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LECTURES ON THE
GEOMETRY OF POSITION
BY
THEODOR REYE / /
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF STRASS...
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f.F
LECTURES ON THE
GEOMETRY OF POSITION
BY
THEODOR REYE / /
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF STRASSBl KG
TRANSLATED AND EDITED BY
THOMAS
F.
HOLGATE,
M.A., Ph.D.
PROFESSOR OF APPLIED MATHEMATICS IN NORTHWESTERN UNIVERSITY
PART
I.
THE MACMILLAN COMPANY LONDON: MACMILLAN AND 1898 All rights reserved
CO.,
LIMITED
GLASGOW
:
PRINTED AT THE UNIVERSITY PRESS BY
ROBERT MACLEHOSE AND
CO.
TRANSLATOR S PREFACE. IN preparing
this
of
translation
Professor
Reye s Geometric der
object has been to place within easy reach of the English-speaking student of pure geometry an elementary and
Lage,
my
sole
of
systematic development creasing interest in
demand
this
modern
The
and methods.
ideas
in
study during recent years has seemed to
a text-book at once scientific
and
sufficiently
to give the student a fair view of the field of
comprehensive
modern pure geometry,
him to investigation. The in all these regards is my work Reye translation as an attempt to satisfy our
.and also sufficiently suggestive to incite
recognized merit of Professor only apology for offering this present needs. It
has
been
my aim
to
s
present in fair readable English the text, rather than to hold myself, at
geometric ideas contained in the .all
points, to a literal translation
;
yet
I trust that I
have not alto
Some changes gether destroyed the charm of the original writing. have been made; the articles have been numbered, the examples set at the
end of the
lectures to
which they are related and a few new
ones added, explanatory notes have been inserted where they seemed I have not necessary or helpful, and an index has been compiled.
deemed
advisable to omit from this edition any part of the original or introduction, even though, at this distance from their prefaces first
it
publication, they might not be
For the most part
I
demanded
established terminology.
A
in their entirety.
hold rigorously to wellfew instances of deviation from this
have endeavoured
to
I have preferred the terms may sheaf of rays, sheaf of planes, and bundle of rays or planes, to the more common though I think less expressive terms, flat pencil, axial pencil, and sheaf of lines or planes instead of the expression
principle, however,
be mentioned.
;
conformal representation as an equivalent for the German confor?ne l
TRANSLATOR S PREFACE.
vi
Abbildnngj
I
have ventured
*
conformal depiction.
The term
ideal
r
has elsewhere been applied to infinitely distant points and lines f with this I have associated the word actual to apply to points and -
lines of the finite region. I
desire
Professor
to
acknowledge my indebtedness to my colleague S. White for valuable assistance; my thanks also-
Henry
and the gratitude of all are due to Dr. M. C. generosity
What
is
made
its
who may
profit by the use of this translation Bragdon of Evanston whose interest and
publication possible.
commonly known
as
Modern
Synthetic Geometry has-
been developed for the most part during the present century. It differs from the geometry of earlier times, not so much by the subjects dealt with and the theorems propounded, as by the processes which are employed and the generality of the results which are attained.
Geometry was
to the ancients a subject of entrancing interest. progress is prominently connected with the names of Thales of Miletus (640-546), Pythagoras (569-500), Plato (429-348), who Its
cultivated
geometry as fundamental
Menaechmus
(375-325), the
first
to
the
study of philosophy,
to discuss the conic sections, Euclid
of Alexandria (330-275), Archimedes (287-212), and Apollonius of Perga (260-200) ; these among others before the Christian era.
Of
numerous writings of Euclid, the Elements? in which was and systematized much of the geometrical knowledge of that time, has remained for two thousand years a marvellous monument to his skill. Whatever may be its defects, and these have been the of much discussion, it subject certainly possesses some excellent to elegance of demon features it accustoms the mind to rigor, and to the methodical stration, arrangement of ideas in these 2 His Porisms, which is of our admiration." it worthy regards are to have contained many of have been said lost, unfortunately the principles that have formed the basis of modern geometry. Ancient geometry reached its highest perfection under Archimedes the
collected
"
;
;
and Apollonius, the former of whom devoted much study problems by means of geometry, and the latter carried 1
For a convenient summary and characterization of Euclid s Elements, see Geometry in the Encyclopedia Britannica.
Professor Henrici s article on 2
to physical his investi-
Poncelet, Proprieties Projectiles, etc., p.
15.
TRANSLATOR S PREFACE. upon the conic
gations
as to leave few
of their
He produced a systematic properties undiscovered. conic sections containing his own discoveries, and
important
on
treatise
sections so far
vii
1 previous knowledge of these curves. and commentator of the early centuries of great geometer In his Mathematicae the Christian era was Pappus of Alexandria.
including also
all
The
toward the end of the fourth century, he collated
Collectiones, written
the
scattered
multitude
earlier celebrated geometers and a theorems from many sources, to which he of original work as to place him among the most
works of the
of curious
added so much
This work is the chief source of geometers. ancient geometry. It comments so fully upon
illustrious of ancient
information
Euclid
s
on
book of porisms
restore the latter, notably
that frequent efforts have been by Chasles in 1860.
made
to
The work of the ancient geometers was fragmentary. Truly remarkable discoveries were made, but general principles were not brought into prominence theorems were announced disconnectedly ;
as though they had been received by their authors ready made ; the method of their discovery was rarely, if ever, indicated ; the demons
were given in the most polished and systematic form, but the relations existing among different theorems were not shown, and no suggestions were offered for further investigation special trations
;
cases of general theorems were as a rule treated as
though they
were separate and independent theorems. But, scattered here and there, throughout the great volume of geometrical knowledge accumulated by these early geometers, is to be found the material upon which the beautiful and symmetrical structure of
modern geometry has been founded.
For example,
the property of perspective triangles of which use is made in the geometrical definition of harmonic points, though usually credited to Desargues, was in fact announced by Euclid. 2 Harmonic division itself
was known
to Apollonius,
ratio of four collinear points
strated by Pappus, 3
is
and the
fact that the
anharmonic
unaltered by projection was
and was probably, known much
demon The
earlier.
theorem upon which Carnot based the theory of transversals was discovered and published by Menelaus early in the second century. 1 An edition of Apollonius Conic Sections, with notes, etc., by T. L. Heath, M.A., has recently been published by Macmillan & Co., London. 2 Pappus, Mathematical Collectiones, preface to book VI I. 3 Mathematicae Collectiones, VI I., 129.
TRANSLATOR S PREFACE.
viii
As has already been by generality both in
its
modern geometry is characterized in its results. and The founding of processes suggested,
modern pure geometry is usually accredited to Monge (1746-1818), whose lectures in the Polytechnic School at Paris were published under the title of Geometric Descriptive. These lectures, by utilizing the theory of transversals and the principle of parallel projection, called attention to the advantages to be gained through the applica and served to revive the interest in
tion of geometrical methods,
pure geometry, which had been dormant for so
But the
.generalizing processes
were begun 1662),
much
earlier
a contemporary
distant points and lines, with and announced the doctrine of continuity.
fragments
of
generality,
and
his
far as
years.
than the time of Monge. Desargues (1593of Descartes, introduced the notion of
infinitely
(1623-1662) too, so
many
which characterize modern geometry
it is
its
far
reaching results, of Pascal
The methods
possible to judge from the few remaining work, partook of the broadest
mathematical
it is fair
to
assume that had not the work of these two
great geometers been almost entirely lost, and had not their ideas been wholly pushed aside through the overwhelming influence of Descartes discoveries, many of the geometrical theories and results
of the present century would have been developed long ago, and the so-called modem geometry would have been of much earlier date.
As it was, however, pure geometry was but little cultivated for over a hundred years before the time of Monge. Geometrical know ledge was truly increased during this period, especially by Newton (1642-1727), Maclaurin (1698-1746), Robert Simson (1687-1768), and Matthew Stewart (1717-1785), but their methods could scarcely be said to partake of the spirit of modern geometry, and differed but little, if at all, from the methods of the ancient geometers. The illustrious names in connection with the development of modern pure geometery are Poncelet (1788-1867), Steiner (17961863), Von Staudt (1798-1867), and Chasles (1793-1880); and if it were permissible to add the names of living men I should mention
Cremona and Reye. Poncelet
s
great work,
Traite des proprietes projectives des figures r
and at once clearly justifies any claim appeared be in his behalf as the leader in the so-called set may up modern methods. In this work the principle of continuity, the
etc.,
in
1822,
that
principle of reciprocity or duality, are the chief factors.
and the method of projection
TRANSLATOR S PREFACE.
ix
There has been much discussion from time
to time
upon the
question of priority in the establishment of the principle of duality. Poncelet used the method of polar reciprocity with respect to a
and thus derived the dual of any geometrical figure, but claimed for Gergonne that it was he who first established The name duality is clearly duality as an independent principle. conic, it
is
*
due
to
The
1
Gergonne. principle of continuity,
which was
first
assumed by Kepler and
by Desargues, demands
of the geometer as well as the analyst the consideration of imaginary quantities. Monge discovered that later
the results obtained from a geometrical construction would not be if in a different phase of the construction certain of the
invalidated
and lines disappeared. Poncelet devoted much attention to imaginary solutions of geometrical problems, but it remained for Von Staudt to build up and to bring to a fair degree of perfection points
a general theory of geometrical imaginaries. A conception of the geometer s notion of imaginary quantities can probably be best obtained from the following quotation from Professor H. "All
in
J.
S.
Smith:
2
attempts to construct imaginaries have
pure geometry
;
but,
by asserting once
been wholly abandoned
for
all
the principle of
continuity as universally applicable to all the properties of figured space, geometers have succeeded, if not in explaining the nature
of imaginaries, yet, at least, in deriving from them great advantages. They consider it a consequence of the law of continuity that if we once demonstrate a property for any figure in any one of its general states,
and
if
we then suppose
the figure to change
its
of course to the conditions in accordance with which
form, subject it
was
first
we have proved, though it may become un meaning, can never become untrue, even if every point and every line by means of which it was originally proved should disappear. The line of demarcation which was visible as early as the time of Archimedes and Apollonius between the geometers whose theories rest upon metric properties and those whose basal notions are traced, the property
"
purely positional was very prominent during the foundation period development of modern pure geometry. Steiner and Chasles
in the
based
their investigations
1
2
A tinales
de
upon metric
properties, defining the pro-
Mattematiques, T. XVI.,
Collected Papers, Vol.
I.
p.
4.
1826.
TRANSLATOR S PREFACE.
X
by means of the anharmonic ratio; Von Staudt, on and after him Reye, define this relation by reference to harmonic division, and this in turn is defined purely geometrically. Upon such a definition of projectivity they have been able to perfect .a complete theory without any reference to metric properties what
jective relation
the other hand,
soever. Cremona avoids metric properties in his foundations by defining two projective forms as the first and last of a series of
forms in perspective. In all the recent development of synthetic geometry the effect of contact with analysis is clearly seen. Through its influence the foundations upon which the science rests have been carefully ex amined, while characteristic methods of investigation have been .acquired.
The tendency toward
.attributed
largely to the influence
of analysis,
some progress had been made analytical methods had attained such other hand, geometry has done much that
interest in analysis, so that
geometry
nor
existence, or
pure
hope
may
it
to attain
its
this
though
it
is
true
direction before the
universal sway. But, on the to enliven and heighten the fairly be said that neither pure
can
analysis
in
likewise be
may
generalization
boast
any longer
an isolated
highest development independently
of the other.
THOMAS
F.
HOLGATE.
EVANSTON, ILLINOIS, December 1897.
A NOTE FROM THE AUTHOR. DEAR
It
SIR,
is
with great
pleasure and satisfaction that I my Geometric der Lage, which
English translation of
greet your henceforth will take translations.
many
I
friends
countries.
I
trust
and
its
place along with the French
that
it
may
investigators
am, yours
help to in
the
win
for
broad
and
Italian
pure Geometry English-speaking
faithfully,
TH. REYE. STRASSBURG, September
1896.
THE AUTHOR S PREFACE TO THE FIRST EDITION OF PART
I.
THE
Lectures upon the Geometry of Position, which the public, have been written at intervals during the
I
now
offer to
two years. I have been induced to publish them by a need which has been felt for a long time in the technical schools of this country, and perhaps The important graphical methods with which in wider circles. Professor
Culmann has enriched
last
the science of engineering,
and
which are published in his work, Die graphische Statik, are based for the most part upon modern geometry, and a knowledge of this
become indispensable
to students of engineer In the present work I attempt to supply the want of a text-book which offers to the student the necessary material
subject has therefore
ing in our institutions. in concise form,
and which
will
be of assistance to
me
in
my
oral
instruction. I
was obliged, as a matter of course, to make use of the termin
follow the ology adopted by Culmann, and, to a certain extent, to as this, title subject matter of that complete work bearing the same Staudt. Von The Geometry of Position, by Professor namely,
The new terms which Von
Staudt added to the older ones of Steiner are so happily chosen that I have preferred a different one in but a instance, the term (Punktreihe\ first used range of points "
"
single
by Paulus (and Gopel) instead of "line in which Von Staudt establishes
The way with
all
.afford
other writers
form"
upon Modern Geometry appears
advantages so important that, laying aside
erations,
I
(gerades Gebilde).
this science in contrast
all
to
me
to
other consid
Permit me, in a few should prefer it to every other. this for reasons preference. my
words, to assign
To
the engineer as well as to the mechanic
and
architect, the ability
to form beforehand a mental picture of his structure as
it
will
appear
THE AUTHOR
xii
in space is
bridge
is
S PREFACE.
of great service in designing
to be built across a stream.
it.
Suppose, for example, a
From among
the different
possible modes of construction, that one must be chosen at the outset which is best adapted to the given conditions. To this end
the engineer compares the long iron girders with the boldly swung arch or the freely hanging suspension bridge, and endeavours to
how the pressure would be exerted at this point and at that and how distributed among the different members of the huge structure. Again and again he examines and compares, goes more and more deeply into all the details, until the whole structure stands complete in clear outline before his mind s eye. And now the second The project is transferred to paper; part of his creative work begins. all details as to form and strength are completely determined. But still the engineer, and everybody else who wishes ta become familiar with his ideas, must continually exert his power of imagination in conceive point,
order actually to see the object intended to be represented by the drawing which is not at all intelligible to the uninitiated.
lines of a
So also the mathematician and
in
fact
any one who concerns
himself with the natural sciences must, like the technologist, bring At one time he tries to the imagination very frequently into play.
understand a complicated piece of apparatus from an insufficient sketch at another, from a scanty description, to make intelligible ;
remotely connected processes of nature or complicated motions. One principal object of geometrical study appears to me to be the
and the development of the power of imagination in the and I believe that this object is best attained in the way in which Von Staudt proceeds. That is to say, Von Staudt excludes all calculations whether more or less complicated which make no demands upon the imagination, and to whose comprehension there is requisite only a certain mechanical skill having little to do with geometry in itself; and instead, arrives at the knowledge of the geometric truths upon which he bases the Geometry of Position by exercise student,
direct
visualization.
It
cannot be denied that
this
method,
like
is more, Von every other, presents peculiar difficulties; Staudt s own work, evidently not written for a beginner, embodies peculiarities which are praiseworthy enough in themselves, but which
and, what
its
essentially increase
the difficulties of the study.
It
is
especially
expression, and a very condensed, almost form of statement laconic, nothing is said except what is absolutely of explanation given, and it is left toa is there word necessary, rarely
marked by a scantiness of
;
THE AUTHOR S PREFACE.
xiii
the student to form for himself suitable examples illustrative of the The theorems, which are enunciated in their most general form. material, however, is very clearly and systematically arranged; for example, the subjects of projectivity, of the collinear and the recipro cal relation, and of forms in involution, are completely treated before the theories of conic sections and surfaces of the second order are
introduced, and Von Staudt thus gains the advantage of being able to prove the properties of forms of the second order all at once ; but, on the other hand, the presentation is so abstract that ordinarily the
These energies of a beginner are quickly exhausted by his study. features, which, unfortunately, appear to have stood in the way of the well-merited circulation and the general recognition of Von Staudt s work, stamp it as a treatise on Modern Geometry of superior merit, to
which we may very appropriately
to Euclid.
In
refer, as
did the ancient geometers
lectures written for beginners, however, I
my
must
avoid such peculiarities in order not to become unintelligible. There is one other difficulty inherent in the course itself which
have purposely not avoided, since
it
must sooner or
later
I
be overcome
by everybody who desires to comprehend the properties of threedimensional figures. I refer to the difficulty already mentioned of getting a mental picture of such figures in space, a difficulty with which the beginner has to struggle in the study of descriptive geometry and analytical geometry of space, the surmounting of which, as I have already remarked, I hold to be one of the principal objects In order to make the accomplishment of of geometrical instruction.
end
this
my
easier for the student, I
lectures.
fact,
Von
Staudt did not
we should not be
far
have added plates of diagrams to make use of such expedients; in
from the truth
similar to those expressed at
one time by
in ascribing to "
Steiner,
him views
that stereometric
ideas can be correctly comprehended only when they are contem plated purely by the inner power of imagination, without any means
of illustration whatever." By disdaining to make use of these instruments of illustration, which so far as planimetric ideas are con cerned, are not at all likely to lead to an incorrect conception, I should unnecessarily have increased the difficulty, on the part of the student, of
comprehending
my
lectures.
Since the method introduced by Von Staudt excludes numerical computations, and investigates the metric properties of geometrical figures apart from the general theory, it presents still another ad
vantage to which
I
should like to
call
especial attention.
That
THE AUTHOR S PREFACE.
xiv it
is,
account most beautifully, in
turns to
all
its
clearness
and
compass, the important and fruitful principle of duality or reciprocity, by which the whole Geometry of Position is controlled. to its full
No method making merit,"and
is
use of the idea of measurement can boast of this
for the simple reason that in metric
geometry this principle must be admitted that geometry so stimulating to the beginner, and which
not in general applicable.
offers
nothing which
is
But
it
ciprocity,
him on and the
The
that this principle stands
so spurs fact
to
independent research as the principle of re he is made acquainted with it the better.
earlier
geometry of three dimensions, was
for
out so clearly, particularly in me a determinative reason for
not separating the stereometric discussions from the planimetric. Metric relations, I must add, especially those of the conic sections,
have by no means been neglected in my lectures on the contrary, I have throughout developed these relations to a greater extent than did either Steiner or Von Staudt, wherever they could naturally present themselves as special cases of general theorems. I proceed to the study of the conic sections and other forms of the second order treated in this first part of my lectures, by a route different
from that of Steiner or
Von
Staudt,
the latter of
whom
based the theory of these forms upon the doctrine of collineation and reciprocity. By introducing the forms of the second order
from a study of projective one-dimensional primitive forms, hope to have made the comprehension of the projective relation
directly I
same
time, I secure the advantage of being able to prepare beginner by degrees for the more difficult study of collineation and reciprocity. In his highly suggestive pioneer work,
easier; at the
the
Systematische the
Entwickelung, etc., Steiner has furnished us with I have preferred to follow in my lectures from the
model which
fifth to the tenth. For reasons already referred to, however, I was obliged to refrain from defining conic sections, as did Steiner, by means of circles.
THE AUTHOR. ZURICH, March
S//i,
1
866.
THE AUTHOR S PREFACE TO THE THIRD EDITION OF PART THIS new edition exceeds the number of pages. The most
first
in extent
significant
I.
by about two-thirds of its made in the second
changes
edition consisted in the addition of a collection of two
hundred and
A
part of this collection was twenty-three problems and theorems. originally to be found in the appendix to the second volume, but this
has been considerably enlarged by the addition of new problems and The first eleven sections of this collection, with the
useful theorems.
exception of the two upon the principle of reciprocal radii and the ruled surface of the third order, correspond to the lectures with
same headings, and problems and theorems which are com paratively easy to be proved have been selected mainly with a view to the
I urgently advise every beginner furnishing exercise for the student. to actually solve the problems of construction graphically, since the
comprehension of the Geometry of Position is made very much easier by the free use of pencil and paper. The last four sections of the problems and theorems contain new investigations which were not found in the first edition, and which in
more than one
essential
feature have been carried out
by means
of synthetic geometry for the first time so far as I know in this book. In order that the investigations upon self-polar quadrangles and
and upon linear nets and webs of conic become too voluminous, I have chosen for them a form of statement as brief as possible. By this means and by the self-polar
quadrilaterals
sections might not
introduction of a single elementary notion, I have been able, within the narrow compass of twenty-one pages, to present the important theories of sheaves, ranges, nets, and webs of conic sections in an entirely new connection. By means of Stephen Smith s theorem, the synthetic proof of which
I
acquired only after
many
fruitless
THE AUTHOR S PREFACE.
xvi
and by the principle of developed in a manner remarkable attempts,
reciprocity,
these
theories are
and clearness. I have replaced the proof of the fundamental theorem of the Geometry of Position as given by Von Staudt by one free from objections, making use of the remarks in that connection of F. Klein and Darboux (Math. Annalen, Vol. XVIL). It was expressly assumed in the second edition by the definition of correlation," that in two for simplicity
"
projective primitive forms a continuous succession .of elements in the one form corresponds to a continuous succession of elements in
the
other.
Staudt It is
s
This ,has
now been proved upon
the basis of
Von
definition of projectivity.
due like
to the kindly co-operation of the its
Italian
and French
new
publishers that this translations, is supplied with
book, engravings of the diagrams inserted in the
text.
THE AUTHOR. STRASSBURG IN ALSACE, September 8M, 1886.
CONTENTS. INTRODUCTION,
LECTURE THE METHODS
OF PROJECTION AND SECTION
FORMS OF MODERN GEOMETRY,
^
LECTURE INFINITELY DISTANT
-
THE Six PRIMITIVE -
-
"-
*
-
-
9
II.
ELEMENTSCORRELATION OF THE PRIMITIVE
FORMS TO ONE ANOTHER,
-
--
LECTURE PRINCIPLE OF
I.
RECIPROCITY OR
--
-
-
-
17
III.
DUALITY
SIMPLE AND COMPLETE -
W-POINTS, W-SlDES, W-EDGES, ETC.,
LECTURE
IV.
CORRELATION OF COMPLETE W-POINTS, W-SIDES, AND ANOTHER HARMONIC FORMS EXAMPLES, -
LECTURE
n-
EDGES TO ONE
-34
V.
PROJECTIVE PROPERTIES OF ONE-DIMENSIONAL PRIMITIVE FORMS
EXAMPLES,
-
-
-
-
....-
.
-_.-..-
52
CONTENTS.
xviii
LECTURE
VI. PAGE
CURVES, SHEAVES, AND CONES OF THE SECOND ORDER
LECTURE DEDUCTIONS FROM PASCAL
S
-
69
VII.
AND BRIANCHON
LECTURE
EXAMPLES,
S
THEOREMS
EXAMPLES,
8z
VIII.
POLE AND POLAR WITH RESPECT TO CURVES OF THE SECOND
ORDER-
EXAMPLES,
97
LECTURE
IX.
DIAMETERS AND AXES OF CURVES OF THE SECOND ORDER EQUATIONS OF THESE CURVES
ALGEBRAIC 112
EXAMPLES,
LECTURE
X.
REGULI AND RULED SURFACES OF THE SECOND ORDER
EXAMPLES,
-
126
-
134
LECTURE XL PROJECTIVE PROPERTIES OF ELEMENTARY FORMS
LECTURE INVOLUTION
-
XII.
EXAMPLES,
149
LECTURE METRIC RELATIONS OK INVOLUTION
ORDER
EXAMPLES,
XIII.
Foci OF CURVES OF THE SECOND
EXAMPLES,
164
LECTURE PROBLEMS OF THE SECOND ORDER
XIV.
IMAGINARY ELEMENTS
EXAMPLES,
181
CONTENTS,
LECTURE
xix
XV. PACK
SYMMETRY FOCAL AXES AND OF THE SECOND ORDER EXAMPLES, A OF CONE PLANES CYCLIC
PRINCIPAL
AXKS AND PLANES
OF
196
APPENDIX. PRINCIPLE OF RECIPROCAL RADII
RULED SURFACES OF THE THIRD
QUADRANGLES AND QUADRILATERALS WHICH ARE SELFPOLAR WITH RESPECT TO CONIC SECTIONS NETS AND WEBS OF
ORDER
CONIC SECTIONS,
INDEX,
-----------
-
207
-
241
INTRODUCTION. Most of my hearers will have heard till now name of the Geometry of Position; for,
1.
than the the
knowledge of
this
significant
creation
of
scarcely
more
unfortunately, recent times, dis
tinguished alike by abundance of contents, clearness of form, and has been diffused but very little; simplicity of development, and notwithstanding the fact that modern geometry must be
accounted among the most stimulating branches of mathematical science and admits of many beautiful applications to the technical and natural sciences, it has not as yet found its way generally into the schools.
Perhaps, therefore, it will not be amiss if I pre word upon the place which the Geometry my of Position occupies among other branches of geometry, and if lectures with a
face
afterwards I mention still
some theorems and problems which
will serve
further to characterize this science for you.
The pure Geometry
of Position is mainly distinguished from of ancient times and from analytical geometry, in geometry that it makes no use of the idea of measurement in contrast 2.
the
;
with this feature the ancient geometry may be called the geometry of measurement, or metric geometry. In the pure Geometry of Position nothing whatever
is
said about the bisection of segments
of straight lines, about right angles and perpendiculars, about ratios and proportions, about the computation of areas, and just as little
about trigonometric ratios and the algebraic equations of curved these subjects of the older geometry assume measure
lines, since all
ment. I
shall
In these lectures, however, at the end of each main division, make applications of the Geometry of Position to metric
geometry, in which I shall assume a knowledge of planimetry as well We as, in a few instances, a knowledge of the sine of an angle.
A
v
2
^EOMf.TRY OF POSITION.
shall be concerned as little with isosceles and equilateral triangles as with right-angled triangles ; the rectangle, the regular polygon, and the circle are also excluded from our investigations, except in the case of these applications to metric geometry. shall
We
treat of the centre, the axes,
and the
foci of so-called curves of the
second order, or conic sections, only as incidental to the general theory
on the other hand,
but,
;
become acquainted with many
shall
properties of these curves more general and more important than those to which most text-books upon analytical geometry are
We
mark out a new way of approach Geometry of Position we dispense with the help of the circular cone by means of which the ancient geometers defined these curves, and also of the algebraic restricted.
shall
be obliged
to
to the conic sections themselves, since in the
equation through which they are viewed by disciples of Descartes. After what has been said the fact that
it is
scarcely necessary for
no computations
appear in
will
my
me
once in a while in the applications to metric geometry
employ the 3.
Of
mention
to
lectures
;
only
shall
we
sign of equality.
the
knowledge of geometry acquired
schools, I shall therefore
make
very
little
in
On
use.
the
elementary
the other hand,
producing mental images of geometric forms without pictorial representations would be of great service to you, inasmuch as it will not be practicable for me to illustrate every theorem by
a certain
skill in
diagrams, I
especially
shall often
if
the
be compelled
theorem to
refers
to
a
form
in
space
;
make demands upon your imagina
tion.
Since the imagination is brought much into play in descriptive geometry, a knowledge of this latter science would likewise stand
you
in
good stead the converse is equally true, that the Geometry makes an excellent preliminary study for descriptive ;
of Position
And in general, I may say that of all branches of the descriptive is most helpful in facilitating the study geometry in the first place, because it is very of the Geometry of Position in the second place, because in and to the related latter; closely
geometry.
descriptive geometry relations of magnitude come less under con sideration than do the positions of forms relatively to one another or to the plane of projection; this relativity, to be sure, often
being defined by the use of circles and right angles.
Above
all,
however, you
will
find
that
projection plays an important part in the
perspective
or
Geometry of
central
Position,
INTRODUCTION. and
many forms
that
3
of expression used in the latter subject are
derived from the former.
Pure geometry stands in a certain antithetical relation to analytical geometry on account of its method, which, as you know from the geometry of ancient times, is synthetic. In our study we 4.
shall start out with a small
number of
primitive forms
;
the simple
which may be established among these will bring us to the so-called forms of the second order, among which are found the
relations
conic sections, and will at the same time permit the principal shall then properties of these forms to be easily recognized. be able to proceed in the same way from the forms of the second
We
order to
other
still
must avoid
new
During our investigations we
forms.
clearly
processes of analysis, that powerful instrument of mathematics, since we make no use of measurement, and
modern
all
be able to compute with forms in space, we should first be obliged to express them in numbers, i.e. measure them. On account of its methods pure geometry as distinguished from
in order to
is often designated by the name synthetic geometry. Since in the pure Geometry of Position metric relations are
analytical 5.
theorems and problems are very general and For comprehensive. example, the most important of the properties of conic sections which are proved in text-books on analytical geometry are merely special cases of theorems with which we shall not considered,
its
become acquainted
later.
A
few
illustrations
characterize the material with which
exactly to
more
serve
will
we
shall
be con
cerned in these lectures. In designing architectural structures, and in drawing generally, not infrequent that a solution is required of the problem To draw a third straight line through the inaccessible point of inter 6.
"
it is
:
section of two Metric geometry fur (converging) straight lines." nishes us with any desired number of points of such a third line
by the
aid, for instance,
are formed in
one
:
Choose some
a and
transversals.
diagonals
in
transversals
intersection
of the property that proportional segments
parallel lines
by any three
transversals
which meet
The Geometry
point.
as follows lines
upon
b
(Fig.
Then
of Position affords a simpler solution, outside the two given straight point
P
and pass through
i)
ascertain
the
point
of
it
any
number
intersection
of
of the
quadrangles formed by two of these the lines a and b\ all these points of
each of the taken with lie
upon one
straight
line
which passes through the
GEOMETRY OF
4
POSITION.
intersection of a and b.* The proof follows very simply from the important harmonic properties of a quadrangle, which may be stated in the following
form
:
FIG.
i.
If we choose three points A, B, C, upon a straight line (Fig. 2), and construct any quadrangle such that two opposite sides pass through A, one diagonal through J3, *and the other two opposite
C, then the second diagonal meets the straight line The points A, B, a perfectly definite fourth point D.
sides through
ABC *I
in
strongly
theorem
recommend
for himself,
the beginner to draw the figure illustrating this according to the statement of the text, without first having
the one drawn by me, and theorems which are not so simple.
seen
especially
A
to
do so
for
the
subsequent
diagram built up by degrees is far easier to be comprehended, and illustrates most of the theorems to be represented far better than does one with all its auxiliary lines ready drawn.
INTRODUCTION.
D
C,
5
are called four harmonic points, and by the points
B
monically separated from
D
A
is
and
said to
be har
C.
By constructing different quadrangles satisfying the stated conditions, you can easily obtain a confirmation of the fact that all second diagonals do pass through this fixed fourth point D.
The problem
above can be "made use of in surveying when a straight line beyond an obstacle, for extend required a inasmuch as it affords a means of evading forest, example, beyond stated
to
is
it
the obstacle. 7.
Of theorems
following
to
relating
triangles
FIG.
If
I
shall
mention only the
:
two triangles
ABC
and
3.
A^C l
l
l
are so situated (Fig. 3) that
the straight lines joining similarly named vertices, viz. AA^ BB^, CCl} intersect in one and the same point S, then the pairs of similarly
named
sides
AB
intersect in three points
u
;
and A^JB lt BC and B^C^ CA and C^A lt B.2 which lie upon one straight line 2
C A 2
,
,
,
and conversely.
The diagram
illustrating
this
theorem
is
worthy of notice as
representing a class of remarkable configurations characterized by a certain regularity of form. It consists of ten points and ten three of the ten points lie upon each of the straight and three of the ten lines pass through each of the points. 8. Another series of theorems is connected with curves of the second order or conic sections. You know from analytical geometry,
straight lines lines,
;
GEOMETRY OF
6
and
will
be able
POSITION.
prove synthetically, that a curve of the completely determined by five points or five But you also know the difficulty which is met with in the
second order
later to
is
tangents. actual computation
and construction of a conic section determined of Position establishes two very im portant theorems concerning curves of the second order, which render it possible for us to construct with ease from five given in
this
way.
The Geometry
points or tangents of such a curve any required number of new Those of points or tangents, and so readily to draw the curve itself.
you who are already acquainted with these two theorems ber
how much
is
demanded
will
remem
for their
preliminary knowledge proof The first of these, originally established by Pascal, states that the three pairs of opposite sides of any hexagon inscribed in a curve of the second order intersect in three points
by analytical methods.
which was
lie
first
upon one straight line according to the second, which enunciated by Brianchon, the three principal diagonals :
the straight lines joining pairs of opposite vertices) of any (i.e. circumscribed hexagon pass through one and the same point. Both theorems may easily be verified in the case of the circle. You will notice that in these theorems nothing
is
said concerning the size of
the conic section, or concerning its centre or its axes or its foci. But just on that account the theorems are of the greatest generality and significance, so that the
whole theory of conic sections can be based
In particular, the important problem of drawing a tangent at a given point admits of solution by means of Pascal s theorem, even when the conic section is given by only five of its
upon them.
points, without supposing the
whole curve
to
The problem of constructing tangents to a order may be solved in many cases by the aid 9.
be drawn. curve of the second of a theorem which
expresses one of the most important properties of these curves, but which is frequently not to be found in text-books on analytical its analytical proof is quite complicated and is scarcely capable of setting forth this property in its true light. (Fig. 4) which lies in the plane of Namely, if through a point
geometry, since
A
a curve of the second order but not on the curve, secants be
drawn
to the curve, any two such secants determine four points, as K, Z, M, N, upon the curve. Each pair of straight lines, other than the secants, joining four such points
and
NK,
line a,
or
which
KM and is
LN,
two and two,
for
example,
LM
intersect in a point of a fixed straight
called the polar of the given point A.
If the point
INTRODUCTION.
7
A lies outside the curve, its polar a intersects the curve in the points to the of contact of the two tangents which can be drawn from curve ; if lies inside the curve, the latter is not cut by the polar.
A
A
You can
apply
this
theorem
in
drawing
tangents
to
a
conic
section from a given point with the use of the ruler only.
FIG.
4.
secant passing through A, there are four points worthy with of notice; first, itself; next, the first point of intersection the with C intersection of the then follows the curve ; polar a point curve. the with intersection the second of and of ; finally, point
Upon any
B
A
D
A
These four points A, B,
C, D are
thus contains every point which by two points of the curve.
and the polar a harmonically separated from A
harmonic is
points,
The important theorems relating to centres and conjugate dia meters of conic sections are merely special cases of the theorems These latter may easily be extended to surfaces of just mentioned. nd order, since such surfaces are in general intersected by planes in curves of the second order. From these few examples, which I might multiply indefinitely, you will doubtless have observed how different from the theorems
the seer
treated in analytical geometry, for instance,
are those with which
GEOMETRY OF
8 the
POSITION.
Geometry of Position is concerned, but certainly they are not I would remind you still further in this connection
less important.
that analytical geometry seeks to determine the positions of tangents to a conic section, especially by means of the angles which they
form with the focal
rays,
or by
means of the
intercepts
which they
determine upon the axes, thus referring the whole matter to metric Of course, reference is made here only of the elements properties. of analytical geometry to which most text-books on the subject are confined, and not of the exceedingly fruitful modern methods, for whose existence we are indebted above all to the ingenious Pliicker.
A
LECTURE
I.
THE METHODS OF PROJECTION AND SECTION. THE PRIMITIVE FORMS OF MODERN GEOMETRY. As
SIX
known, the great number of concepts which are and in analytical geometry are based for the most part upon measurement accord It ingly, they can find no place in the pure Geometry of Position. ought not, therefore, to be surprising that modern geometry has 10.
is
advanced
well
in the ancient geometry, in trigonometry,
;
purposes a considerable number of characteristic will be made acquainted in this and concepts. in the following lecture, and thereafter they must constantly be set forth
for its
With these you
employed.
The
,
and the plane are the simple elements of modern geometry.* As a rule, we shall designate points by capital letters, lines by small italics, and planes by Greek 11.
letters.
point,
the straight line,
Straight lines (or rays, as
and planes
will
the opposite
is
they will frequently be called) in extent unless
always be considered as unlimited expressly stated.
We
are able to
combine these
elements into systems by looking upon one of them as the base or support (Trager) of an infinite number of elements of another
By this means we arrive at the so-called modern geometry. Before explaining these,
sort.
of
primitive
forms
by way of introduction, give a brief account of the important methods of Projection and Section, of which frequent use is made. I
shall,
If we look at an object, say a tree, every (visible) point of sends to the eye a ray which is called the projector, or the of this point. The projector of the whole tree projecting ray 12.
it
compounded out
is
*
It is
line,
or
of
many
rays,
each of which
worthy of notice that no attempt is made to define a A knowledge of these as fundamental ideas plane.
projects point, is
one
straight
assumed.
H.
GEOMETRY OF
10 or
more points
to the eye.
If a
POSITION.
number
of points
lie
in a straight
not passing through the eye, all their projecting rays lie in that plane which can be passed through the eye and this straight line ; every such straight line is projected to the eye by a plane
line
which
is
called the
projector,
Similarly, a curve
line.
is
or the
projecting plane of this by a conical surface.
in general projected
We
can now intercept, or intersect, the projector of the tree by a plane, each projecting ray being cut in a point and each pro jecting plane in a straight line. By this means we obtain in the plane, as the
*
section or trace of this projector, a perspective d of the picture, tree, and this projection evidently projection throws the same projector into the eye as the tree itself, and is therefore quite competent to convey a notion of the latter to us.
