wrist ————————————— The hand has width ————————— As we have often remarked and done by way of example, you should now draw with appropriate lines the body in profile and seen from the front. Then use for the rear view the same silhouette as for the front view. The cleft of the buttocks has length —— The heel has width ————————— The armpits in the rear view are lower, as we have often said; the vertical distance from the top of the throat to them is ——————————————————

0 0 0 0 0 0

2 2 3 2 2 3

6 3 0 3 0 2

0 0 0 0 0 0

0 6 0 0 0 2 2 0

0 4 5 0

In this body, with the arm extended and raised so the fingertips are at the height of the crown, a compass centered at the navel can draw a circle tangent to the extremities of the hands and feet, as can be seen in our examples as well.


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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder Shoulder joints Throat Armpits seen from the front Nipples Lower chest Waist Navel

Top of thigh Private parts Lower buttocks Narrowing of thigh


Lower outer calf Inner


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After the male body, we now add a female one. In the same way as before, measure out the heights next to the first vertical line, as follows: From the crown to the cowlick let it be — To the forehead ——————————— To the lower chin —————————— To the top of the shoulders —————— To the throat ———————————— From there to the armpits —————— To the armpits seen from the back ——— To the nipples ———————————— To the lower breasts ————————— To the waist ————————————— From the waist to the navel —————— To the top of the thigh ———————— To the top of the pubis ———————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer calf ———————— Inner ———————————————— To the top of the foot ————————— To the lower shin and the ankle ———— To the sole ————————————— Make the length of the foot —————— Now come the lengths of the arm: From shoulder to elbow ——————— From elbow to wrist ————————— From wrist to fingertips ———————

132

0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0

0 2 8 9 1 2 3 4 6 0 2 6 8 9 0 4 9 1 1 6 7 3 4 6 9

6 4 4 8 0 8 7 5 0 7 6 2 3 0 5 5 0 6 9 4 3 3 2 2 4

0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0

0 9 4 0 0 8 4 0 0 6 1 0



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After the heights are marked as indicated, add the measurements of thickness and width: first the thickness, in the profile view. At the cowlick let it be —— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip and nape ——————— The neck at the chin will have width — At the top of the shoulders —————— At the throat ———————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— At the buttocks and upper pubis ——— At the lower pubis —————————— At the thigh next to the lower buttocks At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot ————————— At the lower ankle —————————— Then make the arm in profile as follows: Width at the shoulder ———————— At the biceps ———————————— At the elbow ————————————


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 7 7 6 6 4 4 5 7 8 7 6 8 9 8 7 6 6 4 3 3 4 3 3 2 2 3

5 2 4 6 1 0 5 3 5 0 5 2 0 9 2 9 0 4 5 8 9 2 8 6 4 5 9

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 4 6 0 0 3 6 0 0 2 8 0

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At the wrist let it be ————————— And let the open hand have thickness — The widths in the front view are these: At the cowlick let it be ———————— At the forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— Give the next at the chin a width of —— At the top of the shoulders —————— At the throat ———————————— The distance between the shoulder joints is ———————————————— Between the armpits ———————— Seen from the back ————————— Between the nipples ————————— At the top of the chest ———————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— The distance between the hip joints is — At the top of the pubis ———————— At the lower pubis —————————— At the lower buttocks ———————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ———————————

133

0 1 4 0 0 1 8 0 0 0 0 0 0 0 0 1

4 6 5 6 4 3 9 1

8 4 8 2 8 9 5 8

0 0 0 0 0 0 0 0

0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0

9 8 0 5 3 0 9 1 2 9 2 6 6 5 3 3 3

5 7 4 7 2 2 4 7 4 0 8 1 1 2 9 2 1

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0



view 67: page L6v

At mid thigh ———————————— At the lower outer thigh ——————— Inner ———————————————— At the bottom of the shin ——————— At the ankle ————————————— Below the ankle ——————————— The foot at the toes will have width —— There follows the description of the arm. Its width will be: At the biceps ———————————— At the elbow ———————————— Below it ——————————————— Further down yet —————————— At the wrist ————————————— At the hand ————————————— Having thus marked heights and widths, you should now draw the appropriate shapes of the body in profile and seen from the front. The view from the back has the same silhouette as from the front. The cleft of the buttocks has length —— The heel has width —————————

0 0 0 0 0 0 0

3 3 2 1 1 1 3

8 2 8 6 9 9 4

0 0 0 0 0 0 0

0 0 0 0 0 0

2 2 2 2 1 2

7 4 8 1 7 8

0 0 0 0 0 0

0 6 2 0 0 1 9 0

This body too, with the arms outstretched, can be inscribed in a circle like the preceding male one, as can be seen in the drawings.


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view 67: page M1r

Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulders Throat Armpits seen from the front Nipples Lower breasts

Waist Navel Top of thigh Top of pubis Lower publis Lower buttocks

Narrowing of thigh




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135


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136


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Once again I will describe a male body using the same method. Let its measurements be as follows: From crown to cowlick ———————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— Then divide the space between the upper forehead and the lower chin into three parts; in the top part place the forehead, in the next the eyes, nose, and ears, and in the lowermost the mouth and chin. The procedure for drawing the face is always the same; we will not repeat it each time. You can transfer it from one case to another. From the throat to the top of the chest To the armpits ———————————— To the armpits seen from the back ——— To the nipples ———————————— To below the breast ————————— To the lower thorax ————————— To the waist ————————————— From there to the navel ——————— To the curve of the hip ———————— To the top of the thigh ———————— To the lower belly —————————— To the private parts ————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From mid knee to the lower knee ——— To the lower outer calf ———————— Inner ———————————————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole —————————————

137

0 0 0 1 1

1 7 8 0 1

0 6 6 6 5

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1

2 3 4 5 6 7 9 2 4 6 7 8 1 4 9 1 2 6 7 3 4 6

0 9 2 1 1 1 2 9 7 9 9 6 5 4 1 0 5 4 2 9 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0



view 69: page M2v

The length of the foot will be ————— But the arm is as follows: From the shoulder to the elbow —————————— From the elbow to the wrist ————— From the wrist to the fingertips ———— After this, complement the heights with the widths, starting with the profile view: At the cowlick let it be ———————— At the eyebrows ——————————— At the nose ————————————— At the upper lip ——————————— At the chin and neck ————————— The width of the neck itself is ————— At the top of the shoulder blades also — At the shoulders ——————————— At the throat ———————————— At the chest ————————————— At the armpits ———————————— At the nipples ———————————— Below the breast ——————————— At the lower thorax ————————— At the waist ————————————— At the navel ————————————— At the curve of the hips ——————— At the top of the thigh ———————— At the lower belly —————————— At the private parts ————————— At the thigh next to the lower buttocks A bit further down —————————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee cap ————————— At the height of the lower knee ———— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the bottom of the shin ——————— At the top of the foot ————————— Below the ankle ———————————


0 9 5 0 1 0 4 0 0 8 3 0 0 6 6 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 7 6 5 5 3 3 4 5 7 8 8 7 7 7 6 7 7 7 7 5 5 5 4 4 3 4 4 3 3 2 2 3

6 1 3 5 2 9 9 5 6 3 3 4 9 8 0 8 0 6 2 3 8 5 6 2 0 5 0 2 9 5 4 8 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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The arm in profile should be made thus: Width at the shoulder ———————— At the armpit seen from the back ——— At the elbow ———————————— Below the elbow ——————————— A bit further down —————————— At the wrist ————————————— Thickness of the hand ———————— The widths in the front view should be marked as follows: At the cowlick ———————————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— At the neck just below the chin ———— At the upper shoulder blades ————— At the shoulders ——————————— At the throat ———————————— At that point the distance between the shoulder joints will be ——————— At the chest ————————————— Between the armpits ———————— Between the armpits seen from the back Below the breast ——————————— At the waist ————————————— At the navel ————————————— At the curve of the hip ———————— At the top of the thigh ———————— The distance between the hip joints is — At the lower belly —————————— At the thigh next to the private parts —

138

0 0 0 0 0 0 0

5 3 2 2 2 1 1

0 8 5 6 0 4 7

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1

6 6 6 6 5 3 4 0 3

0 6 2 8 2 5 0 7 4

0 0 0 0 0 0 0 0 0

1 1 1 1 1 0 1 1 1 0 1 0

0 5 0 1 0 9 3 0 1 8 1 5

8 2 4 8 9 0 0 7 5 5 4 5

0 0 0 0 0 0 0 0 0 0 0 0



view 69: page M3r

At the lower buttocks ———————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee also ——————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the ankle and top of the foot ———— Below the ankle at the foot —————— The foot at the toes will have width —— Make the arm as follows: At the biceps make the width ————— Above the elbow ——————————— Below it ——————————————— Further down still —————————— At the wrist ————————————— And the hand should have width ——— Then each view of the male body, profile and front, should be drawn appropriately. The rear view has the same silhouette as the front view. The cleft of the buttocks will have length ————— The width of the heel will be ————— This body too can be inscribed in a circle when the arms are extended and the center is placed at the navel, just as we have shown in previous examples and in the figures given below.


0 0 0 0 0 0 0 0 0 0 0 0

5 4 3 3 3 3 3 3 1 2 2 3

3 4 5 5 1 8 2 0 7 2 1 7

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

2 2 3 2 2 3

6 4 0 4 0 4

0 0 0 0 0 0

0 5 5 0 0 2 2 0

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view 70: page M3v

Crown Cowlick Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Armpits seen from the front Nipples Below the breast Lower thorax Waist Navel Curve of hip Top of thigh Lower belly Private parts Lower buttocks

Narrowing of thigh



139


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140


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141


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We shall now describe a female body of similar proportions. These will be the heights from the crown: To the cowlick ———————————— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the throat ———————————— From there to the shoulders —————— To the chest ————————————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To below the breasts ————————— To the waist ————————————— From the waist to the navel —————— To the upper thigh —————————— To the lower belly —————————— To above the pubis —————————— To the beginning of the pubis ————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From mid knee to the lower knee ——— To the lower outer calf ———————— To the lower inner calf ———————— To the top of the foot ————————— To the outer ankle —————————— To the sole ————————————— The length of the foot will be ————— The lengths of the arm are these: From the shoulder to the elbow ———— From the elbow to the wrist ————— From the wrist to the fingertips ———— After that should be marked down the thicknesses and widths of the body. The thicknesses go next to the first vertical line, for the profile view, thus:

142

0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0

0 1 7 9 1 0 2 3 4 5 7 1 1 6 7 8 9 0 1 3 7 0 1 6 7 3 4 6 9

9 8 5 0 0 5 8 7 6 6 2 8 2 1 0 4 5 2 1 8 9 0 0 0 0 1 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 8 3 0 0 5 8 0



view 71: page M5r

At the cowlick let it be ———————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip ——————————— At the chin and neck ————————— The neck at the same place —————— At the top of the shoulders —————— At the throat ———————————— Next to the shoulder joints —————— At the chest ————————————— Next to the armpits ————————— At the nipples ———————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— At the lower belly —————————— Above the pubis ——————————— At the beginning of the pubis ————— At the lower pubis —————————— At the thigh by the lower buttocks —— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee also ——————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the bottom of the shin ——————— At the top of the foot ————————— At the lower ankle and foot —————— Make the arm in profile as follows: Width at the shoulder ———————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— Further down yet —————————— At the wrist ————————————— Make the thickness of the hand ————


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 6 7 6 5 5 3 3 4 5 7 7 7 7 6 7 0 0 9 8 8 7 6 4 4 3 4 4 3 2 2 3

0 4 3 4 8 6 4 7 8 8 4 5 8 2 2 2 5 0 2 7 2 0 7 8 0 8 4 1 6 5 6 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

4 3 2 2 2 1 1

4 8 6 7 0 5 8

0 0 0 0 0 0 0

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Mark the widths by the second vertical line, as follows: At the cowlick ———————————— At the top of the forehead —————— At the middle of the forehead ————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— At the neck by the chin ———————— At the top of the shoulder blades ——— At the throat ———————————— At the shoulder joints ———————— The distance between them will be —— At the chest and upper arms ————— Between the armpits ———————— Between the armpits from the back —— Between the nipples ————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the top of the thigh ———————— Make the distance between the hip joints At the lower belly —————————— Above the pubis ——————————— At the top of the pubis ———————— At the thigh next to the lower pubis —— At the lower buttocks ———————— At the narrowing of the thigh ———— Above the knee ———————————

143

0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0

5 6 6 6 6 5 3 5 0 2 6 3 9 0 6 9 8 1 3 9 3 3 3 6 6 5 4

0 4 5 0 5 0 2 6 6 0 5 7 1 5 8 7 8 2 0 5 2 5 4 6 6 6 2

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 0 1 0 0 0



view 72: page M5v

At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the top of the foot ————————— At the lower ankle and foot —————— The foot at the toes will have width —— Then add the arm with these widths: At the biceps ———————————— Above the elbow ——————————— Below it ——————————————— A bit further down —————————— At the wrist ————————————— Through the hand —————————— After this has been done, the profile and front view of the body should be drawn, suitably and prettily. The view from the rear, as often mentioned, shares the same silhouette as the front view. Give the cleft of the buttocks length —— And to the heel width ———————— This body can be inscribed in a circle centered at the navel, as has been shown for several others, and as can be seen in the examples.


0 0 0 0 0 0 0 0

3 3 4 3 3 1 2 3

8 7 3 9 4 7 1 5

1 0 0 0 0 0 1 0

0 0 0 0 0 0

2 2 3 2 1 3

9 4 0 4 8 0

0 0 0 0 0 0

0 6 0 0 0 2 0 0

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view 72: page M6r

Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Top of shoulders Shoulder joints Chest Armpits seen from the front Nipples Lower breasts

Waist Navel Top of thigh Lower belly Above the pubis Top of pubis Lower pubis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee



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145


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146


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There follows the description of another female body, agreeing with the earlier male one. Its heights will then be as follows: From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the throat ———————————— From the forehead to the eyebrows —— Then divide the eyebrows from the chin by a horizontal line halfway. The upper of the two equal spaces thus created will be for the nose, eyes and ears; the lower for the nose and chin. Then continue measuring. From throat to top of shoulders ———— To the shoulder joints ———————— To the armpits seen from the front —— To the armpits seen from the rear ——— To the nipples ———————————— To the lower breasts ————————— To the waist ————————————— From there to the navel ——————— To the top of the thigh ———————— To the lower belly —————————— To the beginning of the pubis ————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From mid knee to lower knee ————— To the lower outer calf ———————— To the lower inner calf ———————— To the top of the foot ————————— To the outer lower ankle ——————— To the sole —————————————

147

0 0 0 0 1 0

0 1 6 8 0 2

7 6 4 7 2 2

0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1

0 1 4 5 6 8 2 1 6 7 8 9 0 4 8 0 1 5 6 3 3 5

8 5 7 7 5 0 5 6 5 5 7 6 8 5 7 9 5 8 7 0 9 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



view 74: page N1v

The length of the foot shall be ———— The arm should be made as follows: From the shoulder to the elbow ———— From the elbow to the wrist ————— From the wrist to the fingertips ———— Next mark the thicknesses for the profile view, which are these: At the cowlick ———————————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip and nape ——————— The neck at the chin has width ———— At the top of the shoulder blades ——— At the throat ———————————— At the top of the shoulders —————— At the shoulder joints ———————— At the armpits seen from the front —— At the nipples ———————————— At the lower breasts ————————— At the waist ————————————— At the navel ————————————— At the hip —————————————— At the lower belly —————————— At the top of the pubis ———————— At the lower pubis —————————— The thigh at the lower buttocks gets —— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin where it is narrowest At the top of the foot ————————— At the lower outer ankle and foot ———


0 9 0 0 1 0 0 0 0 8 3 0 0 5 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 7 6 5 3 3 4 5 6 7 7 7 6 7 9 9 8 8 7 6 4 3 3 4 3 3 2 2 3

5 1 2 0 7 2 5 2 5 2 7 5 0 1 1 8 2 6 7 3 0 3 7 6 0 6 4 3 5 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Next comes the arm. Make its thickness At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— The hand itself has thickness ————— Then set down the widths of the front view: At the cowlick let it be ———————— At the top of the forehead —————— At mid forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— The neck at the chin has width ———— At the top of the shoulder blades ——— At the throat ———————————— At the top of the shoulders —————— At the shoulder joints ———————— The space between them is —————— Between the armpits ———————— Between the armpits seen from the rear Back at the chest and shoulders the width is ———————————————— Between the nipples ————————— At the lower breasts ————————— At the waist ————————————— At the navel ————————————— At the hip —————————————— At that point the distance between the hip joints will be ————————— At the lower belly the width is ———— At the top of the pubis ———————— The thigh next to the lower pubis has width ——————————————

148

0 0 0 0 0 0

4 3 2 2 1 1

4 7 4 6 5 8

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 0

4 6 6 6 6 4 3 4 8 0 2 9 8 9

9 3 5 0 5 8 3 0 2 5 2 6 9 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 1

3 6 9 8 1 2

5 4 4 4 1 8

0 0 0 0 2 0



view 74: page N2r

Next to the lower buttocks —————— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— At the lower inner calf ———————— At the lower shin where it is narrowest At the top of the foot ————————— Below the ankle at the foot —————— The foot at the toes will have width —— And let the arm be as follows: At the biceps make the width ————— Above the elbow ——————————— Below it ——————————————— Further down yet —————————— At the wrist ————————————— The hand should be given width ——— And so now draw with light strokes a shape agreeing with the measurements that you have marked down, for both the profile and the front view. The rear view has the same silhouette as the front view. The cleft of the buttocks has length —— The width of the heel is ——————— This body too can be encompassed in a circle centered at the navel and touching the tips of the fingers and toes, as was remarked for several earlier ones, and as we have shown in the examples below.

0 0 0 0 0 0 0 0 0 0 0 0

6 5 4 3 3 4 3 3 1 2 2 3

3 3 0 6 5 0 8 4 6 0 0 3

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

2 2 2 2 1 3

7 3 9 3 6 0

0 0 0 0 0 0

0 6 8 0 0 1 9 0

0 9 1 0 1 3 0 0 1 2 9 0 0 6 4 0


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view 75: page N2v

Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Throat Top of shoulders Shoulder joints Armpits Nipples Lower breasts Waist Navel Curve of hip Lower belly Top of pubis Lower pubis Lower buttocks Narrowing of thigh Upper knee Mid knee Lower knee



149


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150


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I will now describe the image of a man of tall stature and slender, as an example for the reader who may wish to draw somewhat taller people. The measurements are as follows: From the crown to the cowlick let the height be ————————————— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— From there to the upper chest ————— To the armpits ———————————— To the armpits seen from the back ——— To the nipples ———————————— To below the breast ————————— To the lower thorax ————————— To the waist ————————————— From there to the navel ——————— To the curve of the hip ———————— To the hip joint or top of the thigh ——— To the private parts ————————— To the top of the buttocks ——————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From there to the lower knee ————— To the lower outer calf ———————— Inner ———————————————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— The length of the foot will be ————— The arm has these lengths: From the shoulder to elbow —————— From the elbow to the wrist ————— From the wrist to the fingertips ———— Having set down all these heights as stated, you should now mark the widths of the profile and front views. First the profile:

152

0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0

0 1 7 8 0 0 1 3 3 4 5 7 0 2 2 5 7 0 4 9 1 1 6 6 4 5 7 8

7 8 0 0 0 9 9 3 8 4 5 0 0 0 5 9 5 0 5 1 8 6 2 8 8 6 3 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

1 0 6 0 0 8 5 0 0 5 4 0



view 76: page N4r

At the cowlick let it be ———————— At the forehead ——————————— At the eyebrows ——————————— At the nose ————————————— At the upper lip ——————————— The neck at the chin will have width — At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— At the chest ————————————— By the armpits ——————————— By the nipples ———————————— At the lower chest —————————— At the lower thorax ————————— At the waist ————————————— At the navel ————————————— At the curve of the hip ———————— At the hip joint ——————————— At the private parts and buttocks ——— Also there the thigh has width ———— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid calf ————————————— At the lower outer calf ———————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot ————————— At the foot and lower outer ankle ——— To this add the arm’s measurements: At the shoulder ——————————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— The hand has thickness ——————— And here are the widths of the front view: At the cowlick ————————————


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 6 5 5 3 3 4 5 7 7 7 7 7 5 5 6 7 7 5 5 3 3 3 3 3 3 2 2 3

7 1 7 8 2 4 7 3 0 1 5 6 1 0 8 8 0 7 0 4 2 8 2 0 6 4 0 0 3 5

0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

4 3 2 2 1 1

0 0 1 2 2 4

0 0 0 0 0 0

0 4 4 0

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At the forehead ——————————— At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— At the neck by the chin ———————— At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— Also there the distance between the shoulder joints will be ——————— At the upper chest let the width be —— Between the armpits ———————— Between the armpits seen from the rear Between the nipples ————————— At the lower chest —————————— At the lower thorax only ——————— At the waist ————————————— At the navel ————————————— At the curve of the hip ———————— At the hip joints ——————————— The distance between them will be —— At the private parts ————————— At the thigh by the lower buttocks —— At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid thigh ———————————— At the lower outer thigh ——————— Inner ———————————————— At the bottom of the shin ——————— At the top of the foot and ankle ———— Below the ankle at the foot ——————

153

0 0 0 0 0 0 0 1

5 5 5 4 3 3 8 1

5 2 8 3 1 6 8 2

0 0 0 0 0 0 0 1

0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

9 3 9 4 7 9 9 8 9 9 0 7 9 4 3 2 2 2 3 2 2 1 1 1

4 2 0 0 0 6 0 1 1 0 0 5 9 7 7 8 6 6 2 8 5 2 6 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



view 77: page N4v

Give the foot at the toes length ———— The arm is done as follows: At the biceps its width is ——————— Below the elbow ——————————— At the wrist ————————————— The arm will have width ——————— And so now you should fill in the three images of the body with appropriate strokes, as we have often taught and shown by example. Again the rear view is bounded by the same outer lines as the front view. Give: The cleft of the buttocks length ———— And the heel width ————————— The attentive reader will observe from these measurements that the widths of the profile, also called thicknesses, are less than the widths viewed from the front throughout the trunk, but are greater in the case of the thigh and calf. Moreover, it does not take much effort to express using the method of this second book the images measured in the first book by means of parts. Keep in mind also the variations of the head that I expounded at various times, for when the head is modified the whole body appears to change.


0 3 0 0 0 0 0 0

2 2 1 2

2 5 4 5

0 0 0 0

0 5 8 0 0 1 6 0

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view 77: page N5r

Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Armpits seen from the front Nipples Lower chest Bottom of thorax Waist Navel Curve of hip Top of thigh Private parts Lower buttocks

Narrowing of thigh




154


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To the male body just described I will now add a matching female body. Let its heights be given as follows. From the crown to the cowlick ———— To the forehead ——————————— To the lower chin —————————— To the top of the shoulder blades ——— To the shoulders ——————————— To the throat ———————————— From there to the upper chest ————— To the armpits ———————————— To the armpits seen from the back ——— To the nipples ———————————— To the lower breasts ————————— To the waist ————————————— From there to the navel ——————— To the hip joint or top of the thigh ——— To the top of the pubis ———————— To the lower pubis —————————— To the lower buttocks ———————— From there to the narrowing of the thigh To above the knee —————————— To mid knee ————————————— From mid knee to the lower knee ——— To the lower outer calf ———————— To the lower inner calf ———————— To the top of the foot ————————— To the lower outer ankle ——————— To the sole ————————————— And the length of the foot will be ——— The arm should be done thus: From the shoulder to the elbow let the distance be ———————————— From the elbow to the wrist ————— From the wrist to the fingertips ———— Now turn to the widths, dealing first with those of the profile view, which are called thicknesses, as often mentioned. At the cowlick let it be ———————— At the forehead ——————————— At the eyebrows ——————————— At the nose —————————————

156

0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0

0 1 6 8 0 1 1 3 3 4 6 1 1 6 8 8 9 4 9 1 1 6 6 4 5 6 8

5 2 7 2 6 2 6 2 9 8 1 2 3 0 2 7 5 5 5 6 6 2 9 0 0 5 4

0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 9 7 0 0 9 0 0 0 5 4 0

0 0 0 0

3 5 6 5

7 5 2 5

0 0 0 0



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At the upper lip and nape ——————— At the chin and neck ————————— The neck itself at the same level ———— At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— At the neck ————————————— At the armpits ———————————— At the nipples ———————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the hip —————————————— At the top of the pubis ———————— At the lower pubis —————————— The thigh by the lower buttocks has width At the narrowing of the thigh ———— Above the knee ——————————— At mid knee ————————————— Below the knee ——————————— At mid thigh ———————————— At the lower outer calf ———————— Inner ———————————————— At the lower shin at is narrowest ——— At the top of the foot ————————— At the lower outer ankle and the foot — Then prepare the arm. Thickness at the shoulder —————— At the biceps ———————————— At the elbow ———————————— Below it ——————————————— At the wrist ————————————— Thickness of the palm ———————— Then go on to the widths in the front view, which are as follows: At the cowlick ———————————— At the top of the forehead —————— At mid forehead ———————————


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 3 4 5 6 7 7 6 5 6 8 7 7 6 5 3 3 3 3 3 2 2 2 5

0 7 8 1 5 0 5 0 0 2 5 7 8 4 0 2 3 8 2 0 5 2 9 0 2 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

3 3 2 2 1 1

6 3 1 3 2 4

0 0 0 0 0 0

0 3 6 0 0 5 1 0 0 5 5 0

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At the eyebrows ——————————— At the ears ————————————— At the nose and mouth ———————— Make the width of the neck at the chin At the top of the shoulder blades ——— At the shoulders ——————————— At the throat ———————————— Also there the distance between the shoulder joints will be ——————— At the chest and shoulders make it —— Between the armpits ———————— Between the armpits seen from the back Between the nipples ————————— Below the breasts —————————— At the waist ————————————— At the navel ————————————— At the hip —————————————— The distance between the hip joints is — At the top of the pubis ———————— At the lower pubis —————————— Give the thigh at the lower buttocks width At the narrowing of the thigh ———— Above the knee ——————————— At mid knee —————————————

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0 0 0 0 0 0 0

5 5 4 2 3 8 9

2 6 5 8 0 5 8

0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0

8 1 7 9 7 8 7 9 1 8 1 1 5 4 3 3

4 5 3 0 0 0 3 9 2 0 4 3 8 8 5 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



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Below the knee ——————————— At mid thigh ———————————— At the lower outer thigh ——————— Inner ———————————————— At the lower shin at is narrowest ——— At the ankle and top of the foot ———— Below the ankle at the foot —————— Give the foot at the toes length ——— The arm is to be done as follows: Width at the biceps ————————— Above the elbow ——————————— Below it ——————————————— At the wrist ————————————— Width of the hand —————————— Now make drawings according to the measurements and appropriate to the female sex. The rear view is to have the same silhouette as the front view. Give: The cleft of the buttocks length ———— And the heel width ————————— All these things can be seen in the example figures below.


0 0 0 0 0 0 0 0

2 3 2 2 1 1 1 2

8 3 9 7 2 6 5 9

0 0 0 0 0 0 0 0

0 0 0 0 0

2 2 2 1 2

2 0 5 3 4

0 0 0 0 0

0 5 5 0 0 1 6 0

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Crown Cowlick Forehead Eyebrows Nose Chin Top of shoulder blades Shoulders Throat Chest Armpit seen from the front Nipples Lower breasts

Waist Navel

Hip Top of pubis Lower pubis Lower buttocks Narrowing of thigh

Upper knee Mid knee

Lower knee



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We conclude this book by describing two male heads in more detail and more precisely than has been done so far. They are drawn as follows. First make a square as in the first book: it will serve for the profile view. Its side will be one seventh of the height of the body. You can also use a square of side one eighth the height, keeping everything else the same. Label the sides of the square: a for the front, b for the back, c for the top, d for the bottom. Our basic unit of measurement will be one eighth the side of the square: we denote it by , and one-tenth of this length by . With these units we will measure the features of the head, first using vertical lines, in the following way. From line a move back 0 9 and draw a vertical line e, which will touch the pupils and the inner nostril. Next draw line f, back from a 1 8, which will touch the tips of the eyebrows by the temples. The outer corner of the eyes will be halfway between e and f. Then draw another vertical line g, back 3 5 from a; it will mark the ears by the temples. A fourth line h, back 1 0 from g, will touch the nape where the meets the occiput. Draw k forward 0 7 from b; it will touch the cowlick and the lower nape next to the vertical d. Then move on to the horizontal lines: 2 6 below c draw a line l, which will mark the hairline above the forehead. Draw m halfway between c and l; it will cross line k at the cowlick. Divide the space ld into three equal parts with horizontal lines n and o; the former will run through the eyebrows and touch the top of the ear, whereas n will touch the bottom of the nose and ears. Draw line q halfway between o and p: the upper lip will lie between o and q, the lower between q and p. Line q bisects the mouth. Draw line s three sevenths of the way up from o to r; it will be tangent to the flare of the nostril at the top. The earlobe goes between s and o. Now draw two slanted lines: the first goes from the intersection of e with c to the intersection of a with o; the forehead runs just above this line, between l and n, whereas the nose lies below it, between n and o. The second slanted line goes from the intersection of a with n to that of e and d, and delimits both lips and the tip of the chin. Having drawn this grid, fill in the shape of the head: the curve of the crown touches line c where it crosses the vertical line h; the occiput lies between l and n and touches side b; the thickness of the neck, measured aslant from where nape and occiput join down to the throat, is 4 5. All these things we have most diligently tried to illustrate in the following figures. But it may be that some will find this head a bit too big for the overall drawing; it can be made smaller as follows. Place the curve of the crown halfway between the horizontal lines c and m, and between the vertical lines l and n, and extend the occiput only as far as k, between the horizontals l and n. Then make the junction of occiput and nape tangent to a line o at a distance 1 8 from b. Curve back so the nape meets the lower edge d where it crosses the vertical line i. I would like you to be

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well acquainted with both variations: I did not introduce them idly. This was one of the two heads. Here is the other: As before start with a square, labeling the front edge a, the back edge b, the top i and the bottom k. Draw the vertical lines of the grid: line c, at distance 1 1 back from a, will touch the pupils and the inner nostril; line d, 2 2 back from a, marks the outer limit of the eyebrows, and where it crosses k the innermost point of the chin. The outer corners of the eyes lie halfway between c and d. Next draw line e back 4 0 from a; it touches the ears where they come out of the temples, and it is also at i that the crown grazes the top edge of the square. Draw f back 5 0 from a; this marks the outer edge of the ear. Place g at a distance 1 0 forward from side b; it will touch the nape at the point where the latter meets with the occiput. The horizontal lines are as follows. Down 0 8 from i draw l, which will cross the vertical line g at the cowlick. Down 1 4 from i is line m, tangent to the hairline at the top of the forehead. Divide the space between m and k into three equal parts using lines n and o: the first of these touches the eyebrows and the top of the ear, whereas the second marks the lower limit of nose and ear. The occiput also blends with the nape at o; the distance from this point to the throat is 4 6. Then draw line p down 1 0 from n; the eyes are enclosed between p and n, and their corners lie halfway. Next draw line r 0 5 above o; it touches the top of nasal flares. The earlobe is to be drawn between r and o. Draw a line s up 1 0 from k; it will mark the top of the chin. Halfway between o and s lies t, which will bisect the mouth. Draw also two slanted lines: one from the intersection ci to the intersection ao, and the other from the intersection an to the intersection ck. The forehead will be drawn above [sic] the first of these lines, between m and n, and the nose below it, between n and o. The second slanted line is a guide for the lips, mouth, and chin. Now, having completed the grid, draw the shape of the head, as we show in the example below. Notice also that the measurements of each individual member can be derived from its height. For example, consider the shin, from mid knee to the lower ankle. Let its height be known; from it you can deduce the width at various points: for instance at the calf it is one fourth the height, tapering to half of that at the ankle. Thus, as I said, you can find all the measurements of individual features at will. Let this be the end of the second book, dealing with the proportions of normal bodies. At this point I apologize to the readers for any mistakes that might be present in the illustrations. For the task of correcting errors in the engravings and preventing all decay in the wood blocks proved to be beyond our abilities. But the writing, which is susceptible of correction, was checked as carefully as possible before publication, and typographical mistakes were corrected in proof. The kind and well-disposed reader will, I hope, regard this work with favor; and as for the others—“Hippoclides does not care,” as the Greeks say.

