Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
Comment on
1r
Is Wrong!
In his note in the Mathematical Intel l ig en cer vol. 23(2001) no. 3, 7-8, Bob Palais gives his reasons why he thinks the symbol 1r should have been be stowed on 27T. It is of interest to note that Albert Eagle in his book The El liptic Functions As They Should Be (Galloway and Porter, Can1bridge, 1958), thinks 7T should have been be stowed on 7T/2. His reasons are given in his preface, and I quote. There are one or two other impor tant innovations I have made which have nothing to do with elliptic functions and the first of which may cause an outcry at first; but its im mense convenience must surely soon be realized. Most mathemati cians must have often wished that there was a single symbol that could be written instead of +7T or 7T/2. When one thinks that for the ellip tic functions we have the conve nient symbols of K and K' for the two quarter periods, it is really too absurd that for the circular func tions, which are employed millions of times as often as the elliptic func tions, we have no symbol for the quarter period at all, and have to ex press it as "half the half period." The letter 7T should have been used to denote but the meaning of 1r can obviously not be changed now. But the Greek letter T, with its one leg instead of two, so closely re sembles 7T cut in two that I have ap propriated that letter exclusively for +7T. Consequently I ask my readers every time they see it to say to them selves "half-pi" or "pi by two" in stead of "tau" or "taw." If a shorter name is desired for it what could be more appropriate to imply half-pi than "hi"? How convenient it is to write T as the upper limit of a trig integral instead of And how ,
+'IT,
f7T!
convenient it is to say "the integral of sin "I:J dI:J from nought to hi" since the integrand is zero at the lower limit, and the upper limit is its high point! It is natural that the practical man, measuring the diameter and the circumference of a cylinder, should want a symbol for the ratio of the two lengths. But a pure math ematician, noting that a diameter of a circle divides the circumference into two halves, would think it more reasonable to introduce a symbol for the ratio of half the circumfer ence to the diameter. And he, per haps rather surprisingly, would be showing better common sense about the matter than the practical man did! Seriously, who can want to have e -!7T or e - 7TI2 printed instead of e-7? Or who won't much prefer to write T than 7T for the upper limit of a trig integral? Those who peruse the numerous formulae in my book will, I think come to see that the finding of a sin gle Greek letter T to stand for +7T was a necessity that was forced upon me. How immensely nicer books on Fourier's Theorem would look with it! (pp. ix-x).
+
Murray S. Klamkin Mathematics Department University of Alberta Edmonton, Alberta T6G 2G1 Canada e-mail: mklamkin@math. ualberta.ca
. . . and again
•
.
.
I agree with Bob Palais's 7TOUS Opinion, although some might think it 27TOus. Charles W. McCutchen Camp Asulykit
Lake Placid, NY 1 2946-9600 USA
© 2002 SPRINGER-VERLAG
NEW YORK. VOLUME 24,
NUMBER 2. 2002
3
R. MICHAEL RANGE
Extension Phenomena in Multidimensional Complex Analysis: Correction of the Historical Record
omplex analysis in several variables is not a mere extension of the familim· theory in the complex plane. Novel extension behavior is found, with deep consequences for the nature of complex analytic functions. This article is a brief introduction to some of these fas cinating phenomena, both local and global. In it, I also take the opportunity to correct major inaccuracies in the histor ical record, as it has appeared in much of the pertinent lit erature since the mid-1950s. In particular, I wish to put on record important, yet generally overlooked or forgotten con tributions by Francesco Severi, Hellmuth Kneser, and Gae tano Fichera. Thus this article may have much to say to his torians, as well as to experts in multidimensional theory. Preliminaries
To set the stage, I briefly review the central concepts. Com plex Euclidean space IC" = [z = (zb . .. , Zn): ZJ E IC} is a vector space of dimension n over IC. The familiar identifica tion of IC with IR2 extends to a natural identification of IC" with IR2" , thereby giving immediate meaning on IC" to all con cepts familiar from multivariable real analysis. A complex valued C1 (i.e., continuously differentiable)1 functionf: U--'> IC on the open set U C IC" is holmnorphic on U iff is holo morphic in each variable separately, that is, if it satisfies the Cauchy-Riemann equations iJf/iJZj = 0 on U in each variable Zj (j = 1, . . . , n) separately.2 The space of holomorphic func1
It is standard that the
4
Hartogs's Theorem
We begin with the following astonishing result discovered by F. Hartogs [Hartogs 1906]. It is fair to say that this dis covery marks the birth of multidimensional complex analy sis as a new independent area of research.
Theorem 1. Suppose D c IC" is open and bounded, with connected boundary iJD. If n � 2, then every fE O(iJD) (i.e., holomo1phic on some open neighborhood of iJD) has a holomorphic extension to D. The crux of the proof is contained in the following sim ple special case.
Lemma 2. Suppose K is compact and n � 2. Then every bounded f E CJ(IC"\K) is constant on the unbounded com pon ent of IC"\K.
C1 hypothesis can be replaced by continuity alone. Much deeper is a result of F. Hartogs that complex differentiability in each variable sepa
a/a�= 112(iilox1 + i a/ay;). for j
rately already implies continuity in all variables jointly. 2Recall that
tions on U is denoted by O(U). Iteration of the one-variable Cauchy integral formula for discs yields an analogous for mula on products of discs (so-called polydiscs), which read ily leads to other standard properties of holomorphic func tions J, such as the fact that such functions are in C"(U), and have local representations by power series.
=
1,
. . .
, n.
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
side. In fact, Hartogs's first example of simultaneous ana lytic extension arose in the following context.
2 Lemma 3. Consider H = {(z, w) E C :lzl < 1, 1/2 < l wl 1 2, l wi < 1}. Then evmy fE C0(H) has a holo < 1} U liz I< / morphic extension to the bidisc P = {(z, w): lzl < 1, lwl < 1}.
1""
E
Figure 1 . Lemma 2: The complex line {(z, w'): z the ball of radius R.
C} does not meet
Clearly this Lemma includes the obvious generalization of Liouville's theorem (K = 0) to n variables. Thanks to the ex tra variable, one easily obtains the stronger statement given here, as follows: ChooseR so large that K C {z: lzl < R}. Then t for fixed w' E p- (here we use n � 2!) with lw'l > R, the complex line {(Zt. w') : z 1 E C) does not meet K, so that by the one-variable Liouville theorem,fCz t. w') is constant in z1. Interchanging variables, it follows that fis constant in each of the other variables as well, whenever the remaining vari ables are fixed with sufficiently large modulus. It follows that jis constant outside a large ball, and hence, by the identity theorem, on the unbounded component of C"\K.• Given the Lemma, the idea of the proof of Hartogs's the orem is now very simple, at least conceptually. Choose domains D1 and D2 with (piecewise) C1 boundaries, with D1 cc Dcc D2, so that fE fJ(D2\Dt) and C"\D1 is con nected. In case n = 1, the Cauchy integral formula gives f(z) = rCz) - F(z) for z E D2\Dt. where j+(z) =
�
2
Lv2 ��df,
f-(z)
=
� ��df.
2
Lv1
Clearly r E C:J(D2), FE V(C\Dt), and limzl_,"' f-(z) = 0. In higher dimensions, a corresponding integral formula (the Bochner-Martinelli formula) yields the same decomposi tion with analogous properties. (I will discuss this more in detail below.) Hencej-is bounded outside a large ball, so if n � 2, the Lemma implies thatj-is constant there, and hence must be 0. The identity theorem implies thatj- ""'0 on C"\Dt. and hence f=j+ on D2 \ D1• So j+ does indeed extendjto D2.• Hartogs's argument in 1906 was actually quite different. First of all, there was no appropriate higher-dimensional integral representation formula known at that time, which would have yielded the decomposition f= j+ - f-in ar bitrary dimensions. More significantly, Hartogs's point of view was directed to the i nterior of D, and not to the out-
-------- ----
lwl
H
lf.J.
0
Jzl
Figure 2. Lemma 3: Representation of H in the (JzJ, JwJ)- plane (the
absolute plane).
f(z)
=
F + (z) - F - (z) for z
E aD.
As in the proof of Hartogs's theorem given above, The orem 4 is now an immediate consequence of this Lemma. In fact, iff is a CR function, then F+ and F- are both holo morphic, and since limz�x F- (z) = 0, Lemma 2 implies that F - """ 0 when n � 2. Sof = F+ on aD, and the proof of The orem 4 is complete. (By refining these ideas, one can show that the holomorphic extension F is even in C1(D). More generally, if aD is of class Ck and f E Ck(aD), then F E Ck(D). This seems to have been noticed first in 1975 [Cirka 1975], [Harvey and Lawson 1975]; a simplified version of their proof is given in [Range 1986].) Bochner's 1 943 Paper
I already mentioned that since the mid-1960s the global CR extension theorem, under varying hypothesis, has been widely attributed to S. Bochner. Among the many exam ples are [Hom1ander 1966], [Andreotti and Hill 1972], [Wells 1974], [Cirka 1975], [Harvey and Lawson 1975], [Polking and Wells 1975] [Weinstock 1976], [Boggess 199 1 ] , [Ja cobowitz 1995]. However, a careful reading of Bochner's 1943 paper, which is their reference, reveals that Bochner did not prove any such result, nor did he ever indicate that he had been contemplating any generalization of Hartogs's theorem in that direction. The published record suggests that in 1943 Bochner was completely unaware of the idea of tangential Cauchy-Rie mann equations, and, in particular, of the earlier work of Wirtinger, Severi, and Kneser. Instead, the fornmlation and proof of Bochner's main Theorem 4 (p. 659, op. cit. ) clearly shows that he was looking for conditions that would imply the extension of harmonic functions defined in an open neighborhood B of a sum B of simplices, e.g. , in case B 10
THE MATHEMATICAL INTELLIGENCER
equals the boundary aD. In the application to the complex case immediately thereafter (Theorem 5, p. 660), a casual reading of the hypothesis an analytic function of several
complex variables defined i n the connected boundary
might tempt one to think of CR functions, but there is sim ply no evidence to back up such an interpretation. Not only was Bochner just using the standard formulation that a function is holomorphic in a point (or in some set) if it is holomorphic in some open neighborhood of that point (or set), but more importantly, Bochner's proof (i.e., the proof of Theorem 4 in his paper) does not make sense unless the function f to be extended is known to be harmonic in an open neighborhood of aD. In Theorems 6-9 thereafter, Bochner specializes to IC" and derives from Green's fommla what is now known as the Bochner-Martinelli formula. The hypothesis on the function f still states explicitly an a nalytic function in B (i.e., an open neighborhood of B) (pp. 662-663). Bochner explicitly proves Theorem 8, to the effect that F(z) = KBJl,J(?, z) is holomorphic in z outside of aD (see the remark made earlier), but for the other results, in particu lar for the main extension Theorem 9 (p. 663), no proofs are given. Instead, these are viewed as analogues of the cor responding theorems proved earlier for harmonic func tions, and hence it would seem that Bochner simply had in mind the obvious modifications of the proofs of the former theorems. Bochner's later papers (for example [Bochner 1954]) confirm that he had not been thinking at all about any generalizations of Hartogs's theorem to the setting of CR functions. The function to be extended is explicitly as sumed to be an analytic solution of an elliptic differential operator on a neighborhood of the bou ndmy (op. cit., p. 3, and Theorem 5, p. 7, 8). I conclude that the known relevant published record 1943-1954 does not support any attribution of the CR ex tension theorem to Bochner. With hindsight, it is of course true that the conclusions of Theorem 9 in Bochner's 1943 paper remain correct if the functionfis only assumed to be a CR function on aD. How ever, the proof in that case, as completed by Martinelli in 1961 (op. cit), does not just follow by "forgetting" the addi tional hypothesis. The jump formula now does require a new proof involving arguments quite a bit more delicate than those used by Bochner in 1943. If this tenuous connection is all that is available to justify naming the CR extension theorem after Bochner, equal credit should then be given to Martinelli, whose 1942 proof of Hartogs's theorem is defacto identical to the proof of Bochner's Theorem 9. However, given the earlier contributions of Severi and Kneser, the lack of any evidence that Martinelli and Bochner were even re motely aware in the 1940s of the concept of intrinsic tan gential Cauchy-Riemann equations, and lastly Fichera's first complete published proof of the general case, any such at tribution would misrepresent the historical record.