Ordinary photographs of three-dimensional objects are essentially such perspective, plane pictures of the objects.
Upon
this
kind of projection, which is known by the name of is based the theory of perspective and all
central projection,
;
other varieties of projection which are in use in descriptive geometry may be looked upon as special cases of this one. In orthogonal projection, for example, in order that the projecting rays
we need only to imagine the eye removed distance. The shadows which objects throw upon
parallel
to
an
may be .
infinite
planes,
when
they are illuminated from a finite or infinitely distant point, are nothing else than projections of these objects in which the illumi nating point takes the place of the eye.
A
simple example may show how we are able to discover, same time can prove, important theorems through mere visualization, with the help of these methods of projection and section. 13.
and
at the
A
system of parallel lines is projected from the eye by planes, all of which intersect in one and the same straight line, namely, in
that parallel which passes through the eye; these projecting planes are intersected by an arbitrary picture plane in straight lines, all passing through one point, namely, through the trace of the line in
which the projecting
perspective
view
of ,a
planes
tree
or
intersect.
other
Consequently, in the
object
the
projections
of
parallel edges converge toward one point, their so-called vanishing point, and only in one particular case, which you will at once We have thus incident recall, are these projections also parallel. ally established
and proved a well-known fundamental theorem of
central projection.
,
THE METHODS OF PROJECTION AND SECTION.
IX
14. Leaving aside all optical references, let us now further employ the expressions just used, viz. to project, to projector, ray, where instead of the we shall an choose intersect, etc., eye arbitrary
and instead of the definite object or tree, an arbitrary of points and straight lines in space. This system ft is system from S a of and projected by system rays planes, namely, each a each and not point by ray ray, passing through S, by a plane. The point S is regarded as the base of all these rays and planes S,
point
12
which together form the projector of the system 0. If we choose space an arbitrary system 2 of planes and straight lines, then
in
any new plane
and
lines
and each
make up
the
We
15.
would
The
plane
e
appears in this case
all
can also project from and make sections by straight lying outside a
determines a plane, or
with
g,
and
similarly,
sected
this
these straight lines and points which together section trace ) of the system -. (the
of
Thus every point
lines.
in a system of straight each plane in a straight line
intersect
straight line in a point.
base
as the
e
points, namely, in general,
by
is
which
every plane
straight line g,
projected does not
The
this straight line in a point.
from
together
g by a contain g is
plane
;
inter
straight line appears,
base of planes which intersect in it; in the second case, as the base of points which lie upon it. 1 6. Through such considerations as these we obtain the following
in the first case, as the
so-called
primitive
which
forms,
occupy an
important
place
in
modern geometry.
The
totality of points lying
of points individual
(Punktreihe) points of the
upon one a
or
line
straight
straight line
form line
is
called a range
(gerades Gebilde) ; the the elements of the
are
We consider these points to be rigidly connected range of points. with one another, so that their relative positions remain unaltered if
A is
the straight line, their base, be moved out of its original position. portion of a range of points bounded by two points of the range called a segment.
The
totality of rays passing
and the same plane we
through one point and lying in one
shall call a
sheaf of rays
(Strahlenbiischel).
The common
point of intersection of the rays is called the centre of the sheaf; the single rays, unlimited on either side of the centre, are the elements of the sheaf. Here again we imagine the elements to
be
rigidly
connected with one another.
Either the centre
or the
plane
in
which the rays
lie
may be
GEOMETRY OF
12
POSITION.
base of the sheaf of rays. A portion of a sheaf of rays bounded by two rays of the sheaf as sides is called a This consists of two complete plane angle. simple In any sheaf angles which are vertically opposite to each other. of rays S (Fig. 5), if four rays a, b, c, d are chosen at
looked upon as the
random,
then
among
these there are two pairs
of separated rays, for instance, a and are separated from each other by
c
and
b
d,
so
we cannot pass
that
the sheaf from a to ing either b or
The
FIG.
5.
of the sheaf.
As
straight .line
unalterable relative positions. bounded by two planes as faces
unlimited in
be called the
be
we consider
rigidly
connected
A
portion of a sheaf of planes is called a complete dihedral
two dihedral angles which are simple each other. Among four planes of a sheaf
consists of
opposite to
vertically
planes, to
is its
in
and
shall
in the range of points, so here,
the elements of the sheaf, that
angle,
d.
totality of planes,
directions,
the axis
in
without cross
which pass through one straight line we shall speak of as a sheaf of planes (Ebenenbiischel) and
all
*
c
two pairs again are separated. If no confusion is likely to be caused, a form which consists only of discrete points and the intervening segments of a straight be called a range of points. In the same way, a form
line will often
which comprises only discrete elements and the included angles of will often be spoken of as a sheaf. In doing so we must constantly bear in mind that, deviating from the ordinary definition,
a sheaf
we have included
angles as part of a sheaf. designate the range of points, the sheaf of lines and the sheaf of planes as one-dimensional primitive forms or primitive 17.
forms
We of
primitive
the first
form,
for
grade.
example
The elements the
planes
of
of a
a
one-dimensional
sheaf,
are
to
be
looked upon as simple elements, i.e. they are to be viewed apart from the forms (geometrical figures, and the like) whose bases they be. In the case of the sheaf of rays this view is facilitated distinguish the straight lines whose totality makes up the sheaf by the name For, by a ray is ordinarily meant a rays.
might
if
we
straight
line
in
itself,
viewed apart from the points lying on
it
THE SIX PRIMITIVE FORMS.
l
$
passing through it. Unfortunately, there is no second for the corresponding designation plane available. Of the primitive forms of the first grade, moreover, we can the
or
planes
Thus imagine any one to be generated by either of the others. is from an outside (Fig. 6) projected
ABCD
a range of points
fa
\U FIG.
S
point
section.
6.
ABCD
is a by a sheaf of rays abed, of which the range In the same way the range is projected by the
ADCB
A
sheaf of planes a/3yS is intersected by any plane not sheaf adcb. passing through its axis in a sheaf of rays abed whose centre lies upon the axis ; every sheaf of rays is projected from a point not Finally, every sheaf lying in its plane by a sheaf of planes. of planes is intersected by a straight line which does not lie in a plane with its axis, in a range of points ; and every range of points is projected from an axis which does not lie in a plane with it, by
a sheaf of planes. to characterize the
From
these relations
of points,
range sheaf of planes as primitive first
grade.
forms
is certainly permissible sheaf of rays, and the of the same, namely, of the
For, from what has been
of points contains just as
many
it
the
said,
it is
clear that a range
points as a sheaf contains rays or
planes.
There are two varieties of primitive forms which are be of the second grade, namely, the plane field and the bundle of rays. The totality of points and lines which are 1 8.
said to
in a plane system or field ; plane we name a and the the base of points system of plane is In the plane field there are contained, consequently, not
contained the lines.
only points and
many
ranges
of
straight lines
points
as
elements,
and sheaves of
rays
but also indefinitely ;
for
all
the
points
GEOMETRY OF
14
on a
lying
straight
and
of points,
all
line
the
POSITION.
of the field taken together form a range of the field passing through one
lines
The plane field is, therefore, justly point form a sheaf of rays. characterized as a primitive form of higher grade than the onedimensional primitive forms. we
Further,
call
the
any
through
point In (Strahlenbiindel).
in
of
totality
space
and planes which pass
bundle of rays centre) a are contained as elements not
(as
there
this
rays
but also indefinitely many only straight lines and planes, sheaves of rays and sheaves of planes. For all planes of the bundle which intersect in one and the same axis form a sheaf and, in the same way, all straight lines of the bundle one and the same plane form a sheaf of rays. Thus the bundle of rays is in reality a primitive form of higher grade
of planes
which
lie
;
in
than the one-dimensional primitive forms. The term bundle, which is appropriate to denote a multiplicity of higher grade than the term sheaf, was very happily chosen
we can, however, name the foregoing primitive by Von Staudt form a bundle of planes (Ebenenbiindel) with the same ;
propriety
as
bundle
a
of
elements as well as rays. straight
lines
come more
designated as a
rays,
since
contains
it
planes
for
According also as the points or the into
field of points
is
consideration, or a
the
plane
field
field of rays.
It is scarcely necessary to mention that in the plane field and in the bundle of rays, we imagine the elements of which they consist to be rigidly connected with one another, so that in the bundle, for
example, the relative positions of the rays, planes, and sheaves con tained therein are unaltered when the centre, which is the base of the bundle, is moved from its original position. may further assert that a bundle of rays contains just as
We
rays
and planes
as a plane field contains points
and
rays,
many
and we are
therefore wholly justified in considering the two primitive forms as of the same, viz., the second grade, since we can imagine the bundle If we of rays to be generated from the plane field, and conversely. project, for instance, a plane field
2 from an
so that outside point and each of a S, ray ray projected by which of 2 by a plane of S, then we obtain a bundle of rays is called a projector of the field 2, and of which the field is a
each point
P
of
2
is
,
SP
section.
To
aid your imagination,
suppose that 2
is
a plane landscape
THE SIX PRIMITIVE FORMS. spread
out at
your
and
feet,
unlimited
in
extent
that the outside point
S
is
!
5
and sparkling in Each your eye.
variegated colours, point of the landscape, then, sends into your eye a ray of light, each straight line of the landscape a plane of light. If now you
consider these rays and planes as unlimited in all directions, you obviously have a bundle of rays as projector of the whole landscape.
We may
further conclude that each range of points of the plane projected from 5 by a sheaf of rays, each sheaf of rays by a sheaf of planes, each curve by a conical surface belonging to field is
the bundle ; or, in other words, the projectors of a range of points, a sheaf of rays, and a curve lying in the plane field are respectively a sheaf of rays, a sheaf of planes, and a conical surface in the
bundle of
rays.
Just so, each segment is projected by a dihedral angle, etc.
by a plane
angle, each plane angle
Conversely,
if
we consider
the
bundle of rays as the original
form and imagine its centre to be, say, a luminous point which sends out coloured rays on all sides, then the field may be looked upon as a section of this bundle. In this case, each ray of the bundle line,
is
cut in a point of the
field,
each plane in a straight
each sheaf of rays in a range of points and each sheaf of
planes in a sheaf of rays. 19* Finally, there exists the space
a
system, or
primitive form
of
the
third grade,
unbounded space with
all possible contains as space system points, elements indefinitely many primitive forms of the first and second grades, since each of its planes is the base of a field, each point
namely,
lines,*
and planes
in
it.
The
the centre of a bundle of rays, each straight line the base of a range of points and at the same time the axis of a sheaf of planes. 20. To each of the six primitive forms which I have just defined It will be readily conceded there corresponds a distinct geometry. that there
must be
just as
rays as for the plane field
complete a geometry
for the
bundle of
corresponding to every plane geome trical figure we immediately construct a form in the bundle by The theorems projecting the plane field from an outside point. ;
for,
which may be enunciated concerning plane
figures
can be carried
* Properly speaking, the space system viewed as consisting of lines dimensions. H.
is
of four
1
GEOMETRY OF
6
over
We
in
some manner
shall
to
POSITION.
their projectors in
the
have occasion to make frequent use of
bundle of
rays.
this process.
It is more difficult to perceive that there is also a geometry for the one-dimensional primitive forms, for example, for the range of points, i.e. the points of a straight line ; but I need only to
recall the
in
order
theorem upon harmonic points cited to
convince you of
this
fact.
in the introduction,
The statement
I
made
there was that the position of a fourth harmonic point is deter mined by three points of a straight line. To show, further, that reality something of a geometry of one-dimensional primitive forms can be established without the aid of measurement, let me
in
the fact that among four rays of a sheaf there are two of rays which are separated by the others. pairs 21. The discussions up to this point make it possible for me now to indicate the principal contents of the Geometry of Position recall
in
a very few words.
That
treats of the six primitive
their
mutual
relations.
is
to
say, the
Geometry of Position this lecture and of
forms mentioned in
LECTURE
II.
INFINITELY-DISTANT ELEMENTS. CORRELATION OF THE PRIMITIVE FORMS TO ONE ANOTHER. 22. In
the
ancient geometry two straight lines are said to be
same plane and have no point in common. planes or a plane and a straight line are parallel if no Modern point of the one lies at the same time in the other. geometry establishes parallelism in a different manner, and it parallel if they lie in the
Likewise, two
will
be
my
next object to make you familiar with this modern Von Staudt has called the perspective view of
conception which parallelism.
We
are brought directly to this
primitive form from
another by or a projector of the second.
considering
when we the
first
derive one a
section
ire FIG.
6.
straight line u (Fig. 6) lies in a plane with a sheaf of without passing through its centre, then it intersects the sheaf in a range of points, namely, each ray a, b, r,,..of S is
If a
rays
S
cut in a point A, B, C,...of //. If now by rotating about a fixed sense abc, any ray describes the sheaf 5, its trace B
5
in
upon
1
GEOMETRY OF
8
POSITION.
same time describes the range of points The point of intersection moves from the position A farther and farther out beyond B until it is lost to view, and then returns, from an infinite distance on the
the straight line u at the u in the sense ABC.
opposite side back to its original position. According to the ancient notion the rotating ray no longer intersects the line u in the one particular position / in which it is said to be parallel
and on
to u, line it
which
is
this
lies
account the general statement that every straight one point in common with
in a plane with u has
In modern geometry the exceptional case is attributing also to two parallel lines a common point,
not allowable.
removed by
namely, an infinitely distant point.
From
23. line
has
the perspective point of view, moreover, any straight only one infinitely distant point, since, in accordance
one of Euclid
with
drawn
s
axioms,
only
one
straight
line
p
can be
u through a given outside point S, and to this attributed but one point in common with u, just as
parallel to
parallel line
is
every other ray of the sheaf S has only one point in common with u* This conception of parallelism presents distinct advantages
over the ancient one in that,
first,
many theorems can be enunciated
would need always to be cited, and second, many apparently different theorems can in accordance with this view be comprised in a You have already become familiar with this single statement.
in a perfectly general form for which, otherwise, exceptions
idea
in
analytical
straight lines
geometry.
which
lie
in
There we are accustomed
one plane
parallel
if
to
call
coordinates
the
of their point of intersection are infinitely great, the point thus lying at
an
infinite distance.
The infinitely distant point of a straight line is approached a by point which moves continually forward upon the line either in the one sense or in the other. Thus the infinitely distant 24.
point lies out in both directions f upon the line, or as properly in the one direction as in the other, and the straight line appears to be closed, its extremities meeting in the infinitely distant
*The assumption of a single infinitely distant point on a straight line and the definition of parallel lines as lines which intersect in a common infinitely With distant point is equivalent to the assumption of Euclid s twelfth axiom. this axiom as starting point, Euclid proves that one and only c straight can be drawn through a given point parallel to a given straight line. H. H. term direction is used here in its colloquial sense. ;>e
line
INFINITELY-DISTANT ELEMENTS.
We
point.
the
forced
are
that
assumption
one
distant
infinitely
to
this
every
conclusion as line
straight
We
point.
shall
9
soon as we admit
contains see
!
later
one and only that the two
branches of a hyperbola are to be considered as connected at
same way. Analysis leads to similar conclusions, out by frequent examples that we can pass from the pointing the to negative not only through zero but also through positive infinity in just the
infinity.
then, we can go from one point of a straight line another by passing over the infinitely distant point of the line, the following statement is true: Among four points of a 25. Since,
to
there
range the
by
are
only
remaining
two pairs whose elements are separated of the quadruple. This is strictly
points
analogous to the fact that among four elements of a sheaf only two pairs of elements can be so chosen that the elements com posing a pair are separated by the other two ; and just as a sheaf is divided by two of its elements into two complete angles (these being supplementary angles), so a range of points is divided points into two segments of which each is called supplement of the other. One of these two segments contains
by two of the
its
the infinitely distant point of the line unless this point
one of the boundaries of the segments. In the of the two segments may be called a half-ray.
itself
latter
forms
case each
In order to distinguish the infinitely distant point of a straight
26.
from the points of the line which lie in the finite region, we shall call the former an actual ideal point and the latter
line
The modern conception of parallelism, explained at the All of this lecture, might also be characterized as ideal. opening the parallel lines which may be drawn in a plane in any one points.
direction have but one infinitely distant or ideal point in common, namely, that point which any one of them has in common with
each of the others.
These
parallels
may
therefore be considered
as forming a sheaf whose centre is an infinitely distant point of the plane, and such a system we shall hereafter designate as a
sheaf of parallel rays, whenever a distinction from other sheaves of Likewise under the name bundle of parallel rays is desirable. rays
are to be
comprehended
all
possible parallel rays in space,
having a given direction, together with all planes passing through them. I would remind you at this point that the statements lines con"parallel "parallel lines have the same direction" and
GEOMETRY OF
20
same
tain the
infinitely distant
POSITION.
point,"
mean
same
exactly the
thing.
determines one infinitely distant point, and conversely, each ideal point in space determines a single direction ; moreover, every actual straight line determines both a direction
Any
given direction
and
an
upon
distant
infinitely
namely,
point,
the
ideal
point
lying
it.
27. It is assumed of all the infinitely distant points of a plane that they lie in an infinitely distant or ideal line.* This line must
be looked upon as a straight
since
line,
it
intersected by every
is
the infinitely actual straight line of the plane in only one point distant point of that line while curved lines may have, in common
with a straight line, more than one point. Another reason for this view is the fact that in accordance with the perspective idea two parallel planes must have all their infinitely distant points in common. For, if these planes are cut by any third plane in two actual straight lines, these lines can intersect
no actual point ; they are therefore parallel, since they lie in one plane, and consequently have in common an infinitely distant In this manner it may be shown that every point of both planes.
in
one plane
point of the
distant
infinitely
the other.
in
also
lies
general, any two intersecting planes have a single straight line in common, we attribute also to two parallel planes
But
in
since,
common
a single 28.
As
so
direction,
straight line.
of
said
is
it
parallel
we
say of parallel
as,
then,
aspect;
just
distant
point,
straight line.
so
for
in
every
every
be
forming a
considered as distant
infinitely
there
is
an
this
distant
infinitely
space of any one aspect pass these
Parallel planes
we
namely,
straight line,
some one of
sheaf of planes
line;
straight
they have the same there lies an infinitely
infinitely distant
through that straight line in which intersected by each of the others.
have the same
they
that
direction
aspect
All parallel planes in
through one and the same
that
lines
planes
shall
may
whose call
planes
axis
a
is
therefore is
sheaf
an of
parallel planes.
29.
Of
assumed *
That
the that
is,
infinitely
they
lie
distant in
an
points
and
lines
infinitely distant
of space it is surface
or ideal
;
If that the infinitely distant points of a plane form a continuum. line of the plane be rotated about one of its actual points
an actual straight
every other actual point will describe a continuous H. of the infinitely distant point.
line.
The same
is
assumed
INFINITELY-DISTANT ELEMENTS. this
2I
must be considered plane, since it is intersected by one point and by every actual
surface
every actual straight line in only
The infinitely distant or ideal plane is plane in a straight line. to all bundles of parallel rays and to all sheaves of
common parallel
planes,
since
it
passes
through the centre of the former
and through the axis of the latter. In the same way, the infinitely
distant line in any plane is a ray common to all sheaves of parallel rays lying in that plane, since it passes through the centre of each of them. No definite
direction can therefore be assigned to the infinitely distant straight line of a plane, but it possesses the direction (contains the infinitely distant point) of every straight line of the plane. 30. Some light is thrown upon the question of infinitely distant or ideal elements by the relations which may be established between
Two
two primitive forms. each other
forms are said to be
correlated
to
with every element of the one is associated an element of the other. Two elements of the forms which so appertain to if
each other are called
or corresponding homologous elements. two primitive forms are correlated to a third, then are they also correlated to each other. For to every element of the third there If
corresponds an element of each of the other two forms, and these two elements are by this means associated with each other. 31.
Two
primitive forms of different kinds are correlated to each
other in the simplest and clearest manner by making the one a section or a projector of the other. For example, if a sheaf of rays S (Fig. 6) lies in a plane with a range of points u not passing through its centre, we may assign to each ray of the sheaf that
To the parallel ray / of S point of the range which lies upon it. the distant corresponds, then, point of u. infinitely 2 if a field is as being a section of a considered Again, plane bundle of rays ,S whose centre lies outside S, then 2 and S are correlated to each other in such a manner that to each point of
2
corresponds the ray of of 2 the plane of
line
parallel to
2 corresponds
S
passing through
it,
and
to each straight
plane of S passing through therefore to the infinitely distant line of it.
The
S
2 and to each ray of lying in this plane corresponds its infinitely distant point lying in 2. To each sheaf of planes in S corresponds the sheaf of rays in which it is cut by 2; the latter is a sheaf of parallel rays if the axis of the sheaf of planes is parallel to 2. If S is a bundle of parallel rays, its centre lying infinitely distant,
GEOMETRY OF
22
POSITION.
then to each actual point of to each ideal element of
2 corresponds an actual ray of 2 an ideal element of S. If 2
the infinitely distant plane and to each ray of S corresponds
S
and
plane,
its infinitely
distant line
distant range of points distant sheaf of rays.
;
and
its
a point of the infinitely
finite region,
distant point
;
to
6",
is
then
each
to each sheaf of rays,
an
infinitely
to each sheaf of planes,
an
infinitely
;
32. Two primitive forms of the same kind may be correlated most simply by considering them to be sections or projectors of one and the same third primitive form. Thus, in two sheaves of
FIG.
rays, or ranges of points,
7.
which are sections of one and the same
sheaf of planes, those two rays, or points, correspond which lie On the other hand, two sheaves in the same plane of the sheaf.
S
of rays
and
S
1
(Fig.
7)
can easily be so correlated that they
and the same range of points u, i.e. so that a and a v b and b lt c and clt ... which intersect
are projectors of one
those pairs of rays of the range are corresponding rays.
in points
If
two ranges of
points u and u^ (Fig. 8) lying in one plane be considered sections of one sheaf of rays S, then it is worthy of notice that to the infinitely distant point
a point
P
l
(Q)
P (Q-^)
of one range corresponds, in general,
lying in the finite portion of the other.
CORRELATION OF PRIMITIVE FORMS.
23
Two plane fields are correlated to each other if they are sections of one and the same bundle of rays. For example, an extended plane landscape and the perspective picture of it which we obtain by intercepting its projector from the eye by any plane, a vertical one, say, are so correlated that those points of the landscape and picture correspond which lie upon the same ray of the bundle having the eye for centre, that is, any two points of the landscape and picture correspond which are found in a straight line with the
To
eye.
each straight
line of the
landscape corresponds a straight
the picture, and the two straight lines lie in a plane with To the infinitely distant straight line of the landscape the eye.
line
l>f
FIG.
8.
(the horizon) corresponds, in general,
an actual
straight line of the
another reason or warrant for considering the
picture,
infinitely
distant line of a plane to be a straight line. Of two plane fields correlated to each other in this manner we appropriately say that
of the other, and the centre of the bundle projection once a projector of both fields is called the centre of projection. If the centre lies infinitely distant, the bundle is
one
is
which
a
is
at
a bundle of parallel rays, in which case the process of projection becomes the ordinary parallel projection of descriptive geometry. Two bundles of rays are correlated to each other if we conceive
Each to be projectors of one and the same plane field. ray of the one bundle intersects, then, the corresponding ray of the other in a point of the field; likewise, every two homothem
GEOMETRY OF
24
POSITION.
logous planes of the bundles have a straight line of the field as their intersection. The projectors of a plane landscape viewed
from two different points constitute such bundles. 33. I must leave the immediate discussion of the correlation of primitive efforts
;
I
forms to each other for the time being to your own remark, however, that the primitive forms may be corre
in other and more complicated ways. For example, it is possible to correlate two plane fields to each other by correlating them to one and the same third field. Referring again to an
lated
which has been frequently used, you may imagine perspective pictures of a landscape to be constructed from two different centres of projection. Two such pictures or plane fields, then, are correlated to each other, since each is correlated to the illustration
landscape
j
and
clearly,
two points of these correspond
they are
if
same point of the landscape. A straight line of the one picture would correspond always to a straight line of But such plane fields, in general, have no longer the the other.
projections of the
particular position with respect to each other which was previously discussed, that namely, in which corresponding lines lie in a plane
and the joining lines of in one point. Later, we
pairs
of homologous points
all
intersect
have to investigate more minutely the mutual relations of the two fields so correlated to each other.
Two
fields
may be
shall
so correlated that
each straight
line
of the
one corresponds to a curve in the other, or that to each point of the one field corresponds a straight line of the other, and conversely, to each straight line of the former, a point of the For the present, however, it is left to your own ingenuity work out these more diverse relations of primitive forms.
latter.
to
LECTURE
III.
THE PRINCIPLE OF RECIPROCITY OR DUALITY. SIMPLE AND COMPLETE w-POINTS, w-SIDES, w-EDGES, ETC. 34. Before developing further the correlations which may be among the primitive forms of modern geometry, I must make mention of a geometrical principle which will occupy
established
an important place
in these lectures. This principle very greatly the of the of Position, in that it divides Geometry simplifies study the voluminous material of the subject into two parts, and sets
each other in such a way that the one part This principle of reciprocity
these
over against
arises
immediately out of the other.
or duality as
it
is
called
by Gergonne,* Poncelet
was f
first
established in an elementary
way
by means of can be con there space
having previously shown
the polar theory that to every figure in structed one which corresponds to it in the dualistic sense.
35. Although the principle of duality cannot be generally applied in metric geometry, yet there are many theorems in metric
geometry which point directly toward this principle, and which I need only to call to mind in order to make you aware of its existence.
In
three-dimensional
space, the
point and the plane
stand
opposed to each other as reciprocal elements, so that every theorem of the Geometry of Position finds its complement in another which we may deduce from the first by interchanging the terms point and range of points plane, and hence also and sheaf of planes, segment and dihedral angle, etc. Ordin theorems side by side arily we shall write two such reciprocal as the two members of one theorem. For example :
* Gergonne, Annales de Mathematiques, T. XVI., t Poncelet, Traite des proprietes projectives des
1826.
figures, Paris, 1822.
GEOMETRY OF
26
Two
and
Two planes a and /? determine a straight line a(3, namely, their
determine
namely, the
them.
straight line
not lying upon
aB which
B
AB^
line
line joining
A
A
points
a straight
it
POSITION.
line of intersection.
a and a point B determine a plane
A
straight line a
and a plane
not passing through
a point a/3 which
passes through both.
lies
it
(3
determine
upon both. which do y,
Three points A, B, C\ which do not lie in one straight line, determine a plane ABC (the joining
not pass through one straight line, determine a point a/3y (the point
plane).
of intersection).
Two
straight lines
have one point one plane ab.
in
a and
6,
common,
Three planes
/?,
Two
which lie
a,
in
lie
in
in
straight lines a and b, which one plane, have one point ab
common.
36. Incidentally you will notice from these few theorems how useful the introduction of the infinitely distant or ideal elements
geometry proves to be. Without these we should not have been able to enunciate all the above theorems in a general form, but must have called particular attention to special cases as exceptions.
into
The
first
theorem on the
right for
example would have read
"
:
Two
planes a and /3 either determine a straight line a/?, or they are parallel," while from the new point of view a straight line is also
determined in the line of the planes.
namely, the infinitely distant straight In the same way we should have been obliged to
latter case,
distinguish several cases of the as the two given points and
A
have read
A
first
theorem on the
left,
B are actual points or not.
according It
would
determined by two given (actual) or one and a from the perspective points by point given direction "
:
straight line
is
"
;
point of view, however, the latter case is included in the former, since among the given points infinitely distant points may also be You can yourselves easily make similar observations considered.
upon each of the other theorems. 37. For incident are
the if
point,
of
they intersect
incident
and a
brevity we shall call two straight lines a straight line, or plane, and a point ; the latter lies in the former ; and finally, a ray, or
sake
if
plane,
if
the latter passes through the former.
Straight
lines not incident are called
gauche. 38. The foregoing theorems lead now to the following problems, the solutions of which we shall always in future consider possible :
Through two straight line.
points to pass
a
To
find the line of intersection
of two planes.
THE PRINCIPLE OF RECIPROCITY OR DUALITY.
To find the point of intersection of a straight line and a plane not incident with it.
Through a straight line and a point not incident with it to pass a plane.
Through three
Through two
the
common
point of
three planes. To find the
common
point of
To
points to pass a
plane.
incident
straight
find
two incident straight
a plane.
lines to pass
27
lines.
39. For the sake of practice, I shall cite a few double theorems
which are
in
frequent use.
yourselves, from the
I
strongly urge
one half of each of
deduce
to
you
for
these, the other reciprocal
half.
If four
given,
points A,
and the
lines
,
D
are
If
and
CD
given,
C
AB
t
planes
a,
(3,
y,
8 are
and the lines of intersection aft and y8 intersect, then the four planes pass through one and the same point, so that the lines ay and (38, as well as aS and /2y, must also
then the four points lie one and the same plane, so that
intersect, in
four
AC and BD^ as well as AD and BC, must also intersect. the lines
intersect.
If of any number of straight one while they do not all
pass through one point, then they
must
all lie in
one plane.
each intersects every other
lines
lie in one plane, then they must pass through one point.
all
Frequently, a theorem is reciprocal to itself as when point and In plane appear in it symmetrically ; for example, the problem a plane, to draw through a given point a straight line to meet a given straight line which neither lies in the given plane nor passes :
through the given point. Here two solutions stand reciprocally opposed to each other:
We may either intersection
of
join the point of the straight line
and plane, with the given point
;
or
may
pass a plane through the and the given point,
straight line and find its
line
of intersection
with the given plane.
The stated
following problem
may
easily
be reduced
to
the one just
:
one and the same plane with the
plane to draw a which intersects two given straight lines not having one and the same point in common
given point.
with the given plane.
Through a given point to draw a straight line which intersects two given straight lines not lying in
In
a
given
straight line
GEOMETRY OF
28
Pass a plane through the given point and one of the given straight lines,
POSITION.
Determine the point of intersection of the given plane with one of the given straight lines,
and the problem becomes identical with the preceding one. The problem, To draw a straight line which intersects three "
We may either given straight lines," is likewise reciprocal to itself. choose a point in one of the straight lines or pass a plane through one of them, and then under the conditions of the preceding problem find a straight line which passes through this point, or lies in this
plane,
and
intersects the other
two given straight
lines.
The
40.
another
;
primitive forms can also be opposed reciprocally to one for example, the plane and the bundle of rays are fi^ld
clearly reciprocals,
namely, the plane and the
since their bases,
point, are reciprocal elements.
are reciprocal to
Hence, range of points, point, the ray considered as joining two points, the sheaf of rays, etc., in the
the
the plane
The
field,
observation will press
the plane, the sheaf of planes, the ray considered as the intersection of two planes, the sheaf of rays, etc., in the bundle of rays.
itself
upon you
as
here,
in
many
previous theorems, that in space of three dimensions the straight In reality, the straight line line (or ray) is reciprocal to itself.
occupies an intermediate position between the reciprocal elements point and plane. 41.
The
following serves as an example of a double theorem in field and the bundle of rays are opposed to each
which the plane
other as reciprocal forms If
two plane
fields
:
are correlated
by considering them of one and the same
to each other
as sections
bundle, then pairs of corresponding elements (points or lines) of the
upon one and the same element (ray or plane) of the bundle. fields lie
The
line of intersection of the
two planes coincides with its responding line, and hence responds to
itself.
true of each point line.
cor-
cor-
The same found in
The two plane
is
this
fields there-
If two bundles are correlated to each other by considering them as projectors of one and the same
then pairs of corresponding elements (rays or planes) of the bundles pass through one and the field,
same element
(point
or
straight
line) of the field.
The common
ray of the bundles, the ray which joins their centres, coincides with its corresponding i.e.
ray,
and hence corresponds
The same
is
true
to
itself,
of each plane
THE PRINCIPLE OF RECIPROCITY OR DUALITY. lore
have a
passing through this ray. bundles therefore have
self-corresponding
range of points.
29
The two *
a selfcorresponding sheaf of planes.
42. If two forms are correlated to each other, and an element of one coincides with (i.e., is identical with) its corresponding element
we say that this element* (double element) is a element in the two forms. self-corresponding As the and the 43. point plane are reciprocal elements in space in the other, then
of three dimensions, so in space of two dimensions, the point and the straight line, also the range of points and the sheaf of rays, the segment and the angle, are opposed to each other as reciprocal
forms
;
and
similarly in the
the sheaf of rays
For example
bundle of
and the plane, are planes, etc., reciprocal forms.
and the sheaf of
:
(ttj) Any two points of a plane determine a straight line. (a3 ) Any two rays of a bundle determine a plane.
A
As
points lying
the
aggregate
upon
(04)
(a 4 )
of
straight lines of a
planes of a bundle
ray.
(ft) As the aggregate of the straight lines (tangents) enveloping
the
it.
FIG.
will
Any two
determine a
it
And you
Any two
plane determine a point.
may be looked upon
plane curve
(ft)
rays, the ray
find that in
the
(Fig. 9).
9.
modern geometry
the latter con
In ception is brought into use just as frequently as the former. the same way, a conical surface (in the bundle of rays) can be
looked upon (ft)
As
the
rays lying upon
aggregate it.
of
the
(ft)
As
the
planes (tangent
ing
it.
aggregate of the planes)
envelop
GEOMETRY OF
30
Of
44.
POSITION.
four theorems related to one another as are those of the
the two relating to the bundle of rays can always be deduced from the other two by projecting the plane field from As a rule, therefore, I shall in future state only any centre. the two planimetric theorems, and will leave you to seek out the In space, where point and plane are re others for yourselves. ciprocal to each other, the first and last (a x and a 4 ), also the second and third (a 2 and a 3 ) of any four such theorems offset last article,
each other as reciprocal theorems.
The
45.
familiar to
a
principle of reciprocity will
become
clearer
in the course of our investigations
you
;
and more
but only after
developments upon the one-dimensional primitive forms, demonstrate that it has general validity in the Geometry
series of
can
I
of Position, or that in reality to every theorem there corresponds
In the meantime, I shall so adjust my a reciprocal theorem. lectures that theorems associated reciprocally with each other will be placed side by side, and I shall so carry out their demon strations
end,
it
that is
dualism
the
necessary
reciprocal ideas, metrical notions
will
stand
I
should
that
To
out very clearly.
develop beforehand
this
some
in particular modify some of those geo which you have brought over with you from
and
metric geometry. 46. I refer here particularly to the conception of the polygon. In modern geometry we understand by a simple plane ;z-point
not as a rule a portion of the plane which is bounded on all sides by n intersecting straight lines, but a set of // points of a plane and the n straight lines or sides, each of which joins two consecutive points or vertices.
We
in a definite order,
and
shall lie in
The
one
simple
look
upon the points as being arranged no three consecutive points
specify that
straight line. ;z-point
might also be named
a
simple
-side,
since a simple n-side is a set of n straight lines of a plane (the sides of the figure), and the ?i points in which two consecutive sides intersect.
The
-point
and
^-side
are
reciprocal
joining two non-consecutive vertices
figures.
To
the
lines
the
(i.e. diagonals) of a of of intersection non-consecutive sides the points simple -point, in the n-side are reciprocal, each to each.
to
47. In metric geometry where by an //-point
of a plane enclosed by ^-sides, the re-entrant
is
meant a portion
-point,
such as the
THE PRINCIPLE OF RECIPROCITY OR DUALITY. pentagon
ABCDE
(Fig. 10), or the
hexagon
ABCDEF (Fig.
n)
generally excluded from consideration.
is
The -point and the /z-side of modern geometry give very little occasion for distinction between re-entrant figures and others, since We may in fact the sides are supposed unlimited in extent. call
any two of the 2n elements
(vertices
and
sides taken together)
elements, which are opposite separated from each other by half the number of remaining elements ih ih elements, these being reckoned consequently, the m and (n + m)
a
of
simple
/z-point
or
/z-side,
;
from any one element round the figure in either order, are opposite to each other.
FIG. 10.
For example, a side
lie
the side
ABCDE
10) a vertex
(Fig.
and
A
B ABCDEF (Fig.
and the
the sides
pentagon
and opposite to each other in pairs, namely, the vertex and DE, C and EA, etc.; in the hexagon or CD,
hexalateral pairs,
in the
FIG. ii.
AB
n), on the other hand, the vertices in A and D, and E, etc., are opposite the vertices
sides in pairs, as for instance the vertices
and
B
DE,
elements. 48.
Modern geometry, however,
deals
not only with
simple
but also with ^-points and w-sides, complete ;/ -points and -sides, and in these figures the principle of reciprocity may again
We
be distinctly recognized.
A
define as follows
A
a set
complete plane n-potnt of n points of a plane together with
of
all
together with
:
straight
lines
(sides)
joining
straight
the
gether with
together
diagonals.
\\-side
complete plane
n
them two and two, or what is the same thing, a simple -point to all its
:
lines
of
their
all
intersection (vertices), or
same
thing,
with
intersection of
a
all its
:
a
a
set
plane
points
of
what
is
simple n-side the
sides.
points
of
GEOMETRY OF In these definitions lie
^-point of the
-side pass
each
In
it
is
upon the same vertex
plete 7z-point,
(n-
assumed
com
the
plete
complete
;z-point is
49.
It
is
number
of sides of the
^ n (n-
readily
-side
i).
seen that
(n\)
For example
many
^ n (n-
simple
A
com
vertices;
i).
and
^-points
-sides
greater than
is
:
FIG. 12.