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I beg you, reader, not to take offense at the errors found, and to note the emendations listed below. The need for your benevolence comes both from a subject matter unfamiliar to the engraver, and from the multitude of distractions competing for the translator’s mind; and above all because each labored alone, with no help whatsoever. For, as Homer puts it in Diomedes’ mouth [Iliad 10.224–227, Alexander Pope’s translation]: By mutual Confidence, and mutual Aid, Great Deeds are done, and great Discov’ries made; The Wise new Prudence from the Wise acquire, And one brave Hero fans another’s Fire.

[Note: Camerarius’ errata have been incorporated into the translation as appropriate. See the Translation Note for further details.—Editor]

Printed in Nuremberg, in the summer of the year 1532 of Christ our Savior, in the house of Dürer’s widow.

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english translation

Albrecht Dürer

Underweysung der Messung Nuremberg, 1538 The Walter L. Strauss translation of Underweysung der Messung, originally published as The Painter’s Manual (New York: Abaris Books, Inc., 1977), copyright © 1977 by Abaris Books, Inc., is used with permission of the publisher. See the Translation Note, following, for more information on the text. Translation edited by David Price.

Using This Octavo Translation note Octavo Editions are imaged as uncropped page spreads, as if the actual open book were being viewed. Each spread view is sequentially numbered in the Acrobat pdf file. This translation contains divisions corresponding to the page breaks of the original text; in most instances, each page of the translation file corresponds to one page of the original book. All illustrations are included in the translation file at 95% of actual size, except for the following translation pages, where they have been reduced to fit: pp. 97–98, 107, 187, 192 (upper), 193, 195, 199, and 201–202. Each page of the translation has the Acrobat view number and the original page number indicated in the upper right corner, thus: view 83: page A1r

“ view 83 ” refers to the Acrobat image number; “page A1r” refers to the specific page number in the book itself, with r denoting a recto or righthand page (lefthand pages are denoted v, for verso). Recto and verso designations are necessary in the case of the present work, since it was paginated by leaf rather than by individual page (i.e., leaf numbers appear on the rectos only, and are in the form of paired letters and numerals 1–6). This translation begins with the title page and continues consecutively until the end of the text. Illustrations are included and caption material within them is translated, but blank pages are not indicated in the pagination references. Further information about this book’s structure and condition can be found in the Binding and Collation statement.


Translation Note The Walter L. Strauss translation of Underweysung der Messung was originally published in a somewhat different format as The Painter’s Manual by Abaris Books, Inc., in 1977. That edition presented a translation of the first edition of 1525, with the substantive changes to the second edition of 1538 included in an appendix. For this Octavo Edition, Strauss’ translation has been revised to conform to the 1538 edition, with the passages originally given in the appendix incorporated into the text where appropriate. However, Dürer made many other alterations and corrections to the 1538 edition that were not indicated in Strauss’ translation. These differences have been resolved and the translation has been modified throughout to reflect the text of the 1538 edition. The format of the translation text has been modified to mirror that of the original edition. Descriptive subheadings added by Strauss have been incorporated, in modified form, into the bookmarks for the translation file.

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A Manual of Measurement of Lines, Planes, and Solids

by Means of Compass and Ruler assembled by Albrecht Dürer revised by himself while he was alive and expanded by thirty-two figures that were all drawn by himself as any artisan will recognize, and now again published for the use of all who love art. 1538

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Albrecht Dürer Underweysung der Messung Nuremberg, 1538

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To my especially dear master and friend, Herr Willibald Pirckheimer, I, Albrecht Dürer wish health and happiness. Gracious master and friend! It has until now been the custom in our Germany to put a great number of talented young men to the task of artistic painting without real foundation other than what they learned by daily usage. They have therefore grown up in ignorance like an unpruned tree. Although some of them have achieved a skillful hand through continual practice, their works are made intuitively and solely to their tastes. Whenever knowledgeable painters and true artists had occasion to see such unplanned works, they smiled—not without reason—about the ignorance of these people. Nothing is more annoying to men of understanding than a blunder in the painting, no matter how diligently it may be executed. The fact that such painters have derived pleasure from their errors has been the sole reason that they never learned the art of measurement, without which no one can become a true artisan. It is the fault of their masters who themselves were ignorant of this skill. It is this skill which is the foundation of all painting. For this reason, I have decided to provide to all those who are eager to become artists a starting point and a source for learning about measurement with rulers and compass. From this they will recognize truth as it meets their eyes, not only in the realm of art but also in their proper and general understanding, notwithstanding the fact that at the present time the art of painting is viewed with disdain in certain quarters, and is said to serve idolatry. A Christian will no more be led to superstition by a painting or a portrait than a devout man to commit murder because he carries a weapon by his side. It must be an ignorant man who would worship a painting, a piece of wood, or a block of stone. Therefore, a well-made, artistic, and straightforward painting gives pleasure rather than vexation. The books of the ancients show the degree of honor and respect in which the Greeks and Romans held the arts, in spite of the fact that they were later lost or lay in hiding for a thousand years. Only during the last two hundred years were they again brought to light by the Italians. The arts are lost only too easily, and it takes a long time and much effort for them to be rediscovered. Therefore I hope that those with knowledge will not find fault with my intention and instruction. It is well meant and intended for everyone desirous of learning about art—not only for painters, but also for goldsmiths, sculptors, stonemasons, and carpenters. All those who use measurements may find it useful. No one is obliged to make use of my instructions. I am sure, however, that those who do make use of them not only as a beginning, but daily, will reach greater understanding and will seek and then find much more than I have said here. Knowing that you, gracious master and friend, are a lover of all arts, and because of the great affection and friendship I have for you, I have dedicated this book to you. It is not my wish to demonstrate to you that I have written anything great or excellent, but rather that you may know and measure my good will and friendly inclination. My works may not be particularly stimulating for you. Nevertheless, my heart is anxious at all times to requite the favors and the love you have shown me.

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The most sagacious of men, Euclid, has assembled the foundation of geometry. Those who understand him well can dispense with what follows here, because it is written for the young and for those who lack a devoted instructor. To begin with, it is necessary for the young to learn the basis of measurement and how to go about utilizing it. It is newly construed and presented here. There are three things to measure: first, a length that lacks thickness and breadth; secondly, a length that has a certain width; thirdly, a length that has both width and thickness. The beginning of all of these are points. But a point is a thing without size, length, width, or thickness. And yet it is the beginning of all corporeal things we may want to construct, or which we may invent in our minds. And as those who are skilled in this art know well, no point takes up space, for it is indivisible and can in our thought and imagination be placed anywhere. I may in my imagination throw a point high up in the air, or drop it into the depths where I cannot reach it with my body. But in order to make it plausible for the young in its everyday application, I shall draw a point as a dot with a pen and write the word “point” next to it, so that the point will mean “point.” If then this point is connected from its source with another, it is called a line; and this line is a length of any thickness or width and can be drawn as long as desired. This line is drawn here Line with a pen, and I mark it “line.” In one’s imagination this line is invisible, but can be understood by means of this straight drawn line. In this manner the inner understanding is demonstrated by external exposition. And for this reason I shall draw all things I describe in this book next to their explanations, so that the young will see visually what is in their minds, and thereby understand things better. Now it should be noted that a line can be drawn in several ways, especially in the following three, which can be put to many uses: first, a straight line; second, a circular line; then another curved line, which can be drawn by hand or from point to point, as shown further on, whereby its form can be varied. But I know of no word for this kind of line other than “serpentine line,” because it can be drawn in many ways, according to one’s wishes. To make this clear, I have drawn these lines below and marked them with their names. A straight line.

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A circular line.

A serpentine line.


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It should be noted that the above-mentioned three lines may be drawn either short or long. And if one so desires, and time permits, the straight line can be drawn out eternally or reduced in the mind to a minimum. It can also be used in three ways: vertically, horizontally, or extending in depth. The circular line can be used in its entirety or partially, but it cannot be extended, because it ends where it was begun. This circular line can be large or small. However, if one attempts to incline or decline its course, it will become a serpentine line. But the serpentine line can be altered without limit, and wondrous things can be derived from it. Whether in length, width, height, or depth, many strange things can be constructed from a line by those who think—things about which those who do not think about such matters know nothing. Although only little is said about these matters here, it can be imagined what can be done with two, three, or more lines, especially if the three types of lines are used in combination in various arrangements. Many lines can be drawn, especially with the aid of different types of lines. And it is necessary to know what parallel lines are. (They are called parallel lines in Latin. I call them “paired lines” in our German language.) They are lines which continue at all times equidistant from each other, whether they are ordinary lines or serpentine, or circular. One needs to know also that lines which do not continue at an equal distance from each other will eventually join at an acute angle. Like two perpendicular plumb lines which hang next to each other, they are not perfectly parallel. For when they meet in the center of the earth, they will form an acute angle. This is the case with all lines which do not continue equidistant from each other. Either they meet in the end, or they continue steadily. Plumb lines are used interchangeably with vertical lines, however, although they do join in the center of the earth, because our sense of sight cannot discern this fact. I have drawn below examples of straight parallel lines, serpentine parallel lines, circular parallel lines, and plumb lines.

Now that we have understood a little better what lines are and how they differ, I shall describe those which have both length and width, whether they are drawn straight or uneven. They are called planum in Latin, but in German I do not know a word for them other than Ebene [surface]—although it is not exactly the same, as we shall see in the following. One could imagine a surface without limits on all of its sides; we shall deal here, however, only with those of limited area, which therefore can be circumscribed with lines, so that they are endowed with some shape. There are several kinds of these, as I will demonstrate. The first type is entirely even, neither high nor low. The second is 6


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round like half a sphere. The third is hollow like a round kettle. The fourth is beveled and high at one end, low on the other. There are also hooplike surfaces, which are alternately turned in or out. All these types can be put to use in works. If that were not so, there would be no reason to break one’s head [thinking about this]. To begin with, I want to draw a proper square. It is done in the following manner: I draw the base line ab and then extend it upward to a point equal to the length of the base line. It then becomes a square surface. But I draw a round surface as follows: I draw a straight line ab and hold it firmly at point a. At the end b I turn the line until I reach the beginning once more. Point b then becomes a round plate, whereas point a becomes the center, and from this point a the distance is the same to all points on the periphery of the round surface. I have drawn it below. A square surface. / A round surface. / A spherical surface. / A hollow surface. / A bent surface. / A beveled surface.

Now that I have spoken a little about length and width, and what they mean, I will speak of the length, width, and depth which form corpora [solids]. Of these I will now show several and demonstrate how they can be constructed. First, take the abovementioned square surface abba and continue above it and next to it, as high as it is wide, and it will become a proper cube of equal sides, equal surfaces, and equal angles. Now take the above-mentioned round surface and draw a straight line from its center point a to the periphery and mark that point c, so that cab is a straight line. Then mark the one side: d outside of the periphery, and opposite it an e. The line cab forms the axis on which the surface can be turned from d to e. In this manner it will form a round ball which measures the same from any point on its surface to the center a. We must make certain when turning it on its axis that the center point does not move. Here then are two perfect corpora. But there is no more perfect solid than a sphere. I have drawn these two corpora below. A squared solid or cube. A round ball or sphere.

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If it has now been demonstrated what is a line, a width, a surface, and a solid, it must be understood that such things, be they large or small, can be measured with the necessary skill, and can be rendered in proper size for close proximity or distance. Now I will draw several other constructed lines which may prove useful in some applications. It is clear that a line can be formed in various ways which can be shown in a drawing. But first I will draw a snakelike line [spiral] by means of a compass. It will be on one plane, because this will always be useful whether it is in the form of a line or a solid. It is done as follows. First I draw a vertical line, marked a on top and b on the bottom, and I divide it at three points, c, d, and e, into four equal parts. Then I divide de in half and mark the center f. I then mark the left side g and the right side h. I then place a compass with its legs on points d and a and draw an arc on side h up to point b. Following this, I place the legs of the compass on points f and c and draw an arc on side g up to point b. And again I place the compass, with its legs on points d and c, and draw an arc on side h up to point e. I place the compass on the vertical line ab with one of its legs on a point in the middle between d and f and the other on point d, and then I draw an arc on side h up to point f. This will complete the line. It can be utilized in many ways—for instance, for the decoration of the capitals of columns. To make this method clear, I have drawn below in the illustration two horizontal lines, ending in points a and c, from which the spiral lines emerge. This spiral line is drawn by means of a compass.

Now I will draw another spiral by a different method which can be put to many uses and is very practical. Much can be learned from it, and it can be used instead of the one described before. It originates from its center point and continues in length as far as desired. The space between the arcs remains constant, except in the first revolution. I construct it by drawing a circle with a compass around center point a with a diameter of the furthest extent of the desired spiral. Then I divide this circular line into twelve parts. From the center point a I erect a vertical line to the periphery and mark that point b. That same point I also mark 12, and then I begin marking the divisions of the periphery 1, 2, 3, etc. toward the left, until I reach 12. I divide the straight line ab with twenty-three points into twenty-four equal spaces, numbering them 1, 2, 3, etc. I then take a ruler and pinpoint these points of line ab, and I mark each point with its number on the ruler. Then I place the point a of the ruler on point a of the diagram, and point b of the ruler on point 1 of the arc. 8


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I then mark 1 at the spot on the diagram indicated by point 1 on the ruler. I then continue around the circle in like manner, always holding point a of the ruler at point a of the diagram, marking off consecutively the numbers of the ruler. If you pay attention to the numbers you cannot go wrong. When the numbers come up a second time, and you reach point 12 on the circular line, but point 23 on the ruler, be careful that the numbers on the ruler continue accurately, for 1 corresponds to 13, 2 to 14, 3 to 15, 4 to 16, 5 to 17, 6 to 18, 7 to 19, 8 to 20, 9 to 21, 10 to 22, and 11 to 23. And one can continue other lines above these points, if required, with a greater number of points on the ruler, leaving the numbers on the circular line unchanged. The spiral is thereby indicated by its numbers. In order to see the spiral line properly and put it to use, we must remove the ruler and the circular line with all its numbers, so that only the spiral line with all its points will be left standing out alone. I have illustrated below, in two diagrams, how this is to be drawn, in accordance with the above explanation. To the remaining spiral line I have merely added two parallel lines horizontally toward the left at the same angles, the upper one from points 12 and b, and the lower one from point 12 of the spiral line, so that one can see the difference from the first method.

The spiral line.

Spiral line only.

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Now I shall alter this previously made spiral once more by means of that demarcated ruler ab which was used to construct it. The spiral will now have its points preindicated in a different manner. It is done by means of two kinds of lines, a straight one and a curved one, which are placed together, because one will be measured by the other one. One is different from the other; yet they are analogous on account of their unequal parts. I proceed as follows: I draw a vertical line of equal length to the ruler which I used to construct the spiral. I mark the line b on top and a on the bottom. I then draw a horizontal line cd, on which the vertical line rests at point a at a right angle. Then I draw a straight inclined line db, and place the legs of a compass on points d and a. I then draw an arc from a to a point on line db and mark that point e. Then I divide this arc ae with twenty-three points into twenty-four equal parts and draw straight lines from each point on line ae to the vertical line ab with each of the connecting lines pointing toward point d. And where these lines meet line ab, I mark each point with a number, starting to count from the top point b, with 1, 2, 3, 4, etc., down to point a. It will become apparent that the spaces between the lines become larger toward the top and narrower toward the bottom. I then engrave the points on a ruler which I will use in the ensuing task. I have drawn all this below. Keep in mind that many other things of which I have not spoken here can also be done with this device. This spiral will not run in parallel lines one above the other. The ruler used for turning out the spiral line has to have its points engraved according to this line ba.

But if you want to increase the spaces between the outer lines of the spiral even more and make those closer to the center narrower, then incline the vertical line ab at its upper end b toward point c. Then redraw the line db. It will reduce the length of the arc ae. Then begin to divide all the lines once more as before. You will then find a considerable change, as is partially shown in the above diagram. Once the altered line ab and its points are marked on the ruler, draw a circle whose center corresponds to point a on the ruler, and whose periphery touches point b of the ruler. Mark point b on the periphery 12; then use the same method shown on the preceding spiral; but whereas before you proceeded from the center to the outside, you will now proceed from the outside toward the center. You will therefore place the numbers on the opposite side and begin to count, 1, 2, 3, 4, etc. Then, you can see the difference between the preceding method and the present one in the following two diagrams, and you will observe that one is better than the next. 10


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The altered spiral line.

The altered spiral line only.

If you wish to place straight lines on a spiral at each numbered point in orderly fashion, proceed as follows. Take a ruler whose one point touches center point a and whose other end touches point 12, from which point you draw a straight line along the ruler toward the outside. Then continue, letting one end of the ruler touch center point a, while its other end moves from point to point of the spiral, i.e., 1, 2, 3, 4, etc., and draw straight lines toward the outside until you reach the center point a, as I have shown below. How properly to place the lines on the spiral line.

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If you wish to determine the proper length of any straight line to be placed on a spiral, then proceed as follows: Place a compass with one of its legs on point 12 and the other on point 1 and draw an arc to a point above. Then place a leg of the compass on point 1 and the other on point 12 and draw an arc to a point above, and where the two arcs cross, mark this point c. Do likewise between all numbered points of the spiral, i.e., between 1 and 2 and 2 and 3, and mark the crossing of each pair of arcs above one after the other, d, e, f, g, etc., all the way through as far as it goes. Then proceed to connect the straight lines cd and de and fg, etc., until you have done so for all the letters. The connecting lines will cut off the vertical lines erected on points 1, 2, 3, 4, etc., and all the other numbers. If you then decide to divide the leaves which derive from the arcs drawn by means of the compass, simply draw a line directed toward the center a from point c, and then from points d, e, f, g, etc., down to the spiral line. Constructed in that manner, everything will appear to be in proper order, as shown below. How long the straight lines erected on a spiral should be.

Henceforth I shall construct a spiral very simply and lead the line from an outer circle to its center a. And again I will place leaves on this spiral line. But the straight lines which are drawn inside the leaves will be erected in a different manner from before. First, draw a circle around center point a, and mark it with numbers as before. Divide the vertical line, which you use for a movable scale, with eleven points into twelve equal parts. Then rotate it as before, marking off the spiral line until you reach the center a. Then you have accomplished it: the line is drawn, and it can be used for many purposes, especially for a bishop’s crosier. In this case, draw a straight line downward from point 6 of the circle. Use only half of the circle, the one with the higher numbers, including the spiral, but eliminate the other half of the circle, with the lower numbers. Then place a 12


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compass with one of its legs on point 9 of the circle and the other on point 7 and draw a circle. Then place one leg of the compass on point 7 and the other on point 9 and draw a circle. Mark the point where the two arcs meet c. Then draw a straight line from point 8 on the circle to point c. Do the same for points 9 and 11, and mark the points where the arcs meet d. Then place the two legs of the compass onto points 11 of the circle and point 1 of the spiral and draw a circular line above. Then reverse the compass on these same points and draw an arc in the opposite direction. Where the arcs join, mark the point e. Then continue likewise along the spiral line with points 1 and 2, 3 and 5, 5 and 7, 7 and 9, and 9 and 11, marking the points where the curves join f, g, h, i, k. Then draw in each leaf a straight line down to the spiral line. These lines will be called e12, f2, g4, h6, i8, k10. There remains a section between 11 and center point a. Connect it also by means of the compass and mark its apex l. And what I have described above I have drawn below, first with all the necessary lines of construction and then without them. This construction has many applications. The spiral line can be used for a crocket, as shown in the diagram. This line is meant for a bishop’s crosier.

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The bare line of a bishop’s crosier.

This line is meant for a crocket.

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It must be remembered that the circle used to construct a spiral may be divided into as many parts as desired. The more parts, the more exactly it can be measured. You must also divide the ruler used for the construction of a single spiral into as many parts as you divide the circle. If you intend to construct a double, triple, or quadruple spiral, your ruler must be correspondingly divided into double, triple, or quadruple scales. If you then proceed around the curve, you will succeed in your endeavor. But if you are afraid you might get confused because of possible uneven numbers, then proceed as follows. After you have divided the circle, say into twelve parts, and you desire to have a spiral line with two, three, or four rotations, then divide your ruler into as many parts as you like and mark them with numbers 1, 2, 3, etc., up to 12. Then count again, 1, 2, 3, etc., up to 12, then once more, and again. Adjust your ruler so that the numbers of the ruler and the circle will always correspond as you proceed around it. In that manner, you cannot get confused. Now that you have drawn a spiral on a flat surface, I want to teach you in what follows how to draw it in projection. Note that in such cases one should place a structure next to its ground plan, whether it is a building or anything else. For that reason one cannot very well draw a spiral in projection unless it is placed on its ground plan. Therefore, I first draw a spiral line, as described above, including the circle from which it was constructed, but without the lines of construction and leaves. But you must reverse the numbers of the points 1–12, counting 1, 2, 3, etc., toward the center, beginning with the original point 12, and changing the former point 1 of the spiral line to point 13 and continuing up to 23. Now that you have drawn the ground plan, draw a vertical line from point 6 through center point a and point 12, and extend it upward as far as you desire. Then draw a horizontal line above the ground plan and mark it cd. Where it crosses the vertical line, mark the point b, and the top of the vertical line a. Divide this line ab into twenty-four equal parts with twenty-three numbered points. But I want to space these in the same progression. Therefore I follow the prior procedure but reverse the two letters. I place a on top and b below, and I begin to count upward, 1, 2, 3, etc. When the vertical line has been divided by points, and letters have been erected on the center of the ground plan, I erect a vertical line on point 1 of the ground plan through the horizontal line cd. I then draw a horizontal line, emanating from point 1 on line ab, to meet the vertical line based on point 1 of the ground plan, and thereby complete a rectangle whose corner I mark 1. This is the first point of the projected spiral. I proceed likewise with all numbers and points of the ground plan and the vertical center axis ab on both sides. Accordingly, the projected spiral is erected point by point beginning at b and ending at a. Then I draw the spiral by connecting the points. If a circular staircase in a tower is made in this fashion, the lowest step should be much longer than the topmost one. Accordingly, the steps must be reduced in length in orderly progression the higher they rise in the tower. And as mentioned above, the steps will grow progressively steeper, as I have drawn in the following diagram with all the lines of construction. Thereafter I have drawn it once more without these preliminary lines. These spirals can be constructed in narrow tiers or rapidly ascending, depending upon the length of the lines of construction. This line can be used for many purposes. I have also drawn the triangle abc, which I used to arrive at the subdivisions of line ab by means of the segment ae. Note that the above-described spirals can also be drawn in angular fashion if every second number and letter is omitted and you connect points b and 2, 2 and 4, 4 and 6, etc., continuing up to point a. 15


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This is the projected spiral drawn on its ground plan with all lines of construction.

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Here you see the bare spiral.

From this triangle abc, the line ab is made, which serves to divide the spiral line.

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Yet another spiral can be made from a proper circle. It is used by stonemasons and is commonly referred to as a “screw-line.” But whatever it is called, it is a useful line, and I want to explain how to construct it. Whoever explores its possibilities will find that it has many applications. First, draw a circle with center point a. Then divide it with a vertical line through center a into two equal parts, and again mark the point at the upper periphery 12, and the bottom 6. Then extend this line upward as far as you wish and mark the topmost point a. Draw a horizontal line near the periphery of the circle and mark it cd. The point at which it crosses the vertical line is point b. Then divide the circular ground plan into twelve equal parts, numbering each point, beginning next to point 12, 1, 2, 3, etc., until you again reach 12. But thereafter the numbers must continue, as far as needed in the projection, i.e., 13 above 1, 14 above 2, etc. Accordingly, one can use the ground plan numbers three, four, or five times, or as many times as one desires to extend the projection. Now that the ground plan has been prepared, mark off the vertical line ab into as many points as you wish, beginning to number them from point b upward, 1, 2, 3, 4, etc. Then draw a vertical line upward from point 1 on the circle and a horizontal line from point 1 on the vertical line ab. Where these two meet, mark the corner of the rectangle 1. Then proceed likewise with all the points on the circle and all points of the vertical line ab, as high as you want to have it, even where the numbers of the ground plan are used repeatedly. Once the points of the spiral have been established throughout, draw in the line by hand, as I have done in the diagram below. This spiral line can also be drawn in angular shape. It can also be constructed with parallel lines. First it is drawn with a straight axis, but it can also be made to be hollow, or twisted, so that one can look through it from the top down to the ground, as stonemasons are wont to do in their plans by moving the bottom steps of their project. One can make double-, triple-, or quadruple-threaded screws by this means, which can be used to lift or to break surprisingly heavy and strong objects.

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This is the spiral and its ground plan, as described above.

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Here is still another way of drawing a simple spiral line. Proceed as follows: Draw a proper quadrant (in German, a quarter of a circle) abc, with b at the center point, a on top, and c on the side. Then divide this part of a circle with eleven points into twelve equal parts, and begin to number them, starting from point c. Then draw vertical parallel lines from all points on the circular line down to the horizontal line cb. Mark the points on the horizontal line beginning from point c in the same manner as those on the circular line, and it will therefore be divided correspondingly. This constitutes the first ground plan. Now draw half a circle around a center point c below the first ground plan and of the same form as the upper quadrant, so that the line that divides the circle is vertical, and mark it a on top and b on the bottom, and the center point c. Then divide the periphery of the half-circle with eleven points into twelve equal parts and begin counting them, starting on top from a. Now draw straight lines from each of the numbered points to the center c. Then take a compass, and place one leg on point b of the quadrant and the other on point 1 of line bc. Preserving the compass width, move the compass to the half circle. Place one leg on center point c of the semicircle, and draw a curve from point a of the vertical line ab to the radial line 1. Mark this point 1 also. Now place one leg of the compass on point b of the quadrant, and the other on the second point of line cb. Keeping the width, bring the compass to the half circle and, placing one leg on center c, draw an arc from radial line 1c to radial line 2, and mark this point 2 also. Then continue likewise for all other numbers. After you have marked all the points in the semicircle, you will have the outline of the spiral line from the periphery of the semicircle at point a to the center point c, point by point, as I have drawn it in the diagram below. Whenever you place the revolving leg of the compass on the vertical line acb, and from there draw to the radial line, this demonstrates something extraordinary, as you see in the diagram.

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Here is yet another method of constructing a spiral, using a compass. Begin with a center point a and draw a circle which you divide, as before, with twelve points into twelve equal parts. From each point draw a straight line toward the center a and number them consecutively. Begin on top with number 12 and number 1, 2, 3, etc., until you reach 12. Then divide the line a12 with thirty-five points into thirty-six equal parts, numbering them downward toward the center, 1, 2, 3, etc. Then take a compass and put one of its legs on center a and the other on point 1 of the line a12. Then draw an arc to the line 1a. Leaving the one leg of the compass on center point a, now move the other to point 2 on the vertical line a12. Then draw an arc from line 1a to line 2a. Continue in this manner until you have connected all points after you have circled the center three times. As you move along, the arc becomes increasingly tighter until you finally reach center a. Then draw the spiral line, beginning at point 12 down to the center a, as I have done in the diagrams below, first with all lines of construction, and then without them.

Yet another spiral. From a center a, draw a complete circle and divide it by six points into six equal parts. Number them consecutively, the 6 at the top. From each point on the periphery, draw a straight line toward the center a. Then divide the line 6a by means of seven points into eight equal parts. As described before, place a compass with one of its legs on the center point a and the other on point 1 of the line 1a, and follow the same method you used with the preceding spiral for all the other numbers. I have also drawn this method below, with the lines of construction and without them.

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Similarly, before I proceed, I want to teach the way to divide a line exactly in half. Do this as follows: Take line ab, which is in horizontal position, and place one leg of a compass on point a and the other on point b. Draw a circular curve with it above and below. Then reverse the compass and draw a curve above and below in the same manner. Where the curves cross on top, mark the point c; where they cross on bottom, mark the point d. Then draw a straight line from c to d. Mark point e where this line crosses line ab: it is the exact middle of line ab. If you want to place a straight line on the center of a circle, follow the same procedure as above. In that case, the portion of the circle is ab, and the straight line is cd. The same method can be used for a hollow part of a circle. It is useful to know how to divide a given line, be it long or short, into three equal parts. Proceed as follows: Place a vertical line ab on a horizontal line cd. Then take a compass and open it to less than the half-length of the vertical line ab. Place one leg of the compass on the horizontal line cd, next to the line ab, and mark off three equal distances upward, and draw a straight vertical line through these, parallel to line ab. Mark this new line e on top and f on bottom. Mark 1 and 2 at the two points which divide line e into three parts. Then draw a line from e to a and extend it downward to the horizontal line cd. Where it meets this bottom line, mark the point g. Then draw rays from point g to points 1 and 2. By this the line ab will be divided into three equal parts. Another method of dividing a line into three equal parts: Draw four horizontal lines, one above the other, equal distances apart, and mark them 1, 2, 3, 4. Then take the aforementioned line ab and place the upper end a on line 1 and the lower end b on line 4. The two horizontal lines in the middle, 2 and 3, will then divide line ab into three equal parts as I have also shown in the diagram below. I have drawn diagrams below to show the method. For the hollow portion of a circle I have added point f to indicate where the line cd divides it into two equal parts.

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Now I will show how to construct a line that resembles an egg. Proceed as follows: Draw a horizontal line ab and divide it with nine points into ten equal parts. Then place a compass with one leg on point 5 and the other on point 3, and draw a circle through point 7, above and below. Place one leg of the compass on point b and the other on point 3, and draw an arc downward. Next place one leg of the compass on point a and the other on point 7, and draw another arc downward. Where the two arcs meet, mark e. Now draw a horizontal line below the circle, parallel to the line ab, and where the arcs cross this new line, place a c below point 3 and a d below point 7. Then draw a vertical line from point 5 down to the angle e, and mark 10 where this line crosses the horizontal line cd. Divide the circular section between 3 and 10 with a point f in its center, and do likewise for the section between points 10 and 7, marking the midpoint g. Place one leg of a compass on point f and the other on point d and draw a curve below across the vertical line 5e. Then place one leg of the compass on g and the other on c and draw a curve below. Where the two curves meet on line 5e, mark point h. Now divide h10 into two equal parts with a point i, and place one leg of the compass on it. Place the other leg on the circular line ch at the point closest to i, and draw a curve until it meets the other curve hd. This then is the method for constructing an egg-shaped line, as shown below with lines of construction and without them.