JaDf(0
How the Confusion Arose and Spread
Having reached this point, it remains to clarify the origins of the erroneous attribution. After reviewing the literature
and consulting with key mathematicians close to the mat ter, it appears that the attribution to Bochner resulted from unfortunate misunderstandings. The earliest reference linking Bochner to the CR exten sion theorem seems to occur in a paper by J. J. Kohn and H. Rossi [Kohn and Rossi 1965], who state in their intro duction, "Bochner's proof of this theorem [i.e. , the classi cal Ha rtogs Extension theorem, ref is to Bochner's 1948 paper] shows thatf can be so extended under the weaker hypothesis that it satisfies only the equations [i.e. , the ta n gential Cauchy-Riem a n n equa tions] on bM . . . " (op. cit., p. 45 1 ] . Since no precise hypotheses are given, the mean ing of this statement is not clear. The best I can extract from Bochner's proof is the conclusion that given f har monic in a neighborhood of aD, iff satisfies the tangential CR equations on aD, then the extension given by Bochner's Theorem 4 is indeed holomorphic on D. After more than 35 years, Kohn and Rossi do not recall whether they had anything else in mind. They were aware of Lewy's local CR extension theorem in the strictly pseudoconvex case, and the corresponding global question seemed an obvious and natural problem. In the cited paper, Kohn and Rossi proved the CR extension theorem (in the e x category) and its gen eralization to ab-closed forms on complex manifolds, as suming at least one positive eigenvalue for the Levi fom1 everywhere on ?_D, by using Kohn's recent deep regularity results for the ii-Neuman problem. Strictly speaking, the rather limited statement of Kohn and Rossi quoted above does not add up to an attribution of the CR extension the orem to Bochner, yet it could easily be so interpreted, es pecially if no references to earlier work on global CR ex tension were known. Around that time Honnander had also been investigat ing the Cauchy-Riemann equations. He was familiar with Ehrenpreis's recent proof of the Hartogs theorem, and also with Lewy's local result. From the perspective of partial differential equations, it was clear to Hom1ander that the proper setting for these results involved the local and/or global Cauchy problem for the Cauchy-Riemann operator a with zero boundary values on hypersurfaces. In particu lar, in his 1964 lectures at Stanford University, he adapted Ehrenpreis's proof to obtain proofs (under not quite opti mal differentiability hypotheses) of both the local and the global CR extension theorem (the latter one seemingly a new result). These proofs were then included in his mono graph, published in 1966. Attempting to recollect events from over 30 years ago, Hormander believes that he talked about his proof of the global CR extension theorem to Kohn, who was also visiting Stanford at that time, and that Kohn suggested to him that he should mention Bochner's work in this context. Honnander followed this suggestion, and in his book attributed the result to Bochner without ha\-ing personally checked Bochner's 1943 paper. 10 In the absence of any knowledge of the work of Severi, Kneser, and Fichera, Hormander could very well have been cred-
1 0This account is based on a porsonal communication by
L. Hbrmander in Fall
1997,
ited with the proof of the global CR extension theorem, so Hommnder's decision to defer to Bochner-inaccurate, as it turned out-reflects a sense of fairness and generosity. Later authors appear to have simply accepted Homlan der's account. Conclusion
Even though Bochner should not be credited with the proof of any version of the CR extension theorem, his 1943 pa per remains a landmark in the history of the Hartogs ex tension phenomenon. His vision to enlarge his horizon from holomorphic functions to certain ham1onic functions set the stage for further generalizations by himself (for exam ple [Bochner 1954]) as well as for Ehrenpreis's investiga tions on related problems for solutions of more general el liptic partial-differential operators. In closing, it should be pointed out that Bochner's 1943 paper, in an ironic twist, includes an important result for which Bochner did not receive any credit until recently [Range 1986, p. 188] . Bochner proved the solution of a on polydiscs (for (0, I)-forms in the real-analytic case, which was the case of interest to him), via the Cauchy transform with paran1eters in dimension one, and by induction on the number of differentials dZj appearing in tile given form (The orem 1 1 , op. cit., p. 665). This result, with essentially the same proof, 10 years later becan1e widely known as the Dol beault-Grothendieck Lemma. But this is another story. . . . Acknowledgments
I am grateful to E. Martinelli, now deceased, for telling me about the work of Severi and Fichera, and of his own proof of the global CR extension theorem. While preparing the new printing of [Range 1986] , I set out to correct the record, and I thank R. Gunning, L. Hormander, J. J. Kohn, and H. Rossi for their assistance. I became aware of Hellmuth Kneser's 1936 paper only in 1999, unfortunately after pub lication of the corrected 2nd printing. Instead of waiting for a 3rd printing, I thought that a separate historical account might be of interest to a wider audience. I thank E. Straube for pointing out this important historic contribution, as well as G. Betsch and K. H. Hofmann for sharing their 1998 preprint and other personal details about Kneser. REFERENCES
[Andreotti and Hill 1 972] Andreotti, A., and Hill, C . D . : E. E. Levi con vexity and the Hans Lewy problem, Part
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[Behnke and Thullen 1 934] Behnke, H . , and Thullen, P . : Theorie der Funktionen mehrerer komplexer Veranderlichen. Springer-Verlag, Berlin 1 934, 2nd. ed. 1 970 [Betsch and Hofmann 1 998] Betsch, G., and Hofmann, K. H . : Hellmuth Kneser: Pers6nlichkeit, Werk und Wirkung. Preprint Tech. Univ. Darmstadt, 1 998. [Bochner 1 943] Bochner, S. : Analy1ic and meromorphic continuation by means of Green's formula. Ann. of Math. (2) 44 ( 1 943), 652-673.
and was confirmed in another communication in July 2000.
VOLUME 24, NUMBER 2, 2002
11
lntegralformel bei Funktionen mehrerer Veranderlichen. Sitzungsber.
AUTHO R
Kongl . Bayer. Akad. Wissen 36 (1 906), 223-242.
[HaNey and Lawson 1 975] Harvey, R. . and Lawson, B.: On boundaries of complex analytic varieties I. Ann. of Math. (2) 1 02 {1 975). 223-290. {Hormander 1 966] H6rmander, L.: An Introduction to Complex Analy· .
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Departme11t of Mathematics and Statist�
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State University of New York at Albany
Alb6rly, NY 1 2222 USA
[email protected] .edu
Germany and raised in M ilano ,
Phys. 43 (1 936), 364-380.
hol omorphic functions from the boundary of a complex manifold. Ann.
R. MICHAEL RANGE
Born in
u.
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Italy, R. Michael Range
earned his Dip/om in Mathematik in Got1ingen, whe re lectures
orem for regular fui1ctions of two
(21
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[Lewy 1 957]
complex variables. Ann. of Math.
Lewy, H. : An example of a
smooth
lial equation wi thout sollJI:ion. Ann. of Math.
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of Hans Grauert got him hooked on Multidimensional Com
[ Martinell i 1 938] Martinelli. E. : Alcuni teoremi integral i per
1 97 1 . He has held academic positions at Yale University and
[Martinel l i
plex Analysis. A Fulbright Exchange Fellowship brought him
to the United States and UCLA, where he received a Ph.D. in
at the University of Washington, as wel l
as
research positions
at various institllles. A frequent visitor abroad. Range is fluent
le funzioni
analitiche di piu variabili complesse. Mem. della R. Accad. d'ftalia 9
(1 938), 269-283 .
per un
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Mart inell i. E.: Sopra una dimostrazione di A. Fueter
teorema di Hartogs. Comm. Math. He/v.
340-349.
1 5 (1 942/43),
in five langu ages, a skill he has often used In lectures in North
[Martinel li 1 96 1 ] Martinelli, E . : Sutla determinazione di
son, he got into ice climbing and alpine mountaineering. He
[Potking and Wells 1 975] Polking, J . . and Wells, R. 0.: Hyperfunction
America and Europe . Range loves mcuntains and Is
an
avid
downhill skrer. A few years ago, inspired and guided by his and his wife Sandrina have three grown cl1i ldre n . He is shown hP..re
on the summit of the Monch (4099 m.), with the Jungfrau
in the background.
una
fun.::ione ana
lltica dl piu varlabili complesse in un campo, asseg natane Ia traccia sulla frontlera. Ann. Matern. Pura e Appl. 55 ( 1 96 1 ) , 1 9 1 -202.
boundary values and a general ized Bochner-Hartogs 'theorem. Proc.
Symp. Pure Math. 30. 1 87-1 94. Amer. Math. Soc., Providence. Rl, 1 977.
(Range 1 986] Range, R. M. : Holomorphic
functions and Integral Rep
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[Serre 1 953] Serre. J. P . : Quelques problemas globaux re lat lfs aux var iates de Stein. Coli. Plus. Var., Bruxel les.
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[Struppa 1 988] Struppa, D.: The first ei ghty years of Hartogs' Theorem.
Bers et al . . Ann. of Math. Studies 33. Princeton Univ. Press. 1 954.
[Boggess 1 991 J Boggess. A.: CR ManifOlds and the Tangential Cauchy
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[Fich era 1 95 7) Fichera. G . : Caratterizazione della traccia. sulla frontiera di un campo. di una fu nzlone analitica di
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[Trepreau 1 986]
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[Fueter 1 94 1 I42) Fueter. R . : Ober einen H artog s · schen Satz in der The erie der analytischen Funktionen von n komplexen Variablen . Comm.
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[Hartogs 1 906) Hartogs. F.: Einige Folgerungen aus der Cauchyschen 12
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M.effli·i§u@hl£11fuiui§ i§4fii.J •ifi
This column is a place for those bits of contagious mathematics that travel from person to person in the commun ity, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Cont;·ibutors are most welcome.
M i c h ae l Kleber and Ravi Vaki l , E d i t o rs
Four, twenty· four, t .
.
.