FIG. 13.
complete quadrangle
A BCD
any two of AB and CD, or and AC and BD, or finally, BC, which do not pass through one and the same vertex are opposite (Fig. 12) has six sides these sides as
the
of vertices of the complete
is
are contained in the complete figures whenever n three.
of
side lie
>>z-side
through these pass the remaining (n-i] sides (through each vertex a second side). Hence the total
Hence
through a second vertex).
number
no three vertices of the and that no three sides
Upon each
sides intersect
These pass through the remain i) vertices (each of them ing ( the total
that
straight line,
through the same point. of
i)
POSITION.
;
AD
*
A
complete quadrilateral abed (Fig. 13) has six vertices ; any two
and cd or ac and &r, which do upon one and the same
of these as ab bd, or finally,
not
lie
side are
y
ad and
of the
opposite vertices
quadrangle, so that in a quadrangle there are three
quadrilateral, so
Moreover, pairs of opposite sides. the complete quadrangle contains
opposite vertices.
three simple quadrangles ABCD, ACDB, and ADBC, the sides of
three acdd,
each consisting of two pairs of op posite sides of the complete figure.
consisting of two pairs of opposite vertices of the complete figure.
sides
50.
of the
The forms
in the
rilateral
complete
that
in
a quad
there are three pairs
of
Moreover, the
quadrilateral
contains
simple
and
quadrilaterals abed, adbc, the vertices of each
bundle of rays which correspond to these
plane figures are most easily obtained by projecting the latter from Each plane #-point gives rise by a point lying outside their plane. projection to an
and each plane
-edged -side,
figure,
or,
more
briefly,
to an ;z-faced figure, or
an
to
an
/z-face.
-edge,
THE PRINCIPLE OF RECIPROCITY OR DUALITY.
,3
Accordingly,
A
complete //-edge
is
A
a set of
n rays of a bundle, together with
complete
//-face
is
a set of
n planes of a bundle, together with of intersection (edges),
all
all their lines
no three of the // rays or one plane.
assuming that no three of the n planes or faces pass through one and the same ray.
planes (faces) passing through them, two and two, assuming that
edges
lie in
would be an easy matter for you to define the simple //-edge and simple //-face, and to develop properties of these forms in the "
It
bundles analogous to those of the corresponding plane figures. 51. I shall conclude this series of definitions with those of the analogous space configurations.
A
*
complete
A
three-dimensional
consists of
complete
three-dimensional
n planes (faces) of which no four pass through one point, the straight lines (edges) in consists of
points (vertices) of which no four lie in one plane, the straight limes each of
//-face
which joins two of the
each of which two of the n planes
//-point
//
//
points,
and
the planes each of which passes through three of the n points.
intersect, in
n planes I
leave the determination of the
and the points
each
of
which
(vertices),
three
of
the
intersect.
number of edges and
faces of
a three-dimensional //-point, as also the edges and vertices of a three-
dimensional
//-face,
to
your own inquiry.
I
remark, however, that
the three-dimensional tetragon and the tetrahedron do not differ from flfch other any more than do the triangle and the trilateral
the plane. That the principle of reciprocity is applicable to the tetrahedron the following theorems among others will show in
:
The
four vertices
and
six
The four faces and six edges of a tetrahedron are intersected by
edges
of a tetrahedron are projected from any point which lies in none of its faces
by the four edges and
any plane which passes through none of its vertices, in the four sides and six vertices of a complete
six
faces of a complete four-edge.
quadrilateral.
You^-ill here observe that complete plane bundle,
//-point
is
in
space of three dimensions the complete //-face in a
reciprocal to the
and the complete plane //-side to the complete and plane are reciprocal elements.
since point
//-edge,
LECTURE
IV.
THE CORRELATION OF COMPLETE ^-POINTS, w-SIDES, AND w-EDGES TO ONE ANOTHER. HARMONIC FORMS. 52. In my lectures thus far I have sought to solve but one of the problems lying before me, namely, to make you acquainted with the most important concepts peculiar to modern geometry. I
have no doubt that you have many times wearied of
this
multitude
of definitions following in quick succession, but it was necessary to place these before you in a connected form, so that later we light with less interruption the rich treasures which the Geometry of Position affords. Let us now proceed to the first real theorem of modern geometry.
might bring to
The very simple propositions heretofore stated have been mentioned as occasion might offer, more with a view to familiarizing you with the all
new concepts and for completeness than because they were necessary for the establishment of our science.
I shall first call particular
and planes general, which
points, rays,
forms in
;
attention to the theorems
upon harmonic
theorems upon harmonic now develop as essentially funda
in a word, to the I
shall
mental in the Geometry of Position. 53. The properties of harmonic forms, of which mention was made in the Introduction, can be proved most simply by making use of some elementary theorems upon the correlation oflfcpoints,
and ^-edges to one another. In a way similar to that by which we have already correlated the primitive forms, we can associate in certain figures, to each vertex, -sides,
side,
corresponding element of another^ One can be correlated to a second by associat example,
or edge of one
quadrangle, for
a
ing with each vertex of the
first
a vertex of the second; and in
HARMONIC FORMS. consequence of this, a side of the second.
35
to each side of the first there will correspond
FIG., 3.
We may now If
state the following self-evident
two correlated triangles
and A^B^CI
ABC
(Fig. 3) lie in different
If
theorem
:
two correlated three-edged
(or
planes,
three-faced) figures belong to differ ent bundles, and each of the three
of homologous sides
pairs of
AC
sect,
and each of the three pairs AB and A-JB^ and A^C^ BC and B^C, inter
sect (necessarily
upon the common
homologous edges inter then the three points of inter section determine a triangle of
of the planes of the two triangles), then the planes of the
which the two three-edges are pro
three pairs of corresponding sides determine a three-edged figure, of
of the three pairs of homologous planes (faces) of the three-edges lie therefore in the plane of this
line
//
which the two triangles are sections. joining lines AA^ BB^ and
The
jectors.
triangle,
The
lines of intersection
whose
sides they form.
CCi of the pairs of homologous vertices intersect therefore in one point, namely, in the vertex
5
of
the three-edge.
54. It would be an easy matter for you to enunciate the converse of either half of this double theorem. By the help of these we find that
two complete quadrangles and A 1 B 1 C1 D 1 (Fig. 14) King in different planes whose line of intersection // passes through If
A BCD
If two complete four-faced figures belonging to different bundles of
rays whose common ray lies in none of the eight faces, are cor-
GEOMETRY OF none of the eight vertices are cor related to each other, and five sides d, c of the one quadrangle ,
Z>,
B
D
In
D
order to
we
this, imagine the segment B^D^ and do not lie to be traversed by a point P. Of the points P^ and P2 which are harmonically separated from P by Bl and 15 and by B and respectively, the first l describes the supplement of the segment B^D^ and the latter /*., a segment B 2 2 contained in this supplement, whose extremities are harmonically separated from and D^ respectively, by B P and in the sense opposite to and D. The points move 1 2 that in which moves, and must coincide at least once, since is describes a segment within which the segment traversed by 2 contained. If, now, we denote by A this point of coincidence and by C the corresponding position of the point P, then A and C and at the same time B l from harmonically separate B from v
upon which
prove
shall
D
B
P
Z>
Z>,
D
B
l
P
P
P
P
D
D
HARMONIC FORMS.
45
METRIC RELATIONS OF HARMONIC FORMS. ought not to close the theory of harmonic forms without developing for you, as was suggested in the Introduction, their 67.
I
most important metric relations. We approach these most simply by means of the following theorem If in a straight line two points A and C have equal distances from :
a third point B, then are they harmonically separated by this point
D
and the mfinitely distant point form a harmonic quadruple.
of the straight
line,
or A, B, C,
I
>
FIG. 17.
ABC choose two infinitely and toward each of them draw
In a plane passed through the line distant points parallel
points
L
lines
K and M
(Fig.
A
and
through
and N.
The
these will intersect in two
;
segment AC.
Of
KL
intersect in
MN
new
as second diagonal of the of the the bisection point passes through the quadrangle then, two opposite sides straight line
LN,
B
ALCN,
parallelogram
and
17),
C
KLMN,
A, two others
LM and NK in
C, the
LN
diagonal passes through B, and the second diagonal, namely, the infinitely distant straight line KM, passes through D, so that
ABCD are in 68. Since fifth
point
6"
fact
harmonic points.
four harmonic
by four harmonic
we draw through the lines, the one d parallel to If
points rays,
ABCD it
are
projected
from a
follows that
of a triangle ASC, two straight base the AC, and the other b toward the vertex
*S
GEOMETRY OF
46
POSITION.
middle point of the base, then are these harmonically separated by the adjacent sides of the triangle. If is isosceles, then b is at right angles to AC, and con
ASC
sequently also to d\ moreover, the supplementary angles formed by c are bisected by b and d. Hence
a and
:
"
"
bisecting two
supplementary adjacent angles are harmonically separated by the boundaries of these angles, and are normal to each other." lines
"The
The converse
of this theorem
may be
stated thus
:
If, of four harmonic rays, two conjugate rays are at right angles, these bisect the angles between the other two." "
"
The proof
of this
statement, which
quoted "
"
"
will
is derived immediately from the following be recognized as the converse of the one just
:
harmonic sheaf abed
If a
one of
its rays,
is
cut by a straight line u parallel to
then one of the three points of intersection with the
remaining rays bisects the segment between the other two." The points of intersection of u with abed are four harmonic points,
of them lies infinitely distant. These 69. theorems, to which similar ones may be stated for harmonic planes, can be utilized for the solution of a series of
and one
Thus,
problems. "
To
construct
for example, the
the
problem
fourth harmonic
:
"
three
points or rays admits a solution very much simpler than by means of the complete quadrangle as soon as the construction of parallels and of equal
segments
is
conceded.
For
if
to the rays
to
b, c,
d
(Fig. 18), the fourth
ray a, harmonically separated from c, is required to be found, we may intersect b and c by any straight line u parallel to d in the points the ray a of the and C, and upon this line make \ equal to
B
AB
sheaf If, Z>,
bed,
passing through A,
is
BC
the one sought.
further, to the three points
A, B,
harmonically separated from B,
is
C
(Fig. 19), the fourth point
required,
we may
lay off
upon
HARMONIC FORMS.
47
through B, two equal segments A^B and then the of the straight lines determine ; point of intersection or a and CCl or c, and draw through this point a line d parallel
straight line passing
any
BC AA
l l
FIG. 19.
to
This straight line
A^BCV
For, since
harmonic
and
C
will cut
ABC in
the required point
D.
and the
A^ B, 1? infinitely distant point of A^B are four .S points, (A^BC-^D) or abed must be a harmonic sheaf,
therefore
ABCD,
a section of
this sheaf,
is
a harmonic range
of points. 70.
If a
to the line
B
AC
and its middle point are given, a parallel segment can be constructed through any point (Fig. 20)
ABC
.A"
with the use of the ruler only, as follows
Draw
KA
the lines
respectively
by any
and
KC
and
:
intersect
straight line passed through
M
them
B
;
in
L
and
N
then determine
the point of intersection of CL and AN, and through this point will -pass the required For, as second diagonal of the parallel.
quadrangle
KLMN^
in a fourth point
which
the line
KM
intersects the straight line
harmonically separated from
lies infinitely
distant since
B bisects
B
by
A
the segment
and
AC.
ABC C, but
GEOMETRY OF
48
POSITION.
conversely, two parallel lines are given, any segment one of them may be bisected by a linear construction.
If,
upon
lying
How
these constructions can be turned to account, in land surveying for instance, will be evident to you without further comment.
B/
-\4
C/^
FIG. 20.
Among
71.
segments which are formed by four harmonic
the
ABCD
on a straight line there exists an interesting proportion. points In order to find this, we project the harmonic points from some centre S by a harmonic sheaf abed (Fig. 19) and then pass through
B
a parallel to the ray
A
points
time
and
l
C
lt
two pairs
similar to
ASD
proportions
This meets the rays a and
d.
which are equally distant from
of
similar
triangles
and
BC^C
similar to
,
we
formed,
We
DSC.
;
c in two-
at the
same
namely, AA^B obtain then the
:
A If
are
B
divide the
1
B~SD
first
BC =
BC^ SD
of these
AB BC
consideration that
a
by the second, and take
into-
we obtain
^
BC~ CD The segment as
A
externally
and
at
AC D,
is
B
divided internally at
and
B
in the
same
ratio
being harmonically separated by
C.
This relation
is
frequently taken as the definition of harmonic as starting point in the study of
and might be chosen the theory of harmonic forms. points,
D
HARMONIC FORMS. As a consequence
AB
D out beyond AD greater than CD
of this relation the point and likewise
BC
Cj greater than on the other hand, while, so that both and BC; if
is
both are nearer to
49
B
D D
C
than to A.
lies
;
A AB is less than A than to C, or else
out beyond are nearer to
lies
if
CD
and
-
DC
72. In the above proportion, since the equal segments are described in opposite senses, we ordinarily write
DC
instead of
thus
so that the proportion reads
CD,
more symmetrically,
:
AD
AB__
BC~ DC
M
If
the middle point of the segment
is
be written
BD,
may
equation (i)
:
AM-BM _AM+MD
BM- CM~ CM+MD or
if
MD
is
BM:
replaced by _
BM- CM~ CM+BM we obtain
Clearing of fractions
after a very
simple reduction
:
BM* = AM.CM, ........................... (2) or the remarkable property, "
BM
"AM
DM}
(and similarly
and
is
mean
a
proportional
between
CM."
This useful property might also be admitted as a definition of
harmonic 73.
If
points.
we draw any
a tangent
MT from
A
and C (Fig. 19), and to this through then by the well-known theorem upon the
circle
M,
segments of secants of a
circle,
AM. CM= TM2
,
T
and hence Tlfl**BM***DM*. tangent
M and
lies
therefore
radius
thogonally at
BM or MD
T, since
Thus, In a plane, the "
"A
and
"whose
C
upon
of the The point of contact the circle which is described with centre
are cut
diameter
its
;
and
radius
this
MT
is
circle
cuts the other
or
tangent to the other at T.
which pass through two given points orthogonally by any circle, the extremities of are harmonically separated by A and circles
BD
C"
D
GEOMETRY OF
5o
POSITION.
The theory of harmonic points thus leads us easily to the study of systems of circles which intersect orthogonally, and might suitably be chosen as the starting point in the investigation of orthogonal systems of spheres. Z?/"*
74.
The
may be
reciprocal of equation (i),
written
*
tomed
/"*
7~)
n = AJ-J AJj ~rj\t
AB - AC _AC - AD AB AD B, C, D, owe the you are aware, we are accus
to this latter equation that the points A,
it is
name
the equation
:
AC-AB _AD-AC AB AD and
i.e.
harmonic
For,
points.
to say of three
as
quantities ft 7, 8 that
they are
in
har-
monical progression, or are harmonic, if the difference between the first two is to the first as the difference between the last two is
to the last, that
is
if
_ 8
ft
and
AD
the equation last written involving the segments has exactly this form.
Performing the division relation
indicated
AB, AC, and
above we have
finally
AC AC AB~ AD which may be written
in the
form
I}
:
.:
(,)
AC AB AD* You
the
:
yourselves will easily be able to clothe this very remarkable it likewise is ; frequently used as the definition
formula in words
of harmonic points. Similar equations might be developed for the angles formed by four harmonic rays or planes. I prefer, however, to present these in the incidentally supplement to the next main division of our subject, since they
75.
You
will
have no great value
observe that the
for us.
principle
of reciprocity
is
not
applicable to metric relations, or at least it applies only in indi vidual cases. One reason for this is that in a sheaf there exists
no element which occupies a distinctive position with reference to measurement similar to that of the infinitely distant point in the
HARMONIC FORMS.
5
,
of points while, on the other hand, we recognize in the no segment which could be so defined and characterized by measurement as can the right angle in the sheaf.
range
;
latter
EXAMPLES. 1. To three given elements in each of the one-dimensional primitive forms construct the respective fourth harmonic elements.
2. Through a given point draw a straight line which if produced would pass through the inaccessible point of intersection of two given lines.
Without the use of
3.
having given a parallel to 4.
line
;
segment
of a straight line,
AC of a straight AC and to draw
AC
;
also to divide
A C into
;/
equal parts.
D, are four harmonic points of a described upon AC as diameter, of which S
5.
AC
this line.
In a plane are given a parallelogram and a segment it is required, without the use of circles, to bisect
a parallel to
is
circles bisect a
If
A, B,
C,
the arc subtending the angle
BSD,
or
its
straight line, is
any
supplement,
is
and a
circle
point, prove that
bisected at
A
or
at C.
Through a given point P draw a straight line meeting two given A and B so that (i) the segment AB shall be bisected at Under what (2) the segment AP shall be bisected at B. 6.
lines of the plane in />,
circumstances
is
the solution impossible
?
two points are each harmonically separated from a third point by a pair of opposite edges of a tetrahedron, they are harmonically .separated from each other by the third pair of opposite edges. [The plane of the three points intersects the tetrahedron in a complete quadrilateral whose diagonals intersect in the three points.] 7.
If
Given two pairs of points, A, B, and A, B^ upon the same straight which do no t separate each other. With the aid of circles, find two points which harmonically separate each pair. 8.
line,
[Choose any point
D
and describe the
outside the given line
circles
DAB and DA B intersecting a second time in E. Let the straight line DE cut the given line in O. From O draw a tangent OT to DAB or to DA^B^. The circle whose centre O and radius OT will cut the 1
1
is
given line in the required points.] 9. A A B^
straight line intersects the sides of a triangle
ABC in
the points
CD and the harmonic conjugates A.2 B.2 C.2 of these points, with respect to the two vertices on the same side are determined, so that AC-^BC^ BA^CA*, CB^AB*, are harmonic ranges. Show that A lt B.2 C.2 B\i C* -A-2 CH A*, B.2 are collinear, that AA^ BB.2 CC, are concurrent, and that AA 2 BB^ CCl AA BB* CCl AA^ BB^ CC2 are also con l9
,
,
,
,
5
,
,
current.
,
;
;
,
;
LECTURE
V.
PROJECTIVE PROPERTIES OF ONE-DIMENSIONAL PRIMITIVE FORMS. 76. In
the
lecture I shall
present
again
take
up and
further
develop an idea which has already been mentioned, namely, the correlation of two primitive forms so that to each element of the
one there corresponds one and only one element of the other. As very simple methods of correlating two primitive forms of the first grade the following have already been mentioned :
A
sheaf of rays or planes and a range of points (Fig. 6), or a sheaf of planes and a sheaf of rays, are correlated to each other if each element of the latter lies upon the corresponding element (1)
of the former. (2)
Two
ranges of points are correlated to each other and the same sheaf of rays (Fig. 8).
if
they are
sections of one (3)
Two
(4)
Two
sheaves of rays are correlated to each other if they are one and the same range of points (Fig. 7), or sections of projectors of one and the same sheaf of planes, or both. of one
sheaves of planes are correlated
and the same sheaf of
We
if
they are projectors
rays.
speak of two one-dimensional primitive forms which are correlated to each other in any of the ways men 77.
shall
hereafter
*
tioned above, as being in perspective position, or more briefly, they shall be spoken of as being so that perspective to each other of two perspective primitive forms of different kinds, the one is ;
always c
a
section
of
the
other,
while
on
the
other hand,
two
perspective primitive forms of the same kind are always either sections or projectors of one and the same third primitive form. 78. If two one-dimensional primitive forms are correlated per-
PROJECTIVE PRIMITIVE FORMS. one and the
to
spectively
same
form
third
53
two
example,
(for
ranges of points to a third), they are also correlated to each other, but in general are not in perspective position with regard to each other. We thus observe a second, the so-called skew position of
two correlated primitive forms, which might be obtained from the perspective position by giving one or the other of the perspective forms a slight displacement; then with each element of the one form there would continue to be associated a particular element
of the other form,
but
in
general
would
forms
the
lose
their
perspective position.
Two
79.
may be
primitive forms
correlated to each other,
how
ever, example, two sheaves of rays are correlated by viewing them as projectors of one and the same curve, no matter what this curve may be. The method of correla
numberless other ways
in
tion in question, however,
is
for
;
or
elements
selected
others in one
all
whether the forms are
so,
skew position namely, from one of the
in perspective
are
from
distinguished
important particular, and just as clearly
any four harmonic two forms, to these
if
;
evidently correspond four harmonic elements in the other form, since projectors and sections of harmonic forms are in turn
harmonic forms.
This
peculiarity
the following definition
is
not
in
found
general
and we are thus led
other methods of correlation
in
enunciate
to
:
Two
l
primitive forms are said to be related projectively to each other, or, more briefly, are said to be projective] when they are so correlated that to every set of four harmonic elements in the one
form
there correspond
Two also
perspective
projective,
four harmonic elements
one-dimensional
perspective
primitive
correlation
only by the
being distinguished expressions conformal
The
and
From
the
definition
of
is
the
forms are therefore
and projective
relative
correlation
of the
positions
forms.
homographic, which Paulus and
Chasles use, have the same meaning as
introduced the symbol 7\ for
in the other.
projective.
projective projective
Von
Staudt has
to.
relation
it
follows
im
they are
pro
mediately that
If two forms are each projective
to
a third form
jective to each other.
For example, if two ranges of points are perspective, and hence one and the same third range, then are they pro
projective, to
jective
to
each other, but
it
is
only in particular cases that they
GEOMETRY OF
54
POSITION.
are in perspective position relative to each other. The same istrue of any two primitive forms of the first grade. 80. In two projective ranges ofpoints, to any four points A, B, C, D,,
of the one range, of which the first two are not separated by the last
always i?i the other range four points Aj, B lr which are to the same condition. subject Cj, 1? For there are, in the first range, two points and by which there correspond
two,
D
A
M
N D
harmonically separated from B, and also C from (Art. 66), and in accordance with the definition of projectivity there corre is
spond
A
to these, in the other range,
two points
B
M and ^ by which l
D
On harmonically separated from Y and also Cl from Y this account it is impossible for the points and B^ to be l If A and C are separated from each other by Cl and l (Art. 65). is
l
A
D
separated by
B
B
for the opposite
and D^,
l
and D, then must
also
A
l
and
C
conclusion would
l
be separated by
bring
us into dis
agreement with what has just been proved. If in the one range any number of points A, B, C, P, Q, are so chosen that no two of them are separated by the one named .
just before
and the one
.
.
.
.
.
just after them, there correspond to these, in
A B
the other range, the same number of points 19 v ... Y C^, ... P^ If the points P, Q, R, ... of for which the same relation holds true. the first range are consecutive points of the range, then must also their
corresponding points range, for if
there
P
and
Pv Q v R,
Q
lt
say,
would be points
/i,
l
...
Q
be consecutive points
in the
second
were not consecutive points of this range, VY which separate them, and it would
P and Q be separated from each other by points U and V, and they could not in that
be necessary then that the
corresponding
case be consecutive in the
points.
Similar
case of sheaves of rays
by arbitrary transversals important theorem
conclusions
and of
may be reached
planes, since these are cut
in projective ranges of points.
Hence the
:
If two one-dimensional primitive forms arc projectively related, then to every continuous succession of elements of the one form there corresponds a continuous succession of elements of the other. 81. Two projective primitive forms of the same kind may alsoor be superposed, that is to say, may have the same base. projective sheaves of planes, for example, may be placed with their axes coinciding, and similarly two projective ranges of
be
cofijective
Two
may lie on the same straight line, so that each point of the must be considered twice, once as belonging to the one range,
points line
PROJECTIVE PRIMITIVE FORMS. and again
as belonging to the other range.
how many
to
elements
self-corresponding
The
may
55 investigation as
exist in
two pro-
which are superposed, that jective one-dimensional primitive forms with their homologous coincide form of one elements how is, many elements in the other,
of great importance in
is
all
that follows.
FIG. 21.
82. In the
have clear If
either
first
one
place, that
or
two
it
such
is
possible for the two forms to
self-corresponding
elements
is
from the following theorem in
a plane there are given
two sheaves of rays Sl and (Fig. 21) which are projectors of one and the same range of points /x, i.e. are perspective, and these are S.>
intersected
by a
straight
line
T/,
determined upon line two projective
If in a plane there are given two ranges of points u^ and u.2 (Fig. 22)
which are sections of one and the
same sheaf 5, i.e. are perspective, and these are projected from a point
T
of the
plane, then
this
becomes the centre of two
then there are
point
this
projective sheaves of rays in which
straight
GEOMETRY OF
POSITION.
T
the lines joining with 5 and with are self-corresponding rays,
of points u and w 2 in which the intersection of v with u
u^u.^
and with S\S2 are self-correspondThese two points ing points.
lies
ranges
,
coincide
left
two points
respectively, correspond to each
The
A
three
if
u^y
passes through uv.
if -S^S^
In the theorem on the in a point
These two rays coincide upon ST.
of
other
if
and A 2 of z^ and z/ 2 S^A^ and S2 A 2 intersect
A
l
,
u.
ranges of points
,
u^
and u 2 therefore have the
intersection point uv self-corresponding, while u^ and u 2 have also self-corresponding the intersection point of v with the common ray
83. If, of two projective ranges of points u and u 19 the first is described by the continuous motion of a point P, the corresponding at the same time moves continuously along the other point t
P
B
A
C
FIG. 23.
u,
C,
B
C FIG. 24.
FIG. 25.
range,
P
and
and
P
if
u and u^
may move
FIG. 26.
lie
upon the same straight the same sense
either in
line,
or
the points
in
opposite In the case of figure 24, senses (Figs. 23 and 24) upon the line. in which the points move in the same sense along the line, we 1
shall call the ranges directly projective, in the other case (Fig. 23)
We shall in the same way call two projective and Sl which are concentric and lie in the same 25 and 26), or two projective sheaves of planes which
oppositely projective.
sheaves of rays
plane (Figs.
*S
PROJECTIVE PRIMITIVE FORMS. have the same projective
axis,
(Fig.
directly
projective (Fig.
according as
25),
57
26), or
oppositely
two homologous elements
in
describing the sheaves would rotate in the same or in opposite senses. In primitive forms of the first grade, which are superposed and are oppositely projective, the two describing elements must necessarily coincide twice ; hence,
grade which are superposed and are have two self-corresponding elements oppositely projective, always all other pairs of are separated by these elements homologous "
"
Primitive forms of the
first
;
^ "
self-corresponding elements." On the other hand, forms which are superposed projective
have
and are
directly
self-corresponding elements only in case a
two
AB
(or an angle) of the one lies wholly within the corresponding segment (or corresponding angle) of the other (Fig. 24); in particular cases they have only one, and may have (Fig. 26)
segment
no self-corresponding element.
Two
projective
ranges of points
u and u l which have three self-corresponding points A, B, C, must be directly projective. 84. We can now prove the following fundamental theorem of the Geometry of Position If two projective one-dimensional primitive forms have three selfcorresponding elements A, B, C, then are all their elements self:
orresponding
and
in
the
the
forms are
first
consequetitly identical.
place, that the projective primitive forms
Suppose, are two ranges of points u and
it^
(Fig. 27).
Then
a point which
harmonically separated from any one of the self-corresponding points A, B, C, by the other two must coincide with its correspond ing point, since four harmonic points of // always correspond to is
four harmonic points of u^ by definition, and the harmonic conjugate to two others is uniquely determined.
of a point with respect
AB
which Suppose further that there is given upon that segment does not contain C, a point of u which does not coincide with
P
its
corresponding point
P
l
of u r
If
now we permit
the range // in the sense ABC, then :same sense, and must coincide with
P will P in B, l
P
traverse
or
to traverse u^
perhaps
in
the
before
GEOMETRY OF
58
B
POSITION.
P
B
in some point should reaching Y If CBA, then l would move in the sense
P
P
move
in the opposite
CBA, and would
sense
coincide
A, or before reaching A in a point A v In this way we A l l which is either equal to or is a part of AB, and of which no point, except the extremities A l and v coincides with its corresponding point. But this is impossible, since
with
in
B
should obtain a segment
AB
B
which is harmonically separated from C by A l and B^ must coincide with its corresponding point and fall in this segment. Hence the ranges u and x must have every point of the segment AB self-corresponding, and therefore every other point Q selfcorresponding, since Q is harmonically separated from some point of the segment AB by A and B. The theorem may be proved in an analogous manner for the case of two projective sheaves of rays or planes which have three self-corresponding elements, or what is simpler, these cases may be reduced to the one just proved by intersecting the sheaves with a straight line. The ranges of points thus obtained would be projective and have three self-corresponding elements consequently all their elements would be self-corresponding hence all the elements of the sheaves must be self-corresponding. that point
;
;
Two projective one-dimensional primitive forms can have then most two self-corresponding elements, unless every element of the An one coincides with the corresponding element of the other. 85.
at
fundamental theorem is the following projective to any sheaf, or a sheaf of rays to a sheaf of planes,, and three elements of the first form lie in the second form, then the "upon their corresponding elements
important deduction from "
If a range of points
this
:
is
"
"first
For
form
is
a section of the
second."
has three self-corresponding elements, with that section of the second form which is made by its base ; hence all their it
elements are self-corresponding, and the the section of the second form. 86.
If
two projective sheaves of
rays S and S\ (Fig. 29) lying in the same plane, but not concentric,
have their
common
self-corresponding, projectors of one
range of points
//,
ray a (or a^ then are they
and the same and are con-
sequently in perspective position.
If
first
form
is
identical with
two projective ranges of points
u and u (Fig. 28) which lie in the same plane, but are not conjective r have
their point of intersection
A
self-corresponding, then are they sections of one and the same sheaf of rays 5, and are conse(or
AJ
quently in perspective position.
PKOJECT1VE PRIMITIVE FORMS. if we join the points B and which any two rays b and c of the sheaf S are intersected by
For in
(7,
their corresponding rays b l and c l of S lt by a straight line //, then are
the two ranges of points in which and S l identi // cuts the sheaves .S"
since
cal,
related,
they
and
are
projectively have three self-corre
sponding points ua, ub, and
uc.
59
For if we join any two points B and C of // with their corresponding points B and C\ of //, by the straight lines b and r, and denote l
the point of intersection of these lines by S, then are the two sheaves
of rays by which the ranges of points u and // x are projected from S identical, since they are pro-
and have three self-corre sponding rays SA, SB, and SC.
jective,
FIG. 29.
FIG. 28.
87. The following theorems from the geometry of the bundle of rays are analogous to those just stated, and may be proved in a similar manner :
two protective sheaves of planes whose axes intersect have If
the plane of the axes self-corre sponding, then are they projectors of one and the same sheaf of rays,
and are consequently perspective. For if we intersect the two
If two projective sheaves of rays which are concentric, and lie in different planes have the line of
of these planes selfcorresponding, then are they sec tions of one and the same sheaf intersection
of planes,
and are therefore per
sheaves of planes with a plane which is determined by the lines
spective.
of intersection of any two pairs of homologous planes of the sheaves,
of rays from the line of intersection of any two planes, each of which is
we
For
if
we
project the two sheaves
obtain two project! ve sheaves
determined by a pair of homologous
of rays which have three self-cor responding rays, and which are
rays of the sheaves, we obtain two projective sheaves of planes which
consequently identical.
have
three
planes,
and which are consequently
identical.
self
-
corresponding
GEOMETRY OF
60
88. Two projective but not concentric sheaves of rays S and S (Fig. 29)
have perspective position
of the points of intersection of pairs of homologous rays, any three, if,
and
C,
/?,
/?,
in
lie
one straight
line u.
the
ranges of points in which the two sheaves of rays are cut by the straight line For,
u have
projective
these three points
/>
,
C,
D,
and consequently
all their points all self-corresponding points of intersection of homologous rays of ;
the two sheaves
upon the
consequently
POSITION.
Two
projective but not conjective
ranges of points
have
and
^l
perspective
u^ (Fig. 28)
position
if,
of
the lines joining their homologous
any three, BB V CC^ and DD-^ pass through one point S.
points,
,
For, the projective sheaves of rays by which the two ranges of points are projected from the point
S
have these three rays
BB^ CC^
DD^ and consequently all their rays self-corresponding
homologous
all lines
;
points
of
joining
u and
,
consequently pass through S.
lie
straight line u.
You can easily enunciate and prove the analogous theorems for sheaves of rays and planes in the bundle of rays. 89. If two correlated sheaves of rays lie in one plane and are not concentric, the points of intersection of pairs of homologous rays form a continuous curve, for if a ray describes the one sheaf by continuous rotation about its centre, the corresponding ray will rotate continuously about the other centre, and will describe the other sheaf; the point of intersection of the two rays will conse quently describe a continuous line. If
now
the two sheaves of rays are projectively related but are
not in perspective position, all the points of intersection of pairs of homologous rays lie upon a curve which, in consequence of the
theorem of Art. 88, has
in
common
with no straight line
more
than two points. On account of this peculiarity we shall call this curve a curve or range of points of the second order the ordinary of shall be from it appear^ wherever this, range points distinguished ;
name range of points of the first order. two projective ranges of points lie in the same plane and are
necessary, by the If
not perspective, then, as in the case above, the straight lines joining homologous points form a continuous succession of rays,
pairs of
more than two pass through any point of the plane. designate the totality of these joining lines as a sheaf of rays of the second order, the ordinary sheaf of rays being hereafter de signated a sheaf of the first order to avoid confusion. of which not
We
PKOJECTIVE PRIMITIVE FORMS.
61
Curves and sheaves of rays of the second order are denned from the nature of their formation, then, in the following manner
:
Two projcctivc sheaves of rays (of the first order), which lie in the same plane and are neither con-
Two projective ranges of points (of the first order], which lie in a plane and are neither superposed
centric
nor perspective, generate a curve or range of points of the second order, each ray of the one
nor perspective, generate a sheaf of rays of the second order, each point of the one range being pro-
sheaf intersecting the corresponding ray of the other in a point of this
jectedfrom the corresponding point of the other by a ray of this sheaf,
1
Not more than two rays of a
curve.
Not more than two points of curve of the second order
any straight
lie
this
upon
sheaf of the
second
order pass
through a?iy point,
line.
Wholly analogous forms of the second order are generated in a bundle by projective sheaves of rays and planes. All these new forms will be investigated more closely in the next lecture. go. The construction of curves and sheaves of rays of the second order depends upon the following important theorem :
Two
o?ie-dimensional primitive
forms may always
be correlated pro-
each other so that any three elements of the one shall to three elements of the other chose?i at ra?idotti ; to any correspond fourth element of the one form, the corresponding element of the other jectively
is
to
then uniquely determined.
The proof definition
of this theorem might be deduced directly from the and the fact that, by three elements of
of projectivity
a one-dimensional primitive form, a single fourth element is deter mined, which is harmonically separated from one of these three by
the
other
two.
From
considerations
similar
to
those of Article
appears that by means of the three given pairs of homologous elements indefinitely many such fourth pairs are correlated to each 84,
it
other,
and
that
no element of the one form
is
without a corre
sponding element in the other. however, proceed to prove the theorem in a different but manner, only for two ranges of points, since all other cases be For instance, if one or each may easily reduced to this one. I
shall,
of the two primitive forms is a sheaf, then instead of this we can substitute its section by a straight line, which is a range of points. 91. Suppose, in the first place, that two ranges of points u and u l lying in one plane (Fig. 28), are so correlated projectively that they
GEOMETRY OF
62
A
have the point of intersection sponding, and that the points
POSITION.
(or
A-^ of their bases
B and C of n correspond,
to the points
B^ and C^
of u^
then must the one range of points
;
be a projection of the other from
BB
l
CCr
and
S, the point of intersection of fourth point of u corresponds, then, from the point S.
D
To any
in u^ its projection
D
l
self-corre
respectively,
Next, suppose that any two ranges of points u and w x (Fig. 30), not lying on the same straight line, are so related projectively that
FIG. 30.
points A, B^ C, of u correspond respectively to the points A^ B^ Cv of u v Let us now choose, upon one of the straight a lines joining two corresponding points, for example upon
the
AA^
point
,S
different
from either
A
and
straight
line
u 2 cutting
u,
Project
now
the range
u upon
B^ C2
or
A^
different z/
2
and draw through from either
from the centre
#j S,
or
A a AA Y
and
l
let
be the projections of A, B^ C, upon u 2 By this means A^, our problem is reduced to that of the preceding paragraph. For we have only to so correlate u^ and u 2 that they may have their point .
,
of intersection
A
(or
A
2)
self-corresponding,
and the points
J3 l
PROJECTIVE PRIMITIVE FORMS.
63
of //j corresponding respectively to B.2 and C., of //.,. The u u and can then be of looked as upon ranges points projections l In order to de of one and the same third range of points // 2 in t/ 1 to any point termine the corresponding point in //, l
and Cj
.
D
we
D
D
upon w 2 by the aid of which we can then find according to the method already pointed out. l Suppose, finally, two ranges of points lying upon the same straight find
first
its
projection
D
so
are
line
2
,
correlated projectively to each other that the points
A, B, C, of u correspond respectively to the points A lt B^ Cv of U Y This case can be reduced to the preceding by first projecting u^ If any two corresponding points upon another straight line u.2 .
coincide, for
example,
through that point,
A
A
and
lt
drawn most advantageously
is
it.
2
and so the problem
referred back to the
is
first
-case considered.
92. During the process of these investigations there has arisen the following theorem Two projective one-dimensional primitive forms may always be :
fonsidered as the first and last of a series of forms, of which each is perspective both to the one preceding it atid to the one following it.