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It may be necessary to find the center or midpoint of a circle. This is done as mentioned before, by placing an upright line on the periphery of the circular section. Given the section ab, place one leg of the compass on a and the other a little distance further along the curve. Draw a curve above and below from both points, and draw a line through the two points where these curves meet, extending it well below. Then do likewise with point b. Where the two extended lines cross, mark c. This will be the center of the circle, as I have shown in the diagram below.

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It is useful to know that it is possible to connect three unevenly placed points so that they form part of the periphery of a circle. That is done as follows: Given the three points a, b, c, connect them with two straight line: ab and bc. Proceed as in figure 21, finding the center of the two lines ab and bc. Extend the two vertical lines which divide ab and bc downward until they cross. Then place a compass with one leg on point d and the other on point a, and draw a complete circle. It will touch the three points a, b, c, as I have shown in the diagram below.

One should also know how to determine the point of contact of a large circle and a tangential line. This is often difficult to do, because the angle is so acute that the lines melt together. It is possible to determine this point by the following method: Draw a section ab of a circle, which touches the horizontal line cd. Then place one leg of a compass on point c and the other a small distance further along the line in the direction of d. Draw an arc upward and downward, and where it crosses the horizontal line, mark it e. Then reverse the procedure and draw an arc from point e both upward and downward through c. Where the curves meet on top, mark f, and where they meet on the bottom, mark g. Then connect points f and g with a vertical line. Where this line crosses line cd mark h, but where it crosses the circle, mark i. Now draw a horizontal line from point i parallel to the horizontal line cd. Where this line crosses the circle at the opposite end, mark it k. Then take the length ik and place it on the line cd, one end on point h, and mark off its length in the direction of d with a point 1. Then connect points 1k and you will have completed a rectangle h1ki. Now if you wish to find where the segment ab touches the straight line cd, place one leg of a compass on point 1 and draw a curve through point h above and below. Then reverse the procedure and draw the curve through point 1, as shown in figure 21. Where the arcs meet on top, mark m, and where they meet below, mark n. Then draw a line mn. Where it crosses the horizontal line cd, mark o. This is the spot where the curved line ab touches the straight line cd, as shown in the following diagram.

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How is it that two lines which meet at an acute angle which is made increasingly smaller will nevertheless never join together, even in infinity? This becomes evident from the following: Draw two parallel lines, the upper ab, the lower cd, and extend both lines at the ends bd—either in fact or in your imagination. If you then divide the upper line ab with numbered points, and connect these points with point c on the lower parallel line,

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there will be an infinite number of lines possible, and the inclined line will come ever closer to the parallel line cd at an ever increasingly acute angle at the point c, but it still will never join it, as can be seen in the diagram below.

It is possible to conceive of an endless line which continually spirals toward a center, and at its opposite end spirals continually outward and never ends. Such a line cannot be drawn by hand because of its infinite size at either end, whether huge or tiny. Neither its beginning nor end can be found, as only reason can tell. But I want to draw it with a beginning and an end, as far as that is possible. I begin with a point a and draw a spiral line toward the center, but at each new tier, I diminish the width between the lines by one half. In the opposite direction, where the line moves outward from point a, I add one half to the width at each tier. By this means, the line moves closer together as it approaches the center, but moves farther apart as it moves outward, and yet it is infinite in either direction, as shown in the diagram below for easier comprehension.

In the following I will teach how to construct a useful line which curves in a special way. First draw a horizontal line cd and divide it with nine points into ten equal parts. On its center point 5 place an upright at right angles. Mark it a on top and b on the bottom. Divide this line ab with nineteen points into twenty equal parts, numbering them from the bottom upward 1, 2, 3, etc. Then mark off on a ruler the length bd, but name it ef. This length will be used to determine all points of the curve to be constructed. Now take one part of line bd and divide it into three equal sections. Extend one part of line bd by one of these sections and open a compass to that width. Place one of its legs on point d and draw a curve above. Then take

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These are the two curves made by the aforementioned method. the length ef which you have marked on the ruler and place end e on the vertical line ab, at point 1, and end f on the circular section just drawn with the ruler. Where they meet, mark a new point 1. Then, after you have first placed the compass into the newly created point 1 and drawn a curve above it, do the same for point 2. Where end f of the ruler meets this curve, mark the new point 2. Continue in this manner until you have marked all the numbers, and then connect these points with a line, point by point. I have drawn two such lines in the diagram below, one with a shorter line eg. They move closer together on the top than on the bottom.

The above line can also be altered by another method. First the line ef must be moved so that it can be measured and divided by sixteen points into seventeen parts. But these parts of the line ef are not to be all of the same size, for they shall increase in size gradually as they approach point e, and decrease in size as they approach point f. To find this gradation in order to mark your ruler accordingly, you must use the triangle abc, which we described in figure 8, and the curve be in that same figure, but incline line ab of that triangle closer to the segment be. After the circular section be has been divided by sixteen points into seventeen equal sections, and you have extended straight lines from point c through the points on be extending to the line ab, the divisions of line ab will be larger as they approach a, but smaller as they approach point b.

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Now that you have marked line ab of the triangle with points and numbers, the first number is placed next to the smallest section at b. Then transfer these to the ruler ef, placing f next to b and a on e, as I have drawn in the diagram below.

If you now wish to construct this other curve, you must again use the vertical line ab, as described above, as well as the curve and the guideline ef, which has been marked on the ruler. Proceed as follows: First, utilize the line ab only up to the seventeen points, and do likewise with the curve, but leave the guideline ef as before. But as you erect the new line ef from the prior one, reduce it by one degree at each number, until at the uppermost point nothing remains of ef. This occurs after seventeen points of both the vertical straight line and the curve. Then draw this new curve by connecting the points, and you will note the difference from the other constructed curves as shown in the diagram here. Now that you know how this curve can be altered in various ways, I will alter it in yet another way, as follows. Erect line ab a quarter shorter than before and divide it, as before, by nineteen points into twenty parts. But these parts must increase in size toward the bottom and decrease toward the higher numbers on top. This must happen in an orderly manner. To achieve this, use the triangle abc of figure 8. However, you will have to use the short straight line ab [of figure 29]. Now, when you construct the triangle, make use of the above-mentioned

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vertical line ab, which has been divided by nineteen points into twenty equal parts. Place it at right angles on the horizontal line bc. This line bc should be one-sixth longer than the vertical line ab. Thus, ab has five parts and bc has six. Then draw the line ac and place end b of the shorter line ab on angle b of the triangle, so that its end a touches line ac. Now draw lines from each point on the long line ab toward point c. Where these lines meet the shorter line ab, place numbers corresponding to those on the longer line ab. In this manner the shorter line ab will be proportionally divided, and the parts will increase in size toward the bottom and decrease in size toward the top. It should be noted that the more one wishes to increase the size of the spaces on the bottom and the more one wishes to decrease the size of the spaces on top, the longer one has to draw lines bc and ac. But if necessary, one can change the length of line ab so that it will reach line ac. I have drawn the described triangle with its changed line ab below.

Now that you have constructed the short line ab, place it at right angles on the first line, described above, in place of the line ab which you used to construct the first curve. Use the above-described measurements and numbers, as shown in the figure, and determine all the points of the new curve. This line will curve over and above the vertical line ab. It can be utilized for leaves, or for decoration of columns or

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for the top of a door (up to point 14). For this purpose I have also indicated the thickness of the masonry with an inner line, so that you will realize that the wall thickness should be thicker on bottom than on top. These measurements will be arrived at by coincidence, as can be seen in the diagram below.

It may be necessary for stonemasons to know how to extend a semicircle or an arc so that they remain constant in height and in other respects, in order to conform to the existing vaulting. Proceed as follows: Draw an overly long rectangle, whose length measures twice its height. Mark it ab on top and cd on the bottom. Then divide the line cd in half with a point e in its center. Place one leg of the compass on point e and the other on c, and draw a circular line above and down to d. This arc will touch the upper line ab. Then divide the line cd by means of eleven points into twelve equal parts. From these points extend straight parallel lines into the arc. Now, next to this rectangle, draw another one, of the same height and position, but even more extended than the present one. Mark the new one fg on top and hi on bottom. Divide it into twelve equal parts by means of eleven vertical parallel lines, as you did in the first instance. From the eleven points on the first arc where the vertical lines meet the arc, extend parallel lines horizontally across to the vertical lines of the longer rectangle. Where these lines cross, draw the longer arc, beginning at h, point by point, until you reach point i, as I have shown below. 31


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The ancients have shown that one can cut a cone in three ways and arrive at three differently shaped sections. One can also slice the cone down the center. In that case the section is shaped like the cone at the base, but we shall not include this section. But the other three sections are each of a particular configuration, and I want to teach you how to draw them. The first of these is called “ellipse” by scholars. It results from cutting the cone at an angle without touching its base. This section may be higher on one side and lower on the other, meaning that on one side it can be closer to the base than on the other and that it therefore is taken at two angles. The second section results from cutting parallel to the side ab of the cone, or cutting on the opposite side. The scholars call it “parabola.” The third section is represented in cross section by a vertical line, parallel to a line that extends from the center of the base of the cone to its apex a. They call it “hyperbola.” I know of no German words for these three sections, but I want to give them names by which they can be called. The ellipse I call eyer linie [“egg line”] because it looks like an egg. The parabola I call brenn linie [“burn line”] because if a mirror is shaped according to this line, it will set things on fire. And the hyperbola I shall call gabellinie [“fork line”]. If I want to draw an ellipse, I will have to draw a cone first in order to show the section, and then place the ground plan below it. I proceed as follows: The apex of the cone shall be on top, marked a. The base on the bottom is marked bcde. I now draw a vertical line downward from a, and I cut an oblique section through the cone and mark it f on top and g on the bottom. Now I divide this section fg with eleven points into twelve parts and begin numbering them below f. Below this cone I draw the ground plan of which a will be the center and points b, c, d, e will denote its circumference, corresponding to the size of the cone. If vertical lines are then extended downward from the cone into the ground plan below, they will traverse the circumference as fg and as the intervening lines 1, 2, 3, etc., all of which I mark there by letter and number. After this is done, I place one leg of a compass on line a of the cone at the level of point 1, and the other leg at the same level on line ad. Then I transfer this distance to the ground plan, setting one leg of the compass on its center a, and I draw a curve, beginning from line 1 in the direction of d, and continuing down until I reach line 1 once more. I then place one leg of the compass once again on line a of the cone, but now at the level of point 2 of section fg, and the other leg on line ad. As before, I transfer this distance to the ground plan by placing one leg of the compass on center point a and drawing an arc beginning at the vertical line 2 until it reaches that line a (continued)

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second time. I continue in this manner up to point 4. When I reach number 5, I turn the compass around and place one leg on point 5 and the other one on line ab. This distance I transfer down to the ground plan, where I place one leg of the compass on center point a and draw a circular curve from line 5 toward d and down until I reach line 5 a second time. I then proceed in the same manner with all the other numbers, transferring the distances from the upper cone to the ground plan below. After this has been accomplished, I can draw the ellipse without the lines of construction, as follows: I draw a vertical line fg corresponding to the length of the section and divide it with eleven points into twelve equal parts. Then I draw horizontal parallel lines through all these points. After this, I transfer the length of line 1 of the ground plan, as far as it is limited by the curve, to line 1 of the new diagram fg, and I mark its extent at both ends.

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I proceed in the same manner with all the other numbers, and after all the points have been marked on both sides, I connect them and arrive at the ellipse as shown below.

Elliptical section

Ellipse

Cone

Ground plan of cone

The parabola is constructed in the same manner as the ellipse. First I draw the cone abcd and the vertical line a. I slice off the parabola from the top down to the base of the cone so that this cut is parallel to the side ab of the cone. I mark this cut f on top and gh on the bottom. Then I divide fgh, which is one line in the illustration, with eleven points into twelve equal parts and draw horizontal lines from all points on fgh. Those which fall on the side closer to line ad, I extend toward ad, but the others I extend to line ab. Then I draw the ground plan to the cone below it in the form of a circle bcde with a center a. Now I extend vertical straight lines from all numbered points on line fgh of the cone down into the circular ground plan and mark

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them with their corresponding numbers in the same manner as I have done in the ground plan of the ellipse. Then I take all the widths in the cone from the vertical line a to both edges of the cone at each height of the numbers and letters and I record them on the ground plan. I then place one leg of a compass on center a of the ground plan and the other on the straight line 1 and draw a circular curve toward d, continuing until I reach line 1 once more. I continue in this manner for all numbered lines until I reach gh. I will then have before me the complete parabola, foreshortened into its ground plan. Having done this, I construct a parabola without lines of construction out of this ground plan. First I draw a horizontal line on which I erect a vertical line f at right angles to the height of cone fgh with their horizontal line numbered. Then I transfer the length of gh to the horizontal base line of the new parabola, so that the vertical line f is in its exact center. I mark this base gh also. Now I transfer the correspondingly numbered distances, which are limited by the curves in the ground plan, to the vertical line f, making sure that this line is always in the middle of each transferred distance, and I mark off lengths on both sides on all the horizontal lines. Finally, I connect all these points and arrive at the parabola shown below. Parabola Parabolic section

Cone

Ground plan of cone

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If you plan to construct a burning mirror of paraboloid shape, the height of the cone you will have to use should not exceed the diameter of its base—or this cone should be of the shape of an equilateral triangle. If you then cut a parabola from this cone and use its curve to form a hollow mirror, you should cut off a small portion at the edge. It will then burn more intensively in its focus where the rays of the sun join in reflection. To understand this, you must know that in any kind of mirror, whatever is exposed to it is also reflected by it, and what is inside of it is turned around. This is why left is turned right and vice versa. For your better understanding, I shall draw it below. First I draw a horizontal line ab. It represents a mirror plane or a water surface that reflects your image. Now I place a source of light c at a raised elevation on one side, and a figure of a man on the opposite side, so that he will look into the mirror or the water. I mark the eye d. In this configuration the light emanating from source c will not meet the eye until the angles of the light ray c and the line of sight of the eye d become equal. This is demonstrated in the following. If you erect a vertical line upward from point e, where the reflection occurs, and then place one leg of a compass in this point e and draw a circle with the other leg from line ab, and you find that the distance of the light ray c and the distance of the line of sight d from the vertical line are equal, then you have located the correct spot where the light is reflected. If you then extend the line of sight through the mirror downward, and you extend a vertical line downward from the source of light c, it will indicate the apparent depth of the light in the mirror or below the water level. The rays of the sun are reflected in the same manner in a parabolic mirror, where they all meet in one focal point and cause intense heat. The cause of this has been explained by mathematicians. Whoever wants to know it can look it up in their writings. But I have drawn my explanation, as outlined above, in the figure below.

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Parabola

Burning mirror

Now I shall show how to construct a hyperbola. It is similar to the preceding constructions and I begin by drawing a cone abcd. I then erect in this cone a vertical line parallel to the vertical line a and mark this line f on top and gh on bottom. In this way a part of line d is trimmed off. I then divide this section of the hyperbola fgh into twelve parts by means of eleven points. I then extend horizontal parallel lines from each of these numbered points to a length of my choice. After this I erect a vertical line f which traverses all of these horizontal lines at a point adjacent to the cone. Then I draw a round ground plan below the cone with a center a and a periphery bcde. I also extend the section fgh of the cone into the ground plan, where I mark it ghf. Then I take a compass, in the same manner I used it before. Taking half the width of the cone at the level of each horizontal line fgh, I transfer it to the ground plan, where I place one leg of the compass on center a and draw an arc with the other toward d and beyond to line ghf. I number each arc. I then transfer the span of all these arcs

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to the vertical line f, marking off these distances, number by number, on either side on the horizontal lines adjacent to the cone, beginning with 1 [on top] and down to line gh. Then I connect all the points and arrive at the hyperbola, as shown below. It is shown in a way that can be understood even without a written explanation.

Hyperbolic section

Hyperbola

Cone

Ground plan of cone

I shall construct yet another line that can be put to many uses. I proceed as follows: I draw a horizontal line ab and divide it by means of sixteen points equidistant from each other and numbered beginning at a. But between the end b

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and point 16 there shall remain a slightly larger distance. I then erect a vertical line at point 13 of the horizontal line ab which shall be of a length equal to a16. This line I also mark with the same numbers, beginning from the bottom. Then I take a ruler and mark the length ab off on it. Placing one end of the ruler on point 1 of the horizontal line ab and letting its edge touch point 1 of the vertical line, I mark the point indicated by the other end of the ruler 1 also. Then I place one of the rulers on point 2 of the horizontal line ab and its edge on point 2 of the vertical line, and then mark the point indicated by its other end 2. I proceed likewise with all the numbers on the horizontal and vertical lines until I reach point 16. Subsequently, I draw the conch line as in the diagram below. This line can be altered in various ways.

A conch line

It is possible to build a device which will make the construction of conch lines easier. Take a length of wood having a square profile and of the desired length to draw a horizontal line. Let a stand for one end and b for the other. Cut a deep groove into its upper side so that another piece of wood can be moved to and fro in it. Then divide this length of wood in as many points and parts as you wish and number them, beginning at a. Then attach two thin rulers at right angles at the center of the base. These rulers should be of equal length to the horizontal baseboard and have a narrow gap between them. Beginning on the bottom, mark and number them in the same fashion as the horizontal piece. Now make a thin rod of a length of your choosing, and attach a wheel to its lower end. This wheel must fit into the groove of the horizontal section so that it can be moved freely back and forth. Now insert the rod between the two vertical rulers,

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with its point toward b and its wheel near a. The closer the wheel is to the vertical rulers, the steeper will be the position of the rod. I have drawn the diagram below to illustrate what I have explained above.

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There is another type of line which I shall call the “spider line,” because when it is fully drawn it resembles a spider. I arrive at this line by a twofold manner of construction. First, I draw a vertical line ab which I extend with another line, ending in c, whereas line ab terminates at point a. I rotate end b in a clockwise direction around center point a, and I mark each position b, as shown in the diagram below. But then from each point b, the line bc shall be extended, anchored in each point b, but rotated at point c. To do this properly, I draw a circle around point a, but below point b, and mark it with numbers [1–12]. This will assure that the lines ab are drawn exactly from point to point. I then do likewise with point b [i.e., draw a circle around it and number twelve subdivisions of its periphery]. And as I progress from point to point with lines ab, I also advance point by point along the periphery of circle b with lines bc. I then mark each newly created point c and connect these points. This will result in the curve shown below.

In the following I shall show how to construct an instrument that can be used to draw a variety of lines, including a serpentine line. This instrument consists of rods which can be turned and twisted or reversed by means of joints which consist of disks. In the center of these disks are the hubs around which the rods can be turned forward or backward in any direction.

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The rods shall be arranged in a manner that they can be advanced by degrees and can be shortened or extended. Each rod would also be reversible, so that the disk showing the degrees can be read from either side. The instrument should also be made with few or many dials or rods, according to the intended application. Its lowest rod shall be placed in an upright position, its lowest portion firmly fastened and not bent. For all parts of the device have to rotate around this rod, but in this process it may be fully rotated around its fulcrum, as desired. For a beginning, I shall describe a device consisting of four rods with four disk joints around which they can be rotated. Each disk shall be numbered by degrees around its periphery. The last and smallest one of these shall have a long needle [stylus] attached to its center, which will indicate the shape of the curve that will be devised. And this stylus shall be made so that it can be shortened or extended [telescopically] by degrees. This instrument can be made in various ways. It can be made large or small. But it is necessary to remember that the rods and disks must be made larger or smaller in proper proportion. The bottom one shall be the largest, and the one at the forward end shall be the smallest. In order to construct this device in proper configuration, thickness, and weight, proceed as follows: First draw a square of the desired size of the largest disk and mark it bcde with a center point a. Then take a compass and draw a circle within this square. This is the size of the first disk. Then draw two straight lines ac and ad and add the corresponding half outside cd with a corner f. You will then have arrived at a new square acfd. Then place a compass on the midpoint of line cd, which is point g, and draw a circle within the smaller square acfd. This will be the size of the second disk. Then indicate a point h in the middle of line bc and draw a straight line ah. It will form a square hcga. Now divide ac in the middle by point i and place one leg of a compass on this point i. With the other leg of the compass draw a circle within the square hcga. It denotes the size of the third disk. After this draw a straight line ih and add a corresponding half beyond hc. Mark the new corner k. You have now constructed a new square hkci. Now divide line hc and mark the center point 1. Place one leg of the compass on this point and draw a circle within the square hkci with the other leg. It will give you the size of the smallest disk. Thus the disks will be of proportional size, for the first will be twice as big as the second; the second twice the size of the third; and the third twice the size of the fourth. Each rod should measure four times the diameter of the disk which is attached to it. Accordingly, the lengths of the four rods will correspond to the lines or diameters of the squares which are of proportional sizes. The width of the rods should also be derived from the squares. This is done as follows: The first and largest rod should measure in width one-seventeenth of its length. At its tip mark off a square. From the center of this square draw two straight lines to the corners and add the corresponding half to form a square of exactly half the size of the original one. From its sides extend lines which will form the second rod. Its length should also measure seventeen times its width. Then proceed in the same manner with the third and fourth square. If you want more rods, continue in the same way. In each case the width of the succeeding rod will be proportionally smaller. The rods can then be turned and moved according to the adjustment of their squares in any direction and will form a fine spiral. If you wish to use the rods of this size to draw something very delicate, it can be done as explained above, because the forms of large and small things will be proportional. When you construct this instrument, you must be careful that the rods will rotate correctly and in a precise manner. The rods must be attached to the very center of the disks; one on the right and the other on the left, one next to the other. The rods should be marked off on both sides as well, and at their midpoints 42


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they can be pulled apart or pushed together, also by degree, so that they become shorter or longer. They can be extended to a considerable length and therefore can be utilized for excellent things by various craftsmen. In the diagram below I have shown how this instrument can be constructed. It can be used in various ways, according to the user’s pleasure or need.

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Straight lines of varying length arranged proportionally have certain properties which I shall explain in the following. They can be put to several uses, especially if they are circumscribed in order to produce planes or solids. In that manner many useful and practical things can be invented.

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First I shall trim off several lines proportionally to their length. This is to be understood as follows: Draw two straight lines ab and be at right angles to one another so that the right angle will be at b. Then divide line bc with four points into five equal parts and erect vertical parallel lines on points 1, 2, 3, and 4 to side ac. The four lines will thereby be cut off in proper proportion and can be used as the base of planes or solids. This is the most modest of the methods I will explain, yet it is useful. Therefore, I have sketched it below.

For the same purpose, but by a different method, I shall show how to extend lines proportionally by a different arrangement of their [base] points. First, draw a horizontal line ab of the desired length and mark off four points on it. Place the first point 1 close to end b. Mark off point 2 as far from 1 as you wish. Place point 3 at twice the distance from 1 to 2 and point 4 at twice the distance from 2 to 3. You can use as many points as you wish and also you can expand the proportion or reduce it. Once you have established these points, erect vertical parallel lines. These may be as long as you desire. Mark them at their upper points lc, 2d, 3e, 4f. Then draw a radial line through c [from b to f ]. Thereby you will cut off all the vertical lines, whether they are long or short, and arrive at the correct proportional length for all of them. And if you wish to construct planes or solids from these lines you will find the corresponding width or thickness at the upper end after drawing horizontal lines from each point on line fb to the vertical line af. These horizontal lines will also be in proper proportion to each other, as I show in the diagram below.

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Now I will teach another method of sorting lines in proportional lengths and extending them proportionally by means of segments of a circle. I do this in the following manner: I draw a quarter of a circle bc from point a and then erect a vertical line ab and a horizontal line ac so that a will be a right angle. Then I divide ac with four points into five parts and erect vertical parallel lines out of these points to the curve bc. Where they meet this curve, I number the points 1d, 2e, 3f, 4 g. The lengths of these four lines are proportional to one another. I then turn the curve so that ba is in a horizontal position and ac becomes vertical. And now I drop vertical lines from the points d, e, f, g down to the horizontal line ab. These lines will also be proportional in length to one another. It should be noted that it is not only the lengths, but also the breadth and thickness which are proportional, as are the configurations themselves. This is shown in the following diagram.

In the following I shall show and explain how to draw proportional lines and extend them by means of a concave curve. I want to show also what shape they will lend to a plane surface or a solid if their width and thickness is properly determined. First I draw an upright rectangle with a line ab on top and cd on bottom. Then I place one leg of a compass on point a and the other on b and draw a curve to side ad. I then divide side cd with four points into five parts and erect vertical lines to meet the concave curve. Where the lines meet the curve I mark the resulting points 1e, 2f, 3g, 4h. Now you can see how these lines are proportional in length. And when they are cut off on top by means of horizontal lines, you can see the width or depth they attain. If I now turn this rectangle over on its side, so that da will be on top and bc on the bottom, and I drop straight lines from the point e, f, g, h on the concave curve down to the horizontal line bc, you will see the proportional lengths of the lines. But if I extend horizontal lines from points h, g, f, e toward the longer line, you will discern the [cross sections of the] plane surfaces and solids which can be derived therefrom. They can be either rounded or angular in shape. I have sketched this below. 46


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In another method of arriving at proportional lengths of lines, draw a quarter of a circle bc with a center a. Divide the arc with four points into five equal parts. Then drop vertical straight lines downward from each one of these points to a horizontal base line. You will find that these lines are proportional to one another, and, at the same time, you will recognize their width and thickness. I have drawn this method below. In sum, all vertical lines which are erected on a horizontal base, whether spaced equally or not, can be limited by three methods: a concave curve, a convex curve, or a ray, whether long or short, and each will result in a different kind of proportion. Although this has been shown above, it will become even more clear in the following. Draw a horizontal line ab and erect four vertical lines on it, numbered 1, 2, 3, 4. At end a, erect a vertical line at right angles. Then place one leg of a compass on a point of this vertical line, and draw an arc with the other end from a point below on line a across the four numbered lines. In this manner they will be cut off proportionally. However, if you wish to cut off the numbered lines at a lesser distance from their base, place the compass higher up on line a and draw the arc out of the same point as before. The lines will then be shorter than before. This then is the

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concave curve method. If you use a convex curve, cut the aforementioned lines in the same manner. To do this, draw a vertical line through end b. This line should extend above and below b. Then place one leg of a compass on line b and with the other leg draw an arc from a through the four lines which you wish to cut off. If you wish to leave these lines longer, move the other leg further up on line b, but draw the arc out of the same point a. The four lines will then again be cut in proportional lengths. The exact placement of the compass on lines a or b is a matter of experience. For Euclid explains in his third book of his Elements, in the tenth theorem, eleventh proposition, that if a small circle is placed within a larger one and their peripheries touch, the centers of the two circles will be on a straight line. And this straight line, which is drawn through the centers of the two circles, will always point to where the two circles touch. This can be understood by the following: Draw a circle 1, 2, 3 with a center a. Place a point b wherever you wish within this circle. Then place one leg of a compass on this point b and the other on the periphery of the circle 1, 2, 3, and draw a small circle. If you then draw a straight line from center a to center b and extend this line to the periphery of circle 1, 2, 3, this straight line will point out exactly where these two circles touch each other. This is what happens also in the method of cutting off vertical lines, although the compass is placed differently. But you can also use a straight line to cut off lines proportionally. In order to do this, place the point that you will use to cut off the lines at the end of the horizontal line a. Then draw rays across the four vertical lines at a greater or lesser height. In this way, you can trim the lines more or less proportionally to each other, as I have sketched below.

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It is clear that straight lines can be cut proportionately in three ways. One cuts the first by using the method previously shown, but the other two types are divided numerically. First there are those based on even numbers based on the square root of 2; second, those based on uneven numbers based on the square root of 3; but some lines are proportional to each other, but which cannot be expressed by a number. But, understand the above-mentioned three types as follows: First draw four lines of each height next to each other and use even numbers to determine their length. Let the first be 2 units, the next 4, the third 8, and the fourth 16. But for the second set of four lines use uneven numbers. Let the first be 3 units, the second 9, the third 27, and the fourth [81]. In this manner one can increase or decrease the length of lines, whether even or uneven, without always doubling their lengths. Anyone who has an inkling of arithmetics can understand this. You can also use a simple number for the purpose of increasing the length of lines proportionately. How this is done, if a length cannot be divided by that number, will be shown in the next section. In the diagram below, I have sketched the method just explained. This method is also useful for stonemasons who build on the basis of ground plans. One can also extend these lines below a horizontal line, but in that case their proportion will be different.

If you are given a longer and a shorter line and you wish to find a third, even shorter line in proportion to the first two, proceed as follows: Place the two first lines, end to end, in a horizontal position. Beginning at the longer line, mark their combined length abc. Then take the length of the shorter line bc and place its point b on point a and incline its end c above the horizontal line abc. 49


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Now connect point c of the inclined line with point b of the horizontal line by means of a straight line. You will then have constructed a triangle abc. Extend the shorter line bc for some distance, as long as you will need it. Then draw a line parallel to side bc of the triangle, emanating from point c of the horizontal line. At the point where this parallel line crosses the extended line bc, place a d. The line cd will then be proportional in length to the two lines abc, and will be the shortest of the three. It will be related to the line of medium length in the same proportion as the line of medium length is related to the longest line of the three, because the two parallel lines cd and bc divide the lines proportionally. This is a very useful thing to know and has many applications.

The smallest The short one The short one

The longest one

There is yet another method of determining lines which are proportional to each other. Draw a horizontal line and erect on it two vertical lines at right angles, one longer and one shorter, but placed a fair distance apart. Mark the longer one a on top and b on the bottom, and the shorter one c on top and d on the bottom. Then place a ruler on points ac and draw a line extending down to the horizontal line and mark that point x. Thereafter, place as many points on line ab as you wish to determine proportional lines between lines ab and cd. These points should be spaced equally apart. In the figure below we shall place three such points on the line ab and mark them 1, 2, and 3. Then draw straight lines from these points to point x. Now draw a diagonal from b to c and where this diagonal crosses the lines emanating from points 1, 2, and 3, erect vertical lines which are parallel to the vertical lines ab and cd. Mark the line closest to ab e at the top and f at the bottom; the second one g on top and h on the bottom; and the third, which is next to cd, i on top and k on the bottom. Then all these lines will be proportional to each other. This figure can be utilized in many ways and is shown below.

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Here is how to divide a curved line using a straight line and [associated] curved lines. Draw an arc ab and then an adjacent straight line. Divide this line with two points 1 and 2 into three parts. Then place one leg of a compass on point b and with the other draw an arc through a. Now, leaving the compass on point b, draw an arc through point 1 across arc ab. Then do likewise through point 2. Thus b remains the center of the three divisions, as shown in the following diagram.