Michael Kleber
E
nglish is blessed with the word 'four,' which stands out from the other, more humdrum digits because it is, in fact, four letters long. There aren't any more number names like this, which is good: if there were too many, we'd be using a system in which the lengths of the names grow linearly in the value of the number. As a big fan of decimal notation, and of logarithmic use of resources in general, I'm happy keeping things short. On a long drive to Manhattan, Kansas ("The Little Apple"), Rick Char-
Rick reported that in the margins of his notes for selected math lectures he ruled out all numbers up to around five million (!). Impatient sort that I am, I set about teaching my computer how to nan1e numbers, and it promptly reported on 84,672. The way I was taught, that's 'eighty four thousand six hundred sev enty two,' which gives 6 X 4 X 8 X 3 X 7 X 7 x 3 = 84,672. Did Rick miss one? No: his name for the same number ends 'and seventy two,' so the product would be off by a factor of three. The Chicago Manual of Style is implicitly on my side: "One hundred ten men and 103 women will receive advanced degrees this quarter," says the 14th edition, and the editors kindly inform me that the 15th ed. (forthcoming, in 2003) will ex plicitly note that 'and' properly does not appear. Rick, who is sufficiently Canadian that he thinks the alphabet
Cal l a n u m ber fortuitous if it is
equal to the p rod uct of the lengths of the words in its nam e .
Please send all submissions to the Mathematical Entertainments Editor,
Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380,
Stanford, CA 94305-2 1 25, USA e-mail:
[email protected] .edu
trand pointed out to me that the next step is to appreciate 'twenty-four': the two words have six and four letters, re spectively, and 6 X 4 is, indeed, 24. So call a number fortuitous if it is lucky enough to be equal to the prod uct of the lengths of the words in its name. If you're feeling industrious, or are trapped in a long car ride, you can search for the next one. It only takes a little work to rule out everything under a hundred. The words in number names are short enough that, aside from seventy-whatever, we can restrict ourselves to numbers with factoriza tions involving only 2, 3, and 5, and poor 5 has its odd multiples ruled out right away, since they end in a four-let ter word. With such tricks you can quickly pare down the numbers to check, even as the range gets larger;
ends with 'zed,' wonders idly whether there might be a question of American versus British usage. I, being American, decline to acknowledge that other us ages exist. In either case, there are still larger fortuitous numbers. If we eschew 'and' entirely, as I do, the next few are 1,852,200, then 829,785,600, then 20,910,597,120, which is around twenty 'billion' (not 'thousand million'; see above comment on being American), then 92,2 15,733,299,200. If you like the word 'and,' and put it after 'hundred' whenever there's something else in that block of 3 digits, then the computer eventually comes across 333,396,000 and 23,337,720,000 and 19,516,557,312,000 and 56,458,612,224,000 and after that 98,802,571,392,000. (If you follow nei ther of these rules, perhaps your name
© 2002 SPRINGER-VERLAG NEW YORK. VOLUME 24, NUMBER 2. 2002
13
for 207,446,400 uses the word 'and' ex properly, this results in only two more actly once.) These exhaust the exam fortuitous nun1bers, 107,8272 and ples under one 'quadrillion, ' when sud 5,3661,7156,6080, below 102-1. denly you need to consider numbers But seriously, . . . with 1 1 as a prime factor as well. The English examples, that is. If you believe Rutherford's physics/ Thanks to Mme. Eisler in seventh stan1p-collecting dichotomy, it's high grade, I am equipped to observe that time for some of the former, and there French is not endowed with an ana is a natural question to ask: do we ex logue of 'four': under the map taking n pect this fortuitous coincidence to oc to the length of its name, the small cur infinitely often? This is meaningful numbers fall into the cycle trois -> only if we have a system for giving cinq -> quatre -> six. But cinquante (let's say) English names to arbitrarily quatre gives 9 X 6 = 54, and the ves large numbers; Conway and Guy's tigial quatre-vingt dLr gives 6 X 5 X Book of Numbers proposes one such. 3 = 90, the best reason I've ever heard Unless we plan a rigorous analysis, for not switching to nonante. The next I suspect it doesn't much matter what example is 2560, deux m ille cinq cent the system is, provided that number soixante, and based on the unreliable names behave about as we expect. hypothesis that I remember how to Which is to say, as a generic N gets construct number names correctly, these three are the only ones below 109, whose name I shall omit-I won't attempt to deal with the mille m il lion I m illiard I billion mess. While we're playing polyglot, other large enough to be worthy of capital alphabets are extra fun: in Hebrew, letter status, its name should grow to l':li t-:: is again four letters, and Dmitry around log N words and consist of sec Kleinbock tells me that in Russian, 54 tions that look something like 'five is miTb,Il;eCHT 'leThipe, the same fortu hundred eighty-nine supergazillion' itous word lengths as in French! Chi the heart of the system is a way to con nese is fascinating: each "word" in the struct arbitrarily large zillion names. nan1e is a single character, which Mark Those zillions will have to get longer, Zeren recommends dealing with by eventually growing as something like counting strokes, making 1, 2, and 3 log log N in any sensible scheme. into fortuitous digits. The name of a So let f be the map taking a natural large number is read off from its deci number to the product of the lengths mal fom1, grouping digits by fours, so of the words in its name, which has that 1234,5678 is (1 1000 2 100 3 10 4) fixed points at four, twenty-four, etc. 10,000 (5 1000 6 100 7 10 8); a new name At least one in five words in the name is introduced for each power of ten of any N are zillion names, mostly long thousand. l But any power of 10 whose ones, sofwill generally grow as around N coefficient is zero is dropped, and there (log log N)10g , better known as JVIog log log N_ is a special 13-stroke character that ap It seems f(N) eventually pears-! just got Bong Lian to explain grows much faster than N does. Have it to me, I hope I get it right-any tin1e we shown thatfhas only finitely many this drop creates an internal gap in the fixed points? sequences of 10" for n = 3, 2, 1, 0, or Hardly. The generic N may have a n = . . . , 16, 12, 8, 4, 0. If I understand name of log N words, but the nan1e
Other al phabets are extra fu n .
'The numbers of strokes in the characters for 1 /2/3/4/5/617/8!9 are 1 /2/3/5/4/4/2/2/2. respectively. For 1 0" with they are 1 2/1 5;6/8/9.
14
THE MATHEMATICAL INTELLIGENCER
length ignores any part of Ns decimal expansion which is a long string of ze ros. We can readily construct copious examples of N in the same ballpark as j(N), where a ballpark is around three orders of magnitude: given any f(N) »N, write N in decimal, and start inserting zeros, three at a time, just af ter the first few digits. This multiplies N by around 103, but has approxi mately no effect on f(N)-it merely changes one zillion name for the next larger one, which is only a few char acters different in length. We can eas ily puff N up with zeros until it is close to j(N). More generally, a large fixed-point f(N) = N could appear as long as one out of every log log log N of the blocks of three digits in N is ',000,' which certainly happens in abundance. But such an N would also need to have its largest prime factor on the order of log log N, the length of the longest words that appear in its name. If we assume having enough zeros and hav ing small enough prime factors are in dependent, my gross estimate is that there are infinitely many N that do both. But if we then think off as tak ing on random values with the right sized prime factors, the probability that any one of these candidate N is a fixed point of f seems to fall off so quickly that I'd reluctantly guess the total for tuity of the natural numbers is finite. Perhaps f might have infinitely many cycles (like the period five 168 -> 525 -> 672 -> 441 -> 420 -> 168, or the one with period four beginning at 1 , 170,993,438,720)? I can't think of how to begin to estimate this, which may be just as well. Perhaps it's time to stop.
Department of Mathematics Brandeis University Waltham, MA 02454 USA e-mail:
[email protected] n
= 1 !213 they are 2/6/3, and with
n
= 4.'8/1 2/1 6120
PETER A. LOEB
A Lost Th eo re m of Ca c u u s •
n setting up the Riema nn in tegral for an application, hmo does one determine that the in tegrand has been correctly chosen ? Why, for example, is f(x) tl.x a good ap proximation in calcula ti ng the a rea under the graph off bu t tl.x a bad approxima tion jor the graph 's length? Decades ago, ma thematicians used Duhamel 's prin-
ciple to answer the question (see [5], [4], or [2], page 35); they later used a substitute by Bliss [1] when applicable. These theorems are now known to only a few in the old est generation of mathematicians. In modem texts, a rig orous treatment of the integral uses upper and lower sums for both the theory and its applications, and most instruc tors are unaware that this method fails as a test for the in tegrand in simple problems that arise even in the first course. I became acutely aware of this inadequacy when, as a visitor at an Ivy League university, I was required over my strong objections to ask the following question of my stu dents in a combined calculus final: Set up and fully justify the integral for the force of water on a circular window of radius R in the wall of a swimming pool, where the water level is just at the top of the window. The students in the course had learned only the method of upper and lower sums, so they could correctly solve the problem for the top half of the window, but not for the bottom half (see Ex ample 2 below). The difficulty that occurs with this test of integrands is somewhat subtle. If a quantity Q is equal to the integral of a function!, then every upper sum off is larger than Q and every lower sum off is smaller than Q. On the other hand, even with some applications occurring at the most ele mentary level, it is not possible to know a priol'i that up per and lower sums bound Q. One knows this only after
showing in some other way that the integral off equals Q. Consider, for example, the area between the graphs of the functions g(x) = 1 + a:· 2 and h(x) = 2.r2 on [0, 1]. While for a small �.r > 0, the maximum of g(.r) - h(.r) on [0, �x] oc curs at 0, no rectangle of height and width .ll· contains the region between the graphs over [0, �x], so it is not clear a pri o ri that 1 ir is larger than the area of that region. Of course there are several methods to justify the integral needed here (see Example 1), but even for this simple ex ample the "universal" method of upper and lower sums fails, and Bliss's theorem also fails, as a test for the inte grand. What is my answer? In calculating a quantity Q as the integral of a continuous function! on an interval [a, b], one approximates on the ith interval of a given partition a part. Q; of Q by the ith tem1 of the Riemann sum off The re sulting errors e; must be small enough so that their sum goes to zero as the partition is refined. We shall have an swered the opening question when we say how small each e; should be. The answer given below in Theorem 2 is a special case of Duhamel's principle derived from Keisler's [3] Infinite Sum Theorem. Keisler's result uses infinitesimal numbers. It says that for a partition by infinitesimal inter vals, each error e; should be an infinitesimal times the length of the corresponding interval. I replace infinitesimals here with functions having limit 0 at 0. To make the answer useful, I must also challenge the widely held belief that uni-
1
·
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 2. 2002
15
form continuity cannot be introduced at the elementary level; it can indeed be introduced there with the relatively easy, equivalent condition given below in Theorem 1. As in most introductory courses, I have implicitly as sumed so far that a desired quantity Q exists as a number and the problem is to correctly calculate it with an integral. This point of view is appropriate for even advanced appli cations of calculus when physical principles and elemen tary considerations define a quantity and one needs to check the correctness of the value given by a particular in tegral. Theorem 2 provides a general method for validating such a calculation. While in more fom1al settings, quanti ties to be considered must first be defined, this is usually accompanied with some justification. Throughout the lit erature one can fmd arguments doing this that approximate parts Q; of a not-yet-de fined quantity Q. As in the area example discussed above, however, the mean ing of any part of Q may be no clearer than that of the whole. The simple mod ification of Duhamel's method for justifying an in tegral definition given in Theorem 3 does not refer to individual pieces of a quantity Q, but rather to intervals J; in which such num bers, if they exist, can arguably be found. Whether one takes the formal or informal approach to setting up an integral, errors can and do occur. Because no one can lmow to what use students will eventually put their training, they should be prepared for totally new applica tions. It seems strange, therefore, that order-of-magnitude checks on the validity of an integral are not included in cur rent calculus texts. These texts are, after all, replete with methods for numerically evaluating integrals and for check ing the accuracy of those evaluations. At the same time, many Computer programs can be used instead to evaluate any integral at lea.•';t numerically, and often symbolically. What these computer programs cannot do is tell the user if the answer is correct for the initial problem. Given the decreased need to evaluate integrals by hand, error esti mates to justify the choice of integrands should take a prominent place in the training of future generations.
ferentiation. Keisler notes that if a < c < b, then using val ues of !l.r; that evenly divide [a, c], one can show that the integral over [a, b] is the sum of the integrals over [a, c] and
[c, b).