Two upon
and last of a series of not more than four ranges of which each is a projection of the adjacent ones. This
of points,
first
fact justifies the
93. At
the
use of the term
If the
:
sides a lt
a*,
...
an
,
of a
simple //-point rotate in order about n fixed points Slt S2
S nj
a.,a3,
while
_!,
...
ni
,
vertices
of the
a^a^
same move
along fixed straight lines // 1} u^ ... #_!, respectively, then the remaining vertex, and likewise ever} other point of intersection of sides of the ;/-point, describes either a curve of the second order or a straight line a straight line certainly in case the ;
fixed centres of rotation
Sn
,
line.
all
of ap
series
of these I shall mention only the
;
variable
...
projective.
same time we obtain very simply a
parently complicated theorems following
may be looked
ranges of points, for example,
projective
as the
lie
upon one
S 52 lt
,
...
fixed straight
If the vertices
A^
A,
A^
of a variable simple //-side in
u lt
...
A
n,
move
order along n fixed straight lines u.2y ... u n while n I of its sides ,
A A2 A A 1
2
,
3,
...
A
H
-iA n
,
rotate
about the fixed points 6\, S^ ... ^-i, respectively, then the remain
A
n Ai, and likewise each diagonal of the //-side, either describes a sheaf of rays of the second
ing side
order or else point
the
;
happens ...
/,
point.
if
all
it
rotates about a fixed
latter
case
certainly
the straight lines intersect
in
u^ u^
one fixed
GEOMETRY OF
64
The about
sides a^ # 2 6\,
^>
of which each points u lt # 2
sheaves of rays
6",,,
forming perspective sections of these sheaves; consequently,
all
of
pairs
happens,
among
This
other ways,
centres of the sheaves
if
all lie
...
An
,
de-
the
to
52
,
...
,
following one, while form the respective
Sn -u
;
consequently,
pairs of these ranges of points, and in particular the first and last, all
are projectively related and generate a sheaf of the second order if
they do not chance to be perspectThis latter case happens, ive. among other ways, if the straight
case
latter
l5
centres of projection
these
sheaves, and in particular the first and last, are projectively related, and generate a curve of the second order if they do not chance to
be perspective.
ive 6\,
-i>
>
A^ A^
vertices
upon z/ z/.,, ... u n ranges of points, of which each lies perspectscribe
perspective to the ranges of
lies
the following one,
The
a m describe
>
...
2,
POSITION.
the
lines
upon
z/ l5
u%,
...
u, n intersect in
one
one straight line for in that case this line would be a self-correspond-
point P, for in that case the ranges
ing ray of all the sheaves, since in the variation of the n-po mt all the
ing, since in the variation of the n-side all the vertices at one time-
;
sides at
all
have
this point self-correspond
coincide with P.
one time coincide with the
line of centres.
These theorems, of which the one on the left is a generalization of a theorem by Maclaurin and Braikenridge, and the other is due to Poncelet, afford us a means of finding any desired number of points of a curve or rays of a sheaf of the second order by linear constructions.
These constructions are simplest when n =
3.
METRIC RELATIONS OF PROJECTIVE PRIMITIVE FORMS OF THE FIRST GRADE. and segments which are formed by any in two projective primitive forms there elements homologous exists an important proportion, which in conclusion I shall now 94.
Among
the angles
four
and a range of with a sheaf of rays four 28 and Any rays a, b, c, */, 29). (Figs. The of u. their of A, B, C, D, corresponding points pass through whose common u of these and two formed rays, by triangles develop.
Let us
set out
points u perspective to
6"
it
-5"
vertex
is S,
have equal altitudes
bases, so that, for
example
tASB ~ AB_ ASD AD But the area of a
;
their areas therefore vary as their
:
triangle
is
a d
kCSB ~ ^CB d CSD CD
equal to half the
product of two-
METRIC RELATIONS OF PROJECT! VE PRIMITIVE FORMS. and the
sides
sine of the included angle.
between two rays
/
and q by
four triangles the values
(pq),
we denote
If
we obtain
65
the angle
for the areas of the
:
&ASB = %AS S3 sin (ab) &ASD = IAS.SD. sin (ad) .
.
;
;
A CSB = I CS SB sin (cb) CSD = lcS.SD.sm (cd); .
.
and the it
these values are substituted
if
common
factors in
;
the above
in
equations,
and
numerator and denominator are removed,
follows that
SB.sm(ab) ~ AB
SD Dividing the
.
sin (ad)
first
,
AD
SB.
sin(^)
SD
sin (cd)
.
~ CB
CD
of these equations by the second
sin^
we obtain
AB CB
s\ncb
:
.
AD C"
Each term of
95. is
this
a ratio
is
proportion
the ratio of the two segments into which
point
C
(which in Fig. 28
lies
outside
for
;
BD
BD), and
s
is
m .
H
example, divided by the
^
is
the ratio
sin (cd)
of the sines of the two angles into which the angle (bd) is divided by the ray c. Thus the left-hand side, and similarly the right-hand side, of this equation is a ratio between two ratios, or a so-called double-ratio,
-
cross-ratio,
or
anharmonic
ratio.
You
will
readily
observe the particular way in which these anharmonic ratios are formed. That on the right, for example, among the segments formed by A, B, D, is obtained by dividing the ratio between
A
and A, respectively the straight lines in M. Show that as v rotates about P, will line which passes through O. [Chasles, loc. cit., in
;
M
Art. 343-1
This proposition If
may
also be stated as follows
the three sides of a variable triangle
:
MA A
rotate about
three
A
and fixed collinear points P, S , S, respectively, while two vertices move upon two fixed straight lines which intersect in O, then the will describe a straight line which also passes through O. third vertex
A
M
In this form
the
proposition
is
upon perspective triangles stated
equivalent to in
Article
7.
Desargues
theorem
State the reciprocal
theorem. 11.
If the
four vertices
A, B,
C,
D, of a variable quadrangle
move
straight lines which pass through one respectively upon of the sides while three AB, BC, CD, rotate about three fixed O, point
four
fixed
collinear points, then the remaining three sides will also rotate about fixed points, these six fixed points forming the vertices of a complete i.e. they lie three by three upon four straight [Cremona, Protective Geometry, Oxford, 1885, Art. in.]
quadrilateral,
lines,
LECTURE
VI.
CURVES, SHEAVES, AND CONES OF THE SECOND ORDER. 99. In the last lecture If lie
we reached the
two projective sheaves of rays one plane, but are neither
in
nor
concentric points
the perspective, intersection of their
of
homologous rays form a curve or range
of
points
of
the
second
following important results
two projective ranges of points one plane, but are neither upon the same straight line nor If
lie
in
perspective, the lines joining pairs
of homologous points form a sheaf of rays of the second order, which
which has not more than two points in common with any
has
straight line.
order.
order,
:
not
common
more than two rays in with any sheaf of the first
In order to give you a definite conception of these forms of the second order, I shall state now, giving proofs later, that the curve of the second order is identical with the conic section, and
hence may be obtained by intersecting an ordinary circular cone with a plane. A sheaf of rays of the second order consists of the system of tangents to such a conic section. 100. To the two preceding theorems from plane geometry, there correspond the following from the geometry of the bundle of rays :
If
two
projective
sheaves
of
planes whose axes intersect are not perspective, the lines of inter section of their
homologous planes
form a cone of the second order, which has not more than two rays in
common
point
with any plane.
of intersection
axes, through which
all
of the
The two
rays of the
If two projective sheaves of rays whose planes intersect are con
centric
not
but
the
perspective,
determined by pairs of homologous rays form a sheaf of planes of the second order, which has not more than two elements planes
in
common
planes
of
with
the
first
any
sheaf
order.
of
The
GEOMETRY OF
;o cone pass,
is
called
the
vertex
of the cone.
POSITION.
common
centre of the sheaves of through which all planes of the generated sheaf pass, is called rays,
the
vertex
of the sheaf of planes.
The cone and the sheaf of planes of the second order may be derived from the curve and the sheaf of rays of the second order by projecting the latter forms from some point not lying in their For, the two projective sheaves of rays S and S1 (Figs. 31 plane. and 33), by which a curve of the second order is generated, are projected from such a point O (your eye, for example) by two projective sheaves of planes which generate a cone of the second order having its vertex at O and passing through the given
In
curve.
the
same way the two
projective ranges of points
u
and
(Fig. 32) which generate a sheaf of rays of the second 7/j order are projected from O by two projective sheaves of rays, and
these in turn generate a sheaf of planes of the second order which passes through the given sheaf of rays and has O for vertex.
Conversely, every cone of the second order is intersected by a not passing through its vertex in a curve of the second
plane order
; for, the two projective sheaves of planes which generate the cone are intersected in two projective sheaves of rays which If, then, more than two rays of the generate the curve of section.
cone were to
lie
of section would
in lie
one plane, more than two points of the curve in one straight line, which, from the theorem
You will quoted at the opening of the lecture, is impossible. easily be able to prove for yourselves the analogous property of a sheaf of planes of the second order, and at the same time you will
recognize the correctness of the following statements
Any
curve or any sheaf of rays
of the second order
is
projected
from a point not lying in its plane by a cone or by a sheaf of planes of the second order.
10 1. You
will
Any cone
:
or any sheaf of planes is intersected
of the second order
by a plane not passing through
its
vertex in a curve or in a sheaf of
rays of the second order.
observe from this that
all
results
which are ob
tained for plane forms of the second order may be immediately carried over by projection to the analogous forms in the bundle
of rays.
I
shall confine myself, therefore, in the first place to
investigation of curves
and sheaves of rays of the second
shall begin with the following observation
:
order,
the
and
CURVES, SHEAVES,
AND CONES OF THE SECOND ORDER.
The curve of the second order k 2 which is generated by two projectile sheaves of rays S and S x (Fig. 31),
The sheaf of rays of the second K 2 which is generated by two
,
Passes through the centres of these sheaves.
or /, of the to the ray lt sheaf i.e. to the line joining the two centres, corresponds in the 6",
S l a ray p l different from Si$, since the sheaves are not per the point of intersection spective sheaf
;
and /i, namely 51? lies there 2 and similarly upon the curve be shown that the centre may
p
fore it
S
is
,
projective ranges of points u and u i (Fig. 32), contains the straight
u and Uj
lines
SS
For
of
order
/
,
also a point of the curve.
ranges of points
upon which the lie.
For to the point uu lt or of the range u, i.e. to the point of intersection of the two straight />,
lines,
corresponds in u a point P l from u\u, since the ranges
different
of points are not perspective the or u^ belongs l ;
joining line therefore to similarly
it
PP
the
sheaf 1C2
may be shown
,
and
that
u
ikewise belongs to the sheaf.
FIG. 31.
102. In the theorem on the
left,
of the preceding article, the
the only ray passing through S1 which has but one point, namely Slt in common with the curve. Any ray a^ of the sheaf Slt different from p^ is intersected by its corresponding ray /j (Fig. 31)
is
2 not coinciding with Sr ray a, in a second point of the curve 1 say therefore that the ray p l touches the curve K in Slt or that /
,
We
it
is
a tangent to
Similarly, in
/
2 .
the theorem on the
the only point of K*, namely, the ray
is
right
the
point
P
l
(Fig.
32)
through which there passes but one ray of x // itself. For through any other point A l T
GEOMETRY OF
POSITION.
A
of u^ there passes a second ray A^A of with and hence A^A differs from u r
2
since
,
P
We
K^
in the ray
To the common ray of two projective sheaves of rays there corre
projective
point of contact of the sheaf
sponds
in
each
sheaf a tangent
To
uv
the
A
call
cannot coincide
P
common
of contact
which
second order which
generated by the sheaves.
:
point
of
two
ranges of points there corresponds in each range a point
to the curve of the second order is
therefore a
l
Hence
of
the
sheaf is
of
the
generated
by the ranges.
FIG. 32.
103. As you know, two one-dimensional primitive forms can be so correlated projectively to each other that to any three elements of the one correspond three elements of the other, chosen arbitrarily,
and the
correlation so established
is unique. construct a curve of the second order by means pf projective sheaves of rays, we may not only choose at random the centres S and Sl of the generating sheaves (Fig. 31), If,
then,
we wish
to
but also three of the points of the curve, namely, the points of intersection of three pairs of corresponding rays of the sheaves. In case one of these three points of intersection coincides with S, the tangent to the curve at the point
and the same thing
is
true for
S may
be chosen
arbitrarily,
Sr
In order to construct a sheaf of rays of the second order by projective ranges of points, we may not only choose at
means of
bases u and u l of the generating ranges (Fig. 32), but also the plane containing u and u v three other rays of the sheaf, namely, the lines joining any three pairs of corresponding points will the
in
CURVES, SHEAVES,
AND CONES OF THE SECOND ORDER.
73
of the ranges. In case one of these joining lines coincides with then the point of contact of the sheaf in the ray u may be //, chosen at will, and the same thing is true for U Y
Consequently the problems
To construct a curve of the second order of which five points, or four points and the tangent at one of them, or three points and the tangents at two of them, are given,
admit of
solution,
l
of corresponding rays a and ;
to
Z>
SD
point
.
the principal diagonals, of the hexagon
are
vertices,
i.e.
the
lines
joining
opposite
AND CONES OF THE SECOND ORDER.
CURVES, SHEAVES,
\Ve accordingly have the following theorems
:
BRIANCHON
PASCAL S THEOREM.*
77
S
THEOREM.
In any simple hexagon which is inscribed in a curve of the second
In any simple hexagon which is formed of six rays of a sheaf of
order, the three pairs of opposite sides intersect in three points of
the second order, the three prin-
one straight
point.
To
1 08.
line.
cipal
diagonals
intersect
one
in
be made perfectly rigorous the demonstration of these
theorems must, I admit, be freed from certain restrictions. The two hexagons which here come into question contain elements which are not chosen freely for on the left, for example, the vertices A, J/, Z x and were chosen at random upon the curve /-, ;
P
,
but the same
is
the centres
and
not true of
6"
and Sr
might be thought that
It
S
of the projective sheaves of rays by which the curve is generated are distinguished from other points of the curve by peculiar properties ; for example, that Pascal s theorem 6"
l
holds true only for those inscribed hexagons of which
two
are
We
vertices.
shall
now remove
this
6"
and
possibility
S
l
by
demonstrating that any two points of the curve whatsoever might be taken as the centres of projective sheaves of rays which generate the curve, and hence that
S and
^
may be
replaced in the Pascal
hexagon by any two points of the curve chosen arbitrarily. The same thing is true of the straight lines u and u 1 which appear as sides in the Brianchon hexagon.
MAL
all
we imagine 109. If in the Pascal hexagon SPS1 1 (Fig. 33) the vertices except to remain fixed while moves along the
A
curve, then
while
L^A
D
the points
dl and d
in
A
or u^ will rotate about Zj l
and
D
move upon
and
MA or
the fixed
DD
//
about
M,
lines
straight
such a way that the straight line l always passes The Pascal theorem therefore holds
through the fixed point
*$.
At
respectively,
projective sheaves of rays, these being projectors of the perspective ranges of points dl and d ; we may thus consider the curve k- to *
Pascal discovered this fundamental property of six points of a conic section 1639 when only 16 years of age. Brianchon published his equally fundamental theorem in 1806 in \hz Journal de V Ecole polytechnique, Vol. xm. in
GEOMETRY OF
78
POSITION.
Z
be generated by the protective sheaves of rays and J/, whose x centres have been chosen arbitrarily upon the curve. Similarly, let us imagine the side SSl of the Brianchon hexagon
SS jRDD Q
to so move as to remain a ray of the sheaf l l (Fig. 34) while the other sides are unaltered ; then Sl describes a range of points S-^-R or rlt and *S describes a range of points SQ l or g
K
l
2
For the point of intersection of the principal and SD moves upon the fixed straight line diagonals S^D^ and describes in this line a range of points u.2 to which q and r^ projective to rv
Q^
We might therefore consider the sheaf of the perspective. second order to be generated by the projective ranges of points
.are
q and r v the Pascal and Brianchon theorems are perfectly general,
Hence
the proof of which has brought us to the following fundamental theorems upon curves and sheaves of rays of the second order :
A
curve of the second order is projected from any two of its points by projective sheaves of rays, those
A
sheaf of rays of the second
order
is
cut by any
two of
its
rays
in projective ranges of points, those
pairs
of rays being homologous which pass through the same point
pairs of points being homologous
of the curve.
sheaf.
1 10.
We
shall
make use
which
lie
upon the same ray of the
of these theorems later in correlating
forms of the second order to each other and to the one-dimensional similar to that in which we have estab primitive forms, in a way lished projective relations
among
the latter forms.
With
this
in
view and upon the basis of these theorems we propose the following definitions
:
Four points of a curve of the harmonic from points if they are projected from and every, consequently any, second order are called
fifth
point
harmonic
of
rays.
the
curve by four
Four rays of a sheaf of the second order are called harmonic if
rays any, fifth
they are
intersected
by
and consequently by
every, ray of the sheaf in four har-
monic
points.
Thus by three points of a curve, or three rays of a sheaf, of the second order, the fourth harmonic is determined unambiguously, and can easily be constructed as soon as it is specified from which of the three given points or rays it is to be separated.
in. Bearing in mind the statement of Article 102 concerning second order and points of contact tangents of the curve of the
CURVES, SHEAVES,
AND CONES OF THE SECOND ORDER.
of the sheaf of rays of the second order, and the 109, we conclude that
79
results of Article
through any point of a curve of the second order there passes one
upon any ray of a sheaf of the
tangent to the curve.
of contact of the sheaf.
second order there
one point
lies
Every curve of the second order, then, is enveloped by a system of tangents, and every sheaf of rays of the second order envelops a series of points of contact. It will be one of the problems of next
my
lecture to
show you
curve of the second order
is
that this system of tangents to a nothing else than a sheaf of rays of
the second order, and that the series of points of contact in a sheaf of rays of the second order constitutes a curve of the second order.
112. Other very important properties of curves and sheaves of rays of the second order are stated in the following theorems, of which we shall hereafter make frequent use :
Two
curves of the second order
coincide if they have in common either Jive points, or four points and the tangent at one of them,
or three points and at two of them. For,
if
we
from one of
S by this is
the tangents
project both curves their
common
points
a single sheaf of rays, to project! ve each of the two
Two sheaves of rays of the second order coincide if they have in com mon and
either five rays, or four rays the point of contact in one of them, or three rays and the points
of contact in two of them. For, to the range of points in which one of the common rays u intersects both sheaves of rays,
projective each
is
of the
ranges of which the two sheaves
sheaves projecting the curves sep from another common
points in of rays are separately intersected
But the latter sheaves point 5 r are identical in any of the three
by a second common ray
cases mentioned, since in each case they have three self-corresponding
identical in
any
mentioned,
since
arately
the rays joining to the three common points of
rays,
S
l
namely:
(i)
the curves, different from and S l (2) the rays joining S\ to the two 6"
if
common
points and to the curves have a common
remaining
5
;
tangent at
5 and ;
(3)
the ray join
ing S l to the third common point, the ray 5X 5 and the tangent at S l if the curves have common tangents
the
latter
*/j.
But
of
points are of the three cases
ranges
each
in
case
they have three self-corresponding points, namely: (i) the points of intersection of /^ with the three
common
rays of the sheaves, differ ent from u and u l (2) the points of intersection of u l with the two ;
remaining common rays and with u if the sheaves have a common point of contact in // and (3) the points of intersection of u l with the ;
GEOMETRY OF
8o at
S and S
v
.
S
Every ray of
there-
POSITION.
third
common
ray,
and with u and
fore intersects both curves a second
the point of contact in n v
time in the same point, and hence the two curves are identical.
have common points of contact in it and 2/ x The rays of the two sheaves, therefore, which
if
the
sheaves
.
pass through the same point of u coincide, and hence the two sheaves are identical.
EXAMPLES. Given five points of a curve of the second order with the aid of s theorem determine upon any straight line passed through one of them, its second point of intersection with the curve, and so construct 1.
;
Pascal
any required number of points of the curve. 2. Given five rays of a sheaf of the second order with the aid of Brianchon s theorem determine the second ray of the sheaf passing through any point of one of the given rays, and so construct any ;
required
number
of rays of the sheaf.
circle is a curve of the second order, and that its 3. system of tangents constitutes a sheaf of rays of the second order. What angle does that segment of a movable tangent to a circle, which lies between two fixed tangents, subtend at the centre ? Is the
Prove that the
angle constant
?
A
variable triangle and ties of the base 4.
A
ASA so moves in its A continually lie upon 1
1
respectively, while the sides
SA
and
SA
1
plane that the extremi
straight lines u and u^ rotate about the fixed point Sy
S remaining of constant magnitude. Show that the base generates a sheaf of rays of the second order, to which u and u l belong. [It will be seen later that the point S is a focus of the curve the angle
AA
l
enveloped by the sheaf of rays.]
two pairs of straight lines a, b and a-^ b lying in the same first pair about S, the second about 6\, so that the angles (ab) and (a^b^ remain of fixed magnitude, and one point of intersection aa^ of a pair of sides traverses a straight line, then each 5.
If
plane rotate, the
of the three remaining points of intersection ab^ a^b, and bb^ describes a curve of the second order passing through S and S^ [Newton s organic method of describing a conic section.] 6. If two concentric sheaves of rays, whose planes intersect at an oblique angle, are so correlated to each other that every ray is per pendicular to its homologous ray, then will they generate a sheaf of
planes of the rotated about
second its
order
vertex, so
;
in
that
other words,
if
one side moves
a right angle be a certain fixed
in
AND CONES OF THE SECOND ORDER.
CURVES, SHEAVES,
81
plane and the other side in a different plane, the plane of the angle envelop a cone of the second order, to which the two fixed planes
will
are tangent. 7. If from any point perpendiculars be let fall upon the tangent planes of a cone of the second order, these will lie upon another cone of the second order.
The tangent planes are generated by two projective sheaves of rays, and from these can be immediately derived two projective sheaves of planes of the first order which generate the second cone. The geometrical
locus of a point S, from which a plane quadrangle is a curve projected by a harmonic sheaf of rays of the second order circumscribing the quandrangle (Art. no). If we draw the fourth harmonic ray to NK, NL, and NM, this will be tangent to the required curve at N* and hence the curve can easily 8.
KLMN
S(KLMN\
is
be constructed. Ascertain whether or not there
is
more than one curve of the second
order satisfying the given conditions. 9.
State the reciprocal of
Remark
Example 8. With Von Staudt we shall
:
a group of four elements
call
of a one-dimensional primitive form, taken in a definite order, a throw Two throws, and abed, are said to be projective when [ Wurf\. the two primitive forms in which they lie can be so related projectively
ABCD
to
each other that
a, b,
d
c,
Suppose abed of the
is
order
first
;
D
of the one correspond to then be generalized as follows a given throw consisting, say, of four rays of a sheaf then all points S, from which a quadrangle
the elements A, B, C,
of the other.
Example
8
may
:
KLMN
projected by a sheaf S(KLMN} projective to abed, lie upon a curve of the second order circumscribing the quadrangle. How would the tangent at be constructed in this general case ?
is
N
The
reciprocal theorem
may
be generalized
D
in a similar
way.
f
two triangles ABC and jE 1 l are inscribed in a curve of 1 the second order k-, they are also circumscribed to another curve of the second order and conversely. The throws A(BCE 1 F } and D^BCE^}, are projective since they If
10.
;
1
consist
A
of corresponding
elements in the projective sheaves of rays curve. If, now, B^C^E^F-^ is the section
and D^ which generate the
and
A (BCE^
made by made D^BCE^F^ by the
of the sheaf
the sheaf
the line
E^,
BCEF
BB^ CQ, order.
and
BCEF
that of
BC, then are also B^C^E^ projective throws, and the six sides of the triangles, BC, EiFn EE^ and FF^ are rays of a sheaf of the second line
The converse may be proved
analogously.
LECTURE
VII.
DEDUCTIONS FROM THE THEOREMS OF PASCAL AND BRIANCHON. The important
properties which have just been proved in the curve and in the sheaf of rays of the hexagon concerning the second order bring us to other theorems no less important con 113.
cerning pentagons, quadrilaterals, and triangles, similarly described. I must preface the deduction of these theorems with a remark
upon the tangents
to the curve,
and the points of contact
in
the
sheaf of rays. 114. Any ray/j which lies in the plane of a curve of the second order and which has only one point Sl in common with it (Fig. tangent to the curve at the point Slt 31), has been called a
and we have found tangent
may be
through
6*!
that through each point of the curve a single Any other ray a 1 of the plane passing
drawn.
If now we rotate cuts the curve in a second point A. Slt its intersection moves along the curve and
A
the ray a t about
approaches indefinitely near to the point S^ while a^ approaches The tangent indefinitely near to the position of the tangent p v thus presents
itself
joins two points of
as the limiting position of a straight line which the curve indefinitely near (consecutive] to each
and this definition clearly applies to tangents not only of Simi curves of the second order, but of any curves whatsoever.
other^
we have named any
P
which passes point l (Fig. 32), through 1 a K* of sheaf of the second order, a point of u^ contact of the sheaf in the ray u lt and have found that upon 2 there lies one, and only one, point of each ray of the sheaf other contact. point A^ of u^ there passes a second Through any
larly,
only one ray
K
ray a of the sheaf.
If
we move
A
l
along u v a traverses the sheaf
THE THEOREMS OF PASCAL AND BRIANCHON.
K
z
and approaches indefinitely near to the position of the ray , In this approaches indefinitely near to the point #! as l r way the point of contact appears as the limiting position of the point of intersection of two rays of the sheaf indefinitely near to each other.
P
A
115.
If,
then,
in
a
hexagon which
is
inscribed in a curve of
the second order, two adjacent vertices approach indefinitely near to each other, the side joining them assumes the position of a tangent to the curve ; and if in a hexagon whose sides are rays of a sheaf of the second order, two adjacent sides approach in definitely near to each other, in the place of the vertex in
we have a point of contact of the become a pentagon, a quadrangle, or a
they intersect
gon
will
sheaf. triangle,
which
The hexa according
/\ FIG. 35.
as
one,
two,
or
three
FIG. 36.
pairs of adjacent
applied to the pentagon, read as follows
elements coincide.
As
the theorems of Pascal and Brianchon
:
In any pentagon inscribed in a curve of the second order, the points of intersection of two pairs of nonadjacent sides lie in a straight line -with that point in which the fifth side is cut by the tangent at the opposite vertex.
/;/
any pentagon formed from
rays of a sheaf of the second order,
two straight lines which join two pairs of non-consecutive vertices
tJie
intersect that straight line which joins the fifth vertex to the point of contact of the opposite side in one
and the same point. This double theorem affords the solution of the following two problems :
Given any five points of a curve of the second order, to draw the tangents at these points with the use of the ruler only.
Given any five rays of a sheaf of the second order, to find their points of contact with the use of the ruler only.
GEOMETRY OF
84 1 1 6.
POSITION.
For the quadrangle and quadrilateral we obtain the following
theorems
(Fig.
37)
:
In any quadrangle inscribed in a curve of the second order, the points
In any quadrilateral formed of rays of a sheaf of the second order,
of intersection of pairs of opposite a straight line with the
the diagonals a?id the straight lines joining the points of contact in op
points of intersection of the tangents at opposite vertices.
posite sides intersect in one point.
sides lie in
FIG. 37.
117.
And
finally for the triangle
The three points in which the any triangle inscribed in a
sides of
we have
the following:
In any triangle whose sides are rays of a sheaf of the second ordery
which join
curve of the second order are inter
the three straight lines
sected by the tangents at the opposite vertices lie upon one straight line.
the vertices to the points of contact on the opposite sides intersect in one
point.
118. All these theorems, which prove very useful for the solution of a series of simple problems (in particular those of Art. 103),
admit of direct deduction without reference to the hexagon.
by way of
upon the
I shall
give you the direct proof for the theorems quadrangle, since this proof discloses important new illustration
THE THEOREMS OF PASCAL AND BRIANCHON. and since
properties of protective primitive forms, of these in deducing other results.
shall
I
85
make use
order readily to find in one of two projective sheaves .S 33) the ray corresponding to any chosen ray of the l (Fig. other, we construct (Art. id^) a third sheaf of rays S* perspective to each of the given sheaves. For this purpose we intersect
In
and
S
S
and
the ranges of points u and // 1} respectively, by two the point of intersection of any passing through two homologous rays a and of the given sheaves ; since these .Sj
straight
in
A
lines
^
two ranges are perspective, the sheaf is the one required. This
is
equally true
so that the points
if
fra
l
S.2 of
which they are sections
u coincides with a^ and u l with a (Fig. 38),
and ^0, or
B
and
B^
ca^
and ^a, or
FIG. 38.
C and C
etc., in which two corresponding rays a and a^ are cut lt by any two others, as b and b^ c and cv alternately, lie in a
straight
line
with a fixed point
S.2
If
.
l
is
ly
with Sj
SD falls
^5
The
Z>
1
-S
the ray
this point S2 a a t and a in the
through
DD passed which cuts the rays points D and D respectively, then are SD and S and 6^. If now we bring D rays of the sheaves straight line
l
1
corresponding
into coincidence
so that to this ray of S corresponds upon SSZ or or q^ which joins the centres of the sheaves 5 and Sr straight line SS2 or q is thus a tangent to the curve of the q>
second order generated by S and Slt and similarly is also 2 or/j a tangent. Consequently the fixed point S2 is the intersection of the two tangents q and p^ drawn at and Sv respectively, i.e. the intersection of the two rays which correspond (in the two sheaves)
S^
6"
to the
common
whether we
let
ray //
x
We
6"^.
and
//
obtain therefore the same point S2 respectively with a and a l or
coincide
GEOMETRY OF
86
POSITION.
with some other pair of homologous rays (as b and b^ or c and c-^, That is to say The straight line which joins the points in which :
any two pairs whatsoever of homologous rays intersect alternately and a^b, ac^ and a^c, etc.), passes through the point S2 (e.g., ab^ On the other hand, in order easil^to find the point in one of .
two projective ranges u and u^
(Fig.
34)
which corresponds to
any point of the other, we determine (Art. 104) a third range u% which is perspective to each of the given ranges in the following way Project u and u^ by two sheaves of rays S and S^ whose centres are chosen upon the straight line a joining any two :
homologous points
A
and
A
l
of the
sheaves are perspective, the range jectors is the one sought.
u.2
given
ranges
;
since these
of which they are both pro
FIG. 39.
If will
A
now we let S coincide with A^ and S\ with (Fig. 39), u^ pass through those two points of u and u^ which correspond
D
and For, two arbitrary points point of intersection. A and from are to each other which projected l correspond l which intersect A respectively by two rays A^D and upon l but obviously this intersection, and con the straight line u 2 if is brought l sequently also D^ coincides with the point u t u 2 or to
their
D
AD
;
P
D
uu 1 or P, so that u 2 passes through one (and and Q which the other) of the two points similarly through l // and ur In other of intersection the to point P, Q v correspond
into coincidence with
P
words,
2/2
joins
the points of contact of the sheaf generated by therefore always obtain lines.
u and u l9 lying on these two
We
THE THEOREMS OF PASCAL AND BRIANCHON. the same
as
any two (e.g.,
line u.2 whether we let the centres St and S and A l or with some other pair of homologous and B or C and Cv The point of intersection of
straight
coincide with points,
87
B
A
straight lines joining pairs of
the lines
The
homologous points
BC^ and ^C^lies upon
the straight line
alternately
u.2
.
119. may be arranged in the following double theorem so as to show their relation to each other results just obtained
:
The two points ab l and a^, in which any two pairs a, a-^ and b, b v of homologous rays in projective sheaves S and 5X intersect alternately, lie in a straight line with the point S% in which the rays
corresponding to the
SS
1
common
ray
intersect.
1 20.
You
will
ABl and any two pairs of points A, A l and of projective ranges u and
The two
straight lines
A^B, which homologous B,
B
join
ul
alternately, intersect upon the straight line u 2 which joins those
points of the ranges corresponding to the intersection point of u and u v
immediately recognize
in
these
theorems those
and
already stated concerning the quadrangle in the curve
sheaf of the second order from the following remarks In the curve of the second order, which is generated by the sheaves S and 6\, the points S, aa lt Slt and bb determine an inscribed quadrangle in which a and b^ also a and &, are opposite sides, while the two tangents which touch the curve in the opposite vertices S and S 1 intersect in
in the
:
In the sheaf of rays of the second which is generated by the
order,
ranges of points u and
AA
j,
the rays
u^ and BB-^ form a quad rangle in which A and B lt also A l and B, are opposite vertices, while the two points of contact in the opposite sides u and u l lie upon u 2 //,
lt
.
S.,.
It is apparent that the theorems of the last Article afford a very convenient method of determining that element in one of two pro jective primitive forms of one dimension which corresponds to any
element of the other. For example, if in two projective ranges of points u and // x (Fig. 39) three pairs of corresponding points are given, the straight line u.2 is obtained immediately from
given
these,
and
this
in
turn
very simply determines
the point of u l
corresponding to any given point of u. ^ 121. The theorems upon the quadrangle in the curve and in the sheaf of rays of the second order, proved here for a second time, may be stated in the following general form :
If foitr points K, L, M, N, of a curve of the second order determine
Iffour rays k, 1, m, n, of a sheaf of the second order determine a
GEOMETRY OF
88
a complete quadrangle and their tangents k, 1, m, n, a complete quad
POSITION.
complete quadrilateral and their points of contact a complete quad rangle, the three pairs of opposite
rilateral, the three
pairs of opposite of the quadrilateral lie upon the straight lines joining the points X, Y, Z, inf which pairs of opposite sides of the quadrangle
sides ofthe quadrangle pass througJi the intersection points X, Y, Z, of
vertices
the straight lines (diagonals) join
ing pairs of opposite vertices of the
intersect {Fig. 37).
quadrilateral {Fig.
37).
For, the theorems as previously stated are true for each of the three simple quadrangles comprised within the complete quadrangle
KLMN on
the
left,
and
each of the three simple quadrilaterals
for
comprised within the complete quadrilateral
klmn on the
right.
,M jn
FIG. 37.
But
in
this
same thing whose sides is
form the theorem on the
as that
on the
right
;
for
left
states
exactly
the
both state that the triangle
join the pairs of opposite vertices of the quadrilateral
identical with the triangle
XYZ,
in
whose
vertices the pairs
of
opposite sides of the quadrangle intersect.
And
conversely,
if
a quadrangle
KLMN have such position relative
klmn circumscribing it, then a curve of the second order can be constructed which touches the straight lines k, /, m, n, to a quadrilateral
UNI THE THEOREMS OF PASCAL AND BRIANCHON. in the points AT, Z,
J/",
N;
or,
89
a sheaf of rays of the second order Z, M, N, as points of
can be constructed which has the points
A",
contact in the rays k, /, m, n. For in consequence of our theorem .a curve of the second order which passes through the points Z, M, N, and touches the straight line k in AT has also the A",
and the sheaf of the second order, n, as tangents as point of contact the which contains rays k, /, m, n, and has Therefore four in k, has also Z, M, IV, as points of contact.
straight lines
m,
/,
;
K
tangents of a curve of the second order and their four points of may always be looked upon as four rays of a sheaf of the
contact
second order and their four points of contact. A curve of the second order, which passes through three points of contact Z, J/, of a sheaf of rays of the second order, and A",
K
two of these points and Z has as tangents the rays k and /, which to these respectively, points of contact belong, passes con in
sequently through every fourth point of contact JV, and has the ray to which this point of contact belongs, as tangent in this point. Conversely, a sheaf of rays of the second order contains
of the sheaf
,
every tangent of a curve of the second order if only it contains three of these tangents, the points of contact in two of them being points of the curve.
We and
thus have the following beautiful relation between the curve the sheaf of rays of the second order :
The tangents of a curve of the second order form a sheaf of rays of the second order. 122.
On
The points of contact of a sheaf of rays of the second order form a curve of the second order.
account of their importance we shall prove these relations
again, and by a method in which a new and interesting property of the curve of the second order is brought to light.
N
Of the four vertices Z, J/, (Fig. 37), of a quadrangle inscribed in a curve of the second order, let any one of them, for example, move upon the curve, while the remaining three A",
K
Then the tangent k their tangents do not alter their positions. of the point AT glides along the curve, and its points of intersection, and A, with the tangents n and /, move along these tangents.
and
E
E
A
It is easy to see that by this motion describe two projective and ranges of points upon n and /, respectively, and hence that the tangent For the two k describes a sheaf of rays of the second order.
diagonals
EB and AD of the
quadrilateral k,
/,
m,
n, intersect
always
GEOMETRY OF
90
Y
in a point
POSITION.
LN, and
of the fixed straight line
describe in con
B
and D, two perspective sheavessequence, about the fixed vertices of rays ; the ranges of points n and / described by the points and are sections of these, and are therefore projective.
E
A
Hence
the moving tangent generates a sheaf of rays of the second order. That the points of contact in a sheaf of rays of the second order form a curve of the second order may be proved in a similar
manner.
The
straight line
MK always
passes through the point
Y, and,,
by the motion of the point K, describes a sheaf of rays about the fixed point M, which is perspective to the sheaf described
B
by
BE, and
From
by E.
Of
hence
is
this
it
projective to the range of points n described follows as traverses the whole curve
K
:
a curve of the second order if there be given any fixed point M,
and any fixed
tangent n,
K
projecting a point
and we
correlate to each
of the curve, that point of n
ray through
M,
through which
passes the tangent at the point K, then the sheaf of rays
M
is
pro-
jectively related to the range of points h.
which Chasles places
This theorem,
on conic
treatise
was
sections,
is,
at
the
beginning of
his-
in the plane, self-reciprocal, since, as
now shown,
or as follows directly from the theorem itself, the tangents to a curve of the second order form a sheaf of rays of the second order. I
just
shall only
add that
The tangents at four harmonic points of a curve of order are harmonic tangents.