I have now drawn numerous types of lines. There are still many others which serve various needs. These lines can be used with amazing results. If you keep thinking about them and experimenting with them, you will find that they have many uses, and you will gain much in experience.

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The Second Book: Of Plane Surfaces. In the foregoing I have shown how to construct various lines. Now, as set forth in the beginning, I shall describe planes and surfaces and explain the many figures which can be made from them. It must be understood that a plane or surface is a thing that can be limited and varied by means of lines, but it is not a solid. Such figures can be drawn with straight or curved lines or combinations thereof. Lines delimit a surface in the same manner as surfaces delimit a solid. Now we know from Euclid that two straight lines do not delimit a plane and therefore do not compose a figure. They do not form a whole. If they are parallel lines they are open on both ends. If they are drawn so that they will join at some point, their back portion will remain open, as shown in the diagrams below—first, two parallel lines ab and cd; second, two other lines ef and gh, which meet at a point.

But if two opposing arcs a and b are drawn so that their ends meet, they will enclose a figure. They will also enclose a figure if they are drawn in the same direction but curved differently. In that case, they will form a figure in the shape of the crescent moon. Likewise, if an arc d is drawn above a straight line c, an enclosed figure will result. A surface can also be achieved by means of a curved line. That surface will have no corners. But a surface made with a straight line cannot be without an angle or a corner. This is true also for solids, as I have drawn below.

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But before I begin to construct the figures, I must say something about angles. First, an angle and a corner share the same lines. But the difference between an angle and a corner is this: if you look at the configuration from the outside, it is called a corner; but if you look from the inside into its depth, it is called an angle. I have drawn this below. The outside, that is, the corner, I have marked a; and the angle I have marked b.

Angle Corner

It must also be remembered that there are three kinds of angles and corners: right angle and corner, wide angle and corner, and acute angle and corner. They are made easily in the following manner: First, the right angle: Draw two straight lines which cross one another. Where they cross, place an a. Take a compass and place one of its legs on point a and with the other draw an arc that crosses the straight lines at three points. Mark these crossings b, c, d, and connect b with c and c with d with straight lines. bcd will form a right angle. Then extend the line cd to point e and erect a line cd inclined slightly toward d. This will form two angles: an acute one bcd and a wide one bce, because what you have taken away from one side you have added to the other, as shown in the diagram below.

One should also know that if two straight lines cross at equal angles, four right angles will be formed. But if they cross at unequal angles, the opposite angles will be equal, whether wide or acute. It must be noted that the three angles of the equilateral triangle abc are equal in sum to the two right angles of a quadrangle ab, as shown in the figure below. Acute Equal angles

Wide

Wide

Acute These three types of angles can also be drawn with curved lines—convex, as when two arcs cross, or concave, as when two cones touch each other. An angle will result also if arcs of larger and smaller diameter meet at an angle. If two arcs are drawn 53


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in a certain way they will form a figure that looks like the tusk of a boar. These can be drawn with equal or unequal arcs. Therefore, angles can be constructed from straight lines or from curves, concave or convex. These types I have drawn below.

It is common knowledge that no surface can be circumscribed with fewer than three straight lines. At the least, three straight lines will enclose a triangle. There are three kinds of triangles formed by straight lines: first, one with three equal sides which has three equal angles or corners; second, with two equal and one unequal side which has two equal angles and one unequal one. The third triangle is formed by three unequal sides and three unequal angles. One should know that if the two sides ab and ac of an equilateral triangle are cut off by a line parallel to the line cb, the remaining triangle ade will still remain equilateral and it will not be different from the triangle abc except in size, but all the angles will correspond. If, however, you cut off the sides ab and ac by means of an oblique line de, the resulting triangle ade will have unequal sides. Various triangles can also be constructed with curves—convex, concave, or serpentine in form. It is also possible to draw figures which resemble triangles but lack sharp corners, formed by arcs or serpentine lines, as in the diagram below.

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Furthermore, one must remember that if four lines of equal length are placed next to each other at right angles they will form figure a [a square; see figure 8, following]. Then there are other figures with four corners but with unequal sides. One is formed by right angles and the lines on opposite sides are of equal length. This figure [a rectangle] is marked b. The third four-cornered figure has four equal sides and is drawn out in the form of a lozenge [rhombus]. The angles on opposite sides of this figure are equal. I have marked this figure c. The fourth four-cornered figure is also a rhombus, but its sides consist of two equally long and two equally short lines [parallelogram]. I have marked this figure d. The fifth four-cornered figure has two right-angled corners, two equal sides of a rectangle, two unequal corners, and two unequal sides. And one of the unequal sides is longer than the other. I have marked this figure e. The sixth four-cornered figure has a right angle, two sides like a rectangle and a second pair of lines which are equally long but can be longer or shorter than the lines which are adjacent to the right angle. The figure has two opposite acute angles and one wide angle. I have marked this figure f. But a four-cornered figure formed by a right angle and three unequal angles is marked g. Then follows a four-cornered figure with four different angles and sides. It is marked h. In order to prove that such a figure can be constructed, draw a wide angle a and opposite it an acute angle b. Then extend their respective sides until they cross at points c and d. All sides of this figure will be unequal if a and b are unequal in their relationship. Accordingly, c and d will also be unequal, and none of the sides will be equal to another. It is possible to construct two four-cornered figures with unequal sides. One has two equal angles of a right triangle abc and cut the corner b with the diagonal line de. That creates a four-corner figure aedc of unequal sides. The two angles a and c remain equal and d and e are not. Make a square abcd and cut the two corners b and c with a diagonal line ef. Thus a and d remain right angles, but e and f are unequal. More four-cornered figures can be drawn with curved lines, or dented, as in the following diagram. For the sake of order I have marked the various figures with the letters of the alphabet.

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Now I shall explain how to construct figures with an even and uneven number of corners, that is, three-, four-, five-, and six-cornered figures, etc. First I shall construct a figure with six corners [a hexagon] because it can be made without changing the opening of a compass. Take a compass and place one leg on center point a and with the other leg draw a circle of the desired size, large or small. Then move the compass, without changing its opening, to the periphery of the circle, and mark off this distance on it. Then continue along the periphery in the same manner. You will then obtain six points which you will mark with numerals 1, 2, 3, etc. Connect points 1 and 2, points 2 and 3, points 3 and 4, etc.,

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with straight lines. You will obtain a six-cornered figure. Therefore, the line from the center a to the periphery [i.e., the radius] is one-sixth, and also each line between the numerals is one-sixth part, as shown in the diagram.

Now follows how to use this six-cornered figure [hexagon] to construct a triangle within a circle. I use the six points of the preceding figure, but connect points 1 and 3, points 3 and 5, and points 5 and 1 with straight lines. This will result in a triangle that touches the periphery and is symmetrical in every respect and position, as shown in the diagram.

Now I shall show a simple way of using the triangle of the preceding figure to construct a seven-cornered figure [heptagon]. I draw a straight line from the center a to point 2 which cuts the side 1,3 of the triangle in half. Where this occurs I mark b. The length 1b will then fit seven times around the periphery, as shown in the figure above [figure 10] and drawn below [without the lines of construction; figure 11]. Now, next I will show how to construct a figure with fourteen sides from one of only seven sides. By means of a circle, divide one of the sides in half. Then mark off this length along the periphery. You will have fourteen points which when connected will form a figure with fourteen corners [sides] as shown in the diagram. One can use the same method to construct a figure with twenty-eight sides or twice as many. Now I shall construct a quadrangle with a compass. First I draw a circle. Through its center a I draw a straight horizontal line. Where this line crosses the periphery I mark the points b and c. Then I erect a vertical line from center a

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up and down to the periphery, at right angles to the horizontal line, and I mark the point where it crosses the periphery. On top I mark d, and on bottom e. Then I connect points b and d, d and c, c and e, and c and b by straight lines. They will form a square placed symmetrically within the circle, as shown in the diagram below.

Now it is proper to show next how to construct an eight-cornered figure [octagon]. Using the preceding figure, retain the side bd and divide the arc above it into two halves. Mark the midpoint f. Then connect f and d and it will give you one side of the eight-cornered figure. To obtain a sixteen-cornered figure, divide the arc fd into two halves and mark the midpoint g. The line connecting g and d will represent one side of the sixteensided figure. This is all demonstrated in the three figures below.

Now it is necessary to construct a pentagon within a circle. To do this, first draw a circle with center a and draw a horizontal line through this center a. Where this horizontal line crosses the periphery of the circle on both sides, mark points b and c. Then draw a vertical line through the center a at right angles to the horizontal line. Where it crosses the periphery on top, mark point d. [Then divide line ac in two halves and mark the midpoint e.] Now draw a straight line ed and place one leg of a compass on point e and the other on point d and draw an arc down to the horizontal line bc. Where the arc crosses this line mark point f. Then connect f and d. This line fd represents one side of the pentagon, whereas line fa is equal to one side of a ten-sided figure. Then divide line ac into two halves and erect a vertical line up to the periphery. Its length is equal to approximately one-seventh of the circle. This is shown in the following diagram.

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Construction of a pentagon without changing the opening of the compass is accomplished as follows. Draw two circles which overlap so that the periphery of each touches the other’s center. Then connect the two center points a and b with a straight line. The length of this line is equal to one side of the pentagon. Where the two circles cross, mark c on top and d on the bottom and draw a line cd. Then take the compass without changing its opening and place one leg on d and with the other draw an arc through the two circles and their centers a and b. And where the periphery is crossed by this arc, mark points e and f. Where the vertical line cd is crossed, mark the point g. Then draw a line eg and extend it to the periphery of the circle. Mark this point h. Then draw another straight line fg and extend it to the periphery of the circle. Mark that point i. Then connect i with a and h with b and it will give you three sides of the pentagon. Then erect two inclined lines of equal length to ih until they meet on top. You will then have constructed a pentagon, as shown below.

By using the construction of a triangle, as outlined above, it is possible to convert this pentagon into a fifteen-sided figure. First draw a circle with a center a, and then draw one side of the triangle, marked b on top and c on the bottom. Then take the length of one side of the pentagon and place one end of it on point b and the other on the periphery of the circle. Where it touches the periphery, mark it point d. Divide the remaining distance between d and c by means of a compass into two halves, and mark the midpoint e. If you then connect points e and c, it will give you the length of one side of the fifteen-sided figure, as shown in the following diagram. 59


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You can construct a nine-sided figure based on a triangle. Draw a large circle with center a. Then without changing the opening of the compass, draw three “fish-bladders” whose upper end on the periphery you will mark b. Mark the others c and d. Within the upper “fish-bladder” draw a vertical line ba and divide this line with two points 1 and 2 into three equal parts. Point 2 should be closest to a. Then draw a horizontal line through point 2 at right angles to the vertical line ba. Where the horizontal line crosses the “fish-bladder,” mark points e and f. Then place one leg of a compass on center a and the other on point e and draw a circle through point f. Line ef will then represent one of nine sides which will compose a nonagon inside this smaller circle, as shown approximately in the diagram below.

To construct an eleven-sided figure by means of a compass, I take a quarter of a circle’s diameter, extend it by one-eighth of its length, and use this for construction of the eleven-sided figure. This is a mechanical [approximate] not a demonstrative method. If a thirteen-sided figure is required, I first draw a circle with a center a. Then I draw 60


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the radius ab and cut it in half at point d. I use this length to mark off thirteen points along the periphery of the circle. But this method is also only mechanical [approximate] and not demonstrative.

I divide an arc of a circle into three parts by the following method: The arc ab is connected by a straight line. And, as I explained before, I divide this line with points c and d into three equal parts. Then I place one leg of a compass on point a and with the other I draw an arc from point c to the periphery. Where it crosses the periphery, I mark point e. Then I place one leg of the compass on point b and draw an arc from point d to the periphery. Where it crosses it, I mark point f. Then I erect uprights from points c and d to the periphery. Where they cross the periphery, I mark points g and h. The three lengths on the arc, ae, gh, and fb will then be equally long. The two short lengths eg and hf will remain. I place one leg of the compass on point a and the other on g and draw an arc down to line ab and mark that point i. Then I place one leg of the compass on point b and draw an arc down to the line ab and mark that point k. I divide ci and kd each by two points into three equal parts, as I have shown before. Now I place one leg of the compass on point a and the other in the point next to i, and I draw an arc up to the periphery. I mark the new point l. Then I place one leg of the compass on point b and the other on the point next to k and draw an arc up to the periphery. That point I mark m. I have thus divided the large arc into three equal parts at points l and m, as shown in the following diagram.

If you want to place circular figures [tiles, etc.] on a floor or a wall you can place them next to each other in two ways: first by squares, and second by lozenges. 61


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For the first method, draw a quadrangle with equal sides and angles [i.e., a square] and divide it with four parallel lines vertically and horizontally into nine small squares. Place a center point into each of these. Then place one leg of a compass on each of these center points, one after the other, and draw a circle that touches all four sides of each square. In each case a four-cornered figure will then be enclosed by each set of four circles. For the second method, after the circles are placed next to each other in a diamond pattern, triangular-shaped areas will be enclosed by each set of three circles. To accomplish this, proceed as follows: First draw a square 1, 2, 3, 4, of a height equal to four equilateral triangles placed end to end and point to point on top of one another, and equal in width to three equilateral triangles placed next to each other so that their corners touch. In that way the entire quadrangle will hold twenty-four triangles, halves and whole ones. Then mark these triangles where the horizontal lines traverse them, a, b, c, up to r. Then place a compass on the points marked with the letters of the alphabet and draw circles around each with a radius equal to one-half of the side of the triangles. This will result in seven complete circles and ten semicircles, or a total of twelve circles. If you assemble a great many of these circles, six of them will always enclose a seventh. One can also divide a circle in many ways and arrive at many patterns. I shall give one or two ways of doing this which can be used as a starting point for more. I draw a circle with a center a and mark the periphery with twelve points into equal parts. From each of these points I draw a circle without changing the opening of the compass through the center a. I then draw a thirteenth circle of the same size around center a. Finally, I draw four further circles outside the center a but within the large circle. These circles will pass through the points where the other circles meet, which are marked a, b, c, d, e, etc. If you intend to draw a six-pointed star inside a circle, do this without changing the opening of the compass, in the following manner: Draw a circle with a center a. Then place one leg of the compass at the top of the periphery on a point b, and with the other leg draw an arc through center a. Where this arc meets the periphery, mark the points g and c. Then place one leg of the compass on point g, and with the other draw an arc through center a down to the periphery, and mark that point f. Now place one leg of the compass on point c, and with the other draw an arc through center a to a point on the periphery which you mark d. Then place one leg of the compass on point d and draw an arc with the other leg from point c through the center a to a point on the periphery which you mark e. Place one leg of the compass on point e and with the other draw an arc from point f through center a to d. Then place one leg of the compass on point f and with the other draw an arc from g through a to e. Now you have completed the figure. After this, you can change the opening of the compass and draw small semicircles between the letters. (continued)

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For another example, draw a circle with a center a and divide the periphery with four points, b, e, c, d, into four equal parts, so that b is on top, c is on bottom, and d and e are on the sides. Then divide bd with point o; be with point q; ec with point p; and cd with point r, all into two equal parts. Then place one leg of a compass on b, and draw an arc through a on both sides of the circle. Then mark f between o and d and mark g between q and e. Place one leg of the compass on c and draw an arc through a to i. Do likewise on point e, and draw the arc man; and then draw the arc kal from point d. From point o draw an arc through a to both sides of the periphery. Mark the point between b and m with the letter s, and between d and h mark the point t. Now place one leg of the compass on p and draw an arc extending to the periphery through a. Mark the point between l and c with the letter v, and the point between the g and e with the letter x. Then place one leg of the compass on r and draw an arc through a extending to the periphery. Mark the point between f and d with the letter y, and the point between c and n with the letter z. Place one leg of the compass on point q and draw an arc through a to the periphery on both sides. Between k and b mark the point a1; and between e and i mark the point a2. Now draw an

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arc inside each of the pointed leaves formed by the curves as follows: Place one leg of the compass on g and with the other leg draw the first arc in the pointed leaf ka1. Then place the compass on point x and draw an arc from center a to the next leaf. Continue likewise for all six pointed leaves, and in this manner seek the centers of these arcs in the points marked with the letters of the alphabet at these pointed leaves. You can also draw two kinds of stars from a pentagon and a heptagon. To do this, draw circles surrounding a pentagon and a heptagon. Then place one leg of a compass on one of the points that touches the periphery and open the other leg to the next point but one [i.e., skip one point]. Then draw an arc. Proceed likewise with all the points along the periphery in five and seven points respectively. You will then find that you have constructed two stars, as shown in the diagram. You can use this method to arrive at many other results. There are also many variations possible in the way in which the compass may be used in drawing parts of a circle.

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Now I shall combine several figures with corners and show how they can be placed on floors. I have already shown how to do this with triangles in circles, but I shall draw them also without circles and combine them in a different manner. First I place six triangles next to one another around a center a, corner to corner. Then I place on the outside of each of these triangles another one. Then I connect the points of these triangles with lines, so that new triangles will form. Then I place more triangles on these sides, and so on. It is also possible to combine triangles in a manner that no area remains between them. This happens if you place the point of each triangle beneath the center of the base line of the one above. If six triangles are placed next to each other by their points, they will form a hexagon. You can then attach triangles to each of its sides.

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If quadrangles are combined, they will remain unchanged. You can place them at an angle or horizontally, or you can alternate them in the manner of stonemasons, as shown below.

Lozenges, which have two wide and two acute angles at opposite sides, can be combined in two ways. First they can be placed in a Saint Andrew’s Cross pattern. Second, you can place two side by side and the third horizontally between them. It will then look like a cube. There is another way to attach the lozenges so that two corners come under one side. You can also form a star by placing eight lozenges so that their points meet. The star pattern can also be achieved with five or six lozenges, as shown in the diagrams below.

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Now I shall combine pentagons, hexagons, heptagons, and octagons, each separately, and each differently. For the first pattern, place three pentagons side by side on a horizontal line, so that their corners touch. Then place three more pentagons below them so that their sides touch the upper row, whereas their corners touch in the lower row. As a result, lozenges will form between them. Then repeat this operation, but place the top of the pentagons in the angle formed by the pentagons in the row above. This is the first method and the most exact one. Then proceed likewise and form the first two tiers of pentagons. But now combine them by having their corners touch, so that each pentagon will touch four others with one side and three corners. This will result in the formation between the pentagons of lozenges of two different shapes, one narrower, between the sides, and the other wider, between the corners. This pattern can be extended to any size. Fifth, you can combine pentagons in the following manner: First draw a pentagon and place pentagons of the same size on each side. Then place five pentagons on their sides, particularly along the two sides. This will result in the formation of five narrow lozenges between them. Then add pentagons in the angles which will have formed, so that these will touch the narrow lozenges with their corners. You can continue in this manner as long as you desire. You can also combine pentagons to form rosettes which touch each other, and then reverse them to link one to the next as often as you desire. The leftover areas you can then fill with whatever you like, as shown in the diagram. (continued)

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I will now show three ways of combining hexagons. First, I dovetail them, so that all sides, corners, and angles touch each other and no empty areas remain. Second, I arrange them in a way that only their corners touch, and then triangles are formed between them. Third, I place them next to one another so that each touches the adjacent ones with two sides and two corners, so that a lozenge with four corners remains between them, as shown in the diagram below.

I will combine heptagons in two ways. First, so that they touch at their corners. In that case, triangular and four-cornered stars are formed by the spaces between them. They can be converted into quadrangles which touch the corners or else connect the corners by Saint Andrew’s crosses. Second, I place them side by side and on top of each other so that their corners touch. In that case the remaining spaces will form quadrangles and stars as in the preceding construction, as shown in the diagram. If you place heptagons next to each other, side by side, so that two corners are enclosed on the outside and one corner on the inside, the aggregate will form a circle. However, that will not happen in the case of pentagons. 70


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Further, I shall combine octagons in three different ways. First, so that their corners and two sides touch each other. In that case, the areas left between them form triangles. Second, I shall place them so that four of their sides touch each other and so that they are placed horizontally and vertically in the shape of a cross. The areas between the octagons will form quadrangles which stand on their ends. Third, I place them in a way that four of their sides touch, but at an angle. In that case, the remaining areas will form upright quadrangles in a cruciform pattern. I have drawn these methods below. These patterns can be used for wooden or stone floors. They can also be combined in various ways or else alternated cleverly. The remaining areas can be filled with many kinds of other things, or they can be subdivided. 71


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You can also combine irregular polygons of unequal sides. If they are placed in proper order many beautiful patterns can evolve. At times it is proper to overlap the aforementioned figures, i.e., the triangles, quadrants, etc., as I have shown in the six figures below. One can also place one figure around another or inside the other. This is done more easily by combining even with uneven configurations. Those figures with the fewest corners are most suitable for superimposing on a circle because those with many corners are easily confused.

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It is possible to construct various ornaments from figures with irregular sides whose corners will not touch the circumference of a circle if they are placed within it. Many handsome things can be made therefrom. One can also combine various figures on a surface, such as triangles, quadrangles, and figures with five, six, or seven sides. They can be used with astonishing effect on walls and floors, as mentioned before. One can also combine irregular and regular figures, and handsome or rarely seen things can be made from them, for example, unusual hallways and corridors. If I were to show all these possibilities, this book would become much too long. Therefore let everyone think these things out by himself. Now it is necessary to know a little about the adjustment of figures, so that they will be of equal area. For example, one must know how to draw a triangle and then convert it into a square of equal surface area, and likewise with other configurations. It is done as follows: Draw a triangle abc and divide ac in equal halves by point d. Now draw a line bd which will divide the triangle into two halves down the middle. If you then place the one half abd in mirror image, on the other half bdc at the line bc, you will have formed a quadrangle acdb with equal angles and the same surface area as the triangle abc. Then convert this rectangle to a square with the same surface areas as the triangle. To do this correctly, proceed as follows: Take one short side and one long side of the rectangle abcd and place them end to end horizontally. Mark this line with three points efg, determine its midpoint, and mark it h. Place one leg of a compass on h and with the other draw a semicircle from e to g. Now erect a vertical line from f to the periphery of the semicircle. Where it touches it mark point i. If you then add the other lines of the same length as if at right angles, they will form a square of the same surface area as the rectangle abcd and the triangle abc. One can also arrive at this result by a handier method. To do this, draw a square and divide one of its sides into two halves. Then extend one of these sides by half of its length and let this line be the base of the triangle to be constructed, as shown in the figure below.

If you wish to construct quadrangles, whether large or small, but of proportional shape, proceed as follows: Draw a rectangle with a top line ab and a bottom line cd. Then draw a diagonal from c to b and extend it beyond b, as required. Extend line cd at d, also as required. If you want to reduce the size of the rectangle, place a point e behind point d, on the horizontal line ad. Then erect a vertical line on point e to the diagonal. Where they meet mark point f. From there draw a line parallel to the base to side ac of the rectangle, and mark that point g. The quadrangle ghce will then be of proportional shape to the quadrangle abcd. If you desire to construct a quadrangle larger than abcd, proceed in the same manner, but outside the first quadrangle, as shown in the following diagram. 73


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Now that you have seen how to construct figures proportional to each other in shape, you may also want to construct figures proportional to one another but with two or three times the original surface area. You will proceed partly as before. Draw an elongated rectangle abcd and place its long side on a horizontal plane. Then draw two more rectangles of that size and place these on each side of the original one. Mark the four corners of this long rectangle ef on top and ef on bottom. Then divide the upper line in the middle by a point g. Now place one leg of a compass on point g and draw a semicircle with the other leg from e to f. Then erect the line cb and extend it to meet the semicircle. Mark that point h. The line bh represents one side of a quadrangle which will have twice the surface area of the rectangle abcd. But to find its proper length so that the configurations will be proportional in shape, you must proceed as follows. Draw a diagonal from point d in the first rectangle to point b and extend it further as far as required. Then take the length bh and place its one end on point d of the line ef. It will exceed the line dc in length; mark i at the point where it ends. Then erect a straight line upward from point i to the extension of the diagonal line db. Where they meet, mark the point k. Then draw a horizontal parallel line from k to the extension of line da, and at the point where they meet, mark l. This will result in the rectangle lkid, which will have twice the surface area of the rectangle abcd and will be proportional also in shape. If then you desire to construct a rectangle of three times the surface area, add yet another rectangle to the three base rectangles of the above method. You will now have formed a long rectangle emem. Now place one leg of a compass on point b on the upper side em and draw a semicircle from e to m. Then erect a vertical line ff and extend it to the semicircle. Where ff meets the semicircle, mark point n. This line fn is the length of the triple quadrant. Now take this length fn and measure this distance from d on line ef beyond i to a new point o. Then erect a vertical line from o to the diagonal emanating from d. Where they meet, mark point p. Then draw a horizontal parallel line to the extension of line dl. Where they cross, mark point q. You will have formed a rectangle qpod, which will have three times the surface area of the first rectangle abcd and will have the same shape, as evident from the diagonal line. Thus you can increase or decrease the size of figures, and put this system to use for constructing many handsome and useful things. What I have explained above is shown in the following diagram.

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So that you will remember the preceding method well, and because of its utility, I will now construct by the same method a square which will be seven times the size of the original one. To do that, I begin with a square abcd and extend its base line to a length eight times its side. The entire line shall be from point a to point e. I divide this line in two halves at f and place one leg of a compass on this point. Then I draw a semicircle with the other leg from d to e. Now I extend side cb of the original square upward to the semicircle. Where it meets the periphery, I mark point g. If I then form the other two sides of equal length, I will have formed a new square with seven times the surface area of the original square abcd, as shown in the diagram below.

If you wish to construct a square of the same surface area as a triangle with unequal sides, proceed as follows. Let ac be the longest side in horizontal position of a given triangle abc. Opposite this line will be the angle b. Now drop a vertical line from this angle b to the horizontal line ac and mark where they meet as point d. Divide line bd in half at point e, ab at point f, and line cb at point g.

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Then draw a straight line through points feg and extend it to the same length as ac. Erect vertical lines on a and c and mark the points where they meet the upper line: point i will be above a and point h will be above c. This long rectangle ihca will have the same surface area as the triangle abc. If, however, you want to convert this rectangle into a square, then use a compass, as explained above. The following is a slightly different method for accomplishing the same. Enclose the triangle with unequal sides abc in a rectangle abcd, so that the longest side of the triangle is at the same time the bottom side of the rectangle, and the wide angle c of the triangle touches the upper side of the rectangle. In this manner the triangle abc is completely enclosed by the rectangle abcd and its surface area is half of that of the rectangle. Now divide the rectangle abcd in half with a vertical line fg. The quadrangle agfc will therefore have the same surface area as the triangle abc. Then convert this rectangle into a square by the method explained before. I have drawn these constructions below.

Now note how to construct a quadrangle of the same surface area as a hexagon. First draw a regular hexagon abcdef within a circle. Then draw three diagonals, ad, [b] e, and cf. These three lines cross at center g and form six equilateral triangles. Then draw a straight horizontal line and place three of these triangles on it so that their corners touch. Now draw a straight line to connect their upper corners. You will have formed five triangles

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which dovetail. Then place half a triangle at each end and you will have formed an oblong rectangle with the same surface area as the hexagon. You can convert this rectangle to a square by the method explained before, and it will have the same surface area as the hexagon, as you can see in the following figure. You can do this with all regular polygons, no matter how many sides they have.

The quadratura circuli, which means squaring the circle so that both square and circle have the same surface area, has not yet been demonstrated by scholars. But it can be done approximately for minor applications or small areas in the following manner. Draw a square and divide its diagonal into ten parts and then draw a circle with a diameter of eight of these parts. I have drawn this below.

If a triangle with three unequal sides has a right angle, then the equilateral figure erected on its longest side will have the surface area of the sum of the surface areas of equilateral figures erected on its two other sides. I have drawn two types of figures to demonstrate this below. First, a triangle abc with triangles erected on its sides; second, a triangle def with squares erected on its sides.

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In order to divide a figure formed by two arcs in the proper manner, proceed as follows: Draw a vertical line marked a on top and b on bottom. Then draw a horizontal line at right angles to it through its center and extend this line to the left as far as you desire and mark it c, and then to the right as far as you desire and mark it d. Place one leg of a compass on point c and with the other draw an arc from a to b. Do the same from point d. Then draw two circles, one above and one below the horizontal lines, so that their peripheries touch the horizontal line, as well as the two arcs. Now draw horizontal lines above and below these two circles. Then draw two smaller circles above and below these lines so that they touch the arcs on both sides. Again draw horizontal lines above and below these two smaller circles. Then continue in this manner, and you will have divided the figure in an orderly manner, as shown in the diagram below. Many things can be made from this.

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The Third Book: Of Solids Now that in the preceding pages I have explained a few matters pertaining to plane surfaces, I shall speak to some extent about solids which have their origin in plane surfaces. To start, I shall take a circle bcde with a center a and project it upward as far as I wish. It will then become a round column. Then I take a triangular plane, a square, a pentagon, and a hexagon, and project these to as great a height as I wish, as I did in the case of the circle. They will become columns with three, four, five, or six facets. If I then overlap the basic ground plans, as explained before, they will form very handsome columns in projection. You can design ground plans with as many edges as you wish, and then diminish the diameter or make them spiraling or fluted to a greater or lesser degree, as I show to some extent in the following diagrams.

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Another [sic] Method of Making Pointed Solids I can also project pointed shapes from the preceding ground plans. They will then form cones with three, four, five, or six edges. They can be made simple or turned with as many sides as desired in the same manner as explained for the preceding columns. The ancients called these configurations pyramids. They are shown in the following diagram.

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If cones of the proper size are placed on top of the aforementioned columns, they will take on the shape of a tower which, however, should be decorated. There are three types of cones. The first has its apex above the center of its base. The second has its apex above the edge of its base. Therefore the cone will have one right angle at its bottom. The third is made so that its apex projects beyond the edge of its base, as shown in the diagram.

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Now that I have begun to speak of columns, I shall explain a little more about them. They can be made in many different ways and according to the load they are to carry, they must be made stronger. Some will construct columns with bases and capitals, but others will use simple pillars which are sunk into the ground. At the top these are then divided into arches of vaulted ceilings, or else the pillars end abruptly and support ornamented arches—for example, a fluted pillar surmounted by an arch decorated with the same fluting, but in reverse. If this is done properly, you will find it strange but workable. It is known well to artful builders. The fluting of such columns can be varied, reversed, or made with different angles and edges. But in each case the design must be based on an orderly ground plan and its projection. One might arrange four pillars of the same size, each having differing ornamentation in its ground plan. Once they are constructed and supporting a vaulted ceiling, the effect will be astonishing. But those who prefer similarity may follow their own predilection. However, because there are many who greatly favor unusual vaultings on account of the resulting sumptuousness, I shall draw an example below. If it does not please you, you need not use it. I shall also sketch various ground plans for pillars. I shall also draw several designs for cornices used on pillars, high or low above the floor. In each case you must use the proper proportions. If I were to explain everything in detail, this book would become much too long. Those ground plans which are drawn with simple lines can nevertheless be enhanced in proper proportion with ornaments or edgings, as shown in the cross section. These pillars can be made with various edges and ornamentation as desired.