We shall want to use the following result. It is equiva lent to the fact that a continuous function! on a closed and bounded interval is unifomlly continuous. The reader can easily establish this equivalence, and instructors can prove the result itself to beginning students for the case that the function f has a bounded derivative. THEORE!It
1 (MAXIMUM CHANGE THEOREM). Let f be a contin uous function on [a, b]. Given .:U· > 0 and the con·e sponding partition of [a, b), let M; and m ; denote the max imum and min imum values of the function f in the ith interval [X; - 1 > x;]. Set £j(!::.).r :
Com puter p rog rams can be
(M;
=
-
maximum, ,;; ; ,;; " lim.l.r o
m;). Then Er(!l.r) 0.
used to eval uate any i ntegral ;
_
=
With this result, one can easily show that the inte gral of a continuous func tion exists and has the usual properties. The map ping f- E1, defined for each continuous j, is nec essary for this development. I also use such functions in formulating and then applying the following simple form of Duhamel's principle.
what they can not do i s tell the user if the answer is correct
Q;
for the i n itial problem .
Duhamel's Principle and Uniform Continuity
Given a continuous function! on an interval [a, b], I follow H. J. Keisler [3] in forming Riemann sums. Each &r > 0 cor responds to a unique partition with n subintervals, where n is the first integer such that a + n&r :2: b: The partition end points are X; = a + i!::..x for 0 :5 i :5 n - 1, and Xn = b. We let &r; denote the length of the ith subinterval of the partition; of course !::u:; = &r except for the last subinter val, which may be shorter than !::..r. Always evaluating at the left, the Riemann sum is 1-;'= 1 f(x; - 1 ) !l.r;, and the in tegral is just the limit as !::..r - 0. It is not necessary to in troduce a limit concept different from the one used for dif16
THE MATHEMATICAL INTELLIGENCER
THEOREM 2 (ERROR-SUM PRINCIPLE). Let f be a continuous function on an interval [a, b]. A quanti.ty Q is equal to the integral f(x)dx if there is a. junction E of t:u· with the following"properties: For each positive !::..r and co/re sponding partiti.on of [a, b ], the number Q can be written as a sum Q = 1.;'= 1 Q; in such a way that for each i be tween 1 a nd n,
r
I Q; - f(x; - 1 ) ·
and lim.l.l·
_,
0
E(oi.r)
= 0.
t::...·r;l
:5 E(t:.x) · !l.r;,
PROOF. The quantity Q is the limit as !l..r - 0 of the cor responding Riemann sums, since for any given !l.x,
I Q - i f(X; - 1) !l..'t·;l i i
=
:5
1
s
i
"
L =
;
1
=
1
J Q; - f(X; - 1 ) .
E(t:u·) .:l:x; ·
=
!::..r; J
E(!::..r)
·
(b - a). D
It should be clear to anyone familiar with the historical literature that the above fommlation of Duhamel's princi ple is more appropriate for beginners than the original from 1856 [2], or Osgood's 1903 revision [4], or any later formu lation using Landau's little o notation. Although it may not be suitable for beginning students, we can modify Duhamel's principle as stated by Osgood [4] and Taylor [5] so that while, as in those results, the total
quantity Q is not assumed
to
exist,
now the
emphasis
is on
the intervals in which its parts might be found. The point
of the modification is to remove any ass umption that we
are given at the outset a real-valued map i
Qi for each partition of {a, b ]. '!'his yields the following version of Duhamel's principle that can be used to justify the use of a particular integral to define a quantity. �
THEOREM 3. Let f be a continuous ju nction on an interval [a, b], and let E be a nonnegative fu nction of tl.t· with lim.u- - o+ £(fix) = 0. Associate 1vith each .l:c > 0 a nd each pa rtition interval Cri - b xd an i n ten;al Ji(;l.c) co n ta in ingf(;ri - l)ill·i a n d having length at most E(�J:) · �.r. Then fo r anu choice of poi n ts qi E Ji(.l.r), 1 $ i s n , set ting Q(.l.r) :'= � qi we have lim .l.i· --" 0 + Q(Q.r) f(x)d.r. .
PROOF. Set S(Ax) : = .u· $ 1 ,
IQ(�l:)
-
t j(..r)d.:rl a
s
:s:
i
I l-i= H t ftX.:- 1) �i -
f
1
''
lqi - .1\..r; - J ) ;;U·11
E(il..r) · (b =
[''
+ l
- a) ·
+
=
r
_((.r)rui . When +
·cr
S(�:r)
S(.ll) _. 0.
D
A corollary of Theorem 2 is the 1914 result of G. A Bliss [ l ] that tests the appropriateness of an integrand for a given application when the integrand is a product of two contin uous functions g and ll-. The test requires that when l; is the ith interval of the partition of [a, b] generated by a .::l.r > 0 and Q1 is the corresponding part of the desired quantity Q, then there should exist a point (s;, t;) in f; X 11 such that Q; g(s;)h(tL)J:r;. Alternatively, one can use Theorem 3 to show that for any choice of(s,, I;) in /; x 1;, lim..l.,- - 0 + !i'-1 g(s;)h(t;)!!..x1 g(J')h(.x)d.r. In either case, a proof sets E(!l.r) Eh(ar)"- mruqa,bJ lfll + E9(tb.:) mruqa,bJ lh l .
not show that the integrand is c orrect by using upper
lower
sums; a simple
and
is given in the introductory
example
paragraphs in this article. On the other hand, our more gen eral methods can be directly
Yi
applied
and z1 be the maximum values of y
respectively,
in
the interval
to this problem. Let
[.:ri- " xi), and let Ui and � be the g(:r) and
=
z =
h(x),
minimum values. With the partition interval [.r;- 1 1 xiJ, associate the inteiVal .J;(w) = [� - z;) · ari, Cii; - �a · AxiJ. This interval contains both the ith tenn of the Riemann swn and, assuming it exists, the corresponding part of the area. Now we can apply either Theorem 2 or The orem 3, since the length of .J;(;l.r) is at most [Eg(�.r) + En(.ix)] a.t·i and lim..1.x - o [Eg(�r) + Eh(ax)] = 0. corresponding
·
ExAMPLE 2. Consider a window of height 1 in the wall of a swimming pool with the surface of the water at the top of the window. The window is symmetric about a vertical axis and at depth .x has width sin 7T'..t·. Let w denote the weight density of the water. We want to show that the water force
F on the window
is given by the i ntegral j
1
w · .r · sin m: ti:r.
Upper and Lower sums work to justify the integral for the
top half of
the
II
window, but they do not justify the integral
for the bottom half. There,
length
as
the pressure
increases, the
of horizontal strips decreases. Bliss's theorem can
A U T H O R
=
=
r
The Fundamental Theorem of Calculus is also a corol =
·
lary of the Error-Sum Principle: Given an antiderivative Y
=
continuous function ] on [a,b] and a positive Ax, there is in each subinterval [X;-1, x;] a point c; such that .1 Y1 : = F(r;) - F(,r,. _J) j(c.) .l..r;; the desired quantity is F(b) F(a) 1: ;'= 1 llY;, and for each i, the absolute value of the error is I J(c;) - j(X; - l) l · .1.1.·; s E,r(O:r) f:J.x;. (This inequality also shows that Riemann sums need not be eval uated at the left end points of partition inteJVals.) Nate that the Riemann stm'\ !�'= L f(x; t).l.t-; is the sum of differen tials 1:�=t d Y1• As tl.l· goes to 0, the difference between the sum !;'�1 � Y1 of the actual changes in Y and the sum '2:�'= 1 d Y1 of the approximations to those changes along tan F(:.r) of a
=
-
·
PETER A, I.OEB
=
Depar1ment of Mathematics
·
University of Illinois Urball6 , IL 61 801
_
gent lines goes to 0. This helps explairl the integral notation. Examples
of the Riemann integral, there are elementary problems for
upper and lower sums cannot validate the choice of an integrand.
be
directly applied to
EXAJ..IPLE: 1. Let g and h be continuous functions with g(:L·) :2: h(x) for all .r in [o,b]. By translating graphs and subtract ing ·the resulting areas, one can show that the area between the graph of g and the graph of h .is given by the integral
f (g(x)
-
h(:r)) d.x.
Peter Loob received his Ph. D. degree u nder H. L Royden at Stanford in 1 964. His seventy publications in real
especially wi1h
I conclude with two examples showing that in applications which
USA
e-mail;
[email protected] Without such manipulations, one can-
measures
in potential
theory and
analysis deal
with Robin
son's no nstand3rd analysis. He addressed the 1 983 lnterna
t iooal Congress on the formaHon of standard measure spaces on
nonstandard
models
that
al low
infinite
stochastic
processes to be treated with the combinatorial tools available
for finite processes. Loeb has taught introductory Cillculus
classes from both 1he nonstandard and the standard view
point. His other interests- electronics. photograpf<w. and oom
puters-.are a natural outgrowth of h is belief that he wlm dies
with the most toys wins.
VOLUME 24. NUMBER 2 . 2002
17
be used for this part of the problem, but I shall directly ap ply Theorem 2 using the fact that force exists as a physi cal quantity and is additive. Integrating on the interval [ U2, 1 ], we have for each il..r > 0 and each subinterval [.r;- t, .:ri], bounds on the corresponding fo rce F1 given by The ith term
of the Riemann sum is also between these be justified using ei ther an argument invoMng pressure or one involving the work that would be done by gravity in bringing water down from the surface to push the part of the window from .1.·;- 1 to Xi out a distance h.) The correctness of the above inte gral now follows from the error-srnn principle using the fWlc tion E(�) = w · 1J.· + w Esrn ,l.:U:), because for each i , th.e difference between the bounds is oll·; multiplied by bounds. ('The bounds on the force can
·
·w(x;
sin 7rX; - t - .:l'i - 1 sin ru·; - 1
+ .r; - L sin 11X; - 1 - J:; - 1 sin
For these problems, as indeed for most
encountered in a calculus course, there are check the correctness of the
application. Students
18
'ff.t';)
::::::
E(iir_).
of the problems special ways to
choice of an integral for each should, however, be given a general
11-<E MATJ-;EMAT!CAL NlHLIG.NCffi
principle that they can remember and apply in their future work Such a general pdnciple is the order-of-magnitude test given in Theorems 2 and 3. With the increasing power of computers to evaluate integrals both numericall y and symbolically, once they are correctly set up, such a test should again play a major role in the calculus curriculum. Acknowledgments
I am indebted to Daniel Grayson and the referee for helpful suggestions. REFERENCES
[I ] G. A- Bliss. A substitute for Duhamel's theorem, Annals of Mathe·
121 [31
matics (2) 1 6( 1 914- 1 5). 45-49.