That
is
to say, they are cut
by any
fifth
the second
tangent in four harmonic
four points of contact are projected from any fifth point of the curve by four harmonic rays. The fact that a sheaf of the second order is cut by any two of
points, since their
rays in projective ranges of points may now be stated thus sheaf of rays formed by the tangents to a curve of the second order is cut by any two of these tangents in projective
its
:
"The
"
"ranges
123.
of
The
points."
reciprocal
relation
existing
between
theorem and the Pascal theorem may be seen former "
"
is
The
stated as follows
three
"the
Brianchon
when
the
:
hexagon circumscribed whose sides are tangent to
principal diagonals of any
to a curve of the
the
directly
second order
curve) intersect in one
(i.e.
point."
THE THEOREMS OF PASCAL AND BRIANCHON. So
also the theorems
9
upon the pentagon, the quandrangle, and
the triangle in a sheaf of rays of the second order may be con veniently stated as referring to figures circumscribing a curve
of the second order.
do more at this point than to reproduce in single one of the previously proved theorems relating forms of the second order lying in a bundle of rays, and its
to
shall
I
124. this
not
new form a
It has already been shown (Art. 100) that any curve or any sheaf of rays of the second order is projected from a point not lying in the same plane by a cone or by a sheaf of planes of
reciprocal.
the second order.
Every tangent to the curve is projected by a which has but one ray in common with the cone, and is there plane
fore called a
tangent plane. Similarly every point of contact of the sheaf of rays of the second order is projected by a so-called con tact ray of the sheaf of planes, through which passes only one plane *
Since, then, every cone and every sheaf of planes of the second order is cut by a plane not containing its centre, in a curve or in a sheaf of rays of the second order, it follows that
of the sheaf.
The tangent planes of a cone of the second order form a sheaf of planes of the second order. The rays of a cone of the
second order are projected from any two of them by projective sheaves of planes (comp. Art. 109). 125.
As
in Article
1
10
they are projected from any, and hence from every, fifth ray of the
cone by harmonic planes.
Pascal and
In any hexagonal pyramid inin a cone of the second
intersect
which
lie
in
pairs of opposite in
two of them
in projective
sheaves
of rays. set
up the following
definitions
:
Four tangent planes of a cone the second order are called harmonic tangent planes if they are cut by any, and hence by every, plane tangent to the cone in four harmonic rays. of
:
scribed
order, the three
The tangent planes of a cone of the second order are cut by any
Brianchon theorems may be stated thus for
the cone of the second order
faces
contact rays of a sheaf of planes of the second order form a cone of the second order.
we may now
Four rays of a cone of the second order are called harmonic rays if
The
The
straight
one plane.
lines
In any hexagonal pyramid circumscribed to a cone of the second order, the three principal diagonal intersect in one straight
planes line.
C)
GEOMETRY OF
2
It
would be a useful exercise
POSITION.
to
transfer
remaining theorems which we have proved corresponding forms in the bundle.
for
the
yourselves
for plane figures, to the
126. At this point I shall return to the curve of the second order to draw some inferences concerning the forms which it takes in the plane, from the fact that it has not more than two points
common
with any straight
In consequence of this property with the infinitely distant line of its plane either no point, or one point at which the line is tangent to the curve, or, finally, two points in which it is intersected by this in
such a curve has
in
line.
common
line.
In the
case
first
points of the curve are in the finite region
all
such of the plane, and all tangents are actual rays of the plane a curve is called an ellipse (see Figs. 33 to 39). In the second case the curve extends indefinitely with two branches towards the point in which it is touched by the infinitely ;
distant Fig.
form
This
line.
straight
called
is
a
In the third case the curve consists of two curved
of which extends with two branches to
which the two curved is
(see
parabola
32).
called
lines are
lines,
each
infinitely distant points at
connected
;
in this case the
curve
an
hyperbola (Fig. 31). Since the infinitely distant straight line of the
hyperbola,
two
all
infinitely
tangents distant
tangents which are called the
to
this
points,
curve,
are
in
plane cuts the particular those at the
The actual rays of the plane. distant infinitely points
touch the hyperbola in asymptotes
of the curve.
127. These three varieties of the curve of the second order may be cut from any cone of the second order whose vertex A plane does not lie infinitely distant. passed through the ,
RU
harmonically separated by the tangents a and
U
and
V
are harmonically separated
two tangents //
L
ue.
and
since the points
SU
is harmonically separated so that the statement (3) above correctness of statement (i) follows immediately from ;
similarly,
by the tangents b and
The
c,
by the points of contact of
d,
theorem upon the circumscribed quadrilateral already proved .
fe
116).
$3.
If the polar
u of a point
U
(Fig. 42)
cuts the curve of
U
c^
with second order, the straight lines which join the point the two intersections are tangent to the curve at these points. For, if either of these lines had a second point in common with
U
the curve, these two points would be harmonically separated by The straight line u u, and the first could then not lie upon u.
and is
therefore the chord of contact for the point U, that
is,
the chord
GEOMETRY OF
100
POSITION.
which joins the points of contact of the two tangents drawn from U.
may be summarized
134. All these results If
through a point
U
which
lies
If
to the
as follows
curve
:
from each one of any number
plane of a curve of the second order, but not upon the
of points of a straight line u which lies in the plane of a curve of the
any number of secants of drawn and we deter
second order, but which does not touch the curve, two tangents be drawn to the curve and we deter
the
in
curve,
the curve be
mine
mine
The
The
points of intersection of the pairs of opposite sides of any simple quadrangle inscribed in the
circumscribing quadrilateral whose opposite sides form a pair of these
curve which has two of these se
tangents
cants as diagonals
point
(1)
;
(2) The point upon any secant which is harmonically separated from U by the points of intersection
with the curve
;
(1)
The common
two tangents
point of the at the intersections
with the curve of any one of these secants
drawn from the
same
;
For any point of z/, the which is harmonically separated from u by the two tan gents to the curve drawn from this (2)
straight line
point (3)
diagonals of any simple
(3)
;
The
straight line which joins
the points of contact of any one of these pairs of tangents ;
;
(4) The points of contact of the tangents which can be drawn to the curve from
U
Then
all
;
these points
straight line
u which
is
upon a
lie
called the
U
with respect polar of the point to the curve of the second order.
(4)
u
If
tangents section
Then
cuts the curve, the two the points of inter
at
;
all
these straight lines pass
through a point
U
which
is
called
the pole of the straight line u with respect to the curve of the second order.
jg
A
lies upon a curve of the second order, and
conjugate to the line joining this point with /, then these rays envelop a curve of the the polar
u
;
of the
curve touches point
/,
the
given line T/, and the tangents at the two points in which the given curve
is
cut
when
by
v, if at
all.
It
is
U lies
upon one of these two tangents, i.e. when u and only
i>
intersect in a point of the given curve, that we obtain a sheaf of
rays of the
first
order instead of
THE POLE AND POLAR RELATION. we
that
of the
107
the system of tangents to a curve of the second order,
obtain a range of points order instead of a curve
first
of the second order.*
Suppose now,
145.
the
in
theorem of the left-hand column of
the last Article, that the given curve of the second order and the remain fixed, then to every point of the plane there will point
U
correspond one conjugate point which it
through
lies
in a straight line
U\
spond in general theorem on the right, the given curve and the
straight line v
fixed, to every ray of the plane there will correspond
ray, intersecting
order
first
with
to every straight line of the plane there will corre a curve of the second order. Similarly, if in the
in a point of v
it
there
will
correspond
We
;
remain
one conjugate
to every sheaf of rays of the
in
general
a
sheaf of rays
of
way two particular cases of the so-called geometric transformation of the second order. 146. From the demonstrations in Arts. 143 and 144 we may derive the following theorems the second order.
obtain in this
:
If a triangle AMBj (Fig. 44) is inscribed in a curve of the second
any straight
order,
to
one side
line
AB
which
is
cuts the
X conjugate other two sides in conjugate points j
If a triangle
UVW
(Fig. 45) is of the
circumscribed to a curve
second order, any point which is is proconjugate to one vertex jected from the other two vertices
W
and
and
by conjugate rays ;
is
if a point is projected from two vertices of the triangle by conjugate rays it lies upon the polar of the
conversely, if a straight line cut by two sides of the triangle
in conjugate points it passes through the pole of the third side.
conversely,
tJiird vertex.
AM
The
01 u and or v (Fig. 44) are perspectranges of points each other if to every point in u is correlated its of // and v is selfconjugate point in v, since the common point
B^M
ively related to
conjugate.
must *
lie
As
tioned
upon
M
of the sheaf of rays generated by // and v the tangent constructed at A, since the intersection A^
The
centre
6"
a special case of the theorem on the
left,
the following
may be men
:
"The
middle points of those chords of a curve of the second order which lie upon another curve
converge toward any given actual point of the plane of the second order."
bisection point straight line v in this case lies infinitely distant, and each harmonically separated by the curve from an infinitely distant point, and hence is its conjugate.
The
is
GEOMETRY OF
io8
POSITION.
of this tangent with v
is conjugate to the point A. the tangent at the point JB^. Consequently the side and every straight line passing
upon
AB^
S
Similarly S lies is the pole of
through
v in conjugate points. an analogous manner
S
cuts u
and
The theorem on ;
its
the right can be proved in correctness, however, follows from the
principle of reciprocity.
FIG. 44.
FIG. 45-
147. I shall close this series of theorems with a proof of the following one, which also admits a reciprocal :
FIG. 4 6.
If a curve of the second order
and
BD
(Fig.
is
cut by
two conjugate rays AC B, C, D, are four
46), the points of intersection A,
THE POLE AND POLAR RELATION. harmonic points, and the tangents harmonic tangents.
The must
pole
lie
Q
of
upon the
conjugate to
AC;
AC,
109
at these points are four
a, b, c, d,
which the tangents a and
in
straight line similarly,
BD,
since this
c intersect,
by supposition, the point of intersection of b and d is,
R
upon AC. If P be the point of intersection of AC and BD, then are P, B, Q, D, four harmonic points, and RQ, b, RP, d, four harmonic rays. Thus, also, CA, CB, c, CD, are four harmonic rays, and the points ca, cb, C, cd, four harmonic points; that is, lies
the points A, B, C, D, are projected from C, and consequently from any other point of the curve, by four harmonic rays, and the
tangents
a,
b,
c,
tangent, in four
d,
are cut by
harmonic
c,
and consequently by any other
points.
148. All these theorems
curve of the second order
which have been enunciated for the may be transferred immediately to the
cone of the second order,
since this is cut by any plane not passing through its vertex in a curve of the second order. I shall here state only the following theorem "If there be given in a bundle of rays a cone of the second "order and a s are ray s not lying upon this cone, and :
through
any number of planes cutting the cone, and there are then determined
"passed "
"
i. "s
is
harmonically separated from
by the cone;
The common
ray of the two planes which touch the cone along the lines of intersection of any plane The lines of intersection of the pairs of opposite faces of a four-edge inscribed in the cone, whose diagonal planes are two of the planes passed through s ; The rays of contact of the two planes, if any, which can be drawn through ^ tangent to the cone "2.
"
In each plane the ray which
;
"3.
"
"
"4.
"
;
these rays lie in one plane o-, which is called the polar plane of the ray s with respect to the cone." Simply adding that s is called the pole-ray of the plane with respect to the cone of the second order, I shall leave you to transcribe for yourselves the remaining theorems of the polar theory so as to be applicable to the cone. "All
"
,
such
w.
at
most
three,
The two
common
poles
tangent
coincide, so that each of them is correlated to all the rays through
to all the points of a straight line.
If
such
All
tangents
through one point have in^common
at least
polars
straight lines through one
point correspond, in general, to the tangents to a curve of the second order.
of
lines
common
most
at
IV.
curves
straight
In such a correlation,
The
line
the
point.
second
order
circumscribe
one
quadrangle, the three pairs of opposite sides of this quadrangle intersect These three points are the vertices, and the in the points /, V, W. straight lines u, v, w, are the sides of a the two given curves. [If the
common
self-polar triangle of
two given curves intersect in four (real) points or in no (real) on the one hand, the curves of the second order which
points, then,
EXAMPLES.
1 1 1
in common, and, correspond to straight lines have three (real) points to which curves points, i.e., whose correspond on the other hand, the three (real) tangents have one to point, through rays tangents correspond the in common triangle common to the two given
self-conjugate
;
If, however, the given curves being in both of these cases wholly real. curves intersect in two and only two (real) points, then of the common vertex and one side are real. H.] self-conjugate triangle only one 1 and a ray u a cone of the second order 6. If in a bundle of
K
rays
be chosen, a one to one correspondence may be established between 2 are conjugate with respect to A^ rays a and a^ of the bundle which and lie in a plane with u. If a describes a plane a, a will generate a cone of the second order which passes through u and through the 2 and which has in common with any rays pole-ray of the plane a, common to K* and a, and to K* and the polar plane of u. State the ,
K
^
reciprocal.
[Remark pertain
to
:
The geometrical
relations expressed in
examples
5
and 6
which the theory will be quadratic transformations, of of these lectures. They can fully in the second volume
given more be utilized in ones.
transforming simple forms into more complicated these transformations that known as the Principle of in the treated is and a merits Radii particular place,
Among
Reciprocal
of great importance not only in is It to this volume. of Mathematical Synthetic Geometry, but also in certain investigations of in the Functions.] Theory Physics and
Appendix
LECTURE
IX.
DIAMETERS AND AXES OF CURVES OF THE SECOND ORDER, THE ALGEBRAIC EQUATIONS OF THESE CURVES. 149.
From
the following "
the general theory of the pole
The middle
"order
lie
and polar we derive
:
points of parallel chords of a curve of the second
upon a
straight
line"
(see Fig. 47).
FIG. 47.
For, all these middle points are harmonically separated by the curve from the infinitely distant point of intersection of the parallel
and consequently lie upon the polar of that Such a line is called a diameter of the curve.
chords,
The
polar of any infinitely distant point of the plane with respect second order is a diameter of the curve.
to a curve of the
A
point.
diameter bisects
all
chords conjugate to
it.
DIAMETERS AND AXES.
113
The
points of contact also of the two tangents conjugate to The this diameter. (if such there are) lie upon chord of at the extremities which meet the curve any tangents
any diameter
diameter
a
to
conjugate
intersect
in
a point
diameter
the
of
(Art. 134).
We
have already found (Art. 137) that the polars of the For the of a points straight line intersect in the pole of that line. diameters of a curve of the second order this statement becomes 150.
"
The diameters
of a curve of the second order intersect in one
point, namely, in the pole of the infinitely distant If the
to
it, "
and
its
is
The diameters
"infinitely
line."
a parabola, the infinitely distant line is tangent point of contact is its pole (Art. 135); hence,
curve
of a parabola are parallel,
distant
and pass through
its
point."
If, on the other hand, the curve is an ellipse or hyperbola, the This is known pole of the infinitely distant line is a finite point. as the centre of the curve, and possesses the following property :
"
Every chord of a curve of the second order which passes
"through
the centre
For the centre
is
bisected in that
point."
harmonically separated from the infinitely distant point of the chord by the two points which the chord has in common with the curve. 151.
The
is
parabola has
no actual
centre,
as
seen from the
is
following theorem If two chords of a curve of the second order bisect each other, :
"
"
point of intersection is the pole of the infinitely distant line, and the chords themselves are diameters of the "straight their
"
curve."
The
correctness of this theorem follows from the fact that this
point of intersection is harmonically separated by the curve from the infinitely distant point of each chord. Since now the pole of the infinitely distant line with respect to a parabola is the infinitelydistant point of the parabola,
there can be
no two chords of a
parabola which bisect each other. The asymptotes of a hyperbola intersect at the 152. "
For the tangents at the points a curve of the second order a;
common fa
?
to a
c
t
centre."
raight line
and
through the pole of the
line.
The
centre of a hyperbola lies outside the curve ellipse, inside the curve (Art. 136).
H
;
that of
an
GEOMETRY OF
114
To
I 53*
POSITION.
each diameter of an
ellipse or hyperbola there is a conjugate diameters, each passes through the infinitely distant pole of the other. Any two conjugate diameters of an -ellipse or hyperbola form
conjugate
diameter
;
two
of
"
the infinitely distant line a self-polar triangle with respect to
"with "
the
curve."
Every chord of the conjugate
diameters
curve
is
which
is
bisected
the
parallel
to
one of two
since
it other, by passes the of the latter. through pole If one of two conjugate diameters cuts the curve,* the tangents at its points of intersection with the curve are parallel to the other
Thus
diameter.
the
conjugate
to
any diameter can
easily
be
constructed.
154. The diagonals of any paralelogram circumscribed to a curve of the second order are conjugate diameters of the curve. The sides of any parallelogram inscribed in a curve of the second order are parallel to a pair of conjugate diameters.
FIG. 48.
The
diagonals of both the inscribed and circumscribed parallelo grams (Fig. 48) are diameters of the curve, since their point of intersection in either case has the infinitely distant line for polar (Art. If
p
134).
we draw thrown
and q *
Ms
Note that of two conjugate
points.
point of intersection two straight lines
parallel to the sides of the inscribed parallelogram, each
H.
lines at least
one must cut the curve in
real
DIAMETERS AND AXES. of these bisects the sides parallel to the other
;
thus
p and
q are
conjugate diameters.
Upon p and parallelogram
;
q
also -the poles of the four sides of the inscribed
lie
p and
or
quadrangle whose
sides
q are the diagonals of that circumscribed touch the curve at the vertices of the
But this circumscribed quadrangle is a quadrangle. the since tangents at the extremities of a diameter parallelogram, .are parallel, and it is evident that this circumscribed parallelogram inscribed
may be 155.
considered perfectly general.
The second theorem of the preceding article may be stated thus:
The two chords which join any point of an to
ellipse
or hyperbola
the extremities of a diameter are parallel to a pair of conjugate
diameters.
Hence, if there be given two pairs of conjugate diameters and one point of a curve of the second order, five other points of the Draw the diameter which passes through curve can easily be found.
P
and determine its second intersection the given point is bisected at the centre. the curve, from the fact that
PQ
Q
with
With
two parallelograms, the sides of each of the to one given pairs of conjugate diameters the two parallel new pairs of vertices of the parallelograms so determined lie upon the curve. In a similar way six tangents to a curve of the second
PQ
as diagonal construct
;
order can easily be obtained if one tangent and two pairs of conjugate diameters are known. 156. Jf all pairs of conjugate dia meters of a curve of the second order are
at right angles the curve is a circle. For under these circumstances the adjacent sides of any inscribed parallel
ogram ogram
are at right angles ; the parallel is thus a rectangle, and its
diagonals are equal.
That
is,
any two
diameters of the curve, and hence
all
diameters, are of equal lengths. That the circle is a curve of the
second
order
follows
from
the
FIG. 49.
fact
that angles at the circumference subtended by the same arc (as and LAS^B, Fig. 49) are equal. By virtue of this pro the sheaves of rays ,5 and Slf circle be two perty may generated by
^ASB
GEOMETRY OF
n6
POSITION.
which are equal, and hence projective. In a similar way it may be shown directly that the tangents to a circle form a sheaf of
The
rays of the second order.
conic section
of ancient
times,
namely, the curve in which a cone having a circular base is cut by a plane, is therefore a curve of the second order, for a circular
cone
is projected from any two of its rays by projective sheaves of planes. We shall show later that not only is every conic section a curve of the second order, but also that every curve of the second
order
a conic section, or that through any curve of the second
is
may be passed which have circular sections. the projective properties of circles are easily obtained, Since 157. most authors who, like Steiner and Chasles, have based the modern order cones
geometry upon metric for
point this
relations,
have chosen the
circle as starting-
the study of curves of the second order.
course, however,
it
becomes necessary
to
show
By choosing that
no other
curves of the second order than the conic sections can be generated For, if others by projective one-dimensional primitive forms.
should to
be
exist,
all
theorems enunciated for
investigated
for
the circle would
them independently.
theorem that a curve of the second order
For
is
example,
need the
projected from any
its points by projective sheaves of rays, and that its is cut by any two tangents in of tangents projective ranges. system of points, would need to be proved separately for curves of the That is to say, second order other than the conic sections.
two of
theorems which have been proved for the circle can be directly extended only to sections of a circular cone. 158. Jf a curve of the second order has more than one pair of conjugate diameters at right angles, it must be a circle. and of any diameter For, if we draw from the extremities
B
A
two normal conjugate diameters, we obtain inscribed in the curve of the second order
straight lines parallel to
a rectangle, which
and
is
also
is
inscriptible
in
a
circle
of which
AB
is
a diameter.
Any second pair of normal conjugate diameters would give rise to Thus the a second such rectangle upon the same diagonal AB.
A
B
and in common circle would have at least four points besides with the curve of the second order, and would therefore wholly coincide with If
it
(Art. 112).
two conjugate diameters are
159. they are called axes,
are
known
and
at right angles to
each other
their points of intersection with the curve
as the vertices of the curve.
DIAMETERS AND AXES. The
circle
117
alone has more than one pair of axes
pair of conjugate diameters being axes. In order to construct the axes of an ellipse or
proceed as follows
;
for
it,
any
hyperbola we
:
AB
of the given curve Construct a circle having any diameter This in general cuts the tangents, and 50) as its diameter.
(Fig.
hence also the curve, of
extremities
Each of the two diameter
this
within
and
the
semicircles
into which the circle
by
at
diameter.
this
divided
is
lies
partly
partly without the
given curve, and therefore has intersection with it.
a second
The
four points of intersection circle with the curve
of the
are the vertices of an inscribed
whose
rectangle
are
sides
the required axes. parallel It follows from this construeto
there
tion that
one
pair of axes for
1 60.
An
axis
which stands to
it,
and are
FIG. 50.
always exists
may
an
ellipse or a hyperbola.
also
be defined as a diameter of the curve
angles to the chords which are conjugate bisected by it. The parabola has only one axis ;
at
right
middle points of all chords perpendicular to the direction of the diameters.
this contains the
common
A
curve of the second order
is
divided into two equal symmetrical
parts by each of its axes. 161. Two conjugate straight lines are harmonically separated by the tangents to the curve, which can be drawn from their point
of intersection (Art. 143). Hence, Any two conjugate diameters of a hyperbola are harmonically "
**
"
""
separated by the asymptotes.
One
cuts the curve, the other does not.
of these diameters, therefore, axes of a hyperbola bisect
The
the angles between the asymptotes (Art. 68). 162. Upon any straight line which is parallel to one of two "
conjugate diameters of a hyperbola, a segment
is
included between
the asymptotes whose middle point lies upon the other diameter If the straight line intersects the curve, or is tangent (Art. 68).
GEOMETRY OF
Il8
POSITION.
to the curve, the middle point of the chord, or, in the latter case, the point of contact of the line, coincides with the middle point of this segment. Hence,
two segments of any secant of a hyperbola which
"The
"between
the curve and
its
asymptotes are
segment of a tangent
"That
equal."
hyperbola which
to a
lie
lies
between
bisected at the point of contact." The first of these theorems furnishes a very simple construction for a hyperbola of which the asymptotes and one finite point are
asymptotes
"the
is
By it the second point of the hyperbola which lies upon any secant through the given point can be found immediately. 163. The hyperbola is intersected by but one of its axes the ellipse has four vertices consequently it has two vertices given.
r
;
since
intersected by both of
is
it
one actual
vertex, being cut
by
its
its
axes
axis a
;
the parabola has only in the infinitely
second time
distant point.
164.
A
hyperbola is said to be equilateral if its asymptotes are each other; the angles between any two conjugate
at right angles to
of an equilateral hyperbola are consequently bisected Accordingly, if a diameter of this (Art. 68) by the asymptotes. curve rotates about the centre, its conjugate diameter will rotate
diameters
about the same point in the opposite sense, the two sheaves described by the diameters thus being equal.
The sheaves of rays by which an equilateral hyperbola is projected from the extremities of any diameter are also equal to each other for any two corresponding rays are parallel to two conjugate ;
diameters
(Art.
155),
since
they
hyperbola.
intersect
in
a
point
of
the
D
The straight line joining the middle point of a chord of a parabola (Fig. 51) to the point of intersection C of the and is a diameter of the tangents constructed at parabola, 165.
AB
A
B
AB
the polar of the infinitely distant point of (Art. 134). But and are harmonically separated by the two points of inter section of with the parabola, and one of these intersections since
it
is
D
C
CD
is
the other, therefore, bisects the segment CD, or, which joins the pole of any chord of a with the middle point of the chord, is bisected by the
infinitely distant "
The
straight
"parabola
;
line,
"parabola."
It
in a similar manner, that each of the two which can be drawn from any point of the plane to-
may be shown,
straight lines
DIAMETERS AND AXES. polar, parallel to the
its
119
asymptotes of a hyperbola,
is
bisected by
the hyperbola.
These are some of the most important of the metric relations which are derived from the polar theory of curves of the second 1 66.
/I)
FIG. 51.
But from the theorems upon inscribed and circumscribed quadrangles and triangles there may be deduced other relations not wholly unimportant in particular from the theorem of Art. 116,
order.
which may be stated as follows "The two diagonals of a quadrangle :
BB^D^D
second order intersect in a point
circumscribed
which
"
to a curve of the
"
a straight line with the points of contact of either pair of opposite
S,
lies in
"sides."
and the two
pairs of opposite sides the by asymptotes and any other two tangents (as in Fig. 52), then S lies upon the infinitely distant and B^D are parallel. straight line, and the two diagonals l If the curve
of the
a hyperbola, quadrangle are formed is
BD
The are
triangles
D^DB
and D^B^B, which have the same base D^B,
consequently equal in area, as are also the triangles
and B^AB, which
differ
from these only by the triangle
Therefore,
The
and a
triangles which are formed by the asymptotes of a hyperbola variable tangent are all of the same area.
GEOMETRY OF
I2O
The
BD^
parallel diagonals
upon
proportional parts
AB AD :
POSITION.
B^D, and
the segments
= AD l
:
AB
l
or
the centre
that
;
AB AB .
A
determine
is,
l
= AD
.
ADY BB
In other words, the product of the segments which a tangent l makes upon the two Asymptotes is constant. Now draw through P, the point of contact of the tangent, a
(or
DD^
straight line
PQ
parallel
to
the
other asymptote
and meeting
it
in Q.
/
u,
FIG. 52. *
Since
P
bisects the
segment
BB^
of the tangent (Art. 162),
QP ory = %A and AQ or x = \AB. AB AB constant for all positions of lt
Since
now
BB^
the
.
l
product xy
is
is
constant wherever the point
chosen upon the hyperbola. Hence If we choose the asymptotes of a hyperbola of coordinates, the equation of that curve
the tangent
P
may be
for axes of a
system
is
xy = a constant.
By
this
m^ans the
synthetic
theory of the hyperbola
is
brought
into touch with the analytical theory.
to
167. In the elements of analytical geometry we are accustomed refer the ellipse and hyperbola to two conjugate diameters as
ALGEBRAIC EQUATIONS.
121
axes of coordinates, and treating our curves of the second order in the same way, we are able without difficulty to prove their
second order represented
identity with the analytical curves of the
by equations.
OX
A
the
O
We
prove the following theorem and Cl l which any tangent of the product segments of the curve of the second order determines upon two parallel
diameter
Y.
shall first
:
"
BB^
B
AB
"The
*
OY
OX
and the conjugate diameters (Fig. 53) at least one, the tangents u and u at say, cuts the curve (Art. 161), and are and parallel to the other points of intersection C^
Of
tangents u and u v
is
constant."
C
A"
FIG. 53.
BB
The ^
and
form with any fourth tangent three tangents //, // 1? and l a quadrangle circumscribing the curve, whose diagonals l and from of the diameter lt B^D intersect in a point
BD
the proportion
which
is
This curve
is
is
an
C^D^ = AD: it
C^ v
follows that
constant.
constant
product
ellipse
ellipse lying
;
and
upon OY,
is
positive
in this case b
as
we
(equal to + #-, say) if the is the semidiameter of the
readily find
by drawing
DD
parallel
l
OX. In
to
AB:
readily obtained,
and hence
to
AC
5"
-
the
case
2
fr
,
say)
of
since
the the
hyperbola
segments
AB C B AB and C^B^ .
l
l
is
negative
have
(equal
opposite
GEOMETRY OF
122 senses
here b
;
POSITION.
the absolute value of the length of the segment
is
which each of the two asymptotes determines upon the tangents u and u r
parallel
The three tangents u, u v and BB^, the latter of which touches the curve in the point P, form a circumscribed triangle, of which are the B) jB l and the infinitely distant point of the diameter O
Y
vertices
which
; consequently, join the straight vertices with the points of contact of the opposite sides intersect
the
three
lines
one point (Art. 117). That is, the point of intersection R of the straight and lies upon the ordinate y or PQ of the point P. in
B^
lines
C
l
But since
AB~ AC^~ BB^~ AB then must
We
QR = RP=\QP=\y.
obtain
immediately
the if
we
multiply
curve
the
of
equation
of
the
corresponding
together
second order sides
of the
equations
AB
^
and
CB
ACi
l
l
A.C
and then make QR = \y, l =P, l OQ = x, hence QC1 = a x and A Q = a + x. This gives us 2
.1*
y__
~_
792
4^ in
which the upper sign
_ a2
x2
4a
is
to
lower sign for the hyperbola. coordinates x, y, of any point
when
it
1 68.
/-\t* Ui
o 2
x _2 a
-4-
9 2
2 y _ 79
AC
__
ft
l
= 2.AO=2a and
T
A
j
be used for the ellipse and the This equation is satisfied by the
P
of the curve of the second order
referred to two conjugate diameters as axes. If the curve is a parabola it is customary to choose as is
OY
an arbitrary tangent (Fig. 54), and as axiswhich passes through O, the point of contact of OY. and which Any other two tangents l
axis of ordinates
of abscissae the diameter
OX
AP
intersect
OY
in
B
and
B^
respectively,
AP
form two opposite sides
OY
and the infinitely quadrangle of which the other two sides; consequently, the point of intersection S of the two diagonals of this quadrangle lies upon the two straight lines and which join the 1 of a
circumscribed
distant line of the plane are
PP
OX
points of contact of the two pairs of opposite sides.
ALGEBRAIC EQUATIONS.
ABSB
Since, therefore, the quadrangle and B^SPl are triangles
BPS
the
ordinates
PQ
Q^,
and
y
l
likewise,
y
If
now ^4A
is
drawn
OQ combine these by
is,
the
BS
O T-l H
F in
A",
we have
AK AB~
-/_
and
x
v
cr .V
(9
_
*-
r
x and
meeting
OQ - BP _ y
.
multiplication,
abscissae
and /\,
l
_ = that
P
SP
- *_y
l
of the points
\\
parallel to 6LY,
AK_
a parallelogram and and, moreover, the
we have
respectively, are parallel,
and
y and
or
is
l
similar,
123
T
__ ~
of two
;
points
P
and P^ of a
parabola bear the same ratio to each other as do the squares of the ordinates. For convenience we write this equation of the parabola in the form y-
= 2px,
where 2p
=
GEOMETRY OF
124 Incidentally
it
appears that
BP _ AB ~~ and from
this
follows
the
AP
and
any two tangents
POSITION.
a
a __ ~~
l
}
theorem already proved, namely, that
AP
l
of a parabola are cut proportion
ally by the remaining tangents (Art.
128).
EXAMPLES. Suppose a curve of the second order is given by five conditions, and the tangent at one of them or three points and the tangents at two of them, draw any required number of diameters of the curve and find its centre. 1.
e.g.
four points
2.
Draw
the chord of a given curve of the second order which
is
bisected at a given point. 3. Prove that the chords of a given curve of the second order which are bisected by any given chord envelop a parabola.
4. Of an ellipse or hyperbola there are given two pairs of conjugate diameters and either one point or one tangent; construct any required
number
of points or tangents.
Of
a curve of the second order there are given ( I ) two points or two 5. tangents and one pair of conjugate diameters, or (2) three points or three tangents and the centre
;
construct the curve.
Construct a parabola, having given three points or three tangents and the direction of its diameters, or two points or tangents and the axis. 6.
Given four tangents of a parabola construct its axis. 8. Construct a curve of the second order, having given three points or tangents and one axis. 7.
9.
;
Construct a hyperbola, having given
its
asymptotes and one point
or one tangent. 10.
In determining a curve of the second order for
how many con
ditions (points or tangents) does each of the following parts count: (i) the centre (2) an axis ; (3) a pair of conjugate diameters (4) a self;
;
(5) a point and polar triangle or lines (7) a diameter ? ;
its
polar
;
(6)
a pair of conjugate points
;
be let fall from a point P upon the planes of a the points in which the perpendiculars meet the planes all lie to a as diameter, and upon a circle which has ,the perpendicular from
n.
sheaf
If perpendiculars ,
P
whose plane
is
normal
to a.
Hence
12. If we pass through the sides a and a\ of an oblique angle all possible pairs of normal planes, these intersect in the rays of a cone
EXAMPLES.
I2 5
rays. Any plane normal and any plane normal to the a curve of the second order of which an axis
of the second order of which a and a l
are
to a or a l intersects this cone in a circle,
plane aa l intersects lies in
13.
it
in
aa v
We
also obtain the cone
mentioned
Kelp of a plane a which is normal to a let a right angle whose vertex is aa l so
in the last
at the point
example with the aa v Thus, if we
move that its plane constantly while one side describes the plane a, then passes through the line the other side must describe the required cone. ,
are dropped from any point S upon the 14. If perpendiculars diameters of a curve of the second order kl these meet the conjugate diameters in the points of an equilateral hyperbola, which passes through ,
and whose asymptotes are parallel & which pass From no point of through 5 lie upon this hyperbola (Apollonius). its plane, therefore, can more than four normals be drawn to a curve and through the centre of
_V
to the axes of k-.
The
k~,
extremities of the normals to
of the second order.
Suppose S and S\ are two points of a curve of the second order upon the same diameter, and that if and u 1 are two straight lines of the plane parallel to a pair of conjugate diameters if, now, the curve is projected from S and S l upon u and u lt respectively, we obtain two similar projective ranges of points u and // : which are there A well-known simple construction for an ellipse or fore proportional. is based upon this theorem. hyperbola 15.
lying
;
If we project a parabola from one of its points upon any diameter and also from its infinitely distant point upon any other straight line we obtain in u and u two similar projective ranges of points. From //!, 16.
//,
l
this there follows a very
simple construction for a parabola.
17. In the plane of a curve of the second order k-, a one to one corre spondence may be established among lines which are perpendicular to
each other and conjugate with respect to the given curve. To the rays pHissing through an actual point S, which does not lie upon an axis of 2 would thus be correlated the tangents to a parabola which is touched ,
z by the polar of 5 and by the axes of k (Example 6, p. 95). The tangents which are common to the parabola and k~ touch the latter at the ex tremities of the perpendiculars drawn from S to k- (comp. Example u).
LECTURE
X.
THE REGULUS AND RULED SURFACE OF THE SECOND ORDER. Thus
far we have obtained only curves, sheaves of rays and cones of the second order, from projective onedimensional primitive forms which either lie in the same plane or belong to the same bundle of rays. Let us now investigate
169.
and
planes,
whether or not other forms of the second order can be generated by means of two such projective primitive forms. In the first place
A
we
find that
sheaf of rays
S generates
with
a range of points u projective to it, the same sheaf of planes of the
A sheaf of rays S generates with a sheaf of planes u projectiv.e to it, the same range of points of the first
or second order as with the
or second order as with the sheaf
sheaf of rays by which u is projected from the centre of S, For,
of rays in which u is cut by the For, the point, in plane of S.
the plane which joins any point of n with the corresponding ray of the
former sheaf passes also through
which any plane of u is cut by the corresponding ray of the former sheaf lies also upon the corre-
the corresponding ray of the latter
spending ray of the
first
latter sheaf,
sheaf.
Two
which lie arbitrarily in space no new forms. For, in general, two immediately) will lie in the not same and hence will plane, corresponding rays have no point of intersection. Similarly, a range of points and a sheaf of planes projective to it generate no new forms, since a projective
sheaves
of rays
generate (at least
point of
the
former and a corresponding
plane
of
the
latter
determine no third element. 170. Consequently new forms are generated only by two pro two projective sheaves of planes which If u and u l are two projective ranges of lie arbitrarily in space.
jective ranges of points or
THE REGULUS AND RULED SURFACE.
127
points not lying in one plane, they generate a system of straight lines V, each line of which joins two homologous points of the No two straight lines of this regulus * lie in one plane ; ranges.
FIG. 56.
FIG. 55 .
otherwise, two points of u and their corresponding points in u v and consequently u and u^ themselves, would lie in one plane All the rays of this regulus lie upon a contrary to supposition. for,
curved surface, called a
which
ruled surface,
is
marked by the
following characteristics, namely The surface is covered by a second system of straight lines 7, each line of the one system being intersected by every line of the other system, while no two straight lines of the same system :
intersect each other.
Every point which lies upon a ray gf one system lies also upon a ray of the other.