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Speaking of entire buildings or of parts thereof, I believe that it is no secret to any famous architect or workman how skillfully and masterfully the ancient Roman Vitruvius has written about the durability, usefulness, and ornamentation of buildings. For these reasons his teachings should be followed in preference to those of others. If I introduce here a way of young workmen, I am taking into consideration the German taste, for it is a general practice for those who set out to build something new 83


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to introduce a new style or fashion that has not been seen before. I shall therefore show something different which can be elaborated upon by anyone according to his whim or wish. I shall also show things which can be made with a plane or a lathe as well as how to proceed so that the decorations will be compatible in size to straight, curved, or oblique sections. At first I shall construct a column measuring in height eight and a half times its thickness above the plinth. But the plinth should be one-eighth wider than the column. At the top, the column should measure one-eighth less in diameter than on bottom. After this has been done and you have drawn the corresponding ground plan below it with a center point a, you can proceed to add ornamentation in the form of a spiral. To do this, use the method given in Book I. You should begin first with single or double spirals, but you can have as many as eight parallel spiraling lines ascending the column. In the ground plan draw these as concentric circles, spaced equally. In the profile you can show how they ascend. If only a single or two spiraling lines are involved, have sixteen lines emanate from eight points. These spiral decorations can be applied to the entire height of the column or to only the lower third. Many variations are possible for the application of spiral decorations. They are left to your imagination. But the threads should be placed close to one another or else drawn out to a crisp length. They can have narrow vales near the bottom and become spaced further apart higher up, based on figure 16 of Book I [view 90]. All this applies to straight as well as to convex columns, whether they are equally thick on the bottom and top or not, as long as one takes these matters into consideration. In order to use the spiral line, described before, for this column, you must begin with a ground plan. Establish as many points as you desire in this circular ground plan, mark each with a number, and draw a horizontal line through its center a. You should begin the numbering 1, 2, 3, 4, etc., at this horizontal line. These numbers shall serve to indicate the spiral pattern in the profile of the column so that it is patterned evenly. To do this, draw straight lines upward from each numbered point in the ground plan up to the upper edge of the plinth of the profile and mark there the corresponding numbers. Then divide the upper end of the column, in the same manner, where its diameter is the smallest, and number each point to correspond to the one on the bottom. Now connect the upper and lower numbers with straight lines. Then divide the length of the column with fourteen points into fifteen equal parts and number them 1, 2, 3, etc. beginning at the bottom. In this manner the entire column will be divided into a grid. Even from this pattern much can be derived, but my purpose here is only to use it for the construction of a spiraling ornament. Now begin at the bottom of the column at point 1 of the vertical line and draw a diagonal line to the corner of vertical line 2 and horizontal line 1. Then draw a diagonal line in the opposite direction from point 2 of the vertical line to the corner of vertical line 1 and horizontal line 1. Proceed in like manner for all the numbers and quadrants of the entire column, or only one side of it, or only the first third, in which case use straight vertical lines for the rest. You can also do this only for the lower third of the column, in which case the vertical lines should extend a little higher. In sum, you can do these things separately or in combination, and you can use the horizontal (continued)

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lines or omit them in order to arrive at many astonishing patterns which can be cut or sculpted into the column. Once you put your hand to it, you will find many variations. Now I shall show how to construct a capital for these columns in six different ways, but it can be done in a hundred other ways. First draw a quadrangle that is equal in thickness to the diameter of the column below its echinus, and half as high. Above this, place a square slab measuring one-third the height of the capital. This slab should be a perfect square. How far it should extend on top in relation to the straight edge above it will be specified below. One can also make this slab in the shape of an eight-pointed star, as shown in the following diagram. If you wish to indicate the sides of the square slab by means of a compass, draw two lines crosswise in the square so that it will be divided into four squares. Where these lines cross, mark point a. Then draw diagonals extending beyond the diameter of the column. Where they cross the periphery, mark points b, c, d, e. Now place one leg of a compass on one of these letters, open it to the width of one side of the square,

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and draw a circle at each corner. Where these circles extend beyond the square, cut them off with a diagonal line which touches the corners of the square. The edge of the slabs can be made in various ways: straight, convex, concave, or otherwise, with a sculpted decoration intaglio or in relief. But whoever wishes to change these things can always make something contrary to my instructions. Initially, lay out the slabs as follows: Divide them according to their thickness into two parts. Divide the upper part in two also. The upper portion of it should have a straight edge, the lower one a concave profile, recessed as far as it is thick. The lower part should again have a straight edge. It should also be recessed as far as it is thick. Or else you can use a concave edge, or a quarter-round profile, but you must alternate them. Or you can divide the thickness of the slab in two and shape the upper half quarter round and endow the lower half with a straight edge, or reverse this procedure. Or you can divide the slab into two parts, the upper one with a straight edge and the lower with a concave edge, recessed as far as it is high. You can reverse the arrangement or you can divide the thickness into two parts, the lower with a concave edge, recessed according to its height, and the upper into two portions, so that the middle edge will be straight, yet recessed to some extent. Or you can divide the thickness into six parts, the uppermost having a straight edge, the next two having concave edges, each recessed as far as it is thick, and the lowest two parts combining into one with a wide concave edge. You can also introduce a serpentine line below a straight edge, or vice versa. Or you can have a narrow concave edge followed by a convex one, and then again a wider concave one. But you should retain a narrow straight edge above and below these curvatures. There are endless variations to this. I do not put these things down here for you to follow exactly, but so that you can select what you require and use it for a starting point to find other variations. Nothing is good only in its parts, but there are many good things which can be made by those who know. One has to seek them out, like the renowned Vitruvius; and others have also searched and found good things. But this does not mean that still others cannot yet be found—particularly those for which no proof exists that they have been made in the best possible way. Now I shall describe the most simple capital, using only quadrangles without ornamentation. The ornamentation can be added subsequently, particularly if the structure is large. In that case, one can decorate it with concave or straight edges which may be sculpted. Proceed as follows. Divide the height of the capital with five points into six equal parts. The topmost part shall have a straight edge and shall project one-sixth of the thickness of the capital. Then mark a point below this straight edge, recessed by half the thickness of the topmost edge. Draw a straight line downward through three parts to denote a wide straight edge. For the remaining two parts, use a concave edge on the upper one and recess it as far as it is high. In that manner, only the lowest part will project slightly beyond the width of the column, resulting in a thin neck. You can also proceed as before for the uppermost edge, again dividing the capital into six parts, but place a point in the middle of the fourth part. Draw a horizontal line there. Provide the narrow portion which then remains above the two lower parts of the capital with a concave edge. It should project as far as it is (continued)

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high. Leave the two lower parts as one slab, but provide the upper part with a concave edge that extends, in profile, as far as it is high. A third method is as follows. Let the projections remain as before, but reduce the thickness of the straight edges to half of the former ones. Do nothing else except to draw a concave curve to the upper edge so that it will project on top. In the following three capitals use the same straight-edged parts, but draw a horizontal line below the third part of the six and draw an arc between the straight edge and this newly drawn line without, however, taking away from the thickness of the capital. Then draw a horizontal line through the center of the fourth

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part which will indicate a narrow concave edge. The arc of this concave edge, extended into a circle, will indicate the next recess. It should be equal to the diameter of this circle. Then divide the lowest of the six parts with two points into three equal parts. Cut off the lowest third with a horizontal line and add a straight edge which is to project to a distance equaling its height. Then draw an arc between this narrow straight edge, so that the periphery of its circle touches the thickness of the capital above and below. The next concave edge above should project beyond this. A fifth method: Draw the uppermost concave edge as before and divide the lowest part of the capital with two horizontal lines into three. Endow the upper two porticos with concave edges, but the very lowest with a straight edge. Sixth method: Divide the capital into three parts and make a large concave edge combining the two upper ones. The lower part also has a concave edge. It should not diminish the thickness of the capital. Now that these [six] capitals have been constructed they can be adorned with ornaments in many ways and according to your desire. I shall give a small indication of this. Take the slab described above, first the octagonal one, and place it on top of the first capital. Below each of the corners place a truncated triangle which should be wider in the back than in the front. These should project a distance equal to one-quarter of the capital’s thickness and should be as thick as the slab. Or else you can ornament a capital by placing a square slab on top of it and by decorating two sides of the straight edges opposite each other with spiraling coilwork. Or thirdly, place the square slab on the capital so that its corners project crosswise. Then add in the middle of the four hollows a round ornament as thick as the slab. It should project far enough to touch the sides of the slab. If the quadrants are to be decorated with sculpted ornaments, the ornaments can be leafwork or heads of animals or birds or anything else, according to one’s feeling. Once the capital is completed, it is placed on top of the column, and each will add to the other’s impression. Then construct the base of the column in the following way. The base of the column should be equal in height to the diameter of the column’s lowest portion. Draw two vertical lines from the straight-edged slab below the bottom of the column-shaft down to the ground. Then divide this base with two horizontal lines into three parts. The lowest part should be square and it should project as far as it is thick. Then divide the middle part with horizontal lines into three equal portions. The upper one should have a straight edge, but combine the two lower portions into one with a convex edge. To do this, place one leg of a compass on the horizontal line of the lowest quadrant at the vertical line, indicating its edge, and with the other leg draw an arc from the straight edge of the upper part downward and outward toward the tip of the quadrant. Now divide the upper third of the base also into three parts. The lowest portion is to have a straight edge. It is to project beyond the one below a distance equaling half of its thickness. The upper two portions should be combined into one by a convex arc, and they should project beyond the one below by the thickness of this arc. It is usual to place columns on a postament. It is done as follows. Draw a rectangle measuring in length a quarter of the height of the column, including its base and its capital. Its width should be equal (continued)

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to that of the lowest slab of the column’s base. We shall also prepare a square ground plan. Then cut off one-fourteenth of this rectangle by means of a horizontal line. This portion shall also project beyond the rectangle as far as it is thick. Then divide this portion with a horizontal line into two, the upper to have a straight edge, and the lower a quarter-round edge placed at an angle so that its upper edge meets the uppermost ledge at the midpoint of its projection. Now cut off one-seventh of the height of the postament with a horizontal line at its base. It should also project this same distance at the bottom. Then cut off one-fifth of this seventh part by means of a horizontal line. This portion should project outward for a distance equal to its height, and it should have a straight edge. The angle above this projection should be shaped into a concave arc. Then divide the part between this straight-edged portion and the lowest slab into two by a horizontal line, and draw a straight edge for it. Divide the projection of the lowest slab into three parts by means of vertical lines, and then draw an arc from the upper straight edge down to the

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lower one, so that the lowest slab will exceed the lower tip of the arc by a third. This postament can be adorned with ornaments in various ways. I have drawn it below together with its ground plan. So that you will know how to ornament it with the help of a plane or lathe, I shall mention six ways of how this can be achieved. First, you can use leafwork, second a concave edge, third a convex edge, fourth a pointed angle, fifth a corner, and sixth a serpentine line, drawn according to your desire. Any or all of these can be used singly or in combination, or in part; they can be shallow or deep, flat, or in relief, wide or narrow, sharp or dull, large or small, one compared to the other, however you wish. But let everyone use modesty and use large ornaments for large things and small ornaments for small things. I shall show this by the example of a long and a short straight edge placed one next to the other. Human reason will tell you that a long member should be relatively thicker than a shorter one. Therefore, a builder who constructs a shelf, a door, or the like, should make the external members larger than those inside. Anything contrary to this will seem incongruous. This is to be understood as follows. Draw an upright straight edge between two lines; the straight edge is cut off on top by a horizontal line ab which also denotes its upper corners. Then Straight edge place a point c to the side of it, approximately Concave edge near the midpoint of the vertical lines. Draw two Convex edge rays to this point, ac and bc. Then indicate the inCorner ner straight edge, which is meant to be shorter, Angle by means of a vertical Serpentine line next to the line ab, line placed as far from it as the width of the desired ledge. Where this line meets the ray ca mark point d. Then draw a horizontal line from d to the other ray cb, and mark this point e. Now drop a vertical line from point e. de will then be proportional to ab. The six constructions explained in the foregoing are shown in the following diagrams together with their ornamentation, capitals, bases, and postaments. 90


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If you wish to construct columns with a convex shaft, there are two methods. First, divide the height of the column into three parts and cut off the bottom third with a horizontal line. At that point mark the thickness of the column to equal the width of its base plate. Then mark three points, the uppermost below the top slab, the lowest above its plinth, and the third, where the column’s width is the greatest. Then draw an arc through these three points, as explained in figure 24 of Book I [view 94] . This will result in a well-formed curvature for the column. In the second method, construct the curvature in the same manner. Divide the height of the column, as before, into three parts, as wide as before. Divide the width extending beyond the straight profile into four parts on each side. Then divide the combined upper two-thirds with three horizontal lines into four equal parts, and do likewise for the lower third. The maximum curvature should now occur in the lowest third. Fill in successive horizontal lines one above the other, divided by the small sections which are marked on the widest horizontal line, and reduce each succeeding line by one of these. Drawn in this way, the bulge of the column will be curved properly. If you then wish to decorate the column, you can do it with straight vertical lines or spirals. Do this first in the ground plan of the thickness of the column. Divide the circumference into thirty-two parts and alternate concave fluting with straight edges. The concave fluting should have the depth of a semicircle. On the straight edge you can put an ascending pattern, half as wide as the edge, and in relief. If you wish to have a spiraling pattern, either singly or countervailing, then arrange it so that they expand as they ascend but contract as they descend, as shown in the triangle abc with its arc c shown in the following diagram.

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For the column described above, construct a capital whose thickness equals that of the top of the column or the slab at its base. Then draw the square top slab to measure in thickness one-sixth of the height of the capital, and to project half as far as the thickness of the capital. Now divide the distance between the top slab and the bottom of the capital into six equal parts. Cut off one of these sixths and form it into the shape of a shallow dish. Its upper end should project half as far as the slab above it and should have a straight-edged rim one-quarter the thickness of that slab. Below the disk should be a straight edge which is a third less thick than the rim on top and projects one-fifth as far as the upper slab. Then draw a concave curve from this straight edge down toward the capital. It will be shaped like the lip of a cup surmounted by a dish. In combination, these parts will present a pleasing form. You can then decorate the capital with flowers, animals, or other things, as you wish. If you wish to place ornaments beneath the four corners of the slab, you can turn the edges downward so that they will coil below the lip of the cup. Its quadrangular outline should be of the same thickness as the upper slab and should project two-fifths the distance of the projection of the slab. Then you can also place some ornaments on the dish and add one below the corners of the slab between the coiled leafwork. If the corners of the slab should project too far, you can cut into them using a compass, and trim them, as shown in the diagram below.

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In the following I shall explain how to construct another round column which will be curved and twisted in a special way and can be used as an ornament or as a pedestal for a memorial bust. This column should measure in height nine times its diameter above its plinth. The height is measured up to the level of its echinus, below the capital. Its diameter on top should be one-sixth less than on bottom. Draw this column with straight lines. The diameter of its echinus should equal the diameter of the column at its base, and its thickness should equal the distance of its projection. Divide this thickness with two horizontal lines into three equal parts. The uppermost should have a convex edge, the other two straight edges. It should project half the distance of the projection of the uppermost ring and the diameter of the column at the top. The edge of the lowest third should be concave and quarter-round, descending to the columnshaft. But the plinth below the column should have a straight edge and project oneseventh the diameter of the column at its bottom, but one-sixth at its upper edge. Divide the height of the plinth with three points into three equal parts and mark off the upper third with a horizontal line. Give this upper third a concave edge, and the lower two-thirds a straight edge. Before you draw the postament and capital of this column you must go one step further. First draw a ground plan to show how this column is to be twisted. Once you have drawn the column described above in profile, add a vertical line down its center and mark it a on the bottom and b on the top. This line ab is to spiral like a snail shell. For its ground plan draw a circle of the same diameter as the column with a center a. Then draw a vertical line through the center a from top to bottom. Now divide the upper half of this line into two equal halves with a point c. Divide the lower half in the same way, and mark its center d. Draw a circle around center d whose periphery touches point c and the bottom of the large circle. Now divide line ac in half with a point e, and draw a circle around it whose periphery touches a and c. Once this is done, number the peripheries of these three circles with sixty subdivisions, beginning from point a: 1, 2, 3, 4, 5, etc. In the smallest circle you should reach number 6 at point c; then, continuing on the middle-size circle, count 7, 8, 9, etc., up to 18, along half its periphery. Start with number 19 on the large circle and continue all around its periphery to number 42, below number 18. This will be at the vertical line cead. Then start with number 43 on the other half of the middle-size circle, and continue until you reach number 54 at point c. Then begin with number 55 in the smallest circle, continuing until you reach number 60 at point a. You can guide these numbers along whichever side you prefer. These points in the ground plan indicate how the column or rod must be twisted. Now that you have completed the ground plan, divide the column in profile along its height into sixty points. This must be done in a particular way. Draw a horizontal line below the column-shaft, above its plinth, and extend it to twice the column’s diameter. Mark its tip f. From this point f draw a straight ray to the top of the column-shaft below the echinus. Then place one leg of a compass on point f, and with the other draw an arc upward from a point h close to the column on the horizontal line to meet point g on the ray that extends from f to the top of the column-shaft. Divide this arc into sixty equal parts by means of points, and then draw rays from f through all these points to the shaft. Then draw horizontal lines from where the rays meet the shaft across the column and mark each with the numbers corresponding to those on the ground plan. These shall begin at the bottom and at the horizontal line on the bottom of the arc gh. You will note how the space between them increases as they ascend. 95


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Then draw a second line ab on the profile of a column with all its horizontal lines and numbers. Take a compass and place one of its legs on the straight line in the ground plan, which has been constructed to indicate the curvature of the column, and with the other leg measure the distance to each of the consecutive numbers. Transfer these distances to the correspondingly numbered lines which traverse the vertical line ab of the column. Do this for both sides of the ground plan and indicate all these distances on the horizontal lines crossing line ab. You will then have indicated the curvature of the column. Then take a compass and transfer the thickness of the column from the profile of the straight column to all the horizontal distances of the twisted column and draw a circle around each. These will indicate the thickness of the column. You will realize that you can cut out a sphere at any point along the column’s horizontal divisions. Wherever you place one of these spheres and move it to and fro in whatever manner, it can always be cut through its center on a horizontal plane. Thus if a round column is curved or twisted it can always be imagined that it is made up of such spheres which can be divided at their centers in the above manner. You can think of each point of a twisted column as the center of a sphere, and you can draw a horizontal line through its center. The diameter of each sphere will be equal to the thickness of the column at that point. If you do this for each point of a twisted column you will find its diameter at every level of its curvature. Once you have drawn all these circles, connect their peripheries with a line and you will see the contour of the column. But if you draw horizontal lines through all the center points of the circles, which represent the cross sections of the spheres, and connect these with a line, you will see the threading of the column, and how the circles are concentrated and narrow in some places. Whereas in a round column the horizontal lines represent only round surfaces and the lines are placed evenly one on top of the other, in the case of twisted columns these lines no longer lie one flat on top of the next; instead, they are displaced, or moved to and fro above or below the next horizontal line, or to one side of it. They have become transformed into oblique or curved shapes. This is shown in its simplest form in the following diagrams.

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The base of the twisted column should be made in the following manner. First draw a rectangle measuring three times its height in length, and three times as high as the straight-edged plinth below the column-shaft. Divide the base with horizontal lines and mark them with the letters of the alphabet. Let the top line of the rectangle be a and the bottom one b. Divide ab with two lines c and d into three parts, but subdivide ac with a line into halves. Then divide ae with four points into five parts, and trim off the top one with a line f. Divide ec with three points into four parts, and trim off the bottom one with a line g. Divide eg with three points into four parts, and trim off the upper one with a line h. Then divide db with five points into six parts, and trim off the bottom one with a line i and the topmost one with a line k. Now that the ornamental lines which are required for the decoration have been drawn, add the vertical lines which will serve to limit each part. Once you have done this on one side, you will be able to see what is required on the other side. The line limiting the base is denoted by l, and the line dropped from the straight-edged plinth under the column-shaft is marked m. Now divide lm into two parts by a line n. This line limits the thick straight-edged middle section between c and d. Then divide nm into two halves with a line o. The section between e and f will have a convex edge, limited by line o. Insert a short line p between c and m to divide om in half. Line p will be limited at its ends by lines a and f. It will indicate the small straightedged ledge above the ring-shaped convex section. It will also indicate the limit of the small straight-edged section g and c, below the concave-edged section between h and g. The latter is limited by line m. The small straight-edged section between d and k should project beyond line n a distance equal to its thickness, but the lowest straightedged section is limited by line l, whereas the concave-edged section right above it is limited by line n which is equivalent to the projection of the thick straight-edged middle section described above. Finally, form a shallow concave edge from that corner down to the lowest slab. You should also place this base on a rectangular slab so that the periphery of the base touches the edge of each side and make it one-third higher than the wide section at the base, as I have drawn in the diagram below.

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Now construct two types of capitals for the twisted column described above. First draw a rectangle corresponding to the bulk of the capital and based on the thickness of the column, as explained before. The width and thickness of the square slab should also be done in the manner explained before. Because of the curvature of the column the slab should protrude so that the capital will be impressive. Make the straightedged member under the slab to correspond in width and to protrude in the same way as prescribed for the preceding capital. Then draw a horizontal line midpoint between the straight-edged member and the bottom of the capital. Cut off the lower third of the upper half by means of another horizontal line. Draw an oblique line from the tip of the upper straight-edged member to the lower corner of the capital. Place one leg of a compass on this oblique line and draw a circle having a diameter equal to the upper two-thirds [of the upper half] so that the semicircle that is closer to the capital becomes a wide concave edge of this part [the upper two-thirds]. Then draw a circle next to the lower third of the upper half of the capital so that it touches the top and bottom horizontal lines of this section, as well as the oblique line. This will result in a narrow concave edge for this section of the capital. Then draw a small [quarter-round] concave edge from the preceding edge to the flat side of the capital. If you wish to decorate this capital, the ornaments below the corners of the slab should be limited in size to the thickness of the portion of the capital with the wide concave edge. It should protrude a little more than half the thickness of the capital. In the section with the narrow concave edge you can place a branch with foliage above it. You can decorate the edge of the slab with roses or some other ornament. Construct the second capital as follows: Leave the slab and the straight-edged section below it as before. Cut off the next sixth with a horizontal line and subdivide it into two equal parts. Give the upper subdivision a concave edge which recedes toward the flat side of the capital. Divide the lower subdivision into two with another horizontal line. The edge of these two should be straight, but the lower portion should protrude the same distance as the section below the top slab. The upper portion, however, should be recessed a distance equal to its thickness. Divide the third sixth in two by means of a horizontal line. For the upper portion use a serpentine line as decoration. It should begin its turns below the lowest straight-edged ledge and its width should be the same as the upper narrow straight-edged ledge. The upper line should bulge upward, whereas the lower line should be indented inward. Then draw a convex line along the edge of the capital. It should not protrude farther than the next corner above it. Add ornaments below the four corners of the slab. These should be no thicker than one-and-a-half-sixths of the capital and should protrude half of the thickness of the capital. Also add a rectangular ornament on the edge of the slab. It can be large or small. I suggest rectangles because they can be made in wood or in stone. You can also ornament the edge of the slab as follows. Divide its straight edge into three parts. Keep the uppermost straight-edged, but decorate the two lower ones with a serpentine line so that its openings are on the bottom. This serpentine line should begin immediately at the edge and should extend across the entire width of the ledge. If you want to prepare a ground plan of the slab, draw two crossed lines on its (continued)

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quadrangular form so that it will be divided into four quadrangles. Draw these lines as far beyond the quadrangle as you wish, and fill in the ground plans of the two capitals [which were described above]. Now place one leg of a compass on the point where the lines cross in the middle and the other leg on a point a outside the ground plan. Draw an arc with the first leg through the sides of the quadrangle. Then do the same on the other sides from the points marked a, b, c, and d. Now draw the decoration on the edge of the slab from these same points shown in the depth of its recess. The platform that is placed on top of the capital should be as thick as the capital itself, but only a quarter of its height. Embellish it with narrow fillets above and below. The sculpture that is to be placed on top of the column should be twice as high as the capital and the platform. I have drawn these capitals in the diagram that follows.

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Now that the twisted column with all its components has been completed, make a quadrangular postament below it. First draw a vertical rectangle equal in width to the lowest straight-edged section of the column and twice as high. Place below it a quadrangular stone measuring in height one-quarter of the vertical rectangle and protruding at each side. With a horizontal line trim off one-quarter of the height of the vertical rectangle near its base. Then draw an oblique line from the tip of this horizontal line down toward the lowest stone. The latter should protrude by half the length of the vertical rectangle. This will enable you to construct a flight of four steps, each to protrude on top twice their diameter, and each with a straight-edged ledge on top measuring a quarter of the height of each step. Below these straight edges draw concave edges for each step. Now trim off an eighth of the vertical rectangle at the top by means of a horizontal line so that it protrudes a distance equal to its height. Draw an oblique line beginning at the upper tip of this protrusion down toward the rectangle to the next horizontal line. Divide this section with a horizontal line into two equal parts. Cut off one-sixth of the upper half by means of a horizontal line and give it straight edges. Its lower corner should project beyond the oblique line. The lower half should have convex edges. This arc should protrude beyond the oblique line, so that its beginning

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is outside the oblique line but its end is on the oblique line where it touches the lower horizontal line. Then divide the lower half in two with a horizontal line and subdivide its upper portion into four sections. The uppermost quarter should have straight edges which protrude beyond the oblique line with its low corners. The next ledge should have deeply carved concave edges and its lower corner should again touch the oblique line. Divide the remaining lowest portion into four sections also. The top and bottom sections are to have straight edges, protruding as far as they are high. Combine the two middle ones with a convex edge protruding more than the width of a semicircle. The uppermost arc can be decorated with leafwork. The other parts of the postament can be ornamented as you like. You might decorate the four sides of the quadrangle with historical scenes or epitaphs. The quadrangle should be given a frame as well, a twelfth of the width of postament, with facets of one-half that width. You can place four men as guards at the four corners of the steps, and on the top corners of the postament you can place four long-tailed monkeys. To determine the correct size of this column, consider that if a man stands on the large quadrangular stone in front of the postament, his head should reach the middle of the postament. I have drawn this postament below. Because its ground plan is easily drawn, I have omitted it.

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If you prefer, you can omit the flight of steps of the preceding postament and use a different arrangement. Draw the stone to a height of two quadrangles and the lower portion equal in height to half the quadrangle. This lower portion should protrude this same distance on each side. From the tip of this protruding section, draw an oblique line upward to the lower end of the main part. It will serve to determine the ornamentation. Now divide the upper portion into three parts. The lowest third will have straight edges, but the other two-thirds will be different. Draw straight edges for the lowest third of the middle third, and combine the upper two-thirds with a concave edge. Divide the uppermost third into three portions also. The lowest portion should also have straight edges, and the next ledge concave edges. The uppermost portion should consist of two straight-edged ledges and a middle layer with convex edges the same width as the straight-edged ledges above and below it. The upper shelving should measure a third less in height than the bottom one. It should protrude a distance equal to its height. Divide it into three parts and subdivide the upper third into two straight-edged ledges of which the upper one is to be wider. Decorate the center subdivision with a serpentine line of the same width as the upper and lower straight-edged ledges, but give this midsection concave edges. Give the lower two-thirds concave edges as well, but add a narrow straight-edged collar at the top of the lower one. Thus the lower concave edge will be smaller than the upper. I have drawn this below.

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It happens frequently that after a victorious battle a memorial or a column is erected at the place where the enemy was vanquished in order to commemorate the event and to inform posterity about what the enemy was like. If the enemy was a mighty one, some of the booty might be used for the construction of the column. First build a quadrangular mound of stone which can support whatever you wish to build on it. It should measure twenty-eight feet on one side and rise nine feet above the ground. Place a cannon ball having a diameter of one and a half feet at each corner. Then add four steps centered on this base and rising three feet in height. The lowest step should measure one and a quarter feet in height and the others should be in proportion, so that the uppermost one will be the smallest, and the middle two will be in harmony, as explained in Book I . The length of the lowest step should measure twenty-four feet, but the uppermost step should be twenty-one feet long. On each corner of this step place a powder keg measuring two and a half feet in height and two feet wide at its bulge. Cut off the corners of the steps at an angle. Then place an oblong stone block centered on these steps. It should measure twelve feet in height and eight feet in width. Its upper ledge should protrude a quarter of a foot and be equally thick. The upper quarter of this ledge should have a straight edge. The lowest quarter should also be straight-edged but should protrude only a distance equal to its thickness—that is, it will be shorter than the one above. Then draw a serpentine line to decorate the two remaining quarters. The lower ledges should protrude two feet on each side, the bottom one rising one foot and having straight edges. Then draw a horizontal line one foot above this ledge. The lowest third of this portion should protrude by one foot and have straight edges. The upper two-thirds are combined by means of a concave edge. They should protrude by one-third the length of the bottom section. Now place a mortar on the stone block. It should measure eight feet in width at the bottom and extend ten feet in height. Its barrel on top should have a diameter of five feet, counting the ornamentation and its wall thickness. But the barrel proper should measure only four and a quarter feet in diameter. The ornamentation and the rings will therefore protrude beyond this. The powder compartment should be three and a quarter feet wide. Now place a large, strong cannon barrel on top of the mortar. This barrel should measure twenty-one feet in length, three feet in diameter at the bottom, and two feet wide at its top. The ammunition shall measure one foot in height. The mouth of the barrel shall be strong, and the ornaments at both ends shall be proportionally thicker. This is known to skillful casters of cannon. Then place a bell on top of the cannon, measuring three feet in diameter and two feet in height. Surmount this with two pairs of andirons, put down crosswise, and four suits of armor placed back to back, so that their leg armor hangs over the andirons. All this will total four feet in height, but the plumes can extend above and to the side as far as you like. Each andiron should be six feet in length. All this is shown in the following diagram, but because these things are large, each should be ornamented individually. It should all be cast in a thin layer of metal or cut from stone.

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If someone wishes to erect a victory monument after vanquishing rebellious peasants, he might use paraphernalia according to the following instructions: Place a quadrangular stone block measuring ten feet in width and four feet in height on a quadrangular stone slab which measures twenty feet in length and one foot in height. On the four corners of the ledge place tied-up cows, sheep, pigs, etc. But on the four corners of the stone block place four baskets, filled with butter, eggs, onions, and herbs, or whatever you like. In the center of this stone block place a second one, measuring seven feet in length and one foot in height. On top of this second block place a strong chest four feet high, measuring six and a half feet wide at the bottom and four feet wide at the top. Then place a kettle upside down on top of the chest. The kettle’s diameter should be four and a half feet at the rim and three feet at its bottom. Surmount the kettle with a cheese bowl which is half a foot high and two and a half feet in diameter at the bottom. Cover this bowl with a thick plate that protrudes beyond its rim. On the plate, place a keg of butter which is three feet high and has a diameter of a foot and a half at the bottom, and of only a foot at the top. Its spout should protrude beyond this. On the top of the butter keg, place a well-formed milk jug, two and a half feet high, and with a diameter which is one foot at its bulge, half a foot at its top, and is wider at its bottom. Into this jug put four rods branching into forks on top and extending five and a half feet in height, so that the rods will protrude by half a foot, and then hang peasant’s tools on it—like hoes, pitchforks, flails, etc. The rods are to be surmounted by a chicken basket, topped by a lard tub upon which sits a sad [melancholy] peasant with a sword stuck into his back, as I have drawn below. If someone wanted to erect a memorial to a drunkard he might use the following suggestion: First, make the drunkard’s casket, inscribed with an epitaph that mockingly praises his easy living. Place a beer barrel on top of it, covered by a board game, topped in turn by two bowls which would hold a glutton’s portion and are joined at their rims. On the bottom of the upper bowl put a low beer mug with two handles and cover it with a plate which serves as the base of a high beer glass turned upside down. The entire structure is to be surmounted by a basket filled with bread, butter, and cheese. In this manner, but with other paraphernalia, one can decorate memorials according to the way of life of the person being commemorated. For the sake of adventure [amusement], I wanted to explain this, and I have drawn it below as part of this section on columns.