J. M. c. Duhamel. Elements de Cafcul lnfinitesimal, Paris.
1 856.
H. J. KeisJer. Elementary Calculus, An tnflflftesimal Approach, Prindle, Weber & Schmidt. Boston. 1 986.
[41 W. F. Osgood. The integral as 1he lim it of a sum. and a theorem of Duhamel's, Annals of Mathematics.
(2) 4( 1 903). 1 61-1 78.
[51 A. E. Taylor. Advanced Calculus. Ginn & Company. Boston, 1 955: third editiOn coauthored with w. Robert Mann, John Wiley and Sons. New YorK 1 983.
'
M a t h e m a t i c a l ly B e nt
Col i n Adams.
ditor
saddlebag. Suddenly out of nowhere,
Homotopy on the Range Colin " The p roof is in the pudding.
Opening a copy of The Mathematical l ntelligencer you
unea.t.'ily, "J.Yhat is
may ask yourself this anyway-a
mathematical journal, or u•hatr Or
you rnay ask, "Where "Who am
tion
is at
open
am /?" Or even
I?" This sense of disorienta
its most acute urhen you
f() Colin Adams's column.
Relax.
Breathe regularly. It's
mathematical, it's a humor tolunm,
and it may even be harmless.
Adams
c "Oh,
ookie, these pancakes taste like " COW p ies .
ah
wouldn't use that
twice, " guffawed
Cookie with
trick
a twin
kle in his eye. His dirt-crusted face split
Sioux in Sioux City.
homology'?"
I ask d.
�oh, Like most everybody else, I
picked it up on th e range.
My first cat
fore you !mow it, I was
I spent eight years learning their ways. Mostly
with 'em,
homotopy-theoretic, completely alge--
braic, no topology. They looked at
copy of Whitehead's Elenumts
We
ing longhorns. First day of the drive, the
foreman rode over and hande � '?E... 1 rlih 04 Q 1 93 0" � zqa5t ..
0
c::::>
c::>
'"
�o Q
d Fe, resp. F.t > F.t' Thus, in the case of fitness equality, the resident and not the mu tant is selected. Here and below, we assume that all inter acting populations are synchronized, thus being all present at t = 0. Proposition
We will show the following: if we allow random mutations, which can be of any size as long as they lead to mutants obeying 1 < F < C (condition K), then a sequence of such mutations will finally lock the cicadas into a stable prime period C. To be more precise, we will show that if C is not a prime then there exists a sequence of mutations that will change C, and if C is prime then no mutation will change it. Note that condition K means that all cycles remain above the main diagonal C = F on the C-F-plane.
Let us assume that C = Cv is not a prime; Fr(F, Cv) has the maximum value 2 gcd(FM, CN)/CN - 1 at the fungus period FM = gd(CN) (gd(x): greatest divisor of .r, excluding x). A sequence of random mutations keeping C = CN constant will eventually lead to FM. However, (FM, CN) is abandoned if mutations lead to CFM, eN ::'::: 1). In fact, gcd(F!If, CN) = F,v, implying that Fc(FM, CN) = - 1; gcd(FM, CN ::'::: 1) can not be equal to FM (the reason is: (CN ::'::: 1)/FM = Cv!FM ::'::: 1/Fp,J, the first term being an integer, but the second not, so that FM is not a divisor of eN ::'::: 1) and gcd(F!If, eN ::'::: 1) can certainly not be larger than FM; thus gcd(FM, eN ::'::: 1) < FM; this implies that Fc(FM, CN ::'::: 1) > - 1 = Fe(FM, CN). Thus, we have shown that there exists a sequence of mutations such that cicadas with a non-prime cycle C = CN are ex tinguished. (Note: (FM, CN) may also be abandoned by mu tations larger than CN ::'::: 1). Stability of Prime Prey Cycles
Assume that C = Cp is a prime; by virtue of condition K, any F is relatively prime to Cp; therefore gcd(F, Cp) = 1, so that starting from (F, Cp), there exist no fungus mutants that are fitter than a resident fungus. On the other hand, for any F, gcd(F, C') 2: 1, where C' is a cicada mutant, as compared to gcd(F, Cp) = 1, so that Fc(F, C') ::; Fe(F, Cp), i.e., no prey mutant is fitter than a resident. In conclusion, any initial random choice of (F, C) and mutations fulfilling condition K will lead and lock to a prime C after a suffi ciently large number of mutations. Note that we cannot loosen condition K, because the points (jCp, Cp), where Cp is prime and j = 1, 2, 3, . . . , are unstable with respect to prey mutations. This can be shown easily: gcd(jCp, Cp) = Cp, while gcd(jCp, Cp- k) with k = 1, 2, 3, . . . cannot be larger than Cp- k; thus Fc(jCp, Cp) < Fc(jCp, Cp- k). This means that convergence to cicadas with period Cp is not possible if mutations to the points
,--- - - - - - - - - - - - - - ���--
18
c
17 16 15
14
13 12
L------------------------
11
10
�--�--+---�--�---+----r---�
1
2
3
4
5
6
7
Figure 2. Evolution of cicadas (C) and fungi (F).
F
a
9
VOLUME 24, NUMBER 2 , 2002
31
(j
p,
p) are permitted. These points are rul
dition K.
Generation of Very
Large
d out by con
Primes
i!l!nd of Blo,1ogit;al Cycles
pr ess, w assumed that mutation and selection of F alternate in successive time tep with mutation and selection of C. In order to get very lar primes, we allow for mutations with unbiologically lar e ycle changes. As an exampl , w onsidered muta tions of th siz 10'1, where 11 is randomly and equally cho sen ln fO, 6]; a r suit i..s shown in Pigur· 1 , starting at F 3, C = 9 and leading to the Eui r-paime E given in the ftgur .. oming back to earth, we now considec a biologically realistic process. For chis we ha'0 to Lake into account that condition K is not plallSible in li ing nature. In fact, this condiLion would mean that the r strtctions for cicada mu tations depend on those for fungus mutations, and vice =
v r a. Such dependence is avoided by r stric:ting mutations
U2, U2
2
within
K on the C-F�plan,
d fin d by:
2 s F :s: number main diagonal C =
:s: C :s L, where L is an arbitrary even
Larger than 6. This square lies above the
F (condition K is fulfilled), and it contains the point that de tabil izes non-prime cicada cy I , nru1nely (F 1, CN). be caus l < £';,, gd(CN) ::::: L/2. Figure 2 shows an evolu tionazy process, \vi:thin such a square (dash d lines; L 18), locking at the prime cicada cycl C 1 7. =
=
=
are
faster
methods for the det
Th
than
the method pre
ti n of large primes which nted
here (see e.g. [ )). work, however, is lhe biological rationale underlying tit prime-generating al gorithm. Our algorithm dis:pla.ys th merging of two seen t ingly u_nr lated subjects: numb r th ory and population biology. Fina!Jy, we hope you do not find our explanation for plim r productive cycles too convincing, for it says noth ing aboul why many species hav non-prime cycles. arc
For mimicking an evolutionary
to a squar
Concluding Remariks
There
remarkable feature of the p11escnt
Acknowledgments W thank the Deutsche Forschung gemeitn:scb.aft and F 1 AP ( hile) for financial upport. Also, we thank Oliver S hulz and David Bressoud for fmitful interactions. References
[1 ] Dylan, B. . "New Morning"; Columbia Records; PC30290. [2) May, R. M., Nature 1978, 277 . 347-349. 131 Karban. R. , Nature 1 980, 287, 32�27. 14] Krttsky, G . . Nature 1 989, 344, 288. [5] Simon, C .. Martin, A . Nature 1 989. 34 1 , 288-289. [6] Bamboo ReseaJch in Asia essard. G .. Chouinard. A.. Eds.), [7)
Dev. Res. Center: Orawa. 1 980.
aye!,
M ..
Dybas,
H..S. EvolutiOn 1 966. 20. 466-505.
181 Rlbenbolm , P. The New Book or Prime Number Records, Springer·
Verlag:
New York. 1 991 .
AUT H O R S
MARIO MARKUS
�-Pianck-Jnstitw fOr Mdekulare
Otto-Hahn·Strasse
11
D-44227 Donmund Germany
ERIC QOLE8
Culter for
Mathen teal Modei!L"'g Univ� de Ohl!e
Pt1ysioi0Q'E1
Mario Markus was born in C11i e. He carne to Germany when received
pmsent he heads
a
his PhD In Physics at Heidelberg At
research group at the Max Planck lnsti
tut In Dortmund and is also a Professor at the
Unrtersity. He
cdlects Rorran coins and produces much-exhlbted computer graoncs: bu mainly. he 'Mites and Pt.1btshes poetry.
32
1 1-lo MATI·ibus regu lmibus. Luca's thi rd book, his able
translation from Latin into the vernac ular of
Euclid's Elemen ts, was also
published in 1 509 in 52
Venice.
11-E I>!ATHCMATICAL INTEI.LJS,.NCER
Plero and Luca
There is no documentaiy evidence to show what
kind
of
relationship Piero
della Francesca and Luca Pacioli had,
whether a friendship., a master-and-pupil
relationship, or even a correspondence
between professionals. Piero knew Luca,
for they were both present in the court of
Montefeltro in Urbina in th e period
1 472-1474.