For instance, suppose that
v,
Every through
plane a
ray
which one
of
passes
system
contains also a ray of the other.
v l}
v.2 ,
are any three ~rays of the
system V, so that each of these rays joins two corresponding points of u and //!, and let u.2 be a straight line which also cuts these three rays v, If, then, we project the ranges of points u and u l from the axis u.2 we obtain two projective sheaves of planes which have the three planes u 2v, u zv lt u z v^ and consequently all their planes self-corresponding (compare Art. 84). Thus every ,
*
The term
lines
* regulus is used by Salmon to denote a single infinity of straight a Geotn. of Three Dimensions 4th ed., p. 417. forming regular system >
H
GEOMETRY OF
128
POSITION.
lie in a plane with # and homologous points of u and 2 therefore u% is intersected by every ray of the regulus V. The same holds true of any other straight line ?/ 3 which cuts the three rays
^
pair of
v,
v2
z>
1}
"
rays "
Hence
.
The
"
of
v,
all
,
U
regulus
v v vz
consists of
of the system
all
straight lines
which cut any three
V
consists Similarly the regulus u lt u 9t of the straight lines which intersect three rays ,
V.
,
U"
"system
Since any straight line which cuts more than two rays of either regulus must cut every ray of that regulus, each is called a regulus of the second order.
ray of one regulus
may
a
is
director
be called the
of
director
of the other.
system 171.
Any
Also either system of rays
the other.
The
ruled surface can thus be traversed in either of two ways
by a straight line sliding along three given fixed straight lines, no two of which lie in the same plane. The three fixed lines are directors of the regulus which
is described by the moving line. The moving each point of a director once, and lies once in each plane which can be passed through a director (compare Art. 39). 172. Two projective sheaves of planes u and u l whose axes do not
line passes over
in the
lie
same plane
V
likewise generate a regulus
of the second
This might be obtained from those projective ranges of points u and u lt the first of which is a section of the sheaf u l by the axis u, and the second a section of the sheaf u by the axis u l order.
each straight line in which two corresponding planes of the sheaves intersect joins two corresponding points of the ranges.
for
173- A regulus of the second order is cut by any two of its directors
in
projective
ranges of
A
regulus of the second order projected from any two of its directors by projective sheaves of is
planes.
points.
w
v and w,2 be three directors of the regulus so that of the latter is intersected by these three rays. We then every ray obtain any desired number of rays of the regulus either by passing
Let
?.
,
w
and joining the two points in which each plane planes through 2 or by choosing points in w% and finding is cut by iv and lt the line of intersection of the planes which each point deter
w
mines with directors
w
w
and
and
w
l
ze/
in
to the sheaf of planes
The regulus is thus cut by the two r two ranges of points which are perspective
w
2
;
at the
same time
it
is
projected from
THE REGULUS AND RULED SURFACE.
129
w
and u\ by two sheaves of planes which are perspective to the range of points w.2l and consequently are projective to each other. 174. Four rays of a regulus of the second order are called harmonic rays if they are cut by any, and hence by every, director of the system in four harmonic points, or are projected from any director by four harmonic planes.
That
is
to
say,
if
w and w w and iv l
the
ranges of points other by the regulus
l
are two directors of this regulus, are related projectively to each
and if any four rays of the regulus are ; intersected by in four harmonic points, their intersections with But at the same time the four rays it\ are also harmonic points. are projected from harmonic by planes since these planes pass l
w
w
through four harmonic points lying in w. 175. To three arbitrary rays a, b, c, in space, of which no two lie in the same plane, a fourth ray d may be determined which harmonically separated from one of the three by the other two.
is
If
upon any
we
line
straight
which intersects the three given rays
harmonic point
find the fourth
the three points of inter
to
d. In general d is a fourth ray of the regulus of the second order to which the rays a, b, c, belong, and considered as such is harmonically separated from b by a and c.
upon the required ray
section, this point lies
176. If a straight line has more than two points in common the ruled surface here considered it lies wholly upon the
with
surface
;
for in that case
reguli lying
upon the
and consequently belongs property the surface
A
it
cuts
surface,
is
to the other regulus.
called a
is
more than two rays of one of the therefore a director of that regulus,
On
account of
this
ruled surface of the second order/
the ruled surface along a ray // of one and regulus lying upon it, consequently (Art. 170) contains a ray v of the other regulus, has no point in common with the surface outside the straight lines u and v. Otherwise, every straight line of such a plane passing through point would intersect u and v in of the and the whole points surface, plane would thus lie upon the Since now a third line of the plane surface, which is impossible. which passes through the intersection of u and v has only this
plane which
cuts
the"
point uv in
common
with the surface,
and we the plane of u and v
at this
point,
to the surface at this
shall
in
say that
it
the point uv
point."
i
is
"the "
or
tangent to the surface surface "
is
touched by
the plane
is
tangent
GEOMETRY OF
130 "The
177. "
"
"
number
POSITION.
of planes tangent to a ruled surface of the
second order, which can be passed through a given straight line, is equal to the number of points which the straight line has in
common
with
the surface.
The
surface
is
thus of the second
"
class."
For since in every such tangent plane there is contained a ray of one regulus lying upon the surface (and also a ray of the other), the given straight line has a point of intersection with this
and no two such points of intersection with rays of the same regulus can coincide, since no two rays of the same regulus lie in one plane. Hence there cannot be more tangent planes passed through the line than the number of its intersections with the At every intersection with the surface the given line surface.. meets a ray of each regulus. The plane of the given line and either of these rays is tangent to the surface at some point along the ray; it thus contains also a ray of the other regulus, and these second rays in the two planes must intersect in a point of ray,
f)
the given
Hence
line.
line is just equal to the line.
From
this
it
number of tangent planes through any number of intersections of the surface and the
follows that
be passed more than
-two
if
through a straight line there can
planes tangent to the surface, the line
wholly on the surface. we consider a regulus of the second order to be gener ated by two projective sheaves of planes and these to be cut by lies
178. If
an arbitrary plane cr, there appear in this plane two projective sheaves of rays such that every point of intersection of homolo gous rays of the sheaves lies upon a ray of the regulus. If, on the other hand, we consider the regulus to be generated by two projective ranges of points and these to be projected from
an
arbitrary point S, there will arise two concentric and projective sheaves of rays such that every plane containing a pair of homo logous rays of these sheaves contains also a ray of the regulus.
From
this follows the first part of
A
regulus of the second order is intersected by any plane a- which
contains none of
its
rays in a curve
of the second order.
which are tangent
The
planes
to the ruled sur-
face in the points of such a curve form a sheaf of the second order.
each of the following theorems
A
regulus of the second order
:
is
projected from any point S which upon no one of its rays by a
lies
sheaf of planes of the second order,
The
points in which the surface is touched by the planes of such a sheaf lie upon a curve of the
second order.
THE REGULUS AND RULED SURFACE.
I3I
In order to prove the second half of the theorem on the right, This a plane through three of the points of contact. cuts the sheaf of planes in a sheaf of rays of the second order,
we pass
of which the three chosen points are points of contact But this curve envelops a curve of the second order.
and which is
identical
which the plane cuts the regulus, since these two curves have the three points of contact and the tangents at these with
that
in
points
in
analogous way surface,
The theorem on
common.
whose
planes which touch the surface
A
the
left
is
proved
in
an
by constructing a sheaf of planes tangent to the centre is the point of intersection of any three at points of the given curve.
second order is called a simple a hyperboloid of one sheet (Fig. 55) if it contains no infinitely distant line but is intersected by the infinitely distant On the other hand, it is plane in a curve of the second order. 179.
ruled surface of the or
hyperboloid
a hyperbolic paraboloid (Fig. 56) if one and conse quently (Art. 170) each regulus lying upon it contains an infinitely distant ray, Each of the two reguli of a hyperbolic paraboloid is called
,
by any two of
cut
directors
in
similar
ranges of elements correspond to each other. Thus a hyperbolic paraboloid is described by a which line slides along two fixed straight lines u and u straight l points,
i.e.
gauche
to
its
in ranges
whose
each other
(i.e.
protective
infinitely distant
non-intersecting),
and remains
parallel
plane not parallel to either u or U Y For, the moving line intersects not only u and u l but also the infinitely distant line to a fixed
of the given plane
which
;
it
describes therefore a ruled surface
upon
one and consequently a second infinitely distant ray. The hyperbolic paraboloid is cut by an arbitrary plane which contains none of its rays in a hyperbola, but when the plane is lies
parallel
to
a
particular
straight
line
the
section
is
a parabola.
curve of section passes through the two points in which the infinitely distant rays of the surface are cut by the plane of section, and these points coincide only in case the plane contains
Any
the
common point of the infinitely distant rays. The hyperboloid of one sheet is not, like
80.
the hyperbolic paraboloid, touched by the infinitely distant plane, but, as we said, is cut by it in a curve of the second order. The tangent planes at the infinitely distant points of the hyperboloid are therefore 1
actual planes which (Art. 178) intersect in one point 5 and form a sheaf of the second order. The cone of the second order
GEOMETRY OF
132
POSITION.
sheaf converges toward the hyperboloid along curve and is called its asymptotic cone/ An arbitrary plane which contains no ray of a hyperboloid of one sheet cuts the surface in a hyperbola, parabola, or an ellipse, this
enveloped by
its
infinitely
distant
according as it has in common with the infinitely distant curve of the surface two, one, or no points, or, what is the same thing,
according as it is parallel to two, or to only one, or to no rays of the asymptotic cone. 181. I
shall
add only the following theorem, which appears
directly from what has already been said If straight lines be drawn through any given point the rays of a regulus of the second order, these all :
"
parallel to
"
"
"
"
one
in
lie
asymptotic plane or upon a cone of the second order, according as the regulus lies upon a hyperbolic paraboloid or a hyperboloid of one
of
sheet."
182.
A
its
two
angles.
hyperbolic paraboloid reguli
are
parallel
Each regulus of an
is
called
*
equilateral
respectively to equilateral
if
two planes
the rays at
right
one
paraboloid contains
ray which is perpendicular to a directing plane ray of the other regulus.
and hence
to
each
EXAMPLES. 1. Show that the locus of the vertex of a cone of the second order, to which the six sides of a gauche hexagon are tangent, is a ruled surface of the second order determined by the three principal diagonals of the
hexagon.
The three principal diagonals of a gauche hexagon whose upon a ruled surface of the second order intersect in one
2.
lie
six sides
point.
3. Suppose that a range of points u and a sheaf of rays of the first order S, not lying in parallel planes, are related projectively to each other if rays be drawn through the points of u parallel to the corre sponding rays of S, they will constitute one regulus of a hyperbolic ;
paraboloid. 4. Suppose a range of points u and a sheaf of planes v are related the projectively to each other, their bases not being at right angles perpendiculars let fall from the points of u to the corresponding planes ;
of
v form one regulus of a hyperbolic paraboloid.
of 5. If at the points of a straight line lying upon a ruled surface the second order, normals are erected to the surface, these form one regulus of an equilateral hyperbolic paraboloid (Example 4).
EXAMPLES.
33
6. If planes are passed through any chosen point normal to the rays of a regulus of the second order, these form a sheaf of the first or the second order according as the regulus belongs to a hyperbolic para
boloid or to a hyperboloid of one sheet. 7. The planes which pass through a fixed point and intersect a given hyperboloid of one sheet in parabolas envelop a cone of the second order, each ray of which is parallel to a ray of the asymptotic cone of
the hyperboloid. 8.
Construct a ruled surface of the second order of which there are
given two rays a and b not lying in the same plane, and either three points outside a and b or three tangent planes not passing through
a
or 9.
b.
What
is
a given point
the locus of a point which is harmonically separated from by a ruled surface of the second order ? What sort
A
of intersection has this locus with a plane passing through
A
?
LECTURE XL PROJECTIVE RELATIONS OF ELEMENTARY FORMS. 183. As has been shown, five forms of the second order can be generated by projective one-dimensional primitive forms.; namely, the curve or range of points of the second order, the sheaf of rays and the sheaf of planes of the second order, the cone of the It will be second order, and the regulus of the second order. convenient for us, with Von Staudt, to group these five forms of the second order and the three one-dimensional primitive forms
under the
common name
elementary forms.
Among
the element
ary forms, then, there are two which consist of points, namely, the ranges of points of the first and the second orders; next, two
which consist of planes, the sheaves of planes of the first and the second orders and finally, four which consist of rays, namely, the sheaves of rays of the first and the second orders, the cone ;
of the second order, and the regulus of the second order. In the present lecture I shall undertake to show you that these
elementary forms can be correlated to each other, two and two, in a manner analogous to that employed with the one-dimensional primitive forms.
By
so doing, the realm of our investigations
is
observe immediately instance, you considerably enlarged; that we can obtain a large number of new forms consisting of for
will
and planes, which possess just as noteworthy proper do those hitherto considered. At the same time we are
points, rays, ties
as
made aware
of other important theorems concerning the forms of
the second order, which by this means may be obtained very easily, but otherwise with considerable difficulty. 184. Let me remind you, at the outset, of the following theorems
which have been previously enunciated, and which may be fixed
PROJECTIVE RELATIONS OF ELEMENTARY FORMS. upon
135
the definitions of harmonic quadruples in forms of the
as
second order
:
Four harmonic points of a curve of the second order are projected
Four harmonic planes of a sheaf of the second order are intersected
from any fifth point of the curve by four harmonic rays (Art. 1 10). Four harmonic rays of a cone of the second order are projected from any fifth ray of the cone by four harmonic planes (Art. 125).
by any fifth plane of the sheaf in four harmonic rays (Art. 125). Four harmonic rays of a sheaf of the second order are cut by
any fifth ray of the sheaf in four harmonic points (Art. no).
Four harmonic rays of a regulus of the second order are cut by any director of the regulus in four harmonic points, and are projected from any director by four harmonic planes (Art. 174).
We may now
185.
for primitive
extend the definition of the projective relation
forms (given in Art. 79) so as to apply to elementary
forms in general, thus
Two
elementary forms are said to be projectively related
to each
any four harmonic eleme?its of the one form correspond to four harmonic elements of the other. It follows from this definition that two elementary forms are projective to each other as soon as they are projective to one and other if they are so correlated that
the
same
third form.
Moreover, we may extend the idea of the perspective relation between two one-dimensional primitive forms so as to apply to Thus, elementary forms in general.
Two
unlike projective elementary
forms are said
position if each element of the one form
of the
lies
to be in perspective
in the corresponding element
other.
A
range of points of the second order, for example, is perspective to a cone passing through it if each ray of the latter is correlated to the point of the former which lies upon it. range of points
A
of the second order
projected from any one of
its points by a sheaf of rays perspective to it; a sheaf of rays of the second order is cut by any one of its elements in a range of points per spective to it ; a regulus of the second order is intersected by
each
of
its
regulus, etc.
is
in a range of points perspective to the elementary forms, the one of which is derived
directors
Two
from the other by projection or section, are obviously projectively related, since four harmonic elements of the one correspond
GEOMETRY OF
136
POSITION.
always to four harmonic elements of the other, and when corre sponding elements are superposed the forms are perspective to
each
other.
each point of a curve of the second order is correlated the tangent at this point, then is the curve related perspectively to the sheaf of rays enveloping it; for, the curve is touched in any four harmonic points by four harmonic rays of the sheaf If to
Two curves of the second order are therefore related (Art. 122). projectively to each other if the two sheaves q^rays enveloping them
are projective to each other. 1 86. Two forms of the second order
lated
*
may be
conveniently corre
by establishing a projective
projectively
between
relation
two one-dimensional primitive forms perspective projective elementary forms may consequently
to (as
Two
them. in
Art.
92)
always be considered as the first and last of a series of elementary forms of which each is perspective to the one following. Moreover, two elementary forms can be so correlated to each that
other,
three
given
elements of the one correspond respec
one way;
tively to three given elements of the other, in only
this
has already been proved (Art. 90) for one-dimensional primitive forms, and the proof holds equally well for elementary forms in general.
For example, if it is required to correlate projectively to each other the two ranges of points of the second order, and k 1 l
&
(Fig.
57),
which
lie
in
one plane, so that
Kl correspond the points A S and T respectively, the
of
19
B^,
to the points
Cv
of &\,
A, B, C,
we may denote
2 2 which are points of K and 1 from and the and then projected ray project the A^ by lt given ranges of points from the centres .S and T^ by two sheaves of rays S(ABC...} and These are projective l (A l l l ...).
by
/
I}
,
AA
A
1
T
the
to
given
ranges
of points
BC
and consequently
to
each other.
moreover, perspective since they have the ray and Any two homologous points corresponding.
They two
are,
D
ranges
respectively, line
of
points -are
by two
rays
therefore
projected
AA^
D
from
S
which intersect upon a fixed
l
self-
of the
and
Tv
straight
u.
If two projective elementary forms of the same kind, e.g., two ranges ofpoints of the second order, are superposed, then all their elements are self-corresponding, or else at most two. Elementary forms which are identical are, at the same time, projective. 187.
PROJECTIVE RELATIONS OF ELEMENTARY FORMS. Two curves of the second order which lie in the same plane and have one point S in common are correlated to each projectively other
if
Two curves of the second order which lie in the same plane and have a common tangent s are correlated to each projectively other if those tangents of the two
those points of the curves
made
are
to correspond,
137
which lie For both
curves
are
made
to
correspond,
curves are then perspective to the sheaf of rays S. Every common
which intersect in a point of s. For the sheaves of the second order which envelop the curves are then
S S
perspective to the range of points s. Every common tangent of the
in a straight line with S.
point of the curves different from is self-corresponding the point ;
is likewise self-corresponding if the curves have a common tangent in this point, i.e. if they touch each
curves
different
corresponding
;
from but s
s
is
self-
itself is self-
corresponding only if the curves have a common point of contact in s.
other at S.
FIG. 57.
Two in
the
different
curves of the second order which are correlated
manner indicated on
corresponding points. points, or three
are
For,
common
they identical
(Art.
the
if
points 112).
left
have
at
they have beside
most three
S
four
and a common tangent Similarly,
the
in
self-
common S,
then
two curves on the
most three self-corresponding tangents. are thus brought to the following reciprocal theorems
right have at 1
88.
We
two projectively related curves of the second order have four selfIf
:
If two projectively related curves of the second order have four self-
GEOMETRY OF
138
corresponding points, then
all
their
POSITION.
corresponding
tangents,
then
all
points are self-corresponding and the curves are consequently iden-
their tangents are self-correspond-
tical.
identical.
The theorem on
the
left
ing and the curves are consequently
may be proved
as
follows
:
The two
curves can be correlated to each other in only one way so that the three points A, B, C, of the one correspond to the same three
But this happens if we relate the two curves perspectively to the sheaf of rays S. Suppose now that the curves have the point S also self-corresponding, then
points A, B, C, of the other.
common tangent at this point and are conse The theorem on the right may be quently identical (Art. 112). derived from that on the left by the application of the principle they must have a
we have demonstrated recommend to you as useful
of reciprocity which
however,
I
for the plane (Art. 138)
;
practice an attempt at a
direct proof. If the
curves
are
projectively related
projected cones,
from an arbitrary centre by two
we obtain
these
for
wholly analogous
theorems. 189. If a curve of the second order
is
projectively related to
a regulus or to a cone of the second order, and more than three points of the curve lie upon the rays corresponding to them, then the curve is perspective to the regulus or cone ; for, it is identical with the section of the regulus or cone which lies in its plane, since it is projective to this section and has with it more than three self-corresponding points. Similarly, a sheaf of planes of the second order is perspective to a regulus or to a sheaf of rays of the second order, which is to it, if more than three of its planes pass If we project, for in through the rays corresponding to them. stance, either of the latter forms from the centre of the sheaf
projectively related
of planes we obtain a second sheaf of planes which is identical with the first; for it is projectively related to the first sheaf and has with it more than three self-corresponding elements. 190.
Two
At this point the following theorems may be introduced Two curves of the second order cones of the second order :
which have different vertices and which are touched by the same
which lie in different planes and which are tangent to the line of
plane along the line ^ joining their vertices, intersect in a curve of the
the
second order.
intersection
same
s
of their
point, lie of the second order.
planes at
upon a cone
PROJECTIVE RELATIONS OF ELEMENTARY FORMS. If
each of the cones
is
correlated
;
s.
order which
curves be correlated perspectively to the range of points s, the ray
even ray s self- corresponding other pair of homologous rays of the cones will intersect since they lie
sheaves of rays of the envelop the
the
If
second
perspectively to the sheaf of planes whose axis is s, they will have the
139
be a self-corresponding ray every other (compare Art. 187) of the of rays homologous pair sheaves determine a plane, since
s will
;
one plane passing through The plane determined by any in
the
the rays intersect in a point of s. From the point of intersection of
corresponding not only these three intersection points but also a point
any three of these planes the two sheaves of rays may be projected by two projectively related sheaves of planes of the second order which are identical, since they have self-
three
of
points
intersection intersects
of
homologous rays in two projectively related curves of the second order which are identical, since they have selfcones
of
corresponding these three planes also a plane passing through s.
s.
and
The proof In
simpler.
through any
these
of
three
may however be made much
theorems
theorem
the
points
on
the
if
left,
common
to
the
a
be
plane
passed
two cones, the
two
curves of section lying in it have these three points in common and also the point of intersection with s since both curves are ;
tangent
at
the
common
latter
to
the
point plane of the cones
tangent of section must coincide (Art. 112).
may be proved
is
in
line
intersected,
which
the
these curves
The theorem on
the right
in a similar way.
191. It follows incidentally that
may be
straight
any curve of the second order For a circle
considered as a section of a circular cone.
and a given curve of the second order can be brought in an unlimited number of ways into such position that they touch each other and lie in different planes, and hence lie upon one and Curves of the second order are thus identical with the same cone. the conic sections of ancient times, and may hereafter be desig nated by that name. 192. If a sheaf of rays of the first lies in the plane of a conic
order S section
k 2 which
is projectively re-
but not in perspective position, then at most three rays lated to
it,
If a range of points of the first lies in the plane of a sheaf of rays of the second order which order is
projectively related to
it,
but not
of the sheaf pass through the points of the cun e corresponding to them,
perspective position, then at most three points of the range lie upon the rays of the sheaf corre-
and
sponding
at least one.
in
to them,
and at
least one.
1
GEOMETRY OF
40
POSITION.
&
Sl perspective to the curve is prothe sheaf S, and with it generates, in general, another curve of the second order, which must have in common with the For every sheaf of rays
jective to
curve every point which lies upon the ray of corresponding If more than three rays of should pass through the 2 points of k corresponding to them, the two curves would have first
to
.5"
it.
beside
S
at least four
1
common
and S would be perspective
points,
&.
hence would be
identical,
every curve of the second order divides its plane into two parts, these two curves, in case they do not coincide, must either touch each other in the to
Since
and at least one other point P, point S} or must intersect in so that each curve lies partly within and partly without the other.
^
In the latter case the rays
and consequently corresponding to
SP and
S^P correspond
it
to each other,
P
SP
of the curve k^ passes through the point in the former case, to the ray SS^ of S, there
;
corresponds the common tangent in Sl and hence also the point Sl of the curve k 2 Thus at least one point of the curve lies .
upon the ray of the sheaf corresponding to it. Wholly analogous theorems are true for the forms of and second orders, in the bundle of rays. 193.
We
the
first
conclude that
If a one-dimensional primitive form and an elementary form of the second order are related projectivelv to each other, and more than three elements of the one
form pass through or
lie
upon the elements
other which correspond to them, then the tivo forms are in of perspective position, that is, each element of the one form passes the
through or If
the
lies upon the element of the other corresponding to it. form of the second order is a regulus and the other
either a range of points or a sheaf of planes of the first order, it can be seen immediately that these are perspective to each other three rays of the regulus pass through the three corresponding points of the range or lie in the three corresponding planes of For the base of the range of points or axis of the the sheaf. sheaf of planes is then a directing ray of the regulus (Art. 170)
if
since
it
194.
intersects three rays of the latter.
The importance
the following
of
these theorems
may be judged from
:
A sheaf of planes of the first order and a regulus or a cone of the second order projectively re-
A range of points of the first order and a regulus or a sheaf of rays of the
second order projec-
PROJECTIVE RELATIONS OF ELEMENTARY FORMS. lated to
it
generate, in general, a
common
41
related to it generate, in general, a sheaf of planes of the third order. At least one and at lively
gauche curve of the third order. This has at least one and at most three points in
j
most three planes of this sheaf pass through any point.
with any
plane.
For, a plane cuts the regulus or the cone in a range of pointsof the second order perspective to it, of which in general and at most three points lie upon the corresponding planes of the sheaf. a range of points u of the order and a range k- of the second order projectively related
If a sheaf of rays of the first order and one of the second order
If
first
projectively related to it lie in one plane, the points of intersection of
one plane, the straight lines joining homologous points form a sheaf of rays of the third order at least one and at most to
it
lie in
homologous rays form a curve of the curve
;
is
three rays of this sheaf pass through
line
of
any point of
and
at
its
plane.
For is any sheaf of rays and consequently protective to if
of the
6"
third
intersected
first
;
this
by any
plane in at
its
most three
line or
order
straight least one
points.
order perspective to
//,
most three rays of pass through the corresponding points of /-, and at least one ray. 195. If the ranges of points u and k- of the first and second orders, respectively, have a self-corresponding point P, then every ray passing through P must be considered as a line joining two k-,
at
6"
(coincident) homologous points, and the sheaf of rays of the third order includes the sheaf of the first order as a part of it. The are theorems therefore not to be considered following exceptions
P
but as particular cases
to,
of,
the theorems just
now
proved.
a range of points u of the order and a range k- of the
a sheaf of rays of the first order and a sheaf of rays of the
second order, projectively related, have two self-corresponding points
second order, projectively related, have two self-corresponding rays,
If first
A
and B, they generate a sheaf of
rays of the
first
order.
Suppose that to the point and let S be that point of
C k-
If
they generate a range of points of the first order. *-.
of n the point
which
is
C
of k- corresponds, projected from C^ by the l
If now we relate u and k- perspectively to sheaves of ray C^C. rays Sj these will be so related projectively to each other that to the three points A, B, C, of u will correspond the three points A, B) Clt respectively, of k-. But since (Art. 186) the projective
relation of u
and
k-
is
determined uniquely by the three pairs
GEOMETRY OF
142
POSITION.
of homologous points, the lines joining pairs of homologous points clearly form a sheaf of rays 6 of the first order whose centre lies 1
the curve k
upon
1 .
A curve of the second order
196.
and two
straight lines
a and b each
having one point in common with the curve, but which neither lie in a plane with the curve nor with each other, determine a regulus of the second order perspective to the curve and of which the two straight lines are directors.
The two sheaves
A sheaf of planes of the second order and two straight lines a and b each lying in a plane of the sheaf, but which neither intersect nor pass
through the centre of the sheaf, determine a regulus of the second order perspective to the sheaf and of which the two straight lines are directors.
of planes a
and
b perspective to the curve generate
the regulus.
The two ranges of points a and b perspective to the sheaf of planes generate the regulus.
The
director system of this regulus contains the rays a and likewise and is perspective to the curve or to the sheaf of planes, as the case may be.
/>,
If a range of points of the first order and a curve of the second
order not lying in the same plane are projectively related and have a self-corresponding, they point generate a regulus of the second
A
order perspective to both.
If
two sheaves of planes of the
and second orders respectively not belonging to the same bundle are projectively related and have first
one self-corresponding plane, they generate a regulus of the second order perspective to both.
In the theorem on the
left, suppose that to the points A, B, C, of the of points correspond the points A, B^ range v of the the curve the and the two then determined curve regulus by not to the is curve lines but only CC^ perspective straight
C
;
BB^
also to the given range of points, since the three points A, B, C, of the latter lie in those rays of the regulus which correspond to
them. 197.
The proof of the theorem on the right is wholly From an arbitrary point not in the plane of the
analogous. curve, the
projected by a cone of the second order, a sheaf of planes of the second order. This regulus, by sheaf of planes is cut by an arbitrary plane in a sheaf of rays of the second order and the regulus, in a range of points of the second
conic of the
last article is
and the
order.
Hence
it
follows
:
If a range of points of the first order and a cone of the second
If a sheaf of planes of the first order and a sheaf of rays of the
PKOJECTIVE RELATIONS OF ELEMENTARY FORMS. order are protectively related, and one point of the former lies upon
second order are projectively reand one plane of the former
lated,
the corresponding ray of the latter, the two forms generate a sheaf of
planes of the second order
I43
per-
contains the corresponding ray of the these two sheaves generate a range of points of the second order latter,
perspective to both.
spective to both.
immediately to the following if we remember that any curve of the second order may be looked upon as a section of a cone of the second order
theorem
This
brings
us
:
If a range of points of the first order and a curve of the second
If first
two sheaves of rays of the and second orders, respectively,
lying in the same plane are pro related and have one self-
order lying in the same plane are projectively related and have one self-
j actively
corresponding point, they generate a sheaf of rays of the second order
corresponding ray, they generate a curve of the second order per-
perspective to both.
spective to both.
Two
projective reguli of the second order abc and ct^\c^ of is the director system of the other, generate a curve
which each
of the second order and a sheaf of planes of the second order, both of which are perspective to the reguli.
The two
can be related projectively to each other in only
reguli
one way so that
to the rays #,
b,
c,
of the one system correspond
But this happens if those respectively, of the other system. two rays are correlated to each other which intersect the plane determined by the points aa^ bb^ cc^ in one and the same point, au
c
^i>
\>
or which are projected from the point determined by the planes cc^ by one and the same plane.
aa v bb^
Of two
198.
projective reguli or cones of the second order, at most
four pairs of ho?nologous rays
intersect unless all such
pairs of rays
intersect.
any two homologous rays in two projective reguli of the second lie in a plane e, the two systems may be projected from their In case the two directors lying in e by two sheaves of planes. If
order
directors
do not coincide, these sheaves of planes generate a sheaf
of rays
of the
plane.
Each and the
first
ray
order, since they have
of
5
intersects
e
as a self-corresponding
two homologous rays of the
reguli themselves are intersected by the plane of *S two projective curves of the second order, which have at most
reguli,
in
6"
three
self-corresponding points unless all their points are selfbut these self-corresponding points are points of corresponding ;
GEOMETRY OF
144
POSITION.
intersection of homologous rays of the reguli, and conversely. In case the two directors coincide, they cut the reguli in two projective ranges of points which have either at most two, or else all, of their
At the same time the reguli are pro points self-corresponding. jected from these two coincident rays by two projective sheaves of planes which have either at most two, or else all, of their planes In each of these planes, as well as in each self-corresponding. of the self-corresponding points, two homologous rays of the reguli
The theorem may be proved
intersect.
for either
or both of
in
an analogous way
if
the reguli a cone of the second order be
substituted.
199.
From
this
it
is
clear that
There are in general and at most four points through which pass
four
homologous planes
in
four projectively related sheaves of planes of the first order which are
Of four projective ranges of points situated arbitrarily- in space,, there are in general and at most four sets of four
which
lie in
homologous points one plane,
situated arbitrarily in space.
The
sheaves of planes taken in pairs generate projectively related second order to which the preceding theorem applicable. Every set of four homologous planes has a common
reguli or cones of the is
point as soon as there exist five or more such 200. Two projective curves of the second order which are superposed either generate a sheaf of rays of the second order perspective to both curves, or else there exists
a point which lies in a straight line with every pair of homologous points of the curves.
Two of
sets.
projective sheaves of rays second order which are
the
superposed either generate a curve of the second order perspective to both sheaves, or else there exists a straight line upon which every pair of homologous rays of the sheaves intersect.
Every regulus perspective to the one curve generates with its director system, which may be related perspectively to the other curve, a sheaf of planes of the second order perspective to all four forms; and according as the centre of within the plane of the curves does the
two cases mentioned
in the
this
sheaf
first
or the second of the
theorem occur.
If,
lies
without or
then, of the straight
of homologous points of the curves any three pass through one and the same point U, all such lines intersect in that point (Figs. 60 and 61, p. 150).
lines joining pairs
PROJECTIVE -RELATIONS OF ELEMENTARY FORMS. 201.
Two projective curves
Two projectively related sheaves of planes of the second order which
of the
ABCD
second order and ABC^D^ which have two self-corresponding points A and B, but which do not lie
same
in the
145
have two self-corresponding planes,, but which are not concentric, generate a form of the second
plane, generate a
form of the second order perspective to both, namely, either a regulus or a cone of the second order.
order perspective to both, namely, either a regulus or a sheaf of rays of the second order.
ABCD
the regulus or cone perspective to the curve and the and is curve also to the rays CC^ containing perspective l The curves generate a cone if their tangents 189). 1 1 (Art. For,
DD
ABC D at
C
and C^
intersect the straight line
AB
in
one and the same
Otherwise they would generate a regulus, and the plane of
point.
these tangents would also contain, besides the ray CC^ a director of the regulus, (Art. 170), and therefore would have common with one or both of the curves a point different from either or C^
C
lying
upon
From
Two
this director,
this
it
and
AB
segment
is
impossible.
follows that
conies which
planes,
which
lie in
Two
different
intercept the same upon the line of
cones of the second order
having different vertices, and lying in one and the same dihedral angle,,
intersection of these planes, can be made to lie upon either of two
intersect in
one or other of two-
conies,
cones of the second order. For, the conies can be correlated projectively in a twofold manner so that they have the extremities of their common chord as selfcorresponding points, while the tangents at two other homologous
AB
in one point. points intersect the line 202. are now prepared to prove the following theorem upon the perspective position of elementary forms of the second order
We
:
If a curve and a sheaf of rays of
the second order, or a cone
and
a sheaf of planes of the seco?id order, or, in fact, any two of these four forms, are projectively related, and five elements of the one form lie in the corresponding elements of the other, then the two forms are in perspective position,
We and
shall choose, as the first form, a curve of the
for the other a sheaf of planes of the
that five points
planes a, this one.
/?,
y, 8, It
second order S-, so situated
A, B, C, D, E, of the former e,
of the latter
;
all
lie in
the corresponding
other cases can be reduced to
need only be shown, then, that a
K
second order u 2 ,
rectilinear
form
GEOMETRY OF
146
POSITION.
can be constructed which
is perspective both to the curve and to the sheaf of planes ; for with this it will be proved that each point of the curve lies in the plane of the sheaf corresponding to it. 2 If the plane o- of the curve is an element of the sheaf then we .S"
obtain in
it
(as
section of
S2)
a sheaf of rays of the
,
first
order
2 which is also perspective to the curve perspective to S since more than three of its rays pass through the points of the curve corresponding to them (Art. 193); the point S therefore lies upon the given curve.
,...
of
"
For of the sign 7\ to denote projectivity. are two ranges of points of the first order
same
plane, but not
the straight lines
upon the same
A B CDE AA V BB^ CCV DD^ EE
ABODE
then
Z>
lt
make use
lie
that to the elements
A v B^ C
shall
example,
which
only
in such a
...
"A
.
1
1
1
1
.
straight line,
.
1
.
l
.
.
pass through
THE THEORY OF INVOLUTION. one and the same point or are tangent u l are also tangent.*
155
to a conic, to
which u and
set of four elements ABCD /;/ definite order; chosen arbitrarily an elementary form (a throw ), is projective to every permutation from elements in which one pair and also the other pair are inter these of
A
changed.
That
is,
Suppose,
ABCD 7\ BADC A CDAB A DCBA. a throw upon a for example, ABCD is
straight line,
other cases being reducible
all
to this
and let show that
Project
ABCD
trary point
^
be
it
one,
quired to
re
from an arbi
upon
63)
(Fig.
a
A,
straight linJIpassing through
denoting the projection by
AEFG.
T
Let
4
be the intersection of
CF
and
DE.
Then,
ABCD AEFG
the projection of
is
*
CTFS
AEFG from
the centre S,
CTFS
D,
CDAB
E.
ABCD 7\ AEFG 7\ CTFS 7\ CDAB, Hence, ABCD 7\ CDAB and consequently, The from "
*
If
abcd~RABCD
The
(Art.
may be proved
other relations
We
in a similar way.
infer
this that
relation
97), that
then also abcdl\BADd\CDAB~j\DCBA."i
ABCD f\
A^B^C-^D^ also
has already been shown
signifies, as
the segments of the straight line
among
the following proportion
:
-AB
CB = A^B^ x
:
fl 2-J
u and
:
there exists
C^B^
.
C- J-S
./i
-i
>
-i
t With the help of this important theorem the following among other remark able relations may be proved :
The
six
of
vertices
any two
polar triangles of a conic
second
conic,
number of first
to
&
lie
which an
self-polar
triangles
ABC
six sides of
of
infinite
conic, to
which an
of the
any two
self- polar
a conic touch a second
triangles
can be inscribed.
For suppose
The
self-
upon a
infinite
number of
self-polar triangles of the first
can be
circumscribed.
and
DEF
to
be the two
self- polar
triangles,
no three
GEOMETRY OF
156 216.
The theorem
POSITION.
proved in a different manner, viz., superposed elementary forms which are prorelated are in involution if any two elements A and A l "jectively of Art.
already
209,
"two
"
correspond last article.
doubly,"
For,
if
from the relation proved in the of the one form the element
results directly
to
B
any element
B^ of the other corresponds, so that to the elements A, A^ B, of the former correspond the elements v A, B^ respectively of the
A
latter,
it
follows, since
virtue of permutation, that to the element J3 l of the
by
B
and
A
first
form
B of
the second corresponds, or that any two elements correspond to each other doubly.
the element JB l
consequence which might have been stated earlier is this range of points of the first order u and a sheaf of rays
:
"A
"projective "
to
it
are in involution
the centre of the sheaf
if
P
S
lies
P
and l of the range, e^ch lies upon u, and, of two points ray of the sheaf which corresponds to the other." For the section of the sheaf of rays made by the straight line
outside
"that
u
is
projective to the range of points since the points and jP, it,
upon u and
is
in involution
P
correspond to each other In a similar way we determine when a sheaf of planes doubly. is in involution with a range of points or with a sheaf of rays.
with
A
217. The fact that any two conjugate elements A, v of an involution are harmonically separated by the double elements
and
M
such appear, may also be proved in an elementary way. Let the involution MNAA^ consist of two superposed projective JV, if
ranges of points of the
first
order.