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After having decided the construction of oblong solids to some extent, I want to teach you how to construct a strong round tower. This will be restricted to the mere shape. Those who build it can then decorate the tower according to their desire, for a great deal of ornament should be placed on such a monument. It will be erected in the most suitable spot of a city and placed in the middle of a market square, so that the entire city can be seen from it and it can serve as a guidepost to strangers, no matter in which street they may find themselves. The market square should be at least large enough so that one of its sides is five hundred feet long. In the middle of this square place a circular flight of stairs rising to a height of ten feet and totaling eighteen steps. Each step should be one foot wide. The diameter of the lowest step will be a hundred feet and that of the top step will be six hundred forty-one feet. From the top of these stairs one can see what is happening in the marketplace and what is being offered for sale. The tower is to be erected on the center of these steps. At the bottom it is to have a diameter of forty feet and a wall thickness of ten feet. This will allow for the inside to measure twenty feet in diameter. This inside diameter will be maintained up to its platform, where the thickness of the wall will be only five feet. As a result, the tower will become narrower near the top, that is, a quarter thinner on top than on bottom. This will look well and make it strong. The entire height, up to the roofing, will be two hundred feet. It will therefore measure five times its base in height. On the inside build a circular stairway along the wails and up to the platform. It should be large enough for one to ride up it on horseback, if necessary. For its construction use the method described in Book I, figure 17 [view 91]. The platform will have a narrow walkway around the tower which will extend, including ledges and ornamentation, no more than eight feet in height and three feet in width. The uppermost ledge may protrude a little further. The roofing is to consist of masonry and should have the shape explained in figure 30 of Book I [view 96]. Draw the inner wall by means of a compass so that it will become increasingly thinner as it nears the top. This roofing should have a height of fifty feet not including the belfry. The latter should measure five feet in width and ten in height. Its upper half should be open between the ledges and supported by columns. The roof on the very top should also measure ten feet in height, shaped by two convex arcs drawn with a compass and joined on top. The rod on top with its finial and flag should also have a height of ten feet. The living quarters of the keeper of the tower should be in the roof, so that he can signal from there and adjust the clock. This tower is shown in the following diagram.

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It might happen that one wishes to know the height of a tower. For this purpose one can use an astrolabe, or Saint James’ rod, but here I shall show a simple method. Mark the top of the tower to be measured b and the base c. Mark your eye [point of sight] a, as far from the tower as you like, but at the same level as c. Then take a right angle gauge and with a nail attach a ruler to its one side so that it can move freely around it. Place the gauge on line ac so that the vertical side is parallel to the tower and the eye behind it is at point a where the movable ruler is attached. Now point this ruler in the direction of b. Once you have determined the correct level, mark the angle gauge accordingly and fix the ruler to that spot. Then, without moving its position, put the gauge and the ruler on the ground and indicate the height of the tower with a point d at the side of the tower. You will then have formed two equal triangles—an upright one, abc, and one on the ground, acd. By this you have determined the height of the tower in the ground line cd. You cannot err if you are careful in your measurements. I have drawn this below.

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It is useful for stonemasons, painters, and carpenters to know how to construct a sundial on towers, buildings, and walls. Therefore I shall show in the following how to construct a simple one for everyday use, with a span of twelve hours. First I must show how to erect a vertical line on a given point, as I explained in the preceding book about lines. But to understand it better, proceed as follows: Draw a line ab and place a point c above it. Place a compass on point c and draw an arc through the straight line ab. Where the arc crosses the straight line, mark points d and e. Then place the compass on d and draw an arc as far as you like across line ab. Do the same for point e without changing the opening of the compass. At the points where these arcs cross, mark f on top and g on the bottom. Then connect c, f, and g with a straight line. Where this line crosses ab mark point h. Line chg will then be a vertical line at right angles. This is the first requirement for the construction of the sundial, because you must know what constitutes a vertical line, as shown below.

Then draw a quarter-circle with a center a and ends b and c. Divide this arc into ninety parts or degrees. First divide it into three parts; then divide each third into three more, then divide each ninth in half. This will result in eighteen parts. Divide each of these into five parts, resulting in ninety divisions of the arc. Place this quadrant in a vertical position so that ab will be horizontal. The degrees should be numbered upward from deb toward c. If you are in a city and wish to construct a sundial, you must first consult a table which shows the city’s latitude. Nuremberg is at a latitude of forty-nine degrees. In this case, count from b upward to the forty-ninth degree and mark this point d. Then draw a line from a to d. This straight line is called in Latin axem mundi. It denotes the axis around which the entire world is turning. Then draw another vertical line, by the method shown above, from d down to the line ab and at right angles to it. Mark this point e. In Latin this line is called muralis, but in German mauerlinie [wall line]. This line determines the position of vertical sundials affixed to walls. Then draw a straight line from e to the line ad at equal angles, using the aforementioned method. Mark the point where the two lines meet f. This is called the equinoctial line. It is the line that divides the earth’s axis in two at its center at equal angles [i.e., the equator]. In the following diagrams these lines are based on the latitude of Nuremberg. The quadrant should be large enough for the size of the sundial you intend to construct, because you will need it constantly for your determinations. The triangle aed can be made of wood or metal to which are attached the handles, or pointers. It will be attached to the wall at point d, whereas a should point toward the pole below the earth [i.e., the south pole], which in Latin is called antarcticus, and d should point to the polum arcticum [north pole]. Line ed will then be on a plane with the equinoctial line. 114


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W or ld

Eq ui

no c

t ia

ll

Horizontal plane

in e

Wall line

ax

is

Zenith

Now you can begin to construct the sundial. Draw a straight line gh and erect a vertical line kl on it through its center at right angles. Where these lines cross mark point m. Then measure the equinoctial line ef on the quadrant with a compass and draw a circle with a radius equal to that distance around point m. This circle will be divided into four quarters by lines gh and kl. At the points where the circle crosses line lk mark n and o, and through these two points draw horizontal lines pq and rs, which will be parallel to gh. Now divide each quarter of the circle with points into six equal parts to a total of twenty-four. Then draw straight lines connecting the opposite points through center m and extend these lines beyond the circumference of the circle to the lines pq and rs. Then carefully determine from the lines of the quadrant where the sundial is to be placed. The foregoing is shown in the diagram below.

Sunset

Eq ui n. Horizon

115

Sunrise

Wal l line

Ax

is

Zenith


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The above is positioned according toward the equinoctial line and noon. The pointer is placed in the center m, rising vertically toward the equinoctial line. Its length should be equal to the distance from m to n or o. On the east side or west side the point should measure in length the same as the radius of the equinoctial circle. Then, by means of a compass, transfer the distance de from the above triangle to a line beginning at n and extending to a point on line lk which you will mark v. Using this point as its center, draw a circle around it without changing the opening of the compass. Now draw straight lines from v to p and to q. Then take the distance ae in the above triangle and transfer it by means of a compass from point o to a new one on line kl which you will mark x. From this point draw straight lines to r and s. Then place one leg of the compass on point x and with the other draw a circle through point o. Accordingly, the upper circle is based on length de and is to be placed on a vertical wall, whereas the lower circle is based on the length ae and is to be placed on a horizontal plane. This is shown in the following diagram together with an indication of the sundial’s position.

This is to be

used on a wall

Morning

Afternoon

Sunset

Sunrise

Forenoon Use this for

Afternoon a horizontal plane

These sundials can be put on variously shaped blocks and can be of several designs. But I shall now explain one for simple use in the shape of a cube which is trimmed in some places. On the newly gained surfaces [resulting from the trimming of the corners] one can place sundials so that they are exposed to the sun. Proceed as follows: Mark the corners of one side of a cube k, l, m, n, and place a point c in its center. Draw a circle around c that touches all sides of the square and divide this circle with two lines into four equal parts at right angles, so that the horizontal line acd will represent the horizon, the vertical line will denote the zenith at the top, and point e on the bottom will be its base. Divide each quarter of this circle into ninety parts by the method explained before, and begin to count, starting at b and going upward toward d as many degrees as the latitude of your location. As I have mentioned already, here in Nuremberg this elevation is forty-nine degrees. Mark that point f. Then draw a straight line fc and extend it to the periphery of the circle. 116


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Mark that point g. This line is the axem mundi along which the world turns. The pointer of the sundial which throws the shadow shall be placed in this direction on the wall. But on the opposite side, when placed horizontally, this rod, or pointer shall be placed according to the equinoctial line or the zenith. Each pointer of the four dials within the circle of the equinox will be in a vertical position in its quarter at the same angle. Now draw two lines crosswise through the center of the circle c to the periphery, so that the lines denoting the axes and these new ones will all have equal angles. At the end of the new line place points h and i. This line indicates the equinox at points h and i. Now that the equinoctial line is placed crosswise, at equal angles, over the line denoting the axem mundi, the axis ending in f will ascend in equal measure to the descent of the equinoctial line h. In the same way, the opposite pole of the latitude, marked g, and the opposite end of the equinoctial line, marked i, will move in correspondence. What is added at the pole is reduced at the equinox. Thus the latitude of Nuremberg, that is forty-nine degrees, will be reflected in all four quarters, and the balance of forty-one degrees to the zenith is the width from the equator to our horizon. In this manner, the cube is divided on its square sides k, l, m, and n into eight corners. You can form the parallel lines properly by extending lines fg, the axis mundi, and hi, the equinoctial line, on both sides through the cube. By using the first rule you will find the crossed lines on top on all four sides h, f, i, g, as shown in the diagram. Along these lines cut off the corners of the cube, which will produce eight quadrangular planes. Through the center of all of these draw the noon line. The two adjacent sides will become octagonal, although irregular. The indicated side of this [truncated] cube should face sunrise; its bottom will then face sunset. Point a should face noon and b should face midnight. Accordingly, position the sundial so that the sun can reach four quadrangular planes and the two adjacent sides toward sunrise and sunset, as shown in the following diagrams.

Hori-

-

W or ld

Noon

no c

ax

Eq ui

is

Zenith

zon t ia ll in e

Midnight

Base

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Zenith

ni gh t

di vi

W or ld

Noon

an d

ax

Da y

is

Noon line

Midnight

sio n

Base

Horizontal plane

Zenith

To w eq ar ui d t no he x

Toward the north pole

ax

is

Vertical plane toward noon

Eq ui

Toward the south pole on the bottom of the earth

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W or ld

Oriental

no c

Occidental on the other side Base

t ia

ll

Vertical plane toward midnight

in e

Toward the equinox on the bottom of the earth


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The following sundial is positioned toward the equinoctial line under the earth, toward midnight [i.e., south]. The length of the pointer is determined by the distance from the center to line eight or four on the middle line. ab is the length of the pointer. The line [sic] a is the point at which the pointer should be set vertically.

If you wish to place a sundial on any wall, no matter where, and to find the proper position on any side, whether the zenith is behind it or in front of it, you can determine this position by making two drawings, one two degrees apart from the other. This is denoted by lines aa and bb, the first horizontal, the second vertical, along which the radii become longer or shorter and the areas between them narrower or wider, as shown in the diagram. Zenith

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Although there are a great many other types of sundials which indicate all the hours, the rising signs, the change in the length of days, the constellations, and the planetary hours, etc., I want to show here only those which indicate the common hours, and to demonstrate the fastest method for their construction.

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It may happen that columns, towers, or high walls need to have lettering placed on them. In order to be able to read the top line of what is written as well as the bottom one, it is necessary that the former be written in larger letters than the one below. To accomplish this proceed as follows: Place the point of sight as far from the tower and at as high a level as you wish, and mark it point c. Using the triangle abc of Book I, figure 16 [view 90], let ab be the height of the tower or of what you wish to write on it. Then divide the arc be with points which are spaced as far apart as are the lines of the inscription, and draw rays from the point of sight c through all the points of the arc be to the side of the tower ab. Now draw horizontal lines from these points along the side of the tower across its width. You will have to position the inscription between these lines. This will show you that the upper lines of the inscription will have to be larger than the lower ones. If you then wish to divide a shorter line proportionally to the side ab, extend all the rays toward the point of sight c, but cut them off with a vertical parallel line fg. In this manner the line fg will be divided in the same way as the line ab which is parallel to it. This can be used and adjusted for larger or smaller applications. This method of division can be used for other purposes, not only for inscriptions, for example, if you wish to decorate a high tower with artwork [paintings or sculptures], so that the upper images seem to be of the same size as the lower ones. The method is shown in the following diagram.

the word of god lasts forever. this word is christ.

blessed are all those who believe in christ

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Now, since architects, painters, and others have occasion to place some inscription on a high wall, it is necessary that they know how to form the letters of the alphabet in a correct manner. Accordingly, I shall explain here briefly how to proceed, first with a Latin alphabet and then with a text font, the two kinds of letters we commonly use for such work. Draw squares in which each letter will be contained. When you place each letter into its square, its thicker limbs are to measure a tenth of the width of the square, but its thin limbs are to be only a third as wide. Keep this in mind for all letters of the alphabet. We will begin with an A. Proceed as follows: Mark the corners of its square [frame] abcd—do this for all the letters of the alphabet—and then divide the square by two bisecting lines, marking the vertical one ef and the horizontal one gh. Now place two points i and k on the bottom line cd, one-tenth of the width of the square from points c and d respectively. Draw the thin limb of the letter upward from i to the square, and then draw the thick limb downward so that their outside edges touch points i and k respectively. This will result in a triangle’s remaining in the middle, but point e will be the midpoint of the upper end of the letter. Then connect the [limbs of the] A below the horizontal line gh. This limb should be a third the width of the heavier one. Now trim off the heavier limb on top, outside the square, with an arc, and finish off the letter with a serpentine line having its concave section above the thinner limb. At the bottom extend both limbs acutely toward the outside so that their tips touch the corners of the square at c and d. Do this by drawing semicircles with a radius equal to one-seventh of the square’s side. On the inside, let the bottom of each limb come to a point at a distance of two-thirds of the width of the heavier limb. The limbs [will follow a curve] shaped by an arc of a circle having a diameter equal to the thickness of the heavier limb. You can also trim off the top of this A flatly, in line with the square, and extend the bottom of the limbs on both sides, as shown below. But the longer limb must be in front and the two limbs must be brought a little closer together at the top. You can use any of the three methods, whichever you like best, but remember that you must use the same kind of pointed ends for all the other letters of the alphabet which are made up of oblique lines, such as V, R, and Y, although with some modifications, as will be explained later. You can also draw the A in another [third] way, that is, with its top ending in an acute angle. In that case, the lines are joined more closely at the apex. Accordingly, move the horizontal limb a little further down and make it as thick as you like. You can also cut off the lines on top bluntly, or have them come to a point in front. I have drawn this letter below. (continued)

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Further, construct the letter B within its quadrangle abcd as follows. First divide the quadrangle by means of a horizontal line ef into two parts, and then divide the rectangle abef by means of line gh into two parts also. After this, draw first the wide vertical limb of the letter into the quadrangle at a distance from the line ac equal to its width. Then erect a vertical line ik behind [to the right of] the vertical limb of the letter at a distance of one-tenth the side of the quadrangle. Where this line crosses line gh, mark point l. Then draw the narrow horizontal lines [which connect the vertical limb of the letter] to where the two convex curves will be placed, that is, along the vertical line ik at the top below line ab, [in the middle] above the line ef, and [on the bottom] above line cd. Place one leg of a compass on point l, and with the other leg draw a semicircle from the narrow horizontal lines so that the semicircle connects these short horizontal lines along the vertical line ik below the line ab and above the line ef. Now divide the small horizontal limb of the letter above line ef in its center at line ik with a point marked m. Mark the width of the curved portion of the letter at line gh with a point n. Draw a short horizontal line, as far as needed above line ef from point m toward point f. Then draw a semicircle from this short horizontal line, touching point n to the line ab on top. Now draw a vertical line through point n. Extend the small horizontal line above cd as far as needed toward d and mark it point q. Divide line mq with a horizontal line op in two parts, and where it crosses line n, place point r. Now draw a semicircle that connects the short line above cf with point q and touches point r. Indicate the width of this curved part of the letter with a point s behind point r on the line op, and draw a semicircle, touching points m and s down to line cd. You will then have created three angles within the letter. Round out the bottom one by drawing a small circle with a radius equal to two-thirds the width of the limb of the letter. The curve of the bottom serif should be formed by a circle having a radius of half the limb’s width. You can also construct the letter B by dividing side ac of the quadrangle into nine parts and cutting off the upper four parts with a horizontal line cf. Then draw the vertical limb as described above, but add the upper bulge between ab and ef and the lower bulge between ef and cd. Then divide the line ab into nine parts and mark point g four parts away from b. Then divide line cd into five parts and mark the point h one part away from d. Now draw a line gh which should touch the outside of the upper and lower bulges of the letter b. These bulges must be drawn in a particular manner by placing the compass on the oblique lines in order to draw the arcs. These two oblique lines should be constructed as follows: Divide the line ae into four parts and mark point i one part away from e. Then mark k at that point which is five parts away from e toward c. Connect i with b and k with f with straight lines. You must place the compass on these two lines to draw the upper and lower bulges which will by this method become wider on top than on the bottom, as if drawn with a quill. And these bulges will never become perfect parts of circles because you must draw them by moving the compass to different positions along the oblique lines and then make some freehand adjustments. This is shown below. (continued)

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Now draw the letter C by first drawing a quadrangle with a horizontal line ef straight through its middle, and place a point i in the center of this line ef. Then draw a circle having point i as its center, which touches the four sides abcd of the quadrangle. Without changing the opening of the compass, put it on a point k which is as far behind point i as the greatest width of the letter, and draw another circle that extends beyond line bd. This will give the desired width to the letter. Now draw a vertical line gh onetenth to the left of the line bd, and it will serve to trim off the unwanted portion of the letter C , as was done before. But here I prefer to cut off the lower end between gh and bd. After this, draw in the thin curved portions of the letter along lines ab and cd by hand, at the point where the two circles cross on the top and bottom. But at the bottom, where the letter extends beyond gh, make the curve a little more abrupt, although its tip should still touch the periphery of the circle. Then do the same at the top. Thus the two circles will give the letter its basic shape. In this second method of constructing the letter C, draw a diagonal cb in the quadrangle and place one leg of the compass on point i. With the other leg draw an outer circle, as before, so that it ends on top at line cb, but on bottom it extends a little further than before. Without changing the opening of the compass, draw the inner circle around a point located a distance above i, equal to the desired width of the letter. In this way the width of the letter will be thinner on top than on the bottom, as if drawn with a quill. You will have to draw the rest by hand, and you will cut off the bottom and top, as shown below.

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Make the letter D by dividing its quadrangle abcd with a vertical line gh and a horizontal line ef into four smaller squares. Where the two lines cross, mark point i. Then draw the wide vertical limb of the letter down from line ab to line cd, and to the right of line ac a distance equal to the desired thickness of the limb. End the top and bottom in points in the corners a and c, as described in the construction of the letter B. Use this same method for the straight limbs of all the other letters of the alphabet. Then draw the two thin horizontal lines near the top of the letter where the curved section of the letter will begin below line ab and extend down to above line cd, limited by the vertical line gh. Connect points gfh with an arc, and mark off the greatest width of the limb with a point k on line ef to the left of f. Reduce the radius of the outer curve of the letter and place the compass, opened to that width, with one leg on k and the other on a new point l on line ef. Now draw the arc of the inner curve with one leg of the compass on point l through k to connect to the narrow lines on top and bottom. The upper angle should be sharp, but the lower one should be rounded out by means of an arc similar to those used for the upper and lower serifs on the outer edge of the vertical limb. The curved section of the D can be made in another way, so that it resembles a penstroke, thicker on top and thinner on the bottom. For this purpose, draw a diagonal line cb, and then draw the outer curve, as above. For the inner curve, place a point m on the diagonal cb at a distance below i equal to the desired width of the limb. Without changing the opening of the compass, draw the inner curve; but where the breadth is to be narrow, on the top and on the bottom, you must draw it in by hand, as shown below. (continued)

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[Note: The solid Ds in the above figure are inverted in this 1538 edition.—Editor.]

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The letter E is also constructed within a quadrangle. In the middle of this quadrangle abcd draw a horizontal line ef. Then draw the vertical limb of the letter in the same manner as that described for the letter D. Now draw the uppermost thin horizontal limb of the letter below the line ab, so that it ends at a point a third of a tenth less than six-tenths of the width of the quadrangle, and so that it comes to a point a tenth part of the length of line ab below it. The diameter of the circle which determines the curve of the tip should also equal a tenth of line ab. After this, draw the thin horizontal middle limb on line ef so that it will be one-tenth of the width ab shorter than the limb on top. Its end, however, should be twice as wide as the top limb, and this end should extend on the bottom and the top, coming to points rounded out by arcs of circles having diameters equal to one-sixth of ef. The lowest line of the thin horizontal limb above line cd, should project at its lower corner beyond its upper horizontal line a distance equal to one-tenth the length of line cd. The pointed end should extend another two-thirds of a tenth part of this length [i.e., cd] to the right, and then a sixth part of the length of cd upward. Its curve should be determined by an arc of a circle having a diameter equal to one-sixth of cd. However, the lowest [inner] angle of the letter should have a concave arc of the same diameter used for the curves of the middle member. The other [inner] angles should be sharp, as shown below.

The letter F is constructed in the same way as the letter E , except that the lowest horizontal section is omitted and that the lower end has a serif like the preceding letters, as shown below.

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The letter G is constructed in the same manner as the letter C above, except that you must draw the wide limb of this letter in front of the line gh, beginning at the curve at the bottom and ending on line ef. Add a serif, as explained before, and retain the angle and the corner at the bottom. Or you can construct the letter G within the aforementioned quadrangle by drawing a diagonal line cb and placing one leg of a compass on point i, and with the other leg drawing an arc from e to the middle of cd and marking this latter point i. In the same manner, extend the arc from e to the line ab and continue to the vertical line gh. Mark the last point z. Then mark a point on line gh a tenth of the distance of its length and connect points l and m by hand by means of a curve. Then draw a line upward from point z equal to the thickness of the letter and slanting between the vertical line gh and the curve of the limb. From this end of the letter draw a curve by hand to line ab where it meets the arc, which was drawn with the compass. Then mark off one-third at the bottom of line gh by a point n. This denotes the upper limit of the broad limb that extends down to point m. The serif on top of this limb should be twice its width. Place one leg of the compass on line cb at a distance above point i equal to the width of the limb, and, opening the compass to the distance of ei, draw a curve with the other leg that will touch the outer circle of the letter on top and will end above point l on the bottom. Now connect this curve by hand to the vertical section on the right at the level of point m. Draw the narrow width of the curve by hand on the top, as well, as shown below.

Construct the letter H by erecting within its quadrangle the two large limbs on each side, reaching from the bottom to the top of the quadrangle. The tip of the serifs should (continued) 130


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touch the four corners of the quadrangle at ac and bd. The serifs are to extend on both sides of each limb on the top and on the bottom, in the manner described above, to a width three times that of the thickness of the limbs. After this, draw the thin horizontal limb between the vertical members along line ef, as shown in the following diagram.

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Construct the letter I by erecting a wide limb in the center of a quadrangle. It is to touch the top and bottom lines of the quadrangle, and the serifs should extend to both sides above and below, as shown in the diagram.

Construct the first limb of the letter K in the same manner as the first limb of the letter H. Then draw the second, thinner limb that rises from the first limb at a point on line ef in the middle of the quadrangle to horizontal line ab, parallel to the diagonal. Its serif should extend on each side for a distance of one-tenth the length of the line ab. The curve under the left serif should be formed by the arc of a circle having a diameter of no more than the thickness of this thin limb, but the diameter of the arc of the right serif should be twice that of the one used to construct the arc under the serif of the broad vertical limb. Then draw the lower wide limb downward from this thin limb, so that it will also be parallel to the diagonal of the quadrangle. It should emanate from the acute angle formed by the narrow limb and the vertical wide limb, and it should extend and come to a point in the corner d. Now, on the line cd, place two points as far apart from each other as a tenth of the length of cd, and extend the wide limb with blind lines into this forward area. For its further extension into a point place a point g on line ef at a distance from bd equal to the width of the narrow limb of the letter. Put one leg of a compass on this point and the other on point d and draw an arc through the broad blind line. This will indicate the lower curve of the tip of the limb. For the upper curvature, divide line fd with a point h in the middle. Placing one leg of a compass on h and the other on point d, draw an arc to the broad limb.

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You can also construct the letter K in another way. Leave the vertical limb and the narrow upper one as before, except that the serif of the latter should extend only on the right side into the corner but not to the opposite side. Then draw the wide oblique line from the corner formed by the upright limb and the horizontal line ef downward to the horizontal line cd so that between d and the bottom of the limb remains a distance equal to the limb’s width. Leave the left bottom corner of this limb as is, but extend it into a point on the right toward d, as shown below.

Now construct the letter L by combining the instructions given above for two other letters—take the wide vertical limb from the construction of the letter I, and then complement it at the bottom with the corresponding limb of the letter E . This is shown below.

There are two methods to construct the letter M, both within the quadrangle abcd. First draw a narrow vertical limb behind the line ac at a distance of one-tenth the length ab. Then draw the second vertical limb, but wider at a distance of a tenth of the length of the line ab in front of the line bd. Both vertical lines should touch the quadrangle on the top and the bottom. Then divide the line cd in the center, between the two limbs, with a point e, and draw a wide limb from the upper left corner of the narrow vertical limb down to point e. After that, draw the thin limb upward from point e to the upper forward corner of the wide vertical limb. The serifs on the top of the vertical limbs should extend only outward, in the same way as has been described for the preceding letters. Keep in mind that if these letters are drawn with a quill, their limbs are drawn in rapid succession, but for the sake of instructing you, I have drawn this letter below according to the above instructions. 133


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For the second method of constructing the letter M, divide side ab of the quadrangle into six equal parts and cut off the two outer parts on both sides with two points f and g. Draw the inner wide limb so that its tip ends on point e, as before, and then draw the thinner limb upward, so that between f and g will be an empty area. Then draw the flanking limbs, the narrower one on the left, the wider one on the right, leaving them on top as before, but on the bottom, move them into the corners c and d. Then extend the ends into points as in the preceding method of constructing the letter M, but this extension protrudes beyond the side dc of the quadrangle. Or you can have sharp corners at the top. It will bring the limbs closer together; or else trim off the limbs flat on the top, whatever you prefer, as shown below.

The N is constructed within its quadrangle as follows. First draw the two thin vertical limbs so that they touch the quadrangle on the top and on the bottom. Their serifs at bottom left and upper right should extend into the corners c and b. Then connect the two narrow vertical limbs by an oblique broad limb, and at the angle a extend it into a point at e toward the upper left, so that it ends in an acute angle. It should protrude beyond the corner a by a fifth of the length of line ab, and should curve downward a fifteenth of the length ab. The arcs should be made by means of a small circle on top and a larger one on the bottom. The diameter of the small circle should be equal to a fifth of ab and should have its center outside the quadrangle, but so that its arc touches the tip of the extension, as well as corner a. Then open up the compass a little more and move the center point so that the periphery of the circle will touch the tip of the extension, as well as the wide oblique limb in the middle between side ac and the thin limb on the left. You can also construct the N so that its left serif remains within the quadrangle, or you can have it end in an acute angle, as shown below. 134


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Make the letter O within its quadrangle as follows. Draw a diagonal cb inside the quadrangle and divide it in the middle by a point e. Then indicate the thickness of the letter by points f and g in the middle of the diagonal, on both sides of point e. Let these two points be the centers of two circles which you will draw around them so that they each touch two sides of the quadrangle. At the points where they cross, you must draw the thin portions of the letter in the correct form by hand, as shown below.

Construct the letter P within its quadrangle as follows. Divide the quadrangle abcd with a horizontal line ef in the middle. Then divide ab and ef with a horizontal line gh in the middle also. Then erect the first wide vertical limb of the letter P in the same manner as was shown for the letter K.

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Then draw a vertical line ik between ab and ef at a distance equal to the width of the limb behind the vertical limb, and remember that in each case the forward corner of the quadrangle within which the letter is constructed is marked a, and the one on the right is marked b. Where line ik crosses line gh, mark point l. Then draw the two narrow horizontal limbs at the top below line ab and above ef from the wide vertical limb to the line ik. Place one leg of a compass on l, and with the other draw an arc from the upper narrow horizontal limb through gh down to the lower narrow horizontal limb of the letter P. At the point where this arc crosses line gh, mark point m. Mark off the width of the limb behind m on line gh with a point n. Put one leg of the compass on point n and the other on a point o on line gh. Then draw an arc having point o for its center, so that it touches lines ab and ef, as well as point n. Or you can draw the hook of the letter as follows. Place one leg of the compass below the horizontal line gh on line ik, midway between the horizontal line ef and the lower line of the upper narrow horizontal limb of the letter. Mark this point p and draw an arc with the other leg of the compass through m downward, as before. In this instance, the tip of the curved portion of the letter will end in an acute angle. The pointed end should terminate on the bottom midway between line ik and the wide vertical limb of the letter. Or draw the curved member of the letter P by moving the compass along the diameter, so that the curve will become wide on top, as if drawn with a quill. This is demonstrated in the following letters of the alphabet.

Construct the letter Q within its quadrangle in the same way as the letter O above. Then draw the diagonal line ad inside the quadrangle. Below it, attach the long tail to the curve of the letter which will pass through the corner d, so that this corner is at the center of the tail’s width. Where the tail is attached to the main part of the letter it should be a little narrower than when it passes the corner d. At d it should reach its proper width. From d the tail should extend a distance equal to the diameter and slant downward, yet with an upward curve. The tip of the tail should end at a point not more than a third of the quadrangle’s height below the level of the bottom line of the quadrangle. The longer the tall, the narrower it should be, so that it will in the end become a thin line. Or you can draw a shorter tail for the letter Q as follows. Open the compass to the length cd and draw an arc which will serve as the tail through point d to a length equaling that of line cd. The tail should curve upward to the level of line cd. Mark this point h. Then move the compass and draw an arc with the other leg, beginning at the round member of the letter, extending below d and ending at point h, so that the tail is widest at its inception, as shown in two versions in the following diagram. 136


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To construct the letter R proceed as for the letter P, described above. Then draw a vertical line qr in the middle of the quadrangle, and where it traverses the outer curved limb of the letter, mark point s. From this point draw a wide limb downward into the angle d as you did with the lower limb of the letter K which was described above. But this line should here be slightly curved toward the middle. For this reason you will have to draw this limb by hand and shape its curve so that it will be well formed, ending in angle d. Or you can construct the R so that its curved limb is broader on top and narrower on the bottom, as if drawn with a quill. You can do this by changing the positions of a compass on a diagonal line. In this version, the curved limb should not touch the vertical limb of the letter, as was shown for the letter P. Also the oblique limb should be slightly curved inward, as shown below.