when
It
has been suggested that
Piero was
losing his sight in his
later years, it was Luca
who was his as-
praises Piero as a painter, or at most as
an
author on painterly subjects, but
not as a mathematician. The Ind ictment The charge of
p lagiarism was first and
virulently alleged by Giorgio Vasari, the
artist and architect who is perhaps best known for his collection of biographies
of the principal artists, sculptors, and
architects of the Italian Renaissance [30]. Indeed, Vasari was so outraged by what he saw as Luca's theft of Piero's
work
that
with it:
he begins the Life of Piero
Truly unhappy are those who, after labouring over their studies to give pleasure to others and to leave behind a name for themselves, are not per m itted either by sickness or death to bring to pe1jection the works they have begun. And it often happens that when such a person leaves behind h im works which are not qu ite fin ished or that are at a good stage of develop ment, they a re usurped by the pre smnption of those who seek to cover their own ass's h ide with the noble skin of the lion. And if Time, which is sa id to be thefather ofTruth, sooner or later reveals 1vhat is true, it is none the less possible that for some period of time the man who has done the work can be cheated of the honour due h is labou rs; this is what happened to Piero della Fmncesca from Bargo San Sepolcro. He was regarded as a n uncommon master of the problems of regular bod ies in both arithmetic and geometry, but the blindness which overtook him in old age and finally his death kept him from completing his brilliant ef forts and the many books he wrote 1vhich are still preserved in Bargo, h is native town. The man who should have tried h is best to increase Fiero's glory and reputation (since he learned everything he knew from him), i nstead wickedly and mali ciously sought to remove his teacher Piero 's name and to usurp for himself the honour due to Piero alone by pub lishing u nder h is own name-that is, Fra Luca del Borgo-all the efforts of that good old man who, besides ex celling in the sciences mentioned above, also excelled in pa inting [30, p. 1 63j. And later, he adds:
As I mentioned earlier, Piero was a most diligent student of his art and frequently practised drawing in per spective; he possessed remarkable knowledge of Euclid, to the extent that he comprehended better than any other geometrician all the curves in regular bodies, a nd thus he shed the clearest light yet upon these matters with h is pen. Master Luca del Bargo, the Franciscan monk who wrote about
regular bodies in geometry, was his student. And when Piem reached old age and died after having written ma ny books, the sa id Master Luca usurped them for his own purposes and had them printed as his own work o nce they had fallen into his hands af ter his master's death [30, p. l67]. Vasari is frequently caught telling in teresting anecdotes lacking basis in fact. One such error in his life of Piero concerns Piero's blindness, which Vasari says came on Piero at the age of 60 (circa 1470), though it is known that Piero could see as late as 1482, because he personally made notes to his will, and because his treatises were written between 1480 and 1490 [ 18, p. 634]. Even given the inclusion of errors, however, Vasari's pen could make or break the reputation of an artist. As the translators of his Lives of the Artists note, "Few artists he criticized have been definitively rehabilitated, and al most all the figures he selected for par ticular praise have remained those most popular with collectors, scholars and visitors to the major museums of the world [30, p. x]." But in spite of Vasari's authority, the case against Luca was generally dis missed for centuries, mainly for lack of proof. Piero's books were supposed to have been in the library of Montefeltro in Urbino, but by the end of the eigh teenth century they were no longer to be found. With the passage of time, Vasari's claims grew weaker and weaker, while Luca's reputation grew ever more sterling. The Damning Proof
Finally, 400 years after the accusation, came the first damning discoveries. J. Deunistonn indicated the presence of the De corporibus regu1aribus manu script in the Vatican Library in the 1850s; in 1903 Guglielmo Pittarelli found and identified the manuscript; Pittarelli verified in 1908 that the Vati can manuscript was identical to Luca's Libellus; in 1915 Girolamo Mancini as certained that marginal notes to the manuscript were in Piero's own hand, and finally published the work in Piero's name [20) . Proving that the manuscript in the Vatican Library was
Piero della Francesca's is tantamount to proving that the Libellus published as part of Luca's 1509 edition of De di vina proportione is a copy. Nowhere does Luca give credit to Piero as hav ing authored or even contributed to the work. This effectively constitutes liter ary theft. As if the truth that Luca usurped the De corporibus regularibus of Piero were not enough, in 1989 Enrico Picutti demonstrated that this was not the only plagiary committed by Luca, but one of several (may we call him a "se rial plagiarist"?). First, in his Summa [24), Luca included 54 problems from Piero's Tmttato d'abaco without cred iting Piero. Later he included the other 138 problems in the Libellus [27, p. 76-77]. Finally, according to Picutti, Piero was not the only victim of Luca's ambition. The section titled Geometria in the Summa is a transcription of the manuscript of another mathematician, a maestro Benedetto of Florence [27, p.76). The discovery that Luca copied the work of Benedetto of Florence helps explain one aspect of Luca's work that his readers often decry: his uneven writing style (Bernardino Baldi in the sixteenth century wrote that Luca's style "makes one nauseous"). Bear in mind that in the Renaissance world of independent city-states, differences in dialect and accent were quite marked between cities and regions, as they are to a lesser extent today. It is difficult to believe that in the few years that Luca lived in Florence he would have come to speak a perfect Florentine, but in copying word-for-word the writings of Benedetto, he would naturally be copying the Florentine style as well. Accident or Intent?
By now I think there is no doubt that Luca copied Piero's work as well as the work of the hapless Benedetto of Flo rence. Was he aware that this was wrong? One thing we know is that he took steps to ensure that the same thing didn't happen to him. In the parallels between the cultural context of the late fifteenth and the early twenty-first centuries are lessons to be learned, for the advent of the printing press created at that time a
VOLUME 24, NUMBER 2 , 2002
53
cultural
rea
d
gulf, Fiero in the world
of
divide
imilar to that
by the advent of electronic publishing today . Piero and ide of th
Luca stood
on eith r
the m an us 'lipt, Luca. in the world ofth
pre s, just as scholars today are divid· d
betwe
n tradi tional print publication
and l:he new electronic publication. Pi ro produced his treatise
and figw-e to th
tro of
·,
,
both text
by hand, cons�gni!ng th m
librari
of his patrof\ Monte� 1-
rbino. Luca, on the other hand,
embrac d lh
new medium, establish
ing a firm relationship
with the
Vene
tian publisher Paganino de' Pagan ini who was books
in
to publish all three of his
print, thereby assuring th m
wide disnibution.
B
in mind that when manuscripts we don t
ar
ab ut
t m1 in today's sense of a draft
of
a
'
w
us v
t xt. Man uscripts in the fift
centwy w re finished documen
Figure
1 1 . The
tall< th
ten by hand, omet im
him elf (an
autograph
sometimes by
a
manuscript),
ionaJ ropyist.
The n ew med i u m of pri nt had enormous
f th
page
corporibu.
ript
manu r
the pro m u lgation of knowledge from the late 1 400s forward .
l'h y
writ-
uscripts." Figur
e re sometim
lavi
h1
illus
trated, described as �illuminated man
manusct1pt copy of Piero's
De
11
how
tihe ftrSt
of
Fiero's De
yularibus in the \ atican
Library . F'igure 1 2 , on the other hand,
shows a
pag
from the print edition of
Luca's Smmn o .
The new m clium o J p1int
mous consequ n e
tion
of lmowledg
forward.
had enor
for the promulga
from the late 1 4 00s came much
consequences for
rsion
nth
profe
by the author
iusti Mites
more
Of the wo1·k of mal"hemalics enjoying the grea test
Euclid,
diffusion, the Elemellls of
abo u t 200 mam.lsc,·ipt codices
M·e kiWWII today; even discounl'i llg
the inevilable losses. it ca n be calcu
tha n 300 copi es of at the d isposition of medieval holars, and those 'Were neither easily a e sible nor equally lated that
not
mo1·e
the Elements l'el'e .
reliable. In conh'a l in lite century be
tween
corporibus regularibus In the Vatican Library.
thejir l p1·inled edit ion (Ven ice
ists. Of course, we all know that errors occur in print d documents as well, but errors in print could be easil cor rected in an natum or in future edi tions. For all th n w advantages of print, it raised new probl ms as well. It is at this point that uintellectual property," which we hear so much about today, becomes an issu Today we talk about how easy il is to go on the Internet and "grab" an image and some text and put it up on a web page as one's own. ln Luca's day, ic was easy to take the ext or figure from a manuscript and pub lish them in print under one's own .
name; the fa t that many manuscripts
were one-of-a-kjnd and tucked away in
private
colle tions meant that such
theft would b
hard to
Another probl m
ory, any publi h
he liked. Th
r
bring to light.
was
could
that, in th
print anything
ftrSt mathematics book in
print, Arithmel ica , published in Turtn
in 1478, was in fact an anonymou work, reQuiring no pennission from an author. Was ttl
for
need of a mechanism
the prot ctton of authors and pub lishers recognized , and did a means of protection exist ln the late 1400s? The
answer to both q_uestions is yes; the means of p rot ction was called a prh'
ileg io. Th
rust
ver plivilegio was
granted in Ven:ic in 1 469 to a single publisher, granting him the exclu.sive rights to all publications in the whole territory of the Republic of Venice.
Upon his d ath, others were granted
similar privUegi. After 1480 pri v ilegi were granted for individual works
Figure 12. A page from the print edition of Luca's Summa.
14 '2) a11d the end of lhe sixte
n lh cen
lw'J) abottt 70 edifi.()ns of the Elem nt nen!
p1·inted, for
pt·udenlly about
.
a
total, calculatiug
00 copies per edi tio n . of
0,000 volum es that cin:ulated
thl'ou,qhout Eu mpe [13, p . l 7/. Another
consequence
was
that
printed works introduced a uniformity
that was reviously llllkno wn. Scholars
in disparate areas of Europe,
ample
for
ex
could now possess uniform val page numbers and figure numbers, sur that fellow s •holars could correctly id ntify urn s t hat allowed them ro cite
what
was
tltis is
a
!being discus
feature that i
d.
(lronically,
Ia king in ome
of today's electronic publ icatio
oc um nts produced in hypertext markup language (html) do not in tude page br·eaks; because the in ertion of page .
breaks is detemtined by the configura
tion of each individual's personal com
puter, it is not possible to cite
pages se
Cllrely.) Yet anot11er cons quence of print was reliabilit-y.
the advent
manuscripts, errors
copyist
[ 1 7]. copyright today derives from the pt•i.vilegi of 500 years ago. Either au thor or publi h r· could apply for a Our
cou1d
alter
transmitted unwitting!
In
onunitted by a
the
text; an undiscovered
of
meaning
of a
rror couJd be
by future copy-
privilegio by writing .a letter to the rul
ing
bod of the
tate; in Venice where
Luca PacioLf· works were published,
the ruling body was the Senate of the
Republic of Veni TI1e old
saying goes,
if you want to
protect your house from burglars, ask
a thief what to do. Luca wrote to the
Senate on 19 December 1508 request
ing a p1i vilegio so that, for the next 20 years, no on ould print without Luca's express p rmission either his
Su mma or D
which was to b
ing
year.
1'h
divfna
proportiO'Jle,
published the follow p1•h iiegio was granted,
of Piero's Tra.tta to d. 'abaco and De cor
armouncement of it was on the last page of De d i vi n a proportione published in 1509 as ,.,.·ell as on the last page of the second edi tion of the Summa published in 1523. and
the
printed
v' LUCA P CIO I
Ironically, Luca's claim on the material
meant that not even the heirs of the le· gitimate authors, Piero and Benedetto, could have published the material that was rightfully theirs (27, p. 76]. But Was It Really Plagiary? Surprisingly, the tendency is
not to
condemn Luca for hls usurpation Piero's work, but
of rather to excuse him.