Then
of the one range correspond the points of whose six vertices
A(BCEF)
and
four rays AB, of the second relation
A, JV, A^ A, of the other,
JV,
Then the sheaves of rays straight line. protectively related (Art. 144), since to the of the first sheaf, the rays DC, DB, DF, DE,
conjugate
with respect
A(BCEF) ~f\ D(CBFE)
it
the
to
follows that
conic
2 >
.
But from the
A(BCEF) ~J\D(BCEF)
and
upon a curve of the second If now D and E are two points of 2 and curve which are conjugate with respect to the first-named conic
therefore the six points A, B, C, order as the theorem on the left asserts. />,
this
Av
are
AC, AE, AF, are
M,
one
in
lie
D(CBFE]
to the points J/,
,
F,
lie
,
DEF
F
the third vertex lies also , consequently vertices of a self-polar triangle upon the conic through A, B, C, D, E, F; for the conic passing through A, B, C, has five points in common with that through A, B, C, D, F, and , , ,
D
F
,
hence coincides with
The theorem on
it.
the right
is
proved
in a similar
way.
THE THEORY OF INVOLUTION. since the double points
A
J/and Ware
correspond doubly.
l
That
now we
If
tion is
by
MRKT,
then
and
A"
MA^NA.
through
J/",
this
related
and
A
from an arbitrary point S (Fig. 64} and denote the projec
l
line passing
protectively
MANA^
MANA
project
straight
and
self-corresponding,
is,
MANA^ upon a
57
to
consequently
MA^NA.
to
But these projective forms have the point self-corre
M
sponding and consequently are in perspective position, that is, the straight lines KN,
RA^
TA, intersect in one and the same point Q. By this means we obtain a quadrangle QRST of which two pairs of opposite sides intersect in A and A^ while the diagonals pass through and thus the points MANA^ are clearly harmonic respectively
M
N
points.
A
218.
involution strict
AA BB
group of three pairs of elements is sometimes called an involution
sense
These
word.
the
of
l
.
.
l
CC^
of an
the original and elements are not inde
six
in
pendent, since in an involution the conjugate to any element is determined as soon as two pairs of conjugate elements are given, and since any two forms which are composed of elements chosen
from
these
A^AB^CV
or
Conversely,
six
symmetrically,
it
follows
from
AA^BC
instance,
and
AA^.BB^.CC;
for,
are projectively related. relation 7\ l
the
that the three pairs of elements
involution
for
as,
AB^C^C and A^BCC^ A, in
AA
AA^BC
A the
B
C
l
C
l
l
form an
projective forms
AA^BC
B,
l
l
;
C,
and A l correspond doubly, and A^AB^Cv the elements B, B^, and C, C\, must correspond consequently and
the
elements
doubly.
A
double element
M
or
elements in the involution
an involution
if
an involution
if
MAA^B MANA^
;
~/\
is
A
N
can take the place of a pair of
thus,
for
example,
MA^ABY
M AA^ JBB M N. AA
Similarly, projective to
ments thus forming a harmonic quadruple.
.
MA^NA,
.
l
.
l
is is
the four ele
GEOMETRY OF
58
We
219.
The
now prove
can
the following theorems
three pairs of opposite ver tices of a complete quadrilateral are projected from any point which
QRST
which
none of
the plane of the
lies in
the
lateral but
passes through vertices, in three pairs
its
in
lies
and
quadrangle
:
The
three pairs of opposite sides
of a complete quadrangle (Fig. 65) are cut by any straight line
POSITION.
plane
of the
upon none of
by three pairs of rays
of points in involution.
quadri
its
sides,
in
invo
lution.
FIG. 65.
Let u (Fig. 65) cut the sides
and RS,
in the points
ively,
and
Then
AT OR
Q
and of
O
let
is
A
RT and
A B lt
and
SQ,
ST
and QR, QT and Cv respect of QS and RT,
B^ C
be the point of intersection a projection both of ACA l jB l from the centre
ABA^C^
from the centre
S,
and
therefore,
ACA B ~KATOR ~R ABA^CV interchanging A with A and B with C 1
Moreover, by
and
l
l
we obtain
ABA^1\A
215) Therefore
(Art.
and hence AA^
1
A CA^ 1\ A .
BB^ CC^ .
is
l
an involution.
two ranges of points lying in u are so related projectively that to the points A, C, A v of the first correspond A v C^ A, in involution since respectively, of the second, the two forms are For,
A
if
A
B
are conjugate elements of and and l correspond doubly ; l CA 1 JB1 ~J\ l C^AB. the involution on account of the relation
A
A
THE THEORY OF INVOLUTION.
now, of a range of points u in involution, two pairs of points
If,
A^
A)
159
and
,
B^
to
and
are given
is
it
fifth
conjugate any point C, use of the form of the second
required to find the point
C
l
we can proceed without making order in
the
following
manner.
Construct a complete quadrangle of which two opposite sides pass
B
and A l respectively, two others through B and then the sixth side respectively, and a fifth side through C through
A
intersect the given line in the required point
Two
quadrangles
QRST
and
Q^R^T^
l
will
;
Cr constructed as above
have essentially different positions with respect to the involution l l CC^ if the sides passing through A, B, C, in the one
AA BB .
.
quadrangle intersect in a point T, while in the other they form .a their sixth side in both cases passing through triangle
Q^^,
C\ (Fig. 65).
If the
two quadrangles
eight vertices form two
in different planes their
lie
QRST-^ and
tetrahedra
Q^S^T
are both inscribed and circumscribed to each other. given the other can infinite number of ways. tetrahedron
If
two
is
points
M
and
A
N
of
be
the
and
and
easily
straight
constructed
line
B
monically separated by by A^ is, by two pairs of opposite sides of the quadrangle also
which one
If the
u
in
an har
are
and B-^
that
QRST,
then
C
are they also harmonically separated by and C\, that is, by the third pair of opposite sides of the quadrangle. For, in this case, and are the double points of the involution l
M
N
AA BB^ CCY .
.
FIG. 66.
220.
"
If a
curve of the second
a simple quadrangle "in
QRST
order
is
circumscribed about
(Fig. 66) the three pairs of points which any straight line u passing through no vertex of the
GEOMETRY OF
!6o quadrangle
and by the two pairs of opposite an involution (Desargues theorem).
cut by the curve
"
"
is
sides of the quadrangle form
chosen
Suppose the
RS, and
ST in
and
line
6*
are
and
u
line
P
and
by which the points P, R,
Q
"
cut
is
the points A, B,
curve in the points
PBP^A
Pr Pv T,
215)
it
by the
A v B^
sides
respectively,
TQ, QR, and by the
Then the two sheaves of rays of the curve are projected from related, and consequently the ranges
projectively in which these sheaves are cut l l l
PA P
u are also projectively related.
(Art.
POSITION.
Since
by the
now PA 1 P1 B
1
straight
7\
follows that \
PP AA
i.e.
.
l
l
.
BB^
is
an involution.
A
wholly analogous theorem holds for the curve of the second order which is inscribed in a simple quadrilateral. 221. Since
an involution
a one-dimensional form
in
pletely determined by two pairs of elements A, we have the following theorems
Av
is
com
and B,
B^
:
The
curves of the second order
about
any
fixed
quadrangle are intersected straight line u which lies
in the
circumscribed
by a
The curves
pairs of tangents to the of the second order in-
scribed in any fixed quadrilateral, which pass through one point S
plane of the quadrangle, but which passes through none of its vertices, in point-pairs of an involution.
lateral
The
The double
touched by two of these curves in the double straight
line
is
lying in the
sides,
plane of the quadri-
upon no one of its form an involution sheaf. but
rays of this involution
touch two of the conies in S.
points of the involution.
There are
either
two or no curves of the second order which
are the
quadrangle
same time
are touched
inscribed to a quadrilateral and at the same time pass through the
line,
given point,
circumscribed to
and
at the
by the given straight
not appear. according as these double elements do or do the problem to construct the conies which pass through four given points and touch a given line, or which touch four
Thus
is reduced to finding given lines and pass through a given point, We shall be concerned with the double elements of an involution. this problem of the second order in a subsequent lecture.
EXAMPLES. in the left
If,
hand theorem above, the
161 straight line
u
lies infinitely
distant, this particular case presents itself: "Through
four actual
two or no
"either
points of a
plane
be passed
there can
parabolas."
EXAMPLES. 1.
If
an involution be chosen upon a tangent
to a conic section
from the pairs of conjugate points new tangents drawn, all such pairs of tangents will intersect
The
of
vertices
given parabola
The
all
a
definite
angles whose sides are tangent to a Also, the vertices of all straight line.
right
upon a fixed whose bases
lie
isosceles triangles
and whose 3.
upon
and be
line.
straight 2.
conic
the
to
upon a fixed tangent to a parabola lie upon a fixed straight line. circumscribed to a conic, whose bases
lie
sides touch the parabola,
vertices of all triangles
remain on a fixed tangent and are bisected by its point of contact A, The straight lie upon the diameter of the curve passing through A. lines which in the several triangles join the points of contact of the other two sides are parallel to the base. 4. If two planes rotate about two fixed straight lines in such a way as always to be parallel respectively to conjugate diameters of a given conic, their line of intersection will describe a ruled surface, or else a
cone of the second order, upon which the two fixed lines lie. 5. Two projectively related sheaves of rays or sheaves of planes being given, how can they be brought into such a position as to form
an involution
Of an
?
& ao
o^o
upon a conjugate points A, A^ and 6.
J5,
plete quadrangle the conjugate
determine the
ticular,
conjugate 7.
Two
CTL****~
e
straight line there are given two pairs of
involution
centre
Construct by means of the
B\.
C
to
l
any
fifth
point
of the involution,
i.e.
C
;
com
and, in par
the point
whose
lies infinitely distant.
pairs of conjugate diameters of a conic section are
draw the conjugate
to
any
fifth
known,
diameter.
8. If through any point straight lines be drawn parallel to the three pairs of opposite sides of a complete quadrangle they form an involution.
Hence
:
two pairs of opposite sides of a complete quadrangle intersect Or, the three perpen right angles, so also does the third pair. diculars let fall from the vertices of a triangle upon the opposite sides intersect in one point, the so-called orthocentre of the triangle. 9.
If
at
10.
All curves of the
vertices
second order which pass through the three
and the orthocentre of a
triangle are
L
equilateral
hyperbolas.
1
GEOMETRY OF
62
An
equilateral
rangle
:
this
11.
The
hyperbola can always be circumscribed to any quad
passes
formed by the
POSITION.
through
the
orthocentres
of the
four
triangles
vertices of the quadrangle.
sides of
any triangle form with the
infinitely distant line of
the plane a complete quadrilateral whose three pairs of opposite ver tices are projected from the orthocentre of the triangle by three pairs
of rays at right angles. 12. The two tangents to a parabola which can be drawn from the orthocentre of a circumscribed triangle intersect at right angles. Con
sequently the orthocentres of lie upon a fixed straight line.
triangles circumscribed to a parabola In particular, the orthocentres (Ex. 2.) of the four triangles formed by the sides of any complete quadrilateral
upon one
lie
all
straight line.
ABCD
is circumscribed to an ellipse or a hyper 13. bola, the tangents which can be drawn from any point vS of the circle through the vertices A, B, C, D, intersect at right angles for the two pairs of opposite vertices of the rectangle are projected from .9 by two
If
a rectangle
;
normal rays. Hence we conclude that the vertices of all right angles whose sides are tangent to an ellipse or hyperbola lie upon a circle." This might be looked upon as a special case of the pairs of
following 14.
"
:
Any two
Any two points of a conic section k z which lie upon two conjugate
tangents to a conic
section k^ which pass through two conjugate points of a given involution range
u
upon another
rays of a given involution sheaf S, are in general upon the same ray of a sheaf of the second order.*
intersect, in general,
fixed conic.
&
An
is touched by the straight line u exception arises if the curve the conic can be circumscribed by a quadrilateral In (Ex. i). general
abed of which two opposite vertices ac and bd coincide with two con P and P1 of the involution n. If now two tangents to k z which pass through two other conjugate points Q and Q of u inter
jugate points
l
and SQ^ together with the SP pair of rays by which two opposite vertices R and R^ of the quadrilateral, different from P and /\, are projected from S form an involution (Art. 220), and conse quently SR and SR^ pass through two conjugate points of the If then we correlate the sheaves R and R l to each involution u.
SQ
sect in a point 5, then these tangents and SP^ and with that third rays
projectively so that homologous rays pass through conjugate points of u, they will generate a curve of the second order which is the locus of the point S.
other
*
This sheaf of the second order degenerates into two sheaves of the first is touched by two conjugate rays of the involution S.
order in case the conic
EXAMPLES.
163
The pairs of tangents The theorem on the left might be stated to a curve of the second order which are harmonically separated by "
:
two given points intersect
Thus expressed
order."
the proof of which
is
in it
left
general upon another curve of the second becomes a special case of the following,
to the student
:
The
The
pairs of points of a curve of the second order, which are con jugate with respect to a second
pairs of tangents to a curve of the second order, which are con-
jugate with respect to a second curve of the second order, intersect in general in the points of a third
curve of the second order,
lie in
general upon the tangents to a third curve of the second order.
curve of the second order.
It is easily seen that the third curve (on the left) passes through the points of contact of each of the tangents common to the first two.
15.
If,
of the three circles which have the diagonals of a complete any two intersect, the third passes through
quadrilateral for diameters, their points of intersection.
The
sides of
any right angle which has
vertex at one of these points of intersection touch a curve of the second order inscribed to the quadrilateral.
its
All hyperbolas circumscribing a quadrangle whose vertices lie the directions of these axes bisect a circle have parallel axes upon the angles which the pairs of opposite sides make with each other. For, the double rays of an involution of which three pairs of conju 1 6.
;
the
three
gate rays are
parallel
quadrangle are
at right angles to
to
pairs
of opposite
sides
of
the
each other.
M
of a given chord of a curve of 17. If through the middle point the second order two secants of the curve be drawn, these determine an inscribed quadrangle whose other two pairs of opposite sides inter
cept segments the point _U.
of
the
given
chord
which are likewise bisected
at
LECTURE
XIII.
METRIC PROPERTIES OF INVOLUTIONS. FOCI OF CURVES OF THE SECOND ORDER.* 222. Let A,
A
B
and
68),
be three pairs
of points of a range in involution, so that (Art. 219)
among -others
l ;
B,
l ;
C, C^ (Figs. 67
FIG. 67.
we have
the relation
*The more lating
AA J3C 1
1
A A^AB^C
;
then, for the segments
important focal properties of conic sections, aside from those re and the fundamental properties which we use as the
to the directrices
definition of foci,
Prop. 45.
et
were known
sequitur.
)
to Apollonius.
(Compare Conicorum, Liber
in.,.
METRIC PROPERTIES OF INVOLUTIONS. which the
determine upon the line we have the following
six points
proportion,
viz.
165
:
A A, BA, A,A B,A :
C,~A,C
AC, Instead of this
we may
write
,C
:
A A, BA = AA, AB,. l
AC, for
A,A = - AA,
~BC,~ CA, CA,, etc.,
A C=
;
1
~CB\ since two such segments are
If all denomina of equal length, but are taken in opposite senses. the equation have tors and the common factor are we removed, l
AA
AB,.BC,. CA, = AC,.BA,. CB, ................ (i) This relation
found incidentally
is
in
the Collections of Pappus,
Liber VIL, Prop. 130.
The
relation
AA,BC,
"A"
A,AB,C
does not lose
The same
two conjugate points are interchanged.
its
validity if
thing
is
true
therefore in equation (i).
By
C
interchanging
AB
l
Cv
and
for
example, we obtain the equation
.BC. C,A, = AC .BA,.
C,B,, ............... (10)
and by making
different interchanges other similarly constructed Wholly analogous equations may equations would result from (i). be expressed for the sines of the angles which six elements of an involution sheaf of rays of the first order form with one another.
and (la) assume a much simpler form if one Its example, C,) becomes infinitely distant. in this case coincident with the C becomes so-called conjugate point centre of the involution, that is, with that point O which is co 223. Equations (i)
of the
six points (for
ordinated to the infinitely distant point. At the same time the ratios
BC, = AC,-AB AC,~ AC, and
,,^,,-A,B,^ CB
CB \
approach increase
\
\
_AB AC, i
A,B,
\
indefinitely near to the value unity, since indefinitely,
AB
while
and A, If, are
Equations (i) and (la) thus take the form
AB^ OA, = BA^ OB^ AB, BO =AQ BA, .
and
.
.
AC, and C,B,
finite
segments.
1
GEOMETRY OF
66
Dividing the
respectively.
first
POSITION.
of these by the second
we obtain
Ak-Q?i ~
BO
AO
OA OA =
or
.
l
That
"
is,
The product
OB OB .
(2)
l
of the two segments determined by the
centre and any two conjugate points of an involution range of the "first order is constant."
"
M and N
If the range has two double points of which two conjugate points coincide,
it
(Fig. 68), in
OA.OA^OM^ON* The points.
centre
M
that
O
thus bisects the segment
each
follows from (2) that (3)
MN between
the double
At the same time this equation expresses the fact (Art. 72) and are harmonically separated by A and ^,-as we
N
already know.
OA OA
According as the product
OM
2
.
l
is
positive (equal to a square
do the double points appear or not. In the } former case A and A l lie upon the same side of the centre O, in the latter upon opposite sides. This same conclusion is reached or negative,
in a previous
theorem
(Art.
212).
be described upon each segment AA^ BB^ CC^ which is determined etc., by a pair of conjugate points of an in as and the radius of any one of these circles, volution, diameter, 224. If a circle
AA
say that upon V be denoted by r, while d denotes the distance of its centre from O, then (Figs. 67 and 68)
OA = OK-AK =d-r OA = OK+ KA^ = d+r;
and
l
consequently,
OA OA = (d-r)(d+r) = d .
l
2
- r2
.
If the involution has double points J/, JV (Fig. 68), that is, is 2 lies outside the circles and d~ - r is equal to hyperbolic, then the square on the tangent /, which can be drawn from to the circle ; for / and r are the sides of a right-angled triangle, of which d is the hypotenuse.
O
O
The
length of this tangent is, from equation (3), the same for constructed upon the segments, and is equal to half the length of the segment between the double points of the all
circles
MN
involution.
METRIC PROPERTIES OF INVOLUTIONS. then, a circle
If,
all
is
circles described It
described upon
upon
AA
lt
may be shown, moreover,
BB^,
167
MN with
centre O,
CCV
at right angles.
etc.,
it
cuts
that the latter circles are cut ortho
M
and N. gonally by every circle passing through When the involution is elliptic, i.e. has no (real) double points 67),
(Fig.
centre
its
and d* - r2
is
O
lies
within the circle described
upon
AA
19
negative.
AA
PQ
of this O, at right angles to V a chord of a side or OQ, forms one circle be drawn, either half of it, r the and side right-angled triangle of which d is the other If
now through
OP
hypotenuse,
so
that
^+OP-=r
2 ,
d2 -
or
r-
= - OP- = -
OQ
2 .
2 But from equation (2) of the last article the expression d- - r is same the remains this half chord of the constant, hence length for all circles constructed upon the segments V etc., lt BB^,
AA
CC
P
and consequently, these circles all pass through the two points and Q. The angles APA^ BPB^ CPC19 etc., are therefore right angles, and we obtain the theorem If an involution range of the first order has no double points, in any plane containing the range there are two points P and Q, from :
which
it
may
be projected by a sheaf of rays in involution, such that
any two conjugate rays of the sheaf are at right angles to each other. That the circles described upon AA lt BB^, CCV etc., all pass through the points of intersection of any two of them may also be concluded from the following statement
:
An
involution in a sheaf of rays is rectangular if any two of its a and b are at right angles to their conjugates a l and r "rays The correctness of this assertion follows from the fact that the "
"
rays of a sheaf
6*
can be paired in involution in only one way so
But b^ are conjugates. to it. the at this happens if to each ray is coordinated ray right angles the theorem we insert may following 225. At this point
that
its
rays a
and a l
as well as b
and
:
If a right-angled triangle inscribed in a curve of the second order be permitted to vary in any manner so that its vertex remains fixed, its hypotenuse will constantly pass through a fixed point.
The points of the curve are paired in involution by the rays of the rectangular involution sheaf S^ (comp. Art. 210). 2 226. If in the plane of a curve of the second order K a sheaf ,
U U
whose centre does not lie upon the of rays of the first order are paired in involution if all pairs curve is given, the rays of of rays conjugate with respect to k~ are correlated to each other
GEOMETRY OF
i68
POSITION.
If the involution thus constituted is rectangular its has a particular significance for this curve, and is called a focus of the curve. We may then state the following definition (Art.
207).
centre
:
A
focus of a curve of the second order is such a point of the plane that any two rays through it which are conjugate with respect to the curve are at right angles. 227.
A
focus cannot
whose centre
lie
outside the curve, for then the involution
would have two double rays, namely, the two to the curve which pass through it. tangents focus lies Every upon an axis of the curve, in other words, the diameter passing through F is an axis, since it is perpendicular to its conjugate chord passing through F. is
it
F
The
straight line joining
curve, since
it
be erected to
is it
conjugate in
two to
F and F
t
foci,
F
and
F-^
is
an axis of the
the two perpendiculars which can its pole thus being the infinitely
distant point of intersection of these perpendiculars (Art. 160). 228. The centre of a circle is a focus. Two rays which are
conjugate with respect to a circle intersect at right angles only
if
one or both are diameters whence it is easily seen that a circle has no focus except the centre. The circle, consequently, will be excluded from the following investigation. ;
FIG. 69.
229. To any straight line p lying in the plane of a curve of the second order there is a conjugate normal straight line p l (Fig. 69), namely, the perpendicular which can be let fall upon it from its pole.
Let now a be an axis of a given conic and
let
it
be cut obliquely
or
UNIVERSITY 2^Ai
METRIC PROPERTIES OF INVOLUTIONS.
P
169
P
and Then each ray by / and p^ in the points respectively. l of the sheaf stands at right angles to its conjugate ray of the sheaf For the sheaves and are projectively related if to each r l
P
P
P
P
ray of the one sheaf the conjugate ray of the other is correlated denote the infinitely distant pole of {Art. 144); but since, if
A
P
the axis a, the three rays a, PA, p, of intersect their conjugate at right angles, the sheaves and rays P^A, a, p v of l a circle as and hence generate upon diameter, l any two conjugate rays of these sheaves are at right angles to each
P
P
P
l
PP
From
other.
theorem
this
there
the
results
first
half of
the
following
:
Corresponding to any point P in an axis a. of a curve of the second arder there is a point P l such that conjugate rays through P and P 15 respectively, intersect at right angles. If all such pairs of points are coordinated they form an involution on the axis a..*
The second siderations
half of the theorem arises from the following con
:
The two sheaves of parallel rays of which the one has the direction of the ray p and the other that of the ray p l are related projectively to each other (Art. 144) by correlating to each ray of the one sheaf the ray of the other conjugate to it. The straight line a is cut by these sheaves in two project! ve ranges of points, but these are in
P
P
involution since any two of its points as and correspond to l each other doubly, f If this involution a has two double points, each of these is a focus of the curve J if a has no double points, then each of the two points from which a is projected by a rectangular involution sheaf (Art. 224) ;
a focus of the curve
is
such points intersect
;
for,
every pair of conjugate rays through
at right angles.
In the latter case the foci
lie
in the axis of the conic different
from a and form the double elements of an involution lying upon *
That there can be but one such involution established upon the axis a follows fact that any ray/ has but one conjugate normal H. /j. Any ray through P is conjugate to its normal through Pr The ray through P
from the t
P
having the direction of/
is conjugate therefore to the ray through l having the direction of fa, and also the ray through having tlje direction of fa is conjugate to the ray through The points / and l therefore l having the direction of/. correspond doubly in the projective ranges of points which 4brm sections of the
P
P
parallel sheaves of rays.
P
H.
+ The involution cannot have more than two double points, hence there cannot be more than two foci upon one axis. H.
1
GEOMETRY OF
70
POSITION.
this second axis, which may be constructed was the first.
230.
No
curve of the
in
the
same way
second order has more than two
as
foci
;
any straight line joining two foci is an axis of the curve (Art. 227), hence all foci must lie upon one axis or upon the other, and it was shown in the last article that not more than two foci can lie upon for
one
That a conic
axis.
in general has two foci involution considered in Article
from the
clear
is
229 has no real double points, then two points exist on the other axis answering the conditions for foci (Art. 224).
fact that if the
That
axis
foci of the
other the
a
of an
curve
lie
ellipse
or
conjugate
hyperbola upon which the two
or
called the
is
minor
principal
or
major
axis,
the
axis.
231. If about a right-angled triangle, whose hypotenuse lies in the minor axis of a curve of the second order, and whose other sides are
polar conjugates with respect to the curve, a circle be circumscribed, intersects the major axis in the foci of the curve. This theorem follows immediately from the preceding article. 232.
The hyperbola
that axis
by which
it
within the curve.
lie
is is
intersected by
its
principal axis
not cut the foci could not
The
an
foci of
ellipse
lie,
;
for
it
upon
since they
or a hyperbola are
equally distant from the centre of the curve (Art. 223); for in the involution a, whose double points the foci are, the centre is conjugate to the infinitely distant point, since the minor axis is
conjugate
to
all
straight lines
normal to
it.
In general, the
harmonically separated by any two conjugate lines which are at right angles to each other.
foci are
If the
a parabola and a its axis, parallel rays of which we have
curve of the second order
is
two project jve sheaves of already spoken have the infinitely distant straight line as a selfcorresponding ray, since it Is self-conjugate, being tangent to the In the involution a, therefore, one double point coincides parabola.
the
with the infinitely distant point, which, accordingly, is looked upon The parabola has therefore as a focus (ideal) of the parabola. only one actual focus which, as second double point of the involu tion a, bisects the segment between any two coordinated points
P and Pv
F
and J\ are the two foci of a curve of the 233. Suppose now second order, of which in the case of the parabola the one lies infinitely distant
in
the direction of the parallel diameters.
Any
METRIC PROPERTIES OF INVOLUTIONS.
SP
171
SP
lines and which are perpendicular to each l and lt other (Fig. 69) are harmonically separated by the points and hence by the lines and SF^ ; they therefore bisect the angles
two conjugate
F
F
SF
SF
between
and
SF-^ (Art. 68).
SP
of the straight lines SP, lt and side the curve, then
SP
If 6*
is
a point of the curve one
lies out touches the curve; if the SPl bisect also angles between 6"
the two tangents which can be drawn from these likewise are harmonically separated by
Thus we obtain
the theorems
6"
to the
curve, since
SP and SP
l
(Art. 143).
:
a curve of the second order forms equal a?igles lines which join its point of contact to the foci two the curve. conies, therefore, have the same foci (are confocal) of If the two tangents at a real poi?it common to both conies intersect at
Any
tangent
to
with the two straight
right angles.
of two tangents to a conic be joined to with one tangent the same angle as the forms other line forms with the other tangent. of a curve of the second order 234. The polar / of a focus
If the point of
its foci,
intersection
the one line
F
is
called a
directrix
of the curve.
There are two
directrices for
FIG. 70.
the
ellipse
parabola.
and the hyperbola, and one actual directrix For the latter this theorem holds true
for
the
:
PA
Two tangents to a parabola and PB are perpendicular to each other if their point of intersection P is on the directrix.
GEOMETRY OF
172
In this case, namely, the polar of
POSITION.
P passes
through the focus
F
(Fig. 70) and contains the points of contact A and B of the two tangents, and each of these tangents forms the same angle with
AB as with an arbitrary diameter. Consequently, in the triangle APB the sum of the angles A and B equals the sum of the angles PA
which
PB
and
make
with the diameter
PC
passing through
P
P; and
since the angles A, B, and be a right angle. together make up two right angles, then must 235. Other properties of the focus of a curve of the second order
P,
equal to the angle
i.e.
P
worthy of note arise from its definition, any two conjugate rays through a focus among them the theorem
in
accordance with which
intersect
at
right angles,
:
"
The segment
"and
a directrix
of a tangent which lies between its point of contact subtended at the corresponding focus by a right
is
"
angle."
That
to say, the straight lines
bounding this angle are conjugate the focus since the rays through F, point in which the tangent is cut by the directrix has the straight line joining to the point is
F
f
of contact as 236. Let
its
TA
second order
polar.
TB
and
(Fig.
be any two tangents to a curve of the
AB
71), so that
then the point of intersection
P
is
the polar of the point is
is pole of the straight line TF, and these two lines are conjugate. At the same time and FP, since harmonically separated by
harmonically separated by
FT P and
mentary angles formed by
FA
FP
(Art.
68)
T;
AB and the directrix f the PF at right angles to TF since of
FA A
and FB are and B are
FT. Consequently the supple and FB are bisected by FT and
or,
;
second order be joined to the points contact of two tangents to the curve, and to their point of intersection, this latter line makes equal angles with the two former." "
If a focus of a curve of the
"of "
A
B
and
be drawn parallel to and B l respectively, then AI and B^ are harmonically separated by P and FT. The angles between FA 1 and FB lt therefore, are also bisected by FT and FP. From this it follows that the triangles A^AF and B^BF are mutually equiangular and hence similar, so that If
through
FT which
(Fig. 71) straight lines
cut -the directrix
The segments
AA
l
f in
the points
A
1
and BB^ form equal angles with the
directrix
METRIC PROPERTIES OF INVOLUTIONS.
f t
be
173
and are therefore proportional to the perpendiculars which can let fall from A and upon /
B
Hence, Since
we have
FA AA. = FB
also,
A
:
and
B
2
:
BB.^
are two points of the curve chosen at
random,
the theorem:
The distances of
a?i
arbitrary point of a curve of the second order the corresponding directrix have a constant
from a focus and from
ratio to each other (Pappus).
FIG. 71.
In the parabola the value of this ratio is the two unity, distances are equal; for the vertex is just as far distant from the focus as from the directrix, since it is harmonically separated />.,
by
them from the
infinitely distant point of the parabola.
use of the vertex in the ellipse
it
and
By making
shown that this ratio is less than unity than greater unity in the hyperbola; and since is
easily
I
GEOMETRY OF
74
a curve of the second order either of
by
and
focus the
its
directrix as
its
1
(Figs.
If then
the other.
for
of the
point 72 and 73),
corresponding directrices
divided into two symmetrical parts has the same value for the one
is
ratio
of any
distances
F and F
axes,
the
POSITION.
d and dl
/ and fv
A
curve its
r and
r^
from the two
are foci
distances from the two
then
-,= -r = constant,
d
wherever the point
A
may
dl
lie.
FIG. 73
Also the quantity
=-l equals
this
same constant
the ellipse and in the hyperbola d-dv lt is the distance between the two directrices.
d+d
it
r -f
must be constant
r^
in
the
constant in the hyperbola, that
The sum of
the distances of
is
ellipse
to say
.
ratio.
But in
is
constant, namely, Therefore the sum
and the difference r - r }
:
any point of an
ellipse
from
the foci is
constant.
The
difference
of the distances of any point of an hyperbola from
the foci is constant. It
may be found
without difficulty that this
constant
sum
or
equal to the segment 2a between the extremities of the major axis, and that the ellipse encloses a greater segment of its major axis than of its minor axis. difference
is
237. If two points are symmetrically situated with respect to a i.e. if the straight line joining them is at right angles
straight line,
METRIC PROPERTIES OF INVOLUTIONS. to
the
given
called the
The
and
line
is
bisected by
following
is
:
F
points inverse to a focus tangents to an ellipse or hyperbola "The
centre
"whose
then either of these
of the other with respect to the straight line. theorem, then, is true for the ellipse and the
inverse
hyperbola 41
it,
175
is
the other focus
with
l
lie
respect to the several circle of radius 2a
upon a
F."
FIG. 74.
If,
namely,
G
is
the inverse of F^ with respect to an arbitrary
F
A, and G tangent whose point of contact is A (Fig. 74), then lie in one straight line, since AF^ and consequently also AG, forms with the tangent the same angle as does And (Art. 233). moreover, since the points F^ and G are equally distant from A, t
AF
FG
then is equal to the sum (or difference) of equal to the constant 20, as was shown above. If
N
FA
the curve bisects the segment F^F\ consequently,
FG
i.e.
be the foot of the perpendicular let fall from /^ upon it is the middle point of of F^G, and the centre
the tangent, to
and J^A,
and equals
\FG
or a
;
that
is
MN
M
is
parallel
to say,
tangents to an ellipse or fall with the let upon them from a focus, perpendiculars hyperbola the a circle which has axis as diameter." "lie major upon "The
points
of intersection
of
all
"
the parabola is regarded as the limiting case of an or hyperbola, for example, as an ellipse one of whose foci ellipse lies infinitely distant, we obtain the following theorem 238. If
:
"The
points of intersection of
all
tangents to a parabola
with
i
GEOMETRY OF
76
let fall
POSITION.
upon them from the focus
"the
perpendiculars
"the
tangent at the vertex of the
F
lie
uporr
parabola."
In order to prove this draw through N, the point of intersection of an arbitrary tangent with the tangent at the vertex (Fig. 75),. to a line parallel to the axis, and draw the straight line l
NF
NF
the focus; then the angles
NF
since
and
l
since
through
passes
SNF^
is
FNA the
and SNF^ are equal
second
a right angle,
FNA
(infinitely
must be
233),
(Art.
distant)
focus
;
also.
s
FIG. 75-
the This theorem gives us a very simple means of constructing an of foci the for A simple construction a
focus of
parabola.
is the following: these the tangents at the extremities of the major axis; are which and two in Q, points intersect any third tangent two axis the of conjugate by major projected from any point to obtain the foci, at either of In then, order, 146). rays (Art. to each other, which these two conjugate rays are perpendicular
ellipse or
hyperbola
Draw
we describe a section of this
P
circle
upon
PQ
circle with the
(Apollonius).
TA
as diameter;
major
axis
the points of inter
are the
required
foci
TB
of a curve of the second and 239. If two tangents and the in v respect third a l points tangent order are cut by an and of three the contact, points and being
A
ively,
A, B,
C
B
F
then for the angles subtended actual focus of the curve (Fig. 76),
METRIC PROPERTIES OF INVOLUTIONS.
F
at
true
ijj
by segments of the tangents the following equations are
:
-B FC+-CFA = (-BFC+ -CFA),
consequently, or
l
l
LB FA = -BFT = 1 TFA. I
I
FIG. 76.
then, the
If,
B^FA^
angle
between the
lying this
third
tangent
upon
first
at
F by the
two,
may be
A
has
and
TB
a constant
chosen.
remain
fixed,
the
segment of the third tangent If
magnitude however
the third
tangent glides
and B^ describe two projective ranges of the fixed tangents, while the rays FA^ and FB^
along the curve, points
TA
two tangents
first
subtended
l
describe projective sheaves of rays about F; therefore, "The projective ranges of points in which any two tangents of curve of the second order are cut by the remaining tangents are "a
projected from either focus of the curve by two equal and directly "projective sheaves of rays." "
This theorem holds true also circle,
or
and one focus be the
the curve be a parabola or a infinitely distant point of the former if
both coincide in the centre of
the
latter.
If,
tangents and a focus of a curve of the second
any required number of other tangents can be If the given curve
is
be given;
easily constructed.
a parabola the moving tangent II
three
then,
order
AB l
l
may
GEOMETRY OF
I7 8
POSITION.
become infinitely distant, in which case, the straight lines FA l and FB l form the same angle with each other as do the fixed The quadrangle B^TA^F may therefore tangents TA and TB. be inscribed in a
circle,
which
circle
"The
is
and hence, circumscribed to any tangent triangle of a
parabola passes through the focus." which If, then, we circumscribe circles about the four triangles are formed of the sides of a quadrilateral, these have one point in common, namely, the focus of the parabola inscribed to the "
quadrilateral.
TA
TB
and
B FA l
(=
l
its
TB
is
the asymptote equal to the angle A^F^B^ which
F
focus v angles and
"
AB l
l
subtends at the second
Consequently, B^FA^ and A^F^B-^ are supplementary
B^FA^\ may
be inscribed
in a circle.
Hence
:
upon a circle with the points hyperbola The in which an arbitrary tangent is cut by the two asymptotes. centre of this circle and the centre of the hyperbola lie upon a two
"The
"
B
FB
equal to one of the two angles which makes with the major axis FT; the other is
BFT)
TB
are
hence
distant points,
infinitely
angle
is a hyperbola and the two are asymptotes, then A and are parallel and the and
curve of the second order
If the
fixed tangents
foci of a
lie
same two
circle with these
"second
intersections."
EXAMPLES. 1.
To
construct the focus of a parabola
The
(a)
focus lies
can be drawn
:
upon that perpendicular to any tangent which from its point of intersection with the tangent
at the vertex (Art. 238).