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Construct the letter S within its quadrangle abcd as follows: First divide the quadrangle in the middle by a horizontal line ef and a vertical line gh. Where these two lines cross in the center, mark point m. Then mark off the thickness of the broadest section of the letter on line gh, so that point m denotes the lowest third of this thickness. Then indicate the thin sections of the letter with two points: below g this will be point i, and above h, point k. Mark off the breadth of the letter near the top with a point n and the bottom with a point l. Now place one leg of a compass on line gh midway between g and l and draw a circle with the other leg through g and i. Then place the one leg once more on line gh, midway between n and h, and with the other leg draw a circle through n and [k] . Now place one leg of the compass once more on line gh, midway between l and k, and with the other leg draw a circle through l and k. After this has been done, cut off the upper limb of the letter vertically so that at this point the limiting line is equal to one and a third of the greatest width of the letter. Its point should be on a level with the midpoint between i and n and should descend down to the first third of the inner and outer circles. Then cut off the curve of the letter on the bottom with a vertical line between the two circles. This severing line should be a quarter longer than the upper one, and its point should be at a level with the midpoint between n and h. The letter S can be constructed in another way. Place a point m in the middle of a horizontal line ef that divides the quadrangle abcd into two equal parts. Then put one leg of a compass on the midpoint between g and m, and with the other draw an arc that touches points g and m toward ac. Then put one leg of the compass on the midpoint between m and h, and with the other draw an arc that touches points m and n toward fd. These two curves constitute the outside curvature of the letter S at the top left and the bottom right. Now draw a diagonal [c]b through the center point m, and mark off the greatest width of the letter by two points p and q in the middle. Connect these two points with straight lines to the arcs above and below. Then draw two lines parallel to the preceding straight lines within both arcs to the level of their centers. Mark off two points below g and above h indicating the narrower width of the letter, and then draw the proper form of the S on top and bottom by hand. You must extend the top toward b and then cut it off so that its lower point will touch the circle, whereas the upper limiting line is equal to a tenth of ab. After this, erect a vertical line rs to the right of ec at a distance of a fifth of [c] d, and where this line crosses the diagonal, mark point k. In this very corner you must draw the [lower] tip of the letter S. Its limiting line should be a third longer than the one at the top. It will therefore have to protrude a little beyond k, as shown in the following diagram.

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Place the letter T vertically in the middle of the quadrangle and extend serifs on the bottom to both sides in the same way you did with the letter I . Mark off two points, the first a tenth of the length of ab, to the right of a, and mark it e, and the second the same distance to the left of b, and mark it f. These points denote the length of the horizontal limb of the letter T below line ab, but both ends of the horizontal limb shall be extended and limited by oblique serifs. The serifs will protrude above line ab and measure a fifth of ab in length. Connect the tips of the serifs to the limbs of the letter by means of arcs of two sizes. For the acute angles use a diameter of two-thirds of the letter’s width, but for the wide angle the diameter of the circle should equal the distance from the wide limb of the letter to the side of the quadrangle. Or you can construct the letter T within its quadrangle as follows. As before, place point e to the right of a and cut off the horizontal limb of the letter with an oblique line. The serif should extend only on one side, so that a simple corner remains on top. Do the same on the other side, except that point f should be half again as close to b as e is to a, and the serif should be more vertical and wider than the one on the left. In all other respects, leave things as before, as shown below.

Construct the letter V within its quadrangle in the following manner: Divide line cd in the middle by a point e. Then place a point f a tenth of the length of ab behind a, and place a point g the same distance to the left of b. Now draw the wide limb of the letter downward from f to e, which denotes its acute point at the bottom, and then continue with the thin limb of the letter upward to g. Add serifs as described before for the bottom serifs of the letter A. This is shown below. 139


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In order to construct the letter X, erect two vertical lines ef and gh a distance of onetenth the length ab behind side ac and before bd. Then draw the two crossed limbs of the letter so that its width will touch point e in the front and point h in the back; however, the thin limb should touch point g on top and point f on the bottom left. The serifs should touch the four corners abcd and the arcs determining their points should have diameters of a fifth of ab for the wide angles, but a diameter two-thirds of the width of the broad limb for the acute angles. Or you can alter the letter X as follows: leave everything as before, but move the narrow limb up by half the width of the wide limb. As a result, the upper part of the letter will become narrower and smaller than the lower one and will change its appearance, as shown below.

Construct the letter Y by placing it in the middle of its quadrangle, as you did the letter I, but split the upper part, so that the forward [left] limb will have two-thirds the thickness of the wide limb of the letter, and the rear [right] limb will have one-third that thickness. These two upper limbs should slant outward to the two sides so that the tips of their serifs will touch the corners a and b. The two large arcs under the serifs will have a diameter equal to the distance between the wide limb and the side of the quadrangle, whereas the diameters of the small arcs should be a little larger than the width of the split limb on their respective sides. This is shown below.

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Construct the letter Z within its quadrangle as follows. Place point e below corner a and point f adjacent to it, each at a distance from corner a equal to one-tenth the length of ab. In the same manner, place two points g and h next to and above point d and then connect points e and f and g and h by straight lines. Now draw the first narrow limb below ab from f to the angle b, and then draw the wide limb obliquely down to c. Then draw the narrow limb from c to g. After this extend the two points by hand. Or you can draw the Z as follows: Trim off the quadrangle abcd by one-fifth with a vertical line ef and draw the letter Z in the same way as before in this [narrower quadrangle]. The two horizontal limbs should be limited on top left and bottom right by the vertical lines ac and ef, as shown below.

All the preceding letters may also be constructed in a proportion of nine to one, instead of ten to one as in the letters above. In that case, each of the quadrangles abcd will be divided into nine parts. For better understanding, I have drawn such letters below. These letters can also be made only five widths high, in the size used for handwriting. 141


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These letters are also used for the versals [elaborate capital letters, such as those used in illuminated manuscripts], which should be of the same proportion and form, but a third larger than the ordinary letters.

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The old textur letters were written to look approximately like the ones which follow, but they are now devised differently, as I shall describe below. And although the alphabet customarily begins with the letter A, I shall, for good reason, begin here with the letter I, because almost all the letters of the alphabet can be derived from it, although something always has to be added to it or taken away. Begin by constructing the I out of squares, by placing three vertically one over the other, and dividing the top and the bottom lines of the stack by two points into three equal parts. Then place a square of the same size obliquely on the first point of the top square, so that its diagonal is in a vertical position. The forward [left] corner of the obliquely placed square will then protrude more than the rear [right] corner. Extend the sides of the original stack upward by vertical lines to meet the sides of the oblique square. Now follow the same procedure at the bottom, except that the top corner of the oblique square must touch the second point at the rear [right] of the bottom line of the stack. Again you must extend the sides of the stack downward to meet the sides of the oblique square. That is how it is done. Using a fine pen, draw a small crescent above it. The letter N is made from two I-like stacks placed so that their upper and lower corners touch. The space between the two stacks will then be narrower than the thickness of either, but you must not place the small crescent above them. The short letters of the alphabet should be constructed in this same size. Construct the letter M in the same manner, by using three instead of two stacks. Construct the letter R in the same manner as the I, adding at the top right another square of the same size that touches the original I with its corner. You can also construct it in another way: leave its foot as before, but place two oblique squares on top so that their corners touch at the center line. Then extend the sides of the bottom square to meet the sides of the oblique one. There are three ways of constructing the U. The first is simply like the N, except that the oblique square on top of the right limb is omitted and replaced by an oblique line. The right upper corner should be at the level of the top corner of the left limb, and the left upper corner at the level of the adjacent corner of the left limb. The second V is used for the beginning of words and is made as follows: Draw the first limb as you did for the I, but at the bottom push the lowest square a little more to the right, so that its left corner is in line with the vertical limb. Then place such a limb next to the former, but trim it off at the bottom by means of an oblique line beginning at the lowest corner of the left limb. Next construct the W like the simple U by simply placing another I-like limb next to it at the left. For the B use the second method to construct the simple U, but omit the top [oblique] square of the left limb and add three more regular squares instead. Trim off half the seventh square on top with a diagonal line. If you turn this B upside down, you will have the Q. (continued)

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The X is derived from the I by appending an oblique square of the same size behind the top and adding an acute tail to the bottom oblique square on the left. At the center draw a wide horizontal line across the vertical limb and cut it off on the left and right with oblique lines. The lower left corner should end a distance of half the width of the vertical limb to the left of it, but the upper corner should touch the vertical limb. On the right, the horizontal member should protrude on top as far as the bottom corner of the oblique square appended to the main limb. Trim it off with an oblique line that is parallel to the one limiting this horizontal member on the left.

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The letter C is also derived from the I . Omit the uppermost oblique square and extend the sides of the limb upward to the correct height of the letter. Trim off the left corner at the top with an oblique line, and add a wide limb horizontally to the main limb, extending it a distance equal to the thickness of the latter. Then trim it off with an oblique line at the bottom at half the distance of its protrusion on top. Construct the vertical limb of the E like that of the C , but draw a wide limb downward from the top and to the right of the main stem, placing it obliquely at the same angle as the oblique limiting line of the main limb. It should measure in length a square and a third. Finally, add a short oblique line to connect the lower corner of this limb with the main stem. The T is made in the same shape as the C, except that the oblique line at the top is lengthened. In this way the T gains an acute angle on top and a protrusion of the same shape next to the main stem below. As a result, the T becomes more elegant near the top and appears less hunched [than the C]. Make the lower part of the L like the I, but stack six squares on top of each other and cut off the seventh on the left with a diagonal line so that the top of the letter ends in an acute angle. The S is constructed like the L except that you must add a horizontal limb adjacent to the top on the right, measuring in length the span of one diameter. Trim off the back with an oblique line, parallel to the one in front. The letter F is constructed like the S, except that you must add a wide horizontal limb at the level of the short letters and twice as long as it is wide. Trim it off on both ends with oblique lines so that it comes to a point on the lower left and protrudes by a distance equal to half of the limb’s thickness. The two oblique lines should be parallel to each other. The first limb of the letter H is constructed like the L and the upper part of the second limb is made like the I, except that you must replace the lowest oblique square by a regular one and add a fifth square which you will cut off obliquely by its diameter on the right. For the K construct the first limb like the L and attach an oblique square to its right side. Then connect the bottom corner of the oblique square with a thin oblique line to the vertical limb. Now draw a wide oblique line downward and trim it off with a diagonal so that the distance between the two bottom points of the letter is no more than that of the diameter of one of the squares. The lower half of the D is constructed like the B, but you must cut off the upper left corner of the left limb with a diagonal line. Then add half a square on top of the stack of three squares which form the right limb, and surmount it with an oblique square like the one at the bottom. Incline the jointed limb on top toward the left limb’s top corner, extending it so that its tip is in line with the left limb’s left side. This will require a little less than three squares placed one next to the other because this adjoined limb is to be trimmed off by an oblique line at equal angles. (continued)

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Construct the letter O like the bottom of the D, forming its top portion on the right side in the same way as the bottom. For the P begin with the L, inverting it to form the left limb. For its right limb use the I, omitting the lowest oblique square and trimming off the lowest regular square with a diagonal line. Then add a wide horizontal limb that is to be trimmed off in front with another oblique line and protrudes with its lower point by the width of half of the thickness of the limb. Construct the lower part of the letter A like that of the N, but cut off the left corner of the middle square of the left limb with a diagonal. Leave the stack of three squares of the right limb as is, but incline the oblique square on top a little more so that when another half square is appended to it it will reach the proper height of the letter. Then cut off this half square with an oblique line so that the lower point protrudes further than the top. Now draw a circle whose periphery will be in line with the left edge of the left limb.

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There are three ways to construct the letter Z. In the first, position an oblique square whose top is in line with the proper height of the letter. Then place a second square next to it, side by side, so that the two form a long rectangle that slants downward. Draw an oblique square a distance of one diagonal below the upper square, and connect the top corner of this square with the closest corner above it by a straight line, or connect them by means of an arc. [In that case] extend the lower oblique square by means of two arcs which come to a point below, so that its lower point reaches the bottom at the proper length of the letter with a thin tip in front of it. Or you can use three oblique lines to achieve the same thing, and draw a diagonal so that this end will be inclined toward the rear. The second method of constructing the letter Z is as follows: Place three oblique squares one on top of each other, and extend the bottom one with a curved tail, as before. Construct the letter G by using the I for its left limb, but add another oblique square on the bottom, and on the top extend the upper right corner upward to the desired height of the letter. Draw a diagonal line downward from that point to the upper corner of the uppermost regular square of the stack. If you want that segment to end at a higher point, you can do that, too. Then draw the right limb to match the length of the left one and at the bottom draw a diagonal line from the bottom corner of the oblique limb at the left to the tip of the right limb. Then cut off the right bottom corner of the bottom square of the right limb with a diagonal line. Then draw a wide horizontal limb across the top of the letter from the first [left] limb to the right one and above so that it protrudes at the right a distance equal to the thickness of the upright member. Cut it off by an oblique line, parallel to the one in front. Make the letter Y like the N, but omit the bottom oblique square, adding instead another regular square below the stack of three and then a fifth one which you must cut off with a diagonal line so that its point is on the left. Then extend this diagonal line a distance equal to the side of the square. To construct the short S, place two oblique squares so that their corners touch in the middle of the vertical axis of the letter. Then erect the wide left limb of the S on top of the left square to the desired height of the letter, and similarly, extend the wide right limb of the letter downward from the right square to the proper base line, in the same manner as the bottom of the I . Cut off both these limbs with oblique lines on top and bottom, so that the two corners point toward the center. Next, draw a broad limb toward the right and downward from the upper point, and, in the same manner, draw a wide limb toward the left and upward from the lower point a distance equal to the width of the other limbs, but extending no further than the outside lines of the wide vertical limbs. Then cut off these oblique limbs with oblique lines, extending both oblique lines to meet in the center of the letter. All this I have drawn below in black and white with the lines of construction, as well as in black, in the proper sequence. (continued) 151


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As I have already said, this is the old form of the letters. Today the textur alphabet is drawn in a more elegant style by placing the oblique squares on the middle of the regular ones. As a result, the letters are not as hunched. Moreover, numerous serifs are added. Some of the limbs are cleft, and four and a half squares are stacked on top of each other. Also, the space between the limbs is increased to equal the breadth of the limbs. I have drawn these newer letters below also, and have added the small versals which are used to begin sentences. The latter are to be drawn a third higher than the regular letters.

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Here ends the Third Book.

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The Fourth Book. In geometry there are three kinds of solids which can be constructed with compass and ruler. Some are of uniform shape and used for columns, etc. Others end in points and can be used for cones, as well as pillars, etc., provided the point is at a high enough elevation. But such points can be trimmed off in correct proportion. For that reason, a supporting column shall never have a point higher than the apex of an equilateral triangle. Thirdly, there are solids which have equal sides, planes, or corners. Euclid calls these corpora regularia and describes five types, because there cannot be more which can be placed within a sphere so that all their points touch its shell. These solids can be used in many ways and will be described below. The first is a triangular solid consisting of four triangular surfaces with equal angles. It has four equal triangular corners and six equal edges, as shown below in a ground plan and in profile.

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The second solid is a diamond with equal angles, six equal corners, and twelve equal edges. I have drawn this below open and closed with its ground plan and in profile.

The third solid consists of twenty equal triangular surfaces with equal angles, twelve equal pentagonal corners, and thirty equal edges, as I have drawn below open and closed with its ground plan and in profile.

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The fourth solid is a four-cornered cube with six square surfaces, eight triangular corners, and twelve edges, as shown below open and closed with its ground plan and in profile.

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The fifth solid is constructed from twelve pentagonal planes and, when folded closed, will have twenty triangular corners of the same angles and thirty edges. I have drawn it below with its ground plan and in profile.

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The sphere or ball, if cut along its meridians and then spread out on a flat surface, takes on the shape of a comb, as I have drawn below in approximation.

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These solids touch a hollow sphere with all their corners, and I have drawn such a sphere here. You can also interlace two such solids of the same size so that in each case one point pierces the corresponding surface of the second solid. In this way you can gain handsome results. You can also raise each surface of these solids to a point or depress the surfaces concavely. You can also do this with irregular solids wherever they touch a hollow sphere with their corners. Every solid with straight surfaces will also have corners and straight edges. These edges are formed whether their surfaces join at wide or acute angles. Solids which are rounded on all sides, whether oblong or spherical, have no corners. Many other solids can be constructed which touch a hollow sphere with all their corners but have irregular surfaces. Some of these I shall draw below, opened up, so that anyone can fold them together by using two layers of paper, glued together, and then cut to the depth of one with a sharp knife. It is then easy to fold it along the edges. Use this method for the following figures. They can be utilized in many ways. The first of these solids with unequal surfaces consists of four hexagons and three [i.e., four] triangles. But the edges are all of equal length where they are joined. This solid is here shown open, but when it is folded closed it will have twelve corners and eighteen edges.

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The second irregular solid consists of six octagonal and eight triangular surfaces. If this solid, shown opened up in the following diagram, is folded closed, it will have twenty-four corners and thirty-six edges.

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The third irregular solid has six square and eight triangular sides. If the open pattern is folded closed it will have twelve corners and twenty-four edges.

The fourth irregular solid, when open, has six octagons and six squares. When folded closed, the solid has twenty-four corners and thirty-six edges.

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The fifth irregular solid, when open, has eighteen squares and eight triangles. When folded closed it will have twenty-four corners and forty-eight edges.

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The sixth of the irregular solids, when opened, has six squares and thirty-two triangles. When folded closed, it will have twenty-four corners and sixty edges.

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The seventh of the irregular solids, when open, has eight octagonal and twelve square surfaces. When folded closed, it will have forty-eight corners and seventy-two edges.

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The eighth irregular solid is constructed from six dodecagons and thirty-two triangles between them. However, these do not all have equal sides, as shown in the following diagram.

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Yet another figure is made from twenty hexagons of equal sides and angles and twelve pentagons having sides equal in size to those of the hexagons. The angles of the pentagons are also of uniform size. The folding pattern in the proper arrangement is pictured below. If it is folded the resulting solid will have sixty-two corners and ninety edges and all corners will touch the surface of a hollow sphere in which it is placed.

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You can construct another solid consisting of twelve pentagons and twenty triangles, all equilateral and having corresponding corners. If folded together the resultant solid will have thirty corners and sixty edges. Its folding pattern is shown in the diagram below. All the corners of this solid will touch the periphery of a sphere that encloses it.

If a sphere is divided by twelve meridians and six lateral lines and all points where these lines cross are connected by straight lines so that flat planes are formed by them it will result in a solid consisting of forty-eight quadrangular—none of which is rectangular—and twenty-four triangular facets. Each of the latter will have an acute angle. The solid will have sixty-two corners and one hundred-thirty-two edges, as shown below in perspective, as well as with its ground plan.

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It is useful to know how to convert the circumference of a circle into a straight line of equal length. Proceed as follows. Place three lines each equaling the diameter of the circle end to end to form line ab, then extend this line by one seventh of the diameter and mark it point c. Then ac will equal the circumference of the circle, as achieved by mechanical means, as shown below.

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If you attach six squares to one another and then attach twelve triangles, each with an altitude equal to the side of a square, they will, when folded closed, form a solid whose opened-up pattern I have drawn below.

If you cut off the corners of the preceding solids and then trim off the remaining corners once more, you will be able to form several other kinds of solids. Many other things can be discovered by transposing parts of these solids, and the resulting formations can be used when carving columns or adorning them with ornaments. At one time, when the city of Athens suffered the scourge of pestilence, the citizens asked Apollo for his advice as to how they might rid themselves of the plague. He replied that if the inhabitants could divide his altar in two [he meant into two cubes, each half the volume of the original cube], they would be saved. The Athenians then ordered a large slab of stone to be hewn of the same size as the altar and placed it on top of it. But when the plague failed to cease, they applied to the deity once more, saying that they had done as he had instructed them. He answered that they had not acted according to his instructions, but had [instead] doubled the size of the altar. When the Athenian artisans failed to find a way of solving the problem they addressed themselves to the scholars and especially the philosopher Plato for advice. He taught them how to find two lines which are proportional to two given lines of unequal length. By these means they were able to double, triple, enlarge, and multiply a cubus, that is, a solid made of squares like a die. This is a very useful skill for all workmen. This method is kept secret and hidden by scholars, but I wish to bring it to light and teach it. It can be used to cast containers and bells and enlarge them or double their size as desired, while retaining their proper proportions and their weight. It can also be used for barrels, chests, wheels, chambers, pictures, and whatever one wishes to enlarge. Therefore, let all artisans pay heed, because to this day, as far as I know, no one has explained it in the German language. (continued)

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First place two equal cubes or dice next to each other [and mark them as below with three points, to form line] abc. Then erect a line of the same length as ac at right angles on a horizontal line de. Draw a semicircle from the center c through points d, a, and e. Then draw a straight line from e through b to the periphery of the semicircle and mark it point f. Then mark a small ruler with a center point and divide both sides evenly with the same scale of numbers beginning from the center with 1 [and continuing in either direction]. By moving the ruler you will find the first line and then the second for the doubled cube. To do this, place one end of this ruler

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on point d and keep it anchored on this point, whether you move the ruler up or down. And if you move the other part of the ruler, its center must always remain on line abc. Now move the ruler until you find the middle between line ef and the periphery of the semicircle, and where the movable member of the ruler crosses line ef mark point g. Where it crosses line abc mark point h, and where it touches the periphery of the semicircle on the outside, mark point i. In this way, gh and hi will become equal in length, and hc will denote the first side of the doubled cube. Next place the line hc horizontally adjacent to line ab, representing the side of the simple cube, to form line agc. Put one leg of a compass on the midpoint of line ac and draw a semicircle from a to c. Then erect a vertical line from h to the periphery and mark point k. This line kh represents the side of the doubled cube, as shown in the diagram below.

If you wish to triple or quadruple the size of a cube [or further multiply it], use the preceding method, but adjust it as follows: First place three cubes next to each other and mark their combined length abcd. Then erect line ad at right angles on a horizontal line fe. From its midpoint d draw a [semi]circle fae. Then draw an oblique line from e across line ad at a point one-third its height. This is the length of the simple cube. Mark that point b. Where the line emanating from e meets the periphery, mark point g. Then place one end of the graduated ruler

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on point f and anchor it there. Move the other end and determine the midpoint between eg and the arc eag on the vertical line ad. Once the ruler is in the proper position, draw a line; where it crosses the line eg mark point h, then where it crosses the vertical line ad, mark i, and finally, where it touches the periphery, mark k. By these means, hi and ik will be of equal length, and id will be the first length which will serve to find the second line of the side of the tripled cube. Next, place the line id horizontally, and extend it by the length of one side (ab) of the single cube. Then put a compass on the midpoint of the composite line abid and draw a semicircle ad. Now erect a vertical line bl at right angles to abid and which touch the arc . This line bl represents the side of the tripled cube based on the single cube ab. This is shown below.

Next, place four lengths of the single cube ab next to each other to form the line abcde. Put this line ae in a vertical position on a horizontal line gf. Now draw a line from f so that it cuts off the bottom quarter of ae, and mark i at the point where this line touches the [semi]circle above g. Then use the graduated ruler and place its midpoint on line ae and its one end on point g, and find the middle between fi and the periphery of the semicircle along the vertical line ae. Mark k at the point where fi is then crossed by that line, l where it traverses ae, and m where it touches the periphery between a and f. Then kl and lm will be equally long, and will determine the length le. It is the first length which will help determine the side of the quadruple cube. You will find it, as before,

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by extending the line le by the length of the single cube ab, placing a compass on its midpoint, and drawing a semicircle to encompass a and e. Then erect a vertical line at right angles from l to the periphery, and mark that point n. Then nl will denote the side of the quadruple cube. This is shown in the diagram below.

Now that you have constructed the four cubes according to the preceding method, you can also construct [proportionally] larger cubes of twice, three times, or four times the first sizes. Proceed as follows: Draw a right angle and mark the angle e, the vertical side f, and the horizontal line i. Then draw the single, double, triple, and quadruple cubes in profile, superimposed on each other, so that one corner of each will coincide with angle e, and two sides of each will coincide with the lines f and i. Mark these cubes according to their volume e 1, e2, e3, and e4, and draw a diagonal or oblique line eh through the four [opposite] corners of all four cubes. Then draw the four sides of the cubes, in the size you require, parallel to those marked e 1, e2, e3, and e4, above line eh.

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If you are now given a greater length for the side of a cube to be constructed in the same proportion, erect this length vertically on line i of the single cube so that its bottom rests on the horizontal line ei, and where it ends on top, mark a. Then draw a ray from e through a and through all the vertical lines 2, 3, and 4, to a point g. At the points where the vertical lines 2, 3, and 4 are traversed by the ray, mark b, c, and d. Then a1 will be the side of the single cube, b2 the side of the double one, c 3 that of the triple one, and d4 that of the quadruple one. They will be proportional in size to the cube of the smaller series. All of this can be put to many uses. By this method all things can be properly enlarged and yet retain their proportions. You can see this in the diagram below.

There is yet another way for you to keep doubling a cube or to keep reducing it to half its volume. But in this case you must start out with two cubes, one having twice the volume of the other. Proceed as follows: Draw a vertical line ab and attach to it the sides of the aforementioned two cubes so that they touch each other. The larger cube is to be on top, the smaller one below. Mark points c and e at the two corners of the upper cube which touch line ab, and mark the two corners on the opposite side f and d. Mark g and i at the two corners of the lower cube which touch line ab, and h and k at the two rear corners on the opposite side. Now connect the two corners f and k with a straight line and extend this line upward as far as you like. Mark the end of the line point r. Then extend the line downward until it meets the line ab and mark that point z [next to b; not shown in the diagram]. If you now draw a straight line from corner d to point z it will traverse the two bottom lines of the two cubes. If you extend this line upward to point x it will help define the enlargement of the cubes, 178


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and below x it will define the reduction in size of the cubes. You must follow this procedure: First extend the horizontal line on top of cube cd to the line zr and mark it point l. Next, erect a vertical line from l to the line zx and mark this point m. Now enclose the cube clmn which will have twice the volume of the cube cdef. This duplication will continue as many times as you can place one cube on top of the other, and you will find this [relationship] definite and correct. But below, the cube will in all cases be one-half smaller down to the acute point of the angle z. And this rate of decrease occurs in the same way as the preceding increases. Proceed as follows: Where the line xz crosses the bottom cube’s side ik mark o, and from there drop a vertical line down to the oblique line zr. Mark that point p. From there draw a horizontal line to the line ab and mark this point q. The cube ioqp will then have half the area of cube ghik. You can continue in this manner down to the acute point on the bottom. It is shown in the diagram below. A triangle remains.

You can also increase or decrease a cube by different ratios, for example, 1:3. Follow the above method for whatever proportion is desired. For the proportion 1:3, use two cubes, the first a single one, the second three times its size, as shown in the following diagram. 179


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Moreover it is useful to know for anyone who wants to enlarge cubes or to multiply them how to determine the length of two lines which are to be proportional to two given lines of unequal length. Proceed as follows: The two given lines are eb and bg. Place these two lines at right angles joining at b. Then extend the two lines beyond point b as far as necessary to points d and c. Form two right triangles: gcd whose right angle is at c, and a second triangle cde whose right angle is at d. The instrument you can use for this purpose will be described below. In his Book VI, proposition 8, Euclid demonstrates that cb is a mean proportional between db and bg, and that db is a mean proportional between cb and be. Therefore, gb: cb = cb: db, and also db: cb. Therefore the two lines bc and bd are proportional to the given lines bg and be. Construct the above-described triangles in the following manner: Use a square ruler rpq whose right angle is at p. Put a slot along side pq and insert through this slot a straight piece ts, which remains at right angles to side pq as you move it up and down, and at the same time remains parallel to side pr at all times. After you have readied this instrument, place its side pr so that it touches point g and so that the right angle p lies on line ec. Place the other side

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of the instrument qp along line db. Now shift the movable piece ts so that the angle s is on line bd and the movable piece touches point e. And if everything is done according to these instructions, then pr will equal cg, and ps will equal cd, and st will equal de. Accordingly, it will become clear that the two triangles gcd and cdc are made and constituted as described earlier. This is shown to you in the diagram below.

You can employ another method for the above purpose without the use of the instrument. Once again place the two given unequal lines ab and bg at right angles at point b and close the rectangle with a point d. Divide its diagonal ag into two halves by a point e. Then anchor a ruler on point b, and move it to and fro until you find where it cuts off dh and dz, so that the lines eh and ez are equally long. The arc will prove that the determination by means of the ruler is correct. Next, drop a vertical line et down to line dg. By this you will have divided line dg into two equal parts, according to Euclid’s Book VI, proposition 2. Accordingly, the rectangle with the sides tz and zg and the square made with the line gt are equal in area to a square made from the side tz. This is to be understood as follows: Construct an oblong rectangle out of dz and zg. Then place a proper square on length tz. This square will have the same area as the oblong rectangle with the sides dz and gz plus the square tg, which completes the figure, as shown in the diagram below.

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If the square made from the side tz is now added, then the quadrangle will be equal in area to the square made from ez. Likewise, the quadrangle made from the lines dh and ha together with the square ae will be equal to the square erected on eh. If, then, the two lines eh and ez are equal, this will also be the case for lines ea and eg. Thus the rectangle made from tz and zg is equal in area to the rectangle with the sides dh and ha. By this it is understood that in the first rectangle, the line dh will be the long line and ha will be the short one; in the second rectangle, the line tz will be the longer one and zg the short one. Therefore, the rectangle dha will have the same area as the rectangle dzg. To prove this conclusively, convert these rectangles into squares by the method explained in Book 2, figure 31 [view 118]. The preceding method is also given by Euclid in Book VI, proposition 15—that is, that line dz is related to line dh in the same proportion as line ha is to line gz, and line dz is related to line dh as line gz is related to line gb. In the same way, line ab is related to line ah. According to Euclid, Book VI, proposition 4, line ab is related to line ah as line ah is to line gz, and also as line gz is to line gb. It is therefore demonstrated that two mean proportional lines have been found between the two lines ab and bg. This is shown in the diagram below. Likewise, the difference between the rectangle and the square is that the square has four right angles and the sides of equal length, but that the quadrangle has two long sides and two short sides and yet four right angles, as mentioned here before.

If you now wish to apply the preceding explanation in the duplication of a cube, you must double the length of the side of a given cube and find the two proportional lines. You will find these correctly by using the aforementioned method. They are called mean proportional lines, and if you use the shorter one of these two, it will give you the side of the duplicated cube.