In part this seems to stem from an un
willingness to project modem ideas on the past. Gino L01ia's cormnents are representath·e of the usual arguments:
fLuca 'sf beha cior; i nconC'eivabie to day, i.s new corifirrnation llwt scien tific honesty is a sen timent with mod
ern orig ins; the a n cien ts commJtte(l without scl'uple every so1·t
and,
when
it came
to
ofplagia ry indicating
sou rces draw11 on, they were overcome
with an invLncible
amnesia; i. t is th us no ma rvel tha t such a free-and-easy systEr�n sh ould be adopted !Jy a m a n who w a s not a n original th inker, but a n i ndefatigable compiler [quoted in 6, p. 32/. Loria's
comments appear to ring true
initially.
case of
For
instance.
consider
the
Euclid and Eudoxus. Euclid is said to be, like Luca, a great compiler. And like Luca, Euclid used the work of
earlier mathematicians such as Ett
doxus without giving credit. Yet no.
body tttinks of Euclid as a
plagiarist. yet, Vasari's accusations seem to belie that justification in the Re naissance: if copying without identify ing the original author were indeed And
Figure 1 3. Via Piero della Francesca, Sansepolcro. 56
Tl-lE MATHEt.AATICAJ.. ONTEcLIGENCH\
Figure 14. Via
Luca Paolo�, Sansepolcro.
thought innocuous,
have been
so
why would Vasari vehement? Further, the
maestro BenedeLto, Luca's other vic
por·ib-us regularibus, it
ensUI'ed the therein a wider diffu sion than they wou ld ever have enjoyed in manuscript form. It may even be that Piem's manuscript in the Vatican li bracy would have gone unidentified and Piero's cont1ibution to Renais sance mathematics unrecognized had the material not been so familiar as part of Luca's publication. Whatever the relationship between Piero and Luca in life, certainly they are forever linked in history. A paint ing by Angiolo Tricca, executed in 1927 and on display in the mayor's office in the municipality of Sansepolcro, illus ideas contained
we all
tim, lived at the same time and in the
trates the myth
work of a (·ertain
solids. Titled Piero della Fmncesca dic
same social context as Luca, yet when
he included in his own treatise the maestro Antonio, he
faithfully credited the original author.
If
copying
were so widespread, why
did maestro Benedetto not be have as
Luca clid? I am forced to the conclusion that Luca
did it to arrogate
credit.
In part, the tendency to excuse Luca may arise from a reluctance to attribute mathematical originality to Piero, an artist, rather than to Luca, a mathemat ics teacher. An example of rhis is Fleur Richter
contesting
Vasart's accusation
of a
learned Piero
want to believe,
willingly passing
on
to a reverent Luca the prtnciples of the latf'.s the rules of geometry to Luca Pa
cioli, the painting depicts Piero as an old
man seated in a large chair, cane in hand,
gesturtng
at
Luca
who, dressed in his
Franciscan habit, Vllrites fonnulas on an
ease l, while tlu-ee
fascinated by the
young men the discussion
two great men.
look on,
between
In Sansepolcro there are two paral
lel stTee!S, Via Piero della Francesca
(Fig. 13) and Via Luca Pacioli (Fig. 14).
of Luca: "Piero knew [ Lucaj well, in
The mathematical tourist can walk up
gether, for how
for him- or herself a complicat.ed sto:ry
deed, the two must have worked to
else
could Piero, with
out the instruction of a
teacher, have
handled such complicated material?"
[28, p. 42j. Such a statement shows a fundan1ental. �udgment of Piero's
mathematical skiUs.
Conclusion Luca almost
managed to pull it off. The truth is finally out that he was a schem ing, ambitious man, capable of lying and plagiarism in hopes of going down in history as a great mathematician. There are still those today who don't want to recognize the �hide of an ass covered with the noble skin of a lion' as just that. The important thing is that Piero has fi nally b€en given the credi.t he deserves as one of th e original mathematicians of the Renaissance, in addition to being Oile of the era's great pa:int{'rs. A fmal odd twist: we owe a debt of gratitude to Luca, for if his De d i vina proportione included whole sections
one and down the other, and perhaps
by journey's end will have sorted out begun 500 years ago. Note
I was
working
on this paper when I
learned about the destruction of the
World 'l'rad{' Center and part of the Pen tagon on l l September 2001. I dedicate it to the memozy of those who lost their liv{'s in the horrifying chain of e\•ents. RI!I:&RIINCIS
1 . Baxar1dall, Michael. 1 985. Painting and Ex· perience
in
Rtteenth-Century
Italy:
A
Primer in the Social History of Pictorial Style. Oxford : Oxford University Press.
2. Bouleau . Charles. 1 996. La geometria seg reta dei pi/tori. Milan: Electa.
3. Clark, Kenneth. 1 951 . Piero della cesca . London: Phaidon Press.
Fran
4. Davi s, Margaret Daly. 1 977. The Mathe matical Treatises of Piero della Francesca. Ravenna: Longo Editore.
AUTHOR
quinque corporibus regularibus. Florence:
Giunti Editore. (Anastalic reproduction of
the manuscript found in the Vatican Library with an accompanying modern print tran scription.)
9. Encliclopedia Bompiani. 1 987.
s.v.
"Pacioli
o Paciolo o Paciuolo, Luca". Scienze pure e
applicate, II: 983. Milan: Bompiani.
1 0 . Evans, Robin. 1 995. The Perspective Cast: Architecture and Its Three Geometries. Cambridge
Piero della Fr 0, define
+
__:_ _ _ _ _
_ _ _
11
+
(313/·
2 _ _ ___: ____: :_ _
_ _
n
+
(2 a)
_:_:_ __:_ :_
_ _
n +
F(
(26)
a
-----2 --13
n
Then
r: + y ft (X, y) = -2 r ) = ( ( y) j]_ . ,y x l/2 .r + y 112 j3(x,y) = x� ) .
0, Re a > 0,
(20) (2 1)
n = 0, 1, 2, . . . , ·where
.
>
) a/3 - V-· 2-,
a + 13
=
1
n
2 (F(a,l3)
(4a)2
+ . . .
+
F({3,a)).
Many more mean families other than /1-p and Mp have been developed, as is thoroughly described in [BMV] . For ex ample, the "quasi-arithmetic" means (see [BMV, p. 2 15ff. ]) are obtained by replacing the functions J{.1·) = :rP and f- 1 (x) :c11P in /1-p with an arbitrary pair of inverse func tions, c.p (x) and c.p - 1 (.r) (where c.p(.r) is defined for positive real numbers), as was studied by Kolmogorov and others [BMV, p. 374]. Many such beautiful results involving means =
3
1.5
0.5
-0 . 5
Figure 4. The graph of y
62
0 . 5 -0 . 5
=
f(x), together with those of y
THE MATHEMATICAL INTELLIGENCER
1.5
1
= x and y
=
(27)
a; here a
=
2
1 and b
=
2 . 5
2, so that A
=
3/2, G
3
=
\
2, and H
=
4/3.
3
2 . 5
1.5 - - - - - - - - -
0 . 5
0 . 5
-0 . 5
Figure 5. The graph of y
-0 . 5 =
m(x) = mx(a,b), together with those of y
m(O} = a, m(a) = H, m(G) = G, m(b) = A, and limx-" m(x) = b.
are familiar only to specialists in certain subfields of math ematics. A Newcomer to the Field
My experience with this historically, theoretically, and practically rich subject started with a memory lapse. Dur ing a lecture at the Mathematics and Design 1998 con ference, at a talk concerning proportions used in architec ture, the speaker presented ( 12), after [Wit, p. 109], in defining the harmonic mean. Not certain of the trivial al gebra leading to (4) from ( 12), I wanted to check the pro portion. After the talk, however, I was uncertain as to the final variable of (12), shown below with an .T: m
2 . 5
2
1.5
1
-a a
b-m X
(28)
I tried .r = m (when in fact .r = b is correct) and discovered that (28) then has solution m = \/ab, the geometric mean. Noting that x = a trivially reduces to the defining relation ship of the arithmetic mean, I realized that I had stumbled upon a unification of the three means: (28) is simply an al ternative way to write (7)-(9) in a single equation. Because no one I asked at M&D-98, nor at the next conference on my agenda, Ne.rus '98: Relati.onships Between Architecture and
= x
and y = b; again a
=
1 , b = 2, A
=
3
3/2, G = '\
2,
and H = 4/3. Note
Mathematics, had recalled seeing this tidbit before, I was en
couraged to investigate further to find where it might lead. Does it make sense that different values for x in (28) lead to different mean formulae? The first task is to un derstand what properties must be satisfied for a function to be considered a mean. Although there is no consensus in the literature, [Bob, pp. 230-231] reviews the properties commonly required of means, the first four of which we may paraphrase as follows: 1. A mean is a real-valued function M of two strictly posi tive real variables a and b such that
min(a,b)
:s
M(a,b) :s max(a,b)
(of course, with the assumption that a
M(a,b) :s b). 2. A mean M is strict provided a
:s
M(a,b) = a
or
:s
(29)
b, this becomes
M(a,b) = b
(30)
if and only if a = b. 3. A mean M is homogeneous provided
M(Aa,Ab) = AM(a,b)
for
(31)
A > 0.
VOLUME 24. NUMBER 2 . 2002
63
4. A mean
M is symmetric provided (32)
M(a,b) = M(b,a).
How does my supposed mean measure up? Fixing a and
b, it is given by m = m�.(a,b) =
a(b + x) . a + .r
(33)
with .1: as parameter. If x > 0, it satisfies (29) and (30). When a and b change, the parameter .r has to change in order to
salvage (31). A typical
Moreover, the two values of z which, when plugged into mz yield (35) and (36) are, respectively, z = b\! Wa and z = b 2/a . The common feature leapt into view: we need z of the form bP + 1/aP ; and this leads to the truly special ex pression (6), the P6lya-Beckenbach family of means. One can imagine the thrill of this (re)discovery, even though I was virtually certain that such a splendid result could not be new! The family mp is only the family mx or mz reparame trized by p E ( - :xo,:xo), but the properties of homogeneity (31) and symmetry (32) have been made manifest.
Mathematical prog ress i n
f(x)
graph of = m.r(a,b) is shown in Figure 4. The three points associated with the three means described above are shown with dashed lines: f(a) = A., the arith metic mean; b) = H, the harmonic mean; and the geometric mean is the (attracting) fixed point of the graph, i.e., f( ) (attracting because < 1; see [Dev, p. 39ff. ] for further reading). Further more, f(O) = b, lim_,. __. "' f(x) = a, and every value between a and b is obtained by some choice of x from the interval (0, :xo) . What of property 4 above? Reversing the roles of a and b in (28) or (33) gives a new proposed family of means with parameter z:
the study of means wou l d
A Seemingly Endless Bounty
To underscore the beauty surrounding the means, I will end by describing one more nice result, rediscov ered while investigating the fact that the compound of the arithmetic and harmonic means is the geometric mean. Recall that for a given n > 0 we can approximate \In by compounding A and H as indicated by (14) and (15), e.g., taking a0 = 1 and b0 = n. Of course, I could not resist trying this with, say, n = 2. The resulting double sequence is
take p l ace from the
lf 'CG) I
E n l i g hten ment onward .
f(
G G =
_
mz(a,b) =
b(a + z) . b+z
(34)
This family is in fact the same family reparametrized, for mz(a,b) = rn.,.Ca,b) provided z = ab/x. I might have noted this at once, but the direction I took instead was still more fruitful.
(a;) = ( 1, 3/2, 1 7/12, 577/408, . . . l (bd = [2, 4/3, 24/17, 816/577, . . . lClearly there is a pattern here; in fact, the b; are completely determined from the a; (or vice versa). Letting c; and d; be the numerator and denominator, respectively, of the a;, it is straightforward to deduce that
Ci + l = cT + 2dT d;+l = 2c;d;.
Sensing a nice relationship here, I found that Perseverance Bears Fruit
It seems reasonable, for a given a and b, to view the m.r (x > 0) or the mz (z > 0) as a famly of means. Perhaps, however, continued analysis may lead to a more compelling form for the fantily. The first question that comes to mind is that three values of x give special members of the fam ily (the three principal means), but other possible values for x > 0 may lead to other special mean formulae. One ob serves that the arithmetic mean, A, is the midpoint of the interval [a,b], and that the geometric and the hamwnic means lie in the lower half subinterval [a, A]. Might the val ues in the upper half subinterval [A, b ] that are reflections of and H (about A) be "special"? Direct algebra yields that these two values are, respectively,
G
a\l'a + b\!b a +b-G= v;;i + V b a+b-H=
64
THE MATHEMATICAL INTELUGEt�CER
a Z + b2 a+b "
(35)
(36)
(c; + V2d;)2 = d + 2d7 + 2V2c;d; = Ci + l + V2di + l·
I then generalized this to any n. I had rediscovered a method for approximating Vn with rational numbers: Take 1 + Vn, square this expres-sion, gather the terms as c + v'nd, then square this expression and gather the terms, etc. The upshot is that the sequence of the c/d approaches \ln. Just as with the family of means (6), I am virtually certain that this result has been seen before. I would be most grate ful for reference to a source. Acknowledgments
I would like to thank Karen Parshall for her expert ad vice on historical topics. Thanks as well to Kim Williams, who has worked hard to bring together those interested in the relationships between mathematics and architec ture, and who provides a constant inspiration for all of us.
REFERENCES
[Alb) Leone Battista Alberti. De re aedificatoria . trans. Joseph Ryln ek)n is weakly (i.e., a(c.4, (.D) convergent to 0. 1. A is lim,
,
Observe that the condition (1) is de scribed purely in terms of the matrix A and condition (2) is described purely in tem1s of the convergence domain cA.. Now note that if A is conull and c.4 = cs, then, as there is at most one FK topology for cA = cs, the matrix B must also be conull. The Bounded Consistency theorem can also be extended to the FK-space setting by replacing the convergence domain of cA with the subspace
WE = {(�r= t .rkek>n is
u(E,E')-convergent to xj
of an FK space E and replacing the bounded sequences with any space X satisfYing the "signed pointwise-gliding hump property" (S-PGHP). Several sequence spaces, including the bound ed sequences, p-sunlffiable sequences (0 < p < �), and null sequences, satisfy the S-PGHP property. Theorem: Let E be a n FK-space, X a sequence space with the S-PGHP such that X n E conta ins the finitely non zero sequences, and B be a matrix. If X n WE c Cs, then, for all X E X n WE, lims x = � k(lim, b , k):rk. ,
For the proof-and how to derive the Bounded Consistency theorem from the above-one may (of course!) con sult Boos's book (pages 463 ff.). The book is divided into three parts. The first part presents the classical as pects of the theory and some applica tions of the theory. In addition to pre senting some Silverman-Toeplitz-type results, this section develops some stan dard summability matrices, Tauberian theorems (i.e., when the convergence of Ar implies the convergence of x), and a classical proof of the Bounded Consistency theorem. This section also investigates some special classes of matrices, such as matrices of type M and the relatively recently introduced potent matrices. The last chapter in this section discusses applications such as analytic continuation, Fourier effective methods, and the numerical solution of linear equations. The second part of the book intro duces functional-analytic methods. In its full development, the functional-an alytic approach to summability theory
draws heavily on the general theory of locally convex spaces. A complication is that there are as many as five differ ent natural duals of a sequence space, and as many special subspaces of a se quence space. Chapters 6 and 7 give a quick but thorough introduction to lo cally convex spaces and topological sequence spaces, which includes ver sions of the Banach-Steinhaus theorem and the Closed Graph theorem for bar relled spaces. There is a nice balance between theorems stated with and without proofs. The provided proofs do a good job of demonstrating standard argun1ents that enhance the reader's un derstanding of the material from the viewpoint of sequence spaces. These chapters also provide a good selection of examples. The third chapter of this section focuses on matrix domains as FK and sequence spaces, and gives a functional-analytic view of some of the topics introduced in the earlier chapters. The third part of the book looks at topics that require the use of both clas sical and functional-analytic methods. The first chapter opens by developing a general fom1 of the consistency the orem using gliding hump techniques in conjunction with modem techniques, and closes with considering the ques tion of when a family of regular matri ces is "simultaneously consistent" on the bounded sequences. The second chapter includes consistency theorems using "mixed topologies" on sequence spaces. The third and final chapter dis cusses the links between summability theory and topological sequence spaces. For instance, weak sequential com pleteness with respect to a weak topology is equivalent to an inclusion between sequence spaces, and there is a similar characterization between in clusion and being barrelled with re spect to a Mackey topology. Boos's book is a welcome and timely addition to the summability lit erature. In addition to the high quality of the exposition, the text also fills the need for a contemporary graduate level text in summability theory. The book is well organized and clear, and the proofs are at an appropriate level for such a text. Also to its credit, the book contains a large set of exercises. Even though the book is not compre-
VOLUME 24, NUMBER 2. 2002
69
hensive, Boos often alerts the reader to
a true "life and work" that describes
ter (pp. 37-38) on why he "left astron
further developments of a topic and de
both under all three hats, with empha
omy forever" (namely, it's not rigorous
scribes how to locate further literature
sis toward the end on his ideas as a
enough) would make most mathemati
on the topic. My only complaints are
philosopher of mathematics, especially
cians smile and be informative for non mathematicians. Also, his "just-so sto
more a reflection of my tastes than
ideas on the interaction between math
flaws in the text. The organization of
ematics and society. And yes, it depicts
ries" are fun; e.g., (p. 53): "Quine [a
the book, while conducive to the math
"the education of a mathematician"
famous logician] gave her [his then girl friend, now his wife] an A in the course.
ematical development of the material,
education in
obscures the historical development
schooling, but individual ponderings,
She told me later that she hadn't a clue
and the interplay between the modem
"friendships with people who follow
as to what was . . . going on . . . Quine
and classical approaches to the sub
unique passions" (from the jacket), and
had given me a B . . . and I had thought
As a result, a couple
every form,
not only
chance opportunities. One such was a
I had known what was going on. And
notebook given to him when he was a
that, dear reader, is why ever since I
seemed unmotivated. Sometimes the
child,
notation interfered with the clarity of
record-keeping of horse
the presentation. None of these com
winding up as a place for math j ot
An important part of his "education"
plaints, though, takes away from what
tings-"secret things about numbers, "
has been meeting and knowing inter
ject.
developed
in
the
of the topics
classical
section
intended
by
his
cousin
for
have disliked the philosophy of for
races,
but
malism."
the book accomplishes, and I have no
as h e called them. That brought to
esting people, almost all of them fa
doubt that it will become a standard
mind fond memories of my own child
mous. Roughly a third of the chapter
reference in the years to come.
Department of Mathematics Ohio University
Athens, Ohio 45701
hood "number tricks, " and the fascina
names (at least in the first two-thirds
tion when I took algebra and discov
of the book) are also people names. To
ered that they weren 't "tricks." ("Or were they?" I now ponder in my older
wit: "Ed Block and the Founding of
age. )
Boas, Jr. ," "Modesty Is Not a Virtue:
I n the Prologue (p.
USA
e-mail:
[email protected] The Education of a Mathematician
by Philip J. Davis
NATICK, MA: A. K. PETERS. 2000, 250 US $29.95, ISBN 1 ·56881 · 1 1 6·0
PP.
P
hilip
Davis
mous: Buttonholding Buber," "Power
him the idea of writing about "the slow
Galore: Talking with Leland," "Top o'
steady progression of day-to-day expe
the Morning to You, Professor Synge, "
riences that added up to a set of opin
"Emilie Haynsworth" (about whom I
ions."
reflect, "Being a 'real character' is not
He
then
proceeds
to
write,
usually in short chapters, about his
the same thing as being deep, or inter
childhood, high school and undergrad
esting. Likewise, being tough."), "What
uate years, math-things ("secret things"
I Learned from Mary Lucy Cartwright,"
about other than numbers, such as his
"An Address that Led to Friendship:
encounters with the Theorem of Pap
Reuben
pus and fan10us fommlas he proved in
from Gian-Carlo, " and "Continuing Edu
graduate
school,
Hersh,"
"Taking
Instruction
first
cation: Isaiah Berlin and Giambattista
employment ("at the National Bureau
Vico." Only rarely do I get the feeling that
known.
of Standards in Washington, a traitor
he nanw-drops; however, he leaves me
ous move away from academia and one
with the impression that most of these
chosen to wear three hats: mathemati
that I'm sure raised eyebrows . . . "
friendships were limited, often tempo
cian, philosopher, and writer. From the
p.
"As
well
Norbert Weiner," "Opinions of the Fa
Throughout his lifetime he has
book jacket:
is
Davis de
scribes a motivating incident that gave
dependently),
REVIEWED BY MARION COHEN
ix)
SIAM," "Writing a Thesis under Ralph
a mathematician his
contributions have influenced many
1 12),
and present employment (in
academia). Many
passages
rary, and peripheral. Davis himself says (p.
about
seemingly
44), "It
happens not infrequently:
what begins as a firm and devoted
areas of applications from approxima
small incidents in his development are
As a
amusing and metaphoric of larger is
ated as time goes on and as circum
philosopher he is . . . drawn to a critical
sues and ideas. "Why I Didn't Take Phi
stances change."
tion theory to computer graphics.
analysis of the development of his sci
friendship tapers off and becomes alien
losophy A" (pp. 35-36) goes into detail
In a "light-hearted anecdotal style"
a writer whose work in
about why he chose, and continues to
(from the dust jacket) he writes con
cludes ["Mathematical Encounters of
choose, not to learn philosophy via for
siderably, although sometimes in an
Lore of Large Numbers, Interpolation and Approxi mation, and] the award-winning The Mathematical Experience (with Reuben
mal courses. He states that he retains
in-groupy manner, and with humor typ
"a deep and abiding suspicion of all em
ically dry and not always very funny,
bracing systems of thought, " and that
about the FUN of professional acade
he picked up "general philosophy be
mic life: collaborations, speaking en
hind the garage. " (I do, however, ask,
gagements,
into shining gems that reflect truth and
"What does he mean by 'embracing sys
abroad, and so on. My favorite anec
ence . . . .
As
the Second Kind," The
Hersh), he manages to tum small events
understanding." His new book is an autobiography, 70
THE MATHEMATICAL INTELLIGENCER
tems of thought'? And are
h is
restaurants
and
hotels
systems
dote is "How I Used Blackmail to Be
of thought non-embracing?") His chap-
come a Professional" (p. 336). This is
a play on the double meaning of "pro fessional"-someone who can get complimentary tickets for his friends on occasions for which audiences pay admission to hear him speak versus the kind of professional he'd been all along for many decades. The "blackmail" refers to his sudden pluck when told he could not get the complimentary tick ets: ''I'm not sure what devil got into me . . . but a second soul that I never re alized . . . took charge, and I said . . . in no uncertain terms, 'In that case, I will not speak tomorrow evening' . . . 'Just wait a moment,' Mueller said . . . In five or so minutes Mueller was back. 'Here are your tickets.' " Davis has worn smaller hats as