() The focus bisects that segment of the axis which lies be tween any two conjugate lines at right angles to each other (Art. 233).
tangent to a parabola forms the same angle with a diameter as with the line joining its point of contact to the
Any
(c)
focus (Art. 233). (d)
Any
circle
sides
which
is
circumscribed about a triangle whose
are tangent to a parabola passes through the focus
(Art. 239). 2.
To
construct the directrix of a parabola
(a)
The
directrix
is
:
the polar of the focus.
EXAMPLES. () Pairs of tangents
179
right angles to each other intersect
at
on
the directrix (Art. 234).
The
(c)
orthocentres of
directrix
(Example
circumscribed triangles
all
12,
upon the
lie
162).
p.
The
directrices of all parabolas which can be inscribed in a triangle pass through the orthocentre of that triangle, and their foci lie upon 3.
If perpendiculars be dropped from any point circle. of this circle to the sides of the triangle, their points of intersection with the sides lie in a straight line, viz. the tangent at the vertex of one
the circumscribed
of the inscribed parabolas. 4.
To
construct the two foci of an ellipse or hyperbola
(a)
As double
(b}
By means
:
points of an involution lying upon the major axis
(Art. 229).
(c)
of the
tangents at the extremities of the major third tangent a segment
These determine upon any
axis.
which subtends a right angle at either focus (Art. 238). By means of the circle which touches the curve at the ex If perpendiculars be drawn to tremities of the major axis. any tangent at the points in which it intersects this circle, they will pass through the two foci (Art. 237). constructing a right-angled triangle whose hypotenuse the minor axis and whose sides are conjugate. The
By
(//)
lies in
circle
circumscribing such a
axis in the (e)
With the
two
triangle
the aid of the theorem that the focal
intersects
the
major
foci (Art. 231).
distances
of
a
point
sum
of the
(or difference) of
curve
is
constant
(Art. 236). 5.
Construct a curve of the second order having given one focus,
the corresponding directrix,
and one point or one tangent.
6. Construct a curve of the second order having given one focus and (i) three tangents, or (2) two tangents and the point of contact in
one of them (Art. 239). immediately determined. 7.
If in
In
either
case
the
second focus
can be
a plane the foci of any curve of the second order inscribed are correlated to each other, an involution of the
ABC
to the triangle
established among the points of the plane, of which are the three singular points that is, if the one focus describes a straight line u the other will describe a conic passing through
second order
A, B, and
A, B, and
is
C
C,
;
and projective
to
u
(Art.
233).
The
straight
lines
angles of the triangle are coordinated to themselves. Every straight line of the plane is an axis of one of the inscribed
bisecting curves.
the
GEOMETRY OF
l8o 8.
POSITION,
Construct a curve of the second order having given the two foci
and one point or one tangent.
P
to the 9. The angles made by the two tangents from a point several curves of the second order of a confocal system are all bisected by two fixed straight lines at right angles, namely, by the tangents to
the two curves of the system which pass through P. 10.
The
g
poles of a straight line
with respect to a system of con-
a straight line perpendicular to g. harmonically separated from g by the two foci.
focal conies
lie
in
This
line
is
11. Each pair of opposite sides of a quadrangle determined by the three vertices of a self-polar triangle of a curve of the second order and its orthocentre, are harmonically separated by the foci of the curve.
F
12. Given two points, A and ./?, and one focus of a curve of the second order, the other focus fi\ lies upon one of the two conies passing is given, through F, of which A and B are the foci. For, since must be of constant magnitude. l l
AF BF
AF
BF
F
The directrix corresponding to intersects the straight line AB in one or other of the two points through which pass the lines bisecting the angles formed by FA and FB. 13.
The product to
an
of the distances of a focus from two parallel tan so also is ellipse or hyperbola is constant (Art. 237)
gents the product of the distances of the two 14.
Construct
an
method proposed
ellipse
with
in Ex. 15, p. 125
at the four vertices
;
(c)
axes
;
foci
from one tangent.
of given
length
(a)
after
the
by making use of the tangents with the aid of the foci, which may be deter ;
(&)
mined when the lengths of the axes are
given.
Construct a parabola having given the focus and the directrix, or the focus, the axis, and one point or one tangent. 15.
16. The centres of all circles which touch two given circles lie upon The twp confocal curves of the second order (compare Art. 236). centres of the two given circles are the foci of these two curves. 17.
line
All points of a plane
and a
circle
lie
upon
which have equal distances from a straight either one or the other of two parabolas
(compare Ex. 16). 18. If one side of an angle of given magnitude constantly touches a curve of the second order while the other rotates about one of its foci, the vertex of the angle describes a circle which touches the curve at two points (Art. 237) if, however, the given curve is a parabola the ;
vertex describes a straight line tangent to the curve.
LECTURE
XIV.
PROBLEMS OF THE SECOND ORDER. Our
240.
admit
IMAGINARY ELEMENTS.
investigations have often brought us to problems which
two solutions, and which cannot be solved by the exclusive application of linear constructions, but require the use of a form of the second order. To this class belong in general of
among
others the problems "to determine the in two projective ranges which are
self-corresponding
superposed,"
the double elements of a form in
and
"to
points
determine
involution."
All such problems
may be reduced to the following: of points k- and k\ of the second order
Two
ranges curve
same
the
determine
their
and are
projectively
related
:
it
is
lie
upon
required
to
self-corresponding
points.
Let A, B, C(Fig.
77),
be any
A
C
three points of 2 , and v B^ v the corresponding points of k\. If we project the range k~^ from
A
and the range
we obtain two
A(A
l
l
C
ing.
The
pairs of
sheaves
from
A
l
projective sheaves
and
l)
have the ray
k-
AA
A^(ABC)
which
} self-correspond points of intersection of
homologous rays of these lie
consequently upon a
straight line u, namely, upon that line which joins the intersection of
section of
AC^ and A^C.
The
FIG.
AB^ and A^B
with the inter
points which // has in common with the curve of the second order are the required points, for in the
1
GEOMETRY OF
82
POSITION.
&
and k\ these points are self-corresponding. According, u as then, cuts, touches, or does not meet the curve do we obtain
ranges
two, one, or
no such self-corresponding points. Pascal s theorem the intersection of the jBCl and B^C also lies upon the straight line u
In accordance with lines
straight for
upon
;
this
line
the
AB^CA^BC^
hexagon
three
which
of
opposite sides of the inscribed in the conic, intersect. pairs
is
We
obtain this same straight line, then, by projecting the ranges k 2 l and & 2 from the points and B^ or from C and Clt respectively. If in general and Q are any two points of and l and Q l
B
P
&
P
the corresponding points of k\, then the points of intersection of and P-^Q lie upon the straight line u. If then three points l of k 2 together with the corresponding points of k* l are given, the
PQ
straight line
u can be constructed without
difficulty.
$
and k\ are in involution, u is the axis of 241. If the ranges involution, and the curve is cut by u in the two double points, if
such there
are.
In
we need
this case
to
A and B, B^ in A v of k correspond
of coordinated points A, for, to the points A, B,
lt
know only two pairs order to construct u 7 -
the points A lt B v A, and u passes through the two points in which the straight lines AB l and AB are cut by the straight lines A^B and A^BV respectively (Figs. 60 and 61). The pole of u, in which AA- and BB^ intersect, is the centre of involution, and if from this point two tangents can be drawn 2
respectively, of k* lt
to
the
curve, these touch
the curve in the double points of the
involution.
The different cases of the following general problem may be reduced to the preceding easily "To determine the self-corresponding elements of two projective forms which are superposed." elementary 242.
:
"
If,
reguli
for
of
example, the elementary forms are two cones or two the second order, they may be cut by an arbitrary
plane in two projective ranges of points which lie upon the same curve of the second order; if they are two sheaves of rays of the second order, then the curves of the second order which are
enveloped by them are also superposed and are projective, so that we need only determine the points which the curves have self-corresponding in order to obtain immediately the two selfcorresponding rays, the tangents at these points. If two projective sheaves of rays of the first order are concentric
PROBLEMS OF THE SECOND ORDER. and
lie
order
183
one plane, and are intersected by a curve of the second passing through their common centre, in two projective in
2 ranges of points k and &\, the two rays which are self-corresponding and 2 x have in the sheaves pass through the two points which
$
self-corresponding. If the two elementary forms
to
y
be considered are ranges of
points v and v^ (Fig. 78) which lie "upon the the case can be reduced to the preceding one
same
straight line,
by projecting the
a conic (for Pass, then, through ranges from any point S. example, a circle), in the plane Sv, and project any three points 6"
FIG. 78.
A, B, C, of v and their corresponding points A lt B^ Cv of z\ from the point S upon this curve. By this means we obtain the B C, and A\, B\, C\, and can determine curve points A immediately the straight line u upon which the points of inter section of AB\ and A\B of B C\ and B\C and of C A\ and and C\A lie. If, then, the conic is cut by // in two points ,
,
,
,
M
N
,
and we project these from the point upon the straight line z, of v which coincide with their we obtain the two points and and N^ of i\. corresponding points l If the conic has only one point or no point common with ?/, there is only one or no self-corresponding point of the ranges v and z\.
M
M
N
GEOMETRY OF
184
POSITION.
For the sheaf and the cone of the second order the general problem might be solved directly, without the use of the curve of the the general problem stated above can therefore be ; solved for the case of any of the one-dimensional primitive forms with the help of any one of the elementary forms of the second
second order
The most convenient
order.
solution,
however,
is
the one just
given, since curves of the second order (circles in particular) easily be constructed.
may
243. As is well known, many of the advances in mathematics are intimately connected with the effort to remove exceptional cases from general theorems and principles, and with the attempt to reconcile different theorems to the same point of view, by immediate extension or by the introduction of new concepts. Thus arithmetic was essentially enriched by the introduction
negative, of irrational, and, finally, of imaginary numbers ; without the latter the important theorem, "an equation of the nth order has n roots," with its numerous applications (for example,
of
In the same way in analytical geometry) would be wholly false. the introduction of infinitely distant elements into modern geometry has proved fruitful in the highest degree.
Problems of the second order have given the the
introduction
into
first
occasion for
of
points, imaginary geometry have founded the purely geometric theory of imaginary elements and to have brought it to a high degree of completeness is undoubtedly one of the greatest services which
synthetic
lines,
and planes; and,
Von
Staudt has rendered.
to
From
the nature of the subject this
theory must, in synthetic as well as in analytical geometry, give up all claim upon the powers of intuition consequently, I shall ;
confine myself at this point to the presentation of only the principles of the theory of geometric imaginaries.
We
244.
proposition
shall
define
which, at the
earlier investigations
Two
have two
sponding elements.
elements
time,
by the summarizes the
following of
results
:
*
forms which are superposed but are not rear or two conjugate imaginary self-corre 1
If the self-corresponding elements are real they
coincide.
We
say
that
the
self-corresponding
whenever they do not only
same
projective elementary
identical
may
imaginary
first
real
really appear.
elements
In
all
elements have been considered.
my An
are
imaginary
previous lectures involution always
PROBLEMS OF THE SECOND ORDER. has from
this point of
Moreover, we can
A
view two (real or imaginary) double elements.
say that
Two
curve of the second order has
two points
in
^5
common
with every
real straight line of its plane
and only when the
tangents to a curve of the
second order pass through every real point of its plane
;
different cases
;
comprised under these theorems do we add
are to be distinguished from each other
These two points are imaginary,
These two tangents are imagin-
or they coincide, according
ary, or real, or they coincide, ac-
or
real,
as the straight line lies wholly outside, or cuts, or touches the curve.
and the
If the curve
always for
regard their
rays
we imagine
straight line are completely given
common
or
points as being determined.
we
shall
But
if,
points A, B, C, D, E, of the curve are to be generated by two projective sheaves of
it
and
B(CDE] these are intersected by the straight two projective ranges of points which have the required
A(CDE)
line in
lies within,
upon the curve.
five
example, only
given,
cording as the point without, or
;
points of intersection with the curve as self-corresponding points. points can be determined by the method given above
The two
{Art 242) if any one curve of the second order is completely known. We can also ascertain by this means to which variety a conic determined by five points belongs ; for, a curve of the second order is a hyperbola, ellipse, or parabola according as the two points which it has in common with the infinitely distant straight line are real and different, imaginary, or coincident. The following problem of the second order the same method
Of
;
with any given real straight
line of its plane.
If
solved by
a curve of the second order there are given
Four points and the tangent at one of them, or three points and the tangents at two of them it is required to determine the two points which the curve has in com-
mon
may be
:
it
is
Four tangents and the point of contact on one of them, or three tangents and the points of contact on two of them it is required to determine the two tangents which ;
pass through any given real point of its plane.
required to determine the
common
points of a straight
and a curve of the second order of which five tangents are given, we find first the points of contact in these tangents and so reduce this problem to the one just solved. line
1
GEOMETRY OF
86
POSITION.
Von
Staudt distinguished between the two conjugate double elements of an elliptic involution AA^ BB^ imaginary ... by connecting with the form a determining sense ABA^ or 245.
A^BA. Without going into the matter more minutely we can enunciate upon the basis of what has been said already the following theorems and definitions :
An
also contains the conjugate imagin-
An imaginary plane always passes through a real straight line through this straight line passes also the
ary point.
conjugate imaginary plane.
lies
imaginary point always
upon a
real straight line
;
this line
;
An imaginary straight line of the first kind always passes through a real point and lies in a real plane with its conjugate imaginary line ; namely, the point and the plane are the bases of the projective sheaves of the
first
order which have the two con
jugate imaginary straight lines self-corresponding. Two projective reguli of the second order which are superposed have two real straight lines, which may however coincide, or
two conjugate imaginary straight corresponding.
These imaginary
of the second kind,
lines
lines
self-
second kind are kind in that they
of the
distinguished from the imaginary lines of the first can be cut by no real plane in real points and can be projected from no real point by real planes. A real plane, namely, cuts the two projective reguli in two projective ranges of points of the first
or second order which
self-corresponding
points
lie
only
upon the same base and have if
the
have
reguli
real
real
self-corre
Thus there exists only sponding lines (passing through them). one kind of imaginary points or planes, but, on the other hand, two kinds of imaginary straight lines. 246. lines,
Whenever we make and planes, we shall
unless the contrary text.
is
mention refer
as
simply
of
points,
straight
heretofore to real elements,
expressly stated or
is
evident from the con
This applies in particular to the following problem of the
second order
:
In a plane there are given two simple polygons it is required to construct a third polygon which is inscribed to one of these "
;
"
"
and circumscribed about the other." Or to speak more definitely, "To construct an ^-point whose vertices
lie
in
order upon
n
PROBLEMS OF THE SECOND ORDER. "
given straight lines u
u, // 3 n given points S^ -S2
"through
.
,
.
.
,
187
u n and whose sides pass of a plane." 3 ... Sn ,
S
in order
,
,
from the point Sl upon the straight Project the range of points from i.e. the projection of u line #.7 ; next, project the range l9
^
.
S2 upon
the straight line
upon the
straight line
?/
//
4,
3
S3
then the range u 3 from the point
,
and so on,
till
we
finally
project the
By this means range // from the point Sn upon the straight line u r we obtain // + i projective ranges of points of which each is a pro jection of the preceding, and of which the first and last lie upon Each point which the first one and the same straight line // r and last ranges have self-corresponding can be chosen as the first vertex of the required -point, and a solution of the problem immediately presents itself. In general there are then
at
most two
/z-points
which
satisfy
the conditions of the problem. have If in particular cases the two projective ranges lying in more than two and consequently all their points self-corresponding,
^
then there
an
is
infinite
number
of solutions of the problem.
The
condition that the straight lines u v u^ // 3 ... //, shall lie in one and the same plane with the points Sv S^ S3 ... Sn is moreover not necessary ; it is sufficient if Sl lies in a plane with ,
,
and w
u^
Sn
,
-5*2
m
,
and so on, till finally a pl ane with // 2 and 3 with u n and it r The two resulting ^-points ,
in a plane
lies
would then be gauche. "
"
247. In this connection belongs also the problem, "To find a which intersects four given straight lines a, b, c, d,
straight line
when no two
of
them
lie
in
one
plane."
Relate the sheaves of planes a and b perspectively to the range of points fj and let d intersect them in two projective ranges of points.
Through each of the points which these ranges have self-corresponding a straight line passes which is cut also by a, b, and c, since it lies in two homologous planes of the sheaves a and b. If the straight c, d, belong to one and the same regulus, then the in general, however, has an infinite number of solutions problem there are but two. This problem might be stated thus
lines
b,
(2,
;
:
"A "
ruled surface of the second order
lines #,
"which
248. is
it
,
ct of
its
common
reguli
;
it
is
with any fourth line of the most important problems of the second order
has in
One
one of
given by three straight required to find the points is
the following
:
d"
GEOMETRY OF
i8S
"
POSITION.
"Two involutions lie upon the same base; it is required to determine two elements which are coordinated to each other in
involutions."
"both
Suppose the two involutions lie on the same curve of the .second order, and, in the one involution, to some two points a and ft of the curve the points 04
and
/3j
respectively are coordin
ated, while in the other involu
A
tion, to the points
and
coordinated the points JB lt respectively ; then
B are and
l
we seek
two centres of involution
the
U and
*\J
A
V,
so that
pairs of
all
conjugate points of the one involution lie in a straight line with U, and all pairs of such points
FIG. 79.
of
the
other
straight line with
lie
V.
If,
then,
X
UV
a
in
X
the straight line intersects the curve in two points and v these points will be coordinated to each other in both involutions.
UV
If
touches the curve, the point of contact
of each involution.
no
if
Finally,
is a double point outside the curve, wholly coordinated either with itself or
UV lies
point which is with a single other point in both involutions. This latter case can happen, however, only there
is
real
two involutions has two
real
double points,
since only then do both points further, when the polars of namely, in the pole of
U and V
U
UV>
and
i.e.
V
when
lie
i.e.
when each of
the
both are hyperbolic,
outside the curve; and within the curve,
intersect
the double points of the
one involution are separated by those of the other. If the problem should relate to two concentric sheaves of rays in involution we may intersect them with a curve of the second order which passes through the common centre, and in a similar
way can reduce any chosen case just now treated. The result last obtained
holds
of the general problem to that
good,
not only
therefore,
for
two point involutions which lie upon the same curve of the second order, but it can be stated for a general case thus If two involutions are superposed there will always be two "elements which are coordinated to each other in the one involution :
"
PROBLEMS OF THE SECOND ORDER. "
as well as in the other
"
only in
case
"elements
;
189
these elements are (conjugate) imaginary are hyperbolic and the double
both involutions
of the one are separated by the double elements of the the two doubly conjugate elements coincide, the
If
"other.
"involutions
have one double element in
common."
Suppose, for example, the involutions are two sheaves of rays of the first order and one of them is rectangular (Art. 224), then this theorem follows as a special case :
any sheaf of rays of the first order in involution there are "always two real conjugate rays which are at right angles; these "In
"are
called the
axes
of the involution
sheaf."
We
have thus proved in an entirely different manner the 249. theorem that an ellipse or hyperbola has two conjugate diametersat right angles,
two axes (compare Art. 159);
i.e.
form an involution
if
for,
its
diameters-
pairs of conjugate diameters are coordinated
to each other.
This result naturally belongs to metric geometry, as does also the allied problem "To construct the axes of a curve of the second order of which :
"
two pairs of conjugate diameters are
given."
3E
FIG. 80.
Through the centre S of the curve (Fig. 80) in which the given diameters intersect, pass an arbitrary circle; let this cut the one and pair of conjugate diameters in the points the other lt
A
pair in the points
paired
in
B
involution
and
BY
Then
by means of
A
the points of the circle are the sheaf of diameters, and
GEOMETRY OF
190
U
POSITION.
BB
AA
the point in which and intersect is the centre of l l involution with which each pair of coordinated points of the circle lie in a straight line. If then we pass a straight line through and the centre of the circle, this will cut the circle in two
U
C
X
X
coordinated points and required axes SX and SX r
U
which are projected from
l
5
by the
exist two real double points these are projected from by two tangents to the circle, but from S by the asymptotes of the hyperbola to which the given conjugate diameters belong (Art. 161). How
If
lies
outside the circle there
M and N;
of the involution
U
could the axes be constructed
if
instead of the circle an arbitrary
curve of the second order passing through S were given? 250. In what follows we shall many times need to solve
problem
a sheaf of rays in involution to determine two
"In "
rays
the
:
conjugate in the
which are harmonically separated by two given points
"
plane."
In order that this problem may not be impossible we assume that and do not lie in a straight line with the the given points centre of the sheaf, and that through neither of them passes a double If then we project the involution range of points, ray of the sheaf.
M
N
M and N
are the double points, from S by a second have only to find those two rays which are coordinated to each other in both sheaves, and these will be
of which
we
involution sheaf,
shall
the rays required.
may be
This problem
and
to other elementary forms Instead of the sheaf S, for example,
carried over
stated in different ways.
MN\
might be given an involution range upon the straight line in this case one of the given points, say N, lies infinitely
if
distant the
problem becomes
:
In a point involution of the first order it is required to determine two conjugate points which have equal distances from a given "
"
"
point
M
of the straight
line."
ABC
A
and an 251. triangle involution sheaf of rays of the first order F, of which no double ray passes
through
a
vertex
of
the
It is triangle, are given in a plane. required so to circumscribe a curve
of the second order about the
tri-
A of
triangle
the
first
and a point involution order, of which no
double point lies upon a side of the triangle, are given in a plane, It is required so to inscribe a curve of the second order to the triangle all pairs of corresponding
that
PROBLEMS OF THE SECOND ORDER. angle that of
rays
all
the
I9 i
points of the involution are
pairs of corresponding involution are con-
con-
jugate with respect to the curve,
jugate with respect to the curve.
may be
In order that the problem
F
the centre of the involution
ABC
the triangle
(Art.
142)
two of the sides of the through
F (Fig.
we assume
therefore that at least
AB
and say each of these sides
triangle,
Upon
81).
;
possible it is necessary that should coincide with no vertex of
AC, do
AB
and
not pass
AC there
\
FIG. 81.
must
one point whose polar with respect
lie
in case the latter exists, passes through
to the required curve,
F; and
since a point is harmonically separated from its polar by the curve, while, on the other hand, all pairs of corresponding rays of are by assumption conjugate with respect to the required curve, these points may be
F
determined as follows
:
F
two corresponding rays p and / 1? Determine in the involution which are harmonically separated by the points A and B, and two others q and q^ harmonically separated by A and C (Art. 250). If these rays are imaginary,
exists
which
happen
if
then no real curve of the second order
given conditions, but this case can only the involution has two real double rays which are fulfils
the
F
by A
and
B
A
and C (Art. 248). Let the straight and Pv / and p^ in the points be cut by q and q l in the points Q
or by separated line be cut by the rays
AB
respectively,
and
let
AC
P
GEOMETRY OF
192
and
Qv
Then we must make one
respectively.
following suppositions (1) (2) (3) (4)
P P P
and and and l P^ and
Q Q Q Q
l
l
Each of these
POSITION. or other of the
:
and q^ respectively; and g, respectively and q v respectively and g, respectively.
are the poles of p l are the poles of p^ are the poles of p
are the poles of
;
;
p
four suppositions yields a solution of the given
If, for example, the problem. seek upon the straight line
supposition is made, we should the point C which is harmonically
first
PC
P
and its polar p^ and similarly upon separated from C by which is harmonically separated from the point by Q and That curve of the second order which passes through its polar q v
QB
B
B
ABCB
C and of which any number of points can be determined by previously given methods, fulfils, then, all the given conditions. For, it is circumscribed to the triangle the five points
,
.
C
since two pairs of points of the curve, A, B, and C, }t and / 15 then is the pole of p l are harmonically separated by or p is conjugate to / 3 and likewise the ray q isand the ray
ABC, and
P
P
FP
conjugate involution 252.
If
,
of corresponding rays of the are conjugate with respect to the curve. has two real double rays, these of the involution
g v so
to
F
that
all
pairs
F
necessity being problem thus
tangent
to
the required curve,
we can
state
the
:
About a given triangle to circumscribe a curve of the second order which shall touch two given
In a given triangle to inscribe a curve of the second order which
lines in the plane.
in the plane.
shall pass
through two given points
The above construction shows that each of these reciprocal problems has four and only four real solutions if, in the one case, the two given straight lines are separated by no two vertices of the triangle, and in the other case, if the two given points are separated by no two sides of the triangle
;
otherwise, there
is
no
real solution.
F
253. If the double rays of the involution problem has four solutions. This case happens, if
the involution
is
among
F
other
ways>
a focus of the rectangular and consequently was supposed in Fig. 8 1 ) incidentally, then,
required curve (as solved the problem
we have
are imaginary the
;
:
PROBLEMS OF THE SECOND ORDER. "
To
193
determine the four curves of the second order which circum-
and have a given point as focus." 254. In conclusion, a problem may here be inserted which is clearly not of the second order, but which is closely related to the one last discussed, namely, "
scribe a given triangle
In a plane are given a triangle
In a plane are given a triangle
ABC and an involution point-range
and an involution sheaf of rays of
of the
first
order
n,
so
the
situated
first
order 5, so situated that
S
that neither double point of if lies on a side of the triangle and
through a vertex of the triangle
u passes through no
and the centre S
neither double
vertex of the
required so to circum scribe a conic about the triangle triangle.
that
of
pairs
of
corresponding
u are conjugate with
Let the points
AB
and
passes
upon no
side
rays of 6 are conjugate with respect
respect to the curve.
lines
lies
of
of the triangle. It is required so to inscribe a conic in the triangle that all pairs of corresponding
It is
all
points
ray
to the curve.
K
BC
and
M
82), in which u is cut by the be coordinated in the involution respectively, (Fig.
FIG. 82.
M
AB B
u to the points K^ and v and let K.2 be that point of which is harmonically separated from by the points A and of the curve, and M.2 that point of which is harmonically separated conic,
from
K^KI
is
M by
K
BC
B
With respect to the required the polar of the point K, since both K^ and K% and
C.
MM
are conjugate to K\ similarly, is the 2 1 polar of M. is the point which is harmonically separated from
C
If then
C
by
K
1
GEOMETRY OF
94
and
A"A*
from
A
A
POSITION.
point which is harmonically separated the required curve passes through, and is determined by, the five points A, B, (7, A, Evidently the and and are points conjugate with K^ (and similarly, J/j)
and
r
the
by M and M^M^
to
respect K^KZ>
C
M
K
curve as was required,
this
since
being harmonically separated from
the
.
K
is
the pole of
straight line
K^K.2
both by the points A, B, and C, C This problem might also be stated as follows (compare Art. 244) .
Through
five
To a
points of a plane
:
plane pentagon, three of
of which three are real and the other two either are real or are
whose sides are real and the other two either are real or are conjugate
conjugate imaginaries, and of which
imaginary lines, to inscribe a curve of the second order,
no three
one straight line, to describe a curve of the second lie
in
order.
Each of the
reciprocal parts of this problem has
one
solution.
EXAMPLES. In a given curve of the second order inscribe a simple w-point sides in order pass through n given points of the plane, none
1.
whose
of which
lie upon the given curve. About a given curve of the second order circumscribe a simple ./z-point whose vertices in order lie upon n given straight lines of the plane, none of which touch the curve. 2.
In a plane there
3.
is
it is required to given a simple pentagon is both inscribed and circumscribed to :
draw a second pentagon which the given one. 4.
5.
it is required to draw two straight lines equal segments upon two given lines.
Through a given point
which
will intercept
Choose a point in each of two given them subtends a right angle at
line joining
straight lines so that the either of two given points.
Upon a given straight line determine a segment which subtends given angle at either of two given points. bisects a given 7. Through a given point draw a straight line which 6.
a
triangle, or
given 8.
its
which forms with two given straight
lines
a triangle of
area.
so that two of and the angle at
Circumscribe a triangle about a given triangle vertices
shall
lie
upon two given
straight lines
the third vertex shall be of given magnitude.
EXAMPLES.
I95
In two projective sheaves of rays find a pair of rays at right angles
9.
in the one,
which correspond, each to each, to a pair at right angles
the other (Ex.
5,
in
p. 67).
10. In two projective ranges of points upon the same straight line find a pair of homologous points which are a given distance apart; or, in two concentric sheaves of rays projectively related find two
homologous rays making a given angle with each other. 11. In two projective ranges of points of the first order
find
a pair
of homologous segments of given lengths. 12. A curve of the second order has in general one pair of conjugate diameters which are parallel to two conjugate diameters of another curve -of the second order lying in the same plane. Two hyperbolas,
the directions
of
whose asymptotes separate each
other,
make an
exception to this statement (Art. 248). 13. About a given quadrangle circumscribe a curve of the second order which shall touch a given straight line (Art. 221) or, to a given quadrilateral inscribe a curve of the second order which shall pass ;
through a given point In the plane of two curves of the second order there is at least which has the same polar u with respect to both curves The two real or conjugate-imaginary points of u which (Ex. 5, p. no). are conjugate with respect to both curves (see Art. 248) form with 14.
one
real point
U
U
a.
common
second
self-polar triangle of the two curves. order lying in the same plane, therefore,
common real side.
self-polar triangle
;
this
Two
curves of the
have in general one has at least one real vertex and one
LECTURE
XV.
PRINCIPAL AXES AND PLANES OF SYMMETRY, FOCAL AXES AND CYCLIC PLANES, OF CONES OF THE SECOND ORDER.
The
255.
polar theory of cones of the second order follows by been said (Art. 148), from that of curVes
projection, as has already
of
second
the
but
order,
it
is
further
concerned with
certain
theorems of metric geometry which can be developed out of the polar theory, though they can in no way be transferred from the curve to
the
cone by methods of projection and section they for the cone, and for it must be ;
are in part of different nature
developed independently. Our previous investigations, however, will frequently serve us as models and furnish suggestions for further developments. 256. If a cone of the second order
is chosen in a bundle of which pass through a ray e of the bundle are con jugate with respect to the cone to a certain plane e also belonging to the bundle, where e is the pole-ray of e. The ray e and that ray of the bundle which is normal to e determine a plane e which is normal and at the same time In general there conjugate to e.
rays, all planes
*
only one such conjugate normal plane for each plane e of the bundle but if a plane a is at right angles to its pole-ray a all its In this case a and a bisect conjugate planes a are normal to it. is
;
the angles which are formed by any two rays of the cone lying in a plane with a ; for a and a are normal to each other and
The rays harmonically separate the two rays (Arts. 68 and 148). of the cone are thus symmetrical, two by two, with respect to the and we may
a a plane of symmetry for the cone its usually designated a principal axis of the cone. pole-ray of plane symmetry for the cone thus stands perpendicular to the
plane
a,
call
;
A
is
corresponding principal
axis,
i.e.
perpendicular to
its
pole-ray.
THE CONE OF THE SECOND ORDER.
197
rotates about a ray s of the bundle, its pole257. If a plane ray e describes a sheaf of rays in the polar plane of s while its normal The two sheaves ray describes a sheaf in the plane normal to s.
of rays
described
thus
are
and consequently
s
planes
projectively also
to
related
each
to the
sheaf of
and generate
other,
in
Hence general a sheaf of planes of the second order. of two normal of the If, bundle, the one t conjugate planes rotates about an axis s, the other e will describe in general a sheaf of the second order which contains the polar plane of s with :
"
"
"
""
respect to the given cone to
and the plane of the bundle normal
s."
The plane
rotates about
e
an
axis s
only
if
the two projective
rays have perspective position, that is, have a selfIn this case s lies in a plane of symmetry corresponding ray a. a of which a is the pole-ray (the corresponding principal axis) and since a js the conjugate normal plane of the plane sa, s
sheaves
of
;
must also
lie
That
in a.
is,
In a plane of symmetry a for a cone of the second order each ray s of the bundle is coordinated to a ray s so that any two "
"
"
"
planes of the sheaves s and s normal to each other are conjugate with respect to the cone."
This theorem It brings
order,
we must
first
really exist for
We
258.
analogue of that by the help of which the foci of a curve of the second order.
us to the so-called focal axes of the cone of the second
but
symmetry
the
is
we obtained
226)
(Art.
shall
show
decide the question whether planes of
such a cone, and
if so,
in the first place that the
how many. cone has at
least
one plane of symmetry. Let ^ be the sheaf of planes which
formed by the conjugate a If planes passing through ray s of the bundle. of the first order, the existence of a plane of symmetry is
normals of this
is
evident.
To
is
all
We
shall
suppose therefore that -
is
of the second order.
sheaf of planes belongs every plane of symmetry for the cone as conjugate normal to the plane passing through s given and the corresponding principal axis. Now the planes 77 which cut this
the cone enveloped by - in two rays are separated from the re of the bundle by the tangent planes of the cone If then t is the straight line in which the (compare Art. 136).
maining planes
conjugate normal of one of the planes 7; and a plane intersect, the conjugate normals of all planes passing through / form a sheaf
GEOMETRY OF
198
T of
the
POSITION.
or second order which has at least two
first
and
at
most
four real planes in common with 2. One of these common planes is the conjugate normal of st ; but every other is normal to two different planes conjugate to it, namely, to a plane in each of the
and hence is normal to its pole-ray in which /, And since any plane normal to its these two planes intersect. pole-ray is a plane of symmetry for the cone, there exists at least sheaves s and
and at most three, planes of symmetry. of the second order there exists then at cone For every 259. least one plane of symmetry a and a principal axis a corresponding but it can easily be shown that in the principal axis a two to it one, but in general
.
;
planes of
Namely, if symmetry always intersect at right angles. and two, those rays of the bundle lying in the plane of symmetry a which are conjugate with respect to the cone, we obtain an involution whose axes b and c are principal axes- of the cone for, the ray b, for example, is conjugate and at the same
we
coordinate, two
;
and
to a ; its polar plane ca is consequently normal In particular, if a plane of symmetry of the cone. the involution a is rectangular each of its rays is a principal axis
time normal to to
,
the tangents
which are conjugate with respect 2 y always lie upon the same ,
2 tangent to y
2 /
Two points
(^)
are
>
,
jugate with respect to the conic not touched by them.
,
of
the pole with respect
,
r
which are conjugate to the curve upon which they do not lie. 2 whose (e) Two tangents to y
real
with
From
k 2 of a tangent to y 2 there can be draw n to one or to both curves two real tangents which are con to
.
2 2 (/ and /j) The curve & lies either wholly outside y or partly of two curves inside neither the outside and partly wholly encloses;
the other.
The
30.
Two
k1
is
following will be sufficient proof of these theorems 2 y to which the conic :
vertices of the self-polar quadrangle of
circumscribed determine, with any third point
second
2 which self-polar quadrangle of y
is
A
inscribed to
of
? /
,
a
K
l
( 23 quadrangle a third can be are the same way, of which two vertices A and chosen arbitrarily upon 2 and this third yields a fourth self-polar
and
25); derived in
from
this
second
self-polar
B
,
2 quadrangle of y inscribed to 1 arbitrarily chosen upon K .
it
(&} follows
2 /
,
which has three vertices A, C, (a), and from B>
This proves theorem
immediately.
Let a be the polar of A with respect to y 2 and if this should have no real point in common with y 2 A lies inside y 2 ; the polar 2 a intersects the opposite sides of the self-polar quadrangle of y ,
,
which
has
one vertex
at
A,
in
pairs
of points
conjugate with
NETS AND IVEBS OF
CONICS.
219
n is respect to y- (22), while the involution so determined upon The conic sections has no real double elements. elliptic, />.,
which can be circumscribed to any one of these self-polar quad rangles must therefore intersect the straight line a in two real conjugate points which, taken with A^ form a real self-polar triangle of y- ; for, these conies form a continuous series, an infinite number
them have two
of
common
real conjugate points in
and
tion a (Art.
it
that
is
with the involu
one of them should
221), impossible imaginary points of intersection, since then between this conic and those having real intersections there would lie a conic Thus tangent to a at a real double element of the involution.
have
theorems
and (f) are proved, and from these
(c)
(d)
and
(e)
follow
easily.
Theorems (a^) to (yj) of 29 can be proved in the same numbered from (a) to (/) as soon as it is shown that to the conic y- some one quadrangle can be circumscribed which is self-polar with respect to k-. In order to show this, choose three points P, Q, R, upon /- whose joining lines lie These determine outside y-, this being always possible ( 29/). a quadrangle PQRS self-conjugate with respect to y- and inscribed to k-, whose opposite sides intersect in three points U, F, U\ lying these three points are conjugate two and two with outside y1 If now from k and two of them lie outside k-. respect to U and V pairs of tangents be drawn to y 2 these harmonicalh .^31.
as those
way
,
,
,
separate the pairs of opposite
PQJZS
143);
(Art.
sides
consequently,
of the these
self-polar
four
quadrangle form a
tangents
quadrilateral whose opposite vertices are harmonically separated by the pairs of opposite sides of the quadrangle PQRS, and since one of the six vertices of the quadrilateral, as is easily seen, lies
k1
it must also be harmonically separated from its opposite Two opposite vertices of the vertex by the curve k- (Art. 220). are with to k- ; this quadrilateral thus respect quadrilateral conjugate 2 circumscribing y is consequently self-polar with respect to k-.
within
32.
,
Of two conic
that
*
is
"the
Compare
and y 2 of which the one k2 ,
quadrangle of the other inscribed to a self-polar quadrilateral of
circumscribed
which y 2
sections k-
to
curve
a
of the
my article
Mathematik, Bd. 82.
7-,
self-polar
second
2 ,
we
or
is
of
say*
order k- supports or carries the
in Crelle-Borchardt
sy