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In the same way you can also determine the size of a tripled cube by finding the two mean proportional lines between the length of the side of one cube and of three cubes. The shorter of the mean proportional lines will then denote the length of one side of the triple cube. Accordingly, you can multiply the size of any cube. I shall give one example in regard to weight: If you have a cannonball weighing one pound you can increase it proportionally by a pound by using the above-described method, for the proportion will also give you the correct weight if you use the same metal. You will draw a cube containing the ball and then enlarge it twice, three times, or four times. Accordingly, the weight and the volume will be proportionally enlarged, as shown below. You can continue this even to a hundred pounds.

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There is yet another way to double, triple, or quadruple cubes consecutively as many times as desired. By the same method they can also be halved. Proceed as follows: First draw a straight horizontal line and erect on it two vertical lines at a right angles to it next to one another to represent sides of two cubes which are in proportion to each other of one to two or two to three. In this instance we shall use the proportion one to two. Mark the longer line a on top and b on the bottom, and the shorter one c on the top and d on the bottom. Now place a ruler on ac and extend the line down to a point e where this line meets the horizontal line. Mark the opposite end of the oblique line f, and the opposite end of the horizontal line ge. If you now wish to find the side of the cube which is double the size of the cube ab, proceed as follows. Draw a diagonal cd, and then draw two lines parallel to line cb in front of it [to the right of it] and behind it [to the left of it]. Draw the first line from point a toward the horizontal line bg and mark point h where it meets this line; then draw the second one upward from point d to the line ce, and mark point i where it meets ce. Then erect two vertical lines, parallel to ab and cd—the first from point h to the line fa and marked point k where it meets that line, and the second toward point i and marked l where it meets the bottom of horizontal line ge. If you then construct a cube with the side kh, its volume will be twice that of a cube with the side ab. The cube made with the side il will have half the volume of a cube made with the side cd. You can continue this toward point e on the right or toward points fg on the left. Toward the left it will double in volume and toward the right it will decrease by halves. All the lines of this figure are proportional to each other and therefore have definite relationships, because in the same way that the vertical lines are proportional to one another, the oblique lines are also related, and likewise the distances between k, a, c, and i are proportional to those between h, b, d, and l. This has been properly proven and demonstrated by many others, and it is good and useful, as shown in the diagram below.

(continued)

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Having demonstrated how to construct various types of solids, I also wish to teach you how to render them in a painting. For this purpose I shall choose the simplest solid, that is, the cube, but this will demonstrate how to render other solids as well. I shall also speak of light and shadow and how they are to be used in relation to one another. Whatever is seen must be in the line of sight of the viewer’s eye and it must be in the light, because nothing can be seen in darkness; moreover, there must also be a medium between the eye and the object that is seen, as explained below.

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Every light radiates by means of straight lines, as far as its rays will reach. If an opaque surface is interposed, it will repel the rays of light and cast a shadow to the extent that the rays are held back. I will show this in the following diagram. For this purpose, first draw the ground plan of the cube by drawing a square efgh. Now place a cube on this plane surface. It will in its ground become a square area because the upper four corners coincide with the four bottom ones. Therefore, each corner is marked doubly: the bottom four corners are a, b, c, and d, but the upper ones are 1, 2, 3, and 4. Accordingly, a1, b2, c 3, and d4 belong together. This completes the ground plan. Then you must project the square ground and the cube on it as stonemasons do. Do this as follows: Above the square area efgh draw a parallel horizontal line as long as the square, and mark its beginning at e, h, and its end at f, g. This line is equivalent to the lower plane efgh, and for that reason it is marked with the same letters. Next, erect vertical lines from the ground plan of the cube a1, b2, c 3, d4 through the horizontal line ehfg to the correct height of the cube, and you will have the cube on top of the aforementioned horizontal line. The bottom side of this cube, which rests on the horizontal line ehgf, will then be ad at one end and bc at the other, but the top will be 1, 4, and 2, 3. In this way the four corners of the projected cube—1, 4; 2, 3; b, c; and a, d—are understandably placed above its ground plan below. Now it is necessary to position the source of light. Again I will do it both for the ground plan and for the projection. Therefore, I will have to indicate a source of light for each of these. For the projected one I can place the light source high up or lower; but for the ground plan it will be in the middle or on the side. However, in this instance, place the source of light o for the projected cube as high and as distant as you like. For the ground plan, place the source of light p on either side, but both sources of light must be the same distance from the ground plan and the projection of the cube. Now draw two straight light rays from point o through the upper corners of the projected cube 1, 4, 2, 3 and where these rays meet the line ehfg place the points i and k. These indicate the extent of the shadow. Then draw rays from the lower source of light p through the corners of the cube’s ground plan which are marked b2, c 3, and d4, and extend these rays to where they meet vertical lines dropped down from the points i and k. They will delimit the length and width of the shadow. Mark the corners of this shadow l, m, and n. And then close in with straight lines b2, 1, d4, and n, which will designate the entire area of the shadow. But remember that the two sources of light are really one and the same, and so are the lines eh and fg and the plane efgh, and also that the ground plan and the projection coincide. They are all shown doubly for the sake of this demonstration, in the following diagram.

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Light

Upright cube

Light

Ground plan of cube

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If you now wish to transfer the described cube with its area of light and its shadow to a painting, you will need to know how to do this and by what means. First you will have to determine the point of sight of the eye. Second, you will require the object, whether it is seen head on or in profile. Third, you require light, without which nothing can be seen, as mentioned before. The eye sees the object placed before it only by straight lines. It cannot perceive by curved lines. Thus, if two opaque objects are placed one behind the other, the eye will see only the one in front, but not the one behind it. If, therefore, many objects are to be viewed, they must be spread out, so that they are in the line of sight of the viewer. There must also be a proper distance between the eye and the object to be viewed. The object should not be placed so closely to the eye that it becomes invisible or blurred, for even many large objects can be perceived by a small part of the eye if they are located at the proper distance. But neither should the object be so far away that it becomes lost to the eye, because if an object is too distant the lines of sight flow so closely together that the eye cannot perceive the small area of sight between them. This is shown in the adjoining diagram for your guidance, as you will later hear. This is to be understood as follows. Point a denotes your eye. Very close to it, indicate a line bc, and then connect point a with radials to points b and c. Line bc will cover your eye. Then remove this line bc and place a second line, marked de, a good distance away. Draw radial lines from a to the points d and e. Your eye will be able to see this well. Now remove line de and draw a line fg at a considerable distance away. And again draw radial lines to points f and g. These will converge as they approach point a, so that the eye can no longer discern the area of sight between them. For this same reason, it is not possible to recognize the face of a man at a great distance. It is not because of a weakness. Accordingly, things which are to be seen properly must be placed at a reasonable distance. But this does not apply to landscapes, which can be viewed from a distance of even six or seven miles. This is a different matter. Now note that between the eye and the object viewed a transparent pane is placed which will slice off all the lines of sight which emanate from the eye. This transparent pane may be close to the eye or farther away, and therefore closer to the object. If the transparent plane is taken close to the eye, the area of the object to be painted will appear small; but if you move the pane farther away from the eye, the object will appear larger. This is to be understood as follows. Place two oblique lines b and c to form an acute angle a. Between these two lines, draw

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two vertical lines. Mark the one further away from a, de, and the one closer to a, fg. And from the points where these two lines de and fg are cut off by the two lines b and c, erect squares. You will find that the more distant square de is larger and the closer one fg is smaller, including everything within its frame. This is shown in the following diagram.

Whatever is enclosed by the two forked lines ab and ac and touches them; be it near or far, vertical, oblique, or curved, everything appears to the eye in a certain size, as shown below.

Now I shall begin the procedure. I take the aforementioned cube, as it stands on its quadrangular plane, and render it together with its source of light and its shadow. I shall draw both its ground plan and its projection, and I shall also render it as it appears to the eye. To make this understandable, I shall designate the eye by a point and divide this point of sight in two, as before, as well as the object in its ground plan and in projection. These two points will be designated by eyes and I will draw them on one side, as far from the objects as I wish, and then draw a vertical line through them, parallel to the sides of the ground plan. Of the points of sight, marked on this vertical line, one will be used for the projected rendering, high or low, according

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to how the object is to be viewed, and I will use it to measure its height, depth, or distance. The second point of sight will be below the upper eye on the same vertical line, on whichever side of the ground plan I wish, or else in the middle. Accordingly, I measure the width on both sides, whether near or far. Then I draw rays from the upper point of sight to all points of the object which can be seen and which are marked with letters or numbers. I draw the first ray from the point of sight, relating to the projected view, to the source of light o, and the second I draw to the bottom of the light’s arc. Next I draw two lines to the top four corners of the cube. In front, the corners 2 and 3 coincide, in the rear, the corners a and d correspond also. Then I draw two rays below to the two corners of the cube, the one on the left meets at bc and the one on the right at ad. Then I draw two rays to points k and i. I also draw two lines to the sources of the two horizontal lines on which the cube rests, that is, to the quadrangle marked gh in front and eh in the back. Accordingly, the projected rendering is properly connected to the point of sight by rays, as it should be. Now I draw similar rays from the second point of sight to the ground plan and to all its points, as well as to the plane on which it is drawn. First I connect the point of sight by rays with the four corners of the plane fghe. Then I connect it with the four corners of the cube’s ground plan, which are equivalent to eight corners, wherefor they are marked b2 and e3 in front, and d4 and a1 in the rear. I then draw three further rays to the three corners l, m, and n of the cube’s shadow, and I have drawn all the necessary lines to both plans of the cube. Now that the eye has reached all the points within its range of sight, the object can be placed in a painting. This is accomplished by interposing a transparent plane that slices off all the lines of sight, as has been mentioned before. This is represented in a drawing by a vertical line, close to and parallel to the projections, between the point of sight and the object, which will appear large [or small] depending on distance, as explained above. This line must also be at right angles to the two lines of sight which meet the front of the object. If necessary, the line has to be tilted forward or backward, but it must slice off all the rays. Then I draw two horizontal lines at equal angles from the eyes to the vertical line, denoting the transparent pane, and I mark the two points where the horizontal lines cross the pane with two eyes. All four eyes actually denote only one eye, but they are here divided to simplify the procedure.

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In what follows I shall teach a second and shorter method of bringing the aforementioned object into a painting in the proper perspective. This is the method: Draw a horizontal line equal in length to the preceding lines efgh of the projected rendering, which here represents a quadrangular surface. Then place the near point of sight on the side above the line f in the same manner as for the two crossed lines. Once this is done, I draw two straight lines from the eye to both ends of the line efgh. This will result in the forming of three sides and the two bottom corners of the quadrangle which I show here in perspective with four corners. Now I will have to know how to do the side in the rear, and how high it is to be. I proceed as follows. I place a second eye at a point more distant, as in the earlier construction, but here I place it on the same level as the near eye. I draw two straight lines from this [second] eye to both ends of the existing lines. Then I erect a vertical line aa bb which touches the forward corner. Where this line crosses the long ray emanating from the distant eye to the acute angle, I mark cc. From this point cc I draw a horizontal line parallel to the base through the two rays which connect the near eye with the two ends of the bottom line. The points where these two rays are traversed denote two corners, and in this manner the quadrangular plane is rendered properly in perspective, as in the preceding method. Therefore, I also mark its corners with the four letters efgh, as before. Before preceding further, I have drawn this below.

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Now, after this quadrangular base has been shown in perspective, I place a cube on it in the same manner as in the earlier rendering. Proceed as follows: Take the length of one side of the cube on the near side of the ground plan and mark it off with two points x and y on the lower horizontal line fg. It should be as far from point f as it is from the line ef in the earlier ground plan. Then draw two rays from the eye to x and y. The bottom quadrangle of the cube must be placed between these two lines in the back. The distance to the back is determined by a line eg which passes through the two corners 1a and 3c in the previous diagram. For this reason, you will also draw this line in the present diagram, and where it crosses the two rays x and y, mark point a on x and point c on y. Now draw two horizontal lines from points a and c, and where the horizontal line a crosses ray y, mark d. And where the horizontal line c crosses x, mark point b. Now the quadrangular base of the cube is properly situated with its four corners on the perspectively rendered base efgh, exactly as in the earlier method. Now erect four vertical lines from the four corners abcd. The height of the forward lines should equal the length of the horizontal line bc. Draw a horizontal line at that height from one line to the other and mark the two corners above b with the numeral 2, and above c with the numeral 3. Next, draw two rays from the eye to the corners 2 and 3, and where they cross line a and d, mark numeral 1 above the a, and numeral 4 above the d and draw a horizontal line connecting 1 and 4 and 2 and 3. This completes the proper construction of the cube in perspective, like the one constructed earlier by a different method. Before I continue, I have drawn a diagram of it below.

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Light

Upright cube

Light Ground plan of cube

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Now that you have the preceding explanation before your eyes and have understood it, take another sheet of paper and draw two lines which cross at right angles on it. At the point where they cross, indicate an eye, which here takes the place of the four eyes used in the preceding drawing. All levels, on both sides, whether high or low, deep or wide, will have to be placed according to and related to this point of sight like the rays in the preceding drawing. Take therefore, to avoid errors, two compasses, and use one for the ground plan and the other for the projected rendering. Now place the compass which you intend to use for the projected rendering with one of its legs on the eye of the line in the preceding drawing which denotes the transparent pane belonging to the projected cube, and the other leg on the ray which connects the more distant eye with the source of light o. Keep this opening of the compass. Then take the second compass and place its one leg on the other eye on the line denoting the transparent pane, but belonging to the ground plan, and the other leg, again on this line of the transparent pane, where it meets the ray connecting the lower eye to the source of light p. Now transfer these two distances by means of the two compasses to the following crossed lines. However high point o is placed above the eye, the second compass will show you how far to the side point p will be placed. These two points coincide in this drawing and their point of coincidence is therefore marked with both letters o and p. Follow this method for all the rays which pass through the transparent pane, using the first compass for those related to the upper eye, and the second compass for those related to the lower eye. Transfer all the distances from the eyes to the transparent pane to the eye of the crossed lines, and all points from the projected as well as the ground renderings will be combined and coincide by numbers and letters to the proper height or width. Then connect all the indicated points by straight lines and you will see the result. It will show you where the corners of the object to be painted are to be placed. Whatever cannot be seen by the eye is here indicated by blind lines [lines of construction]. I have drawn this next to the crossed lines below, in order to show what will happen and then how it will look without the lines of construction, where I indicate the shadow by some cross-hatching. This is the proper basis for the rendering of objects in painting [in proper perspective]. Light

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Now that the cube has been rendered in proper perspective on its base, shine a light on it and draw the shadow the cube casts on its base. To determine this, proceed as follows: Place the source of light above the level of the eye on whichever side you prefer, and mark this source of light o. Here I will place it in the same spot as before. After you have done this, drop a vertical line from o down to a point p which denotes the reflecting surface below the source of the light. If the light is to be moved, I will move the source of light higher up along this vertical line. If I wish to move the light closer, I move point p further down. But here I have chosen the same distance as in the earlier construction. Now that the two points o and p have been established, draw straight rays from p through the three upper corners of the cube 2, 3, and 4, and on down to the base. Then draw straight lines from p through the lower three corners b, c, and d of the cube, and where these three lines cut off the three rays emanating from the source of light o, mark three points l, m, and n. Then connect bl with lm, and mn with nd. This will give you the shadow of the cube in proper perspective, as before. So that you can see this clearly, I have drawn it below with all the details, and you will see that the result is the same as before. I have also drawn what will appear in the painting without the lines of construction, and I have shaded it, so that you will understand it better.

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One can render other solids in perspective in the same manner as I have done with the cube by first drawing the ground plan and projection of each solid. In this way these solids can be made suitable for use in paintings. If you wish to find a given point in an object which is rendered in perspective, such as a square, proceed as follows. Draw a square abcd so that ab is the top horizontal line, and extend this proper square on top with another square in foreshortened perspective and mark it abgf. Mark the corresponding vanishing point, indicated by an eye, point o. Then place a point e inside the proper square at a place of your choosing. Next, draw a diagonal ac and a corresponding diagonal bf, the latter in the foreshortened square. Then erect a vertical line on point e, parallel to the side of the square and extending up to line ab and mark point h at the spot where they meet. From there draw a straight line inside the foreshortened square toward the eye point o. Where this line meets line fg mark point m. Next draw a horizontal line inside the proper square from point e to the diagonal ac, and at the spot where these lines meet, mark point i. From there erect a vertical line, and where it meets line ab mark point k. Then draw a line from this point inside the foreshortened square toward point o. Where this line meets the diagonal fb mark point l, and from there draw a horizontal line, parallel to ab, to the line hm, and mark that point n. That is the point you have tried to determine in the perspectively rendered square, and it corresponds to point e in the proper square on the bottom. I have drawn this diagram below. Thus, any given point can be determined in a cube that has been rendered in regular, as well as foreshortened view by dividing up the cube at the location of the given point, as shown above.

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Now I will teach a way of tracing an object. If you wish to trace what you see before you, you can construct an apparatus for this purpose as follows: Place a clean flat pane of glass into a quadrangular frame. Then take a board of the same width as the frame, but longer. Hinge the frame to it as for a board game, so that you can fold the frame down to the board if you wish. Then attach two lugs at the midpoints of the board’s sides, and attach iron links to them which can move freely and can reach the glass frame at equal angles on both sides. Form hooks at the end of the two links which can be locked into two lugs on the sides of the frame. Next, take a four-cornered length of wood, longer than the width of the board, and groove it so that it will slide horizontally on the board, closer or farther away from the frame. Now cut a square slot along the entire length of the sliding piece, sparing its ends, but drill a round hole through each of the ends. Then insert a long screw through these holes, but do not screw it into them. Now cut another wooden piece to half the frame’s height, with a tongue on the bottom, so that it will fit into the slot of the horizontal piece, and so that the ledges on the bottom of the vertical piece slide on the horizontal one. The vertical piece can therefore be moved to and fro at right angles to the horizontal one. Drill a round hole into the lug on the bottom of the vertical piece and insert a nut into it which corresponds to the long screw. Now insert the long screw through the side of the horizontal piece and screw it into the nut that is embedded in the vertical piece until the screw reaches the other end of the horizontal piece. From now on you will be able to move the vertical piece to and fro by means of this screw. Now drill a round hole down the middle of the vertical piece and cut a narrow slot on one side of it.

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On this same side, cut many small notches. Then insert a round rod through the top of the vertical piece so that it fits snugly. Affix a lug to the bottom of this rod so that it will protrude through the narrow slot and can be retained by the notches at any desired height. At the top of this rod attach a small thin board and drill a hole through it, so that you can look through it at the frame with one eye. Now draw whatever you wish to paint on the pane of the glass. This is very suitable for portraiture—especially for those [painters] who are not sure of themselves. If you wish to use this method to paint a portrait, let the subject rest his head so that he will not move it until all the needful strokes are completed. After that has been accomplished you may also use color, but you must be sure to have a steady light. If you nail two horizontal strips of wood to this board to which the glass is hinged, and drill two holes in each, you can insert into these holes four round rods with iron fittings which will convert the apparatus into a table. This can then be easily taken apart, so that it can be carried and taken along. I have drawn this below.

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A second method. You can render anything within reach in correct perspective by means of three threads and draw it on a tablet as follows: If you are in a large chamber, hammer a large needle with a wide eye into the wall. It will denote the near point of sight. Then thread it with a strong thread, weighted with a piece of lead. Now place a table as far from the needle as you wish and place a vertical frame on it, parallel to the wall to which the needle is attached, but as high or low as you wish, and on whatever side you wish. This frame should have a door hinged to it which will serve as your tablet for painting. Now nail two threads to the top and middle of the frame. These should be as long, respectively, as the frame’s width and length, and they should be left hanging. Next, prepare a long iron pointer with a needle’s eye at its other end, and attach it to the long thread which leads through the needle that is attached to the wall. Hand this pointer to another person, while you attend to the threads which are attached to the frame. Now proceed as follows. Place a lute or another object to your liking as far from the flame as you wish, but so that it will not move while you are using it. Have your assistant then move the pointer from point to point on the lute, and as often as he rests in one place and stretches the long thread, move the two threads attached to the frame crosswise and in straight lines to confine the long thread which is stretched through the frame. Then stick their ends with wax to the frame, and ask your assistant to relax the tension of the long thread. Next close the door of the frame and mark the spot where the threads on the frame cross on the tablet. After this, open the door again and continue with another point, moving from point to point until the entire lute has been scanned and its points have been transferred to the tablet. Then connect all the points on the tablet and you will see the result. You can use this method for drawing other things. I have drawn this device below.

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Remember further an easier and better method that derives from the aforementioned explanation of how to trace. It was invented by Jacobus Keser and is very useful and more can be done with it than with the preceding method. It is also more practical. Therefore, for the common good and to honor God for having endowed Jacob Keser with so inventive a mind, I shall explain this method below. Flat things are suitable for tracing on a glass pane from the near point of sight. But other solids become distorted if the eye is too close to them during the process of drawing them. For the same reason, objects which are too close to the eye appear too large compared with objects at a greater distance. If an object is far away from me, when I trace it on the glass pane it will become small on the pane because I cannot place the glass pane farther away from me than I can reach with my hand. But if I place the glass very close to the object to be traced, but retain the distance of the eye, I cannot reach the pane with my hand. For that reason one has to resort to another method which will save much trouble and work and loss of time, and one can use it to render large and small objects. Anyone who wishes to draw a large object in its original size has to position the glass pane on which it is to be traced very close to the object. And the eye must be at a reasonable distance to avoid distortion. But for the hand to be close to the glass and the eye to be far away is impossible. For that reason Jacob Keser’s invention has to be used in this case. To fill this need, he has invented a particular instrument which I will now describe, because it is exact and good. First, place a point representing the eye on one end wherever you prefer [for instance, on a wall], and mark it o. Attach to this point a thin string of pure silk, as long as you require it. This point o is nothing less than the position of a real eye, as you will understand from what follows. At the other end of the string attach the scanner, which you will construct in the following manner. Cut a piece of clear boxwood having a profile in the shape of an oblong triangle, and measuring a span and a half. The two longer sides of the triangle should be equal to one-twentieth the length of the wood, but the bottom width [of the triangle] should be half as wide as the others. Then drill a straight hole lengthwise through the wood, slightly above the small side of the triangle, so that the string can be pulled to and fro through it. Now place on the ridge of this triangular board a pointer as high as the triangle, and mark its topmost point a. At its bottom, where the string emerges, is point f, and at the opposite end point d. Now you must place a small board with a small hole in it at the end above d at a height at which it can be used for scanning. This part can be moved to and fro at the appropriate height during the scanning process. Then proceed as follows. Attach the string to the instrument and pull it taut, so that points f, d, and o form a straight line. Then draw a line ao and erect a vertical line from d. Where this vertical line meets ao mark point b. This point indicates the correct level of the small hole used to scan the object pointed to by the pointer a. In other words, bd divides the triangle afo.

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Sight line

Eye

Line of the rope Here following you see a drawing of the instrument. This is the point on which to set a sight

Line of the rope

Sight line

This is the small hole according to which the eye sets its sight

The wood should have this height but it should be made as long again as is here indicated, since I was unable to put the appropriate length here on the paper

Now that you have completed the instrument and everything is in good order you can begin the task, as follows: Put the object in the desired position and place the glass pane in front of it. Attach the string to point o and connect it to the instrument, which is placed so that its pointer is near the pane but its scanning hole is closer to point o. Take the string that has been threaded through the instrument in your left hand and move it to and fro along the string at whatever distance from the pane you wish to have it. Keep it in the desired position by placing your thumb on the string at point f. Move the instrument with your left hand toward your nose so that you can look through the scanning hole with your right eye as you would sight a shotgun, using the pointer to look through the glass at the object. Now that the instrument has been positioned with your left hand and the string is taut, take a pencil or pen with your right hand and mark off the points corresponding to the indications of the pointer a. In this manner you can mark off not only the points, but also the principal lines. It is all as practical as if you were to use a piece of paper to trace a design from another sheet. All this is shown in the following diagram.

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There is yet another method of copying an object and of rendering it larger or smaller according to one’s wish, and it is more practical than using a glass pane because it is less restricted. In this method one uses a frame with a grid of strong black thread. The spaces or quadrangles should be about two fingers wide. For scanning, one must prepare a pointer whose height should be adjustable to be at eye level, which is marked o. Then place the object to be drawn a good distance away. Move it or bend it as you like, and view it from level o to ascertain that it is in the proper position, so as to please you. Then place the grid or frame between the object and the pointer. If you prefer to use fewer spaces of the grid, move it closer to the object. Check how many spaces of the grid will be utilized to accommodate the width and height of the object and then draw a grid, large or small, on a sheet of paper or a panel on which you wish to draw. Now begin to scan the object with your eye—point o —placed above the pointer, and where it points on the grid in the frame, mark it off on the grid on your sheet of paper. It will be good, and it will be correct. But if you prefer to drill a small hole into your scanner, it will serve the same purpose equally well. I have drawn this method below.

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If you need to paint a tall giant, based on a small picture, on the wall of a high tower, it would be wrong and impractical to glue together many sheets of paper, sufficient for so large a grid. Instead, cut large squares of paper, as large as the quadrangles of your grid should be. Then prepare one quadrangle after the other, as explained before, and mark each quadrangle [so you will know] how they fit together. Then stack them like a deck of cards, and when you work on the wall, you can draw one sheet at a time, and you will not have had to draw the giant in his entirety, as would otherwise be necessary. To the friendly reader. Now we have reached the end of this book. The honorable Albrecht Dürer, when he was still alive, promised in his dedication of this book to the honorable and wise Willibald Pirckheimer that in time he would, God willing, also issue books about human proportion and other matters pertaining to it. And before his death he did that, and he accomplished it with diligence. These books were then also published in Latin by his widow Agnes Dürer, so that others, not familiar with German, might gain knowledge of her husband’s art and labor. For that reason warning is here given that this book, upon pain of punishment, may not be copied in any fashion assured by his Imperial Majesty’s copyright. Let everyone pay heed. Praise to God in eternity. Amen. Printed in Nuremberg by Hieronymus Formschneyder. Anno. M. D. XXXVIII.

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Underweysung der Messung

PH-Messung: Der Leitfaden für Praktiker

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Einkaufsstress: Messung, Determinanten und Konsequenzen

Lebensmittelphysik: Physikalische Kenngrößen - Messung und Anwendung

Gesundheitsprofilierung von Lebensmittel-Markenartikeln: Messung, Determinanten und Implikationen

Der Informationsbedarf von Patienten hinsichtlich der Qualität von Krankenhäusern: Eine empirische Untersuchung zur Messung des Involvements und der Informationspräferenzen

Der Ruf von Branchen: Eine empirische Untersuchung zur Messung, Wechselwirkung und Handlungsrelevanz des Branchenrufs

Antwortreaktionszeiten in Survey-Analysen: Messung, Auswertung und Anwendungen

Kooperation von Zentralcontrolling und Bereichscontrolling: Messung ? Auswirkungen ? Determinanten

Kommunikation in Forschung und Entwicklung: Konzeption, Messung und empirische Analyse

Kooperation von Controllerbereich und externem Rechnungswesen: Messung - Wirkungen - Determinanten

Die Bedeutung der Emotionen beim Besuch von Online-Shops: Messung, Determinanten und Wirkungen

Messung und Steuerung der Markenbeziehungsqualität: Eine branchenübergreifende Studie im Konsumgütermarkt (Basler Schriften zum Marketing Band 26)

Webbasierte Datenintegration - Ansatze zur Messung und Sicherung der Informationsqualitat in heterogenen Datenbestanden unter Verwendung eines vollstandig webbasierten Werkzeuges

Практические сведения по вакуумной технике: Получение и измерение низких давлений. (Praktikum der hochvakuumtechnik: Erzeugunq und messung niedriger drucke)

Innovationsorientierung von Unternehmen: Messung, Determinanten und Erfolgswirkungen (Kundenmanagement & Electronic Commerce)

Sound-Engineering im Automobilbereich: Methoden zur Messung und Auswertung von Geräuschen und Schwingungen

Commitment in Kundenbeziehungen: Eine multipartiale Messung und Analyse von Determinanten und Erfolgswirkungen

Sound-Engineering im Automobilbereich: Methoden zur Messung und Auswertung von Gerauschen und Schwingungen

Der Sand der Seele

Der Fluch der Meerjungfrau

Der Gott der Mar'osianer

Der Schläfer der Zeiten

Der Sand der Seele

Der Stern der Gewalt

Der Fall der Muschel

Der Fluch der Opale

Der Fluch der Kosmokratin

Der Fluch der Meerjungfrau

Der Tempel der Ewigkeit

Underweysung der Messung

PH-Messung: Der Leitfaden für Praktiker

Allianzfähigkeit - Konzeption, Messung, Determinanten, Auswirkungen

Einkaufsstress: Messung, Determinanten und Konsequenzen

Lebensmittelphysik: Physikalische Kenngrößen - Messung und Anwendung

Gesundheitsprofilierung von Lebensmittel-Markenartikeln: Messung, Determinanten und Implikationen

Der Informationsbedarf von Patienten hinsichtlich der Qualität von Krankenhäusern: Eine empirische Untersuchung zur Messung des Involvements und der Informationspräferenzen

Der Ruf von Branchen: Eine empirische Untersuchung zur Messung, Wechselwirkung und Handlungsrelevanz des Branchenrufs

Antwortreaktionszeiten in Survey-Analysen: Messung, Auswertung und Anwendungen

Kooperation von Zentralcontrolling und Bereichscontrolling: Messung ? Auswirkungen ? Determinanten

Kommunikation in Forschung und Entwicklung: Konzeption, Messung und empirische Analyse

Kooperation von Controllerbereich und externem Rechnungswesen: Messung - Wirkungen - Determinanten

Die Bedeutung der Emotionen beim Besuch von Online-Shops: Messung, Determinanten und Wirkungen

Messung und Steuerung der Markenbeziehungsqualität: Eine branchenübergreifende Studie im Konsumgütermarkt (Basler Schriften zum Marketing Band 26)

Webbasierte Datenintegration - Ansatze zur Messung und Sicherung der Informationsqualitat in heterogenen Datenbestanden unter Verwendung eines vollstandig webbasierten Werkzeuges

Практические сведения по вакуумной технике: Получение и измерение низких давлений. (Praktikum der hochvakuumtechnik: Erzeugunq und messung niedriger drucke)

Innovationsorientierung von Unternehmen: Messung, Determinanten und Erfolgswirkungen (Kundenmanagement & Electronic Commerce)

Sound-Engineering im Automobilbereich: Methoden zur Messung und Auswertung von Geräuschen und Schwingungen

Commitment in Kundenbeziehungen: Eine multipartiale Messung und Analyse von Determinanten und Erfolgswirkungen

Sound-Engineering im Automobilbereich: Methoden zur Messung und Auswertung von Gerauschen und Schwingungen

Der Sand der Seele

Der Fluch der Meerjungfrau

Der Gott der Mar'osianer

Der Schläfer der Zeiten

Der Sand der Seele

Der Stern der Gewalt

Der Fall der Muschel

Der Fluch der Opale

Der Fluch der Kosmokratin

Der Fluch der Meerjungfrau

Der Tempel der Ewigkeit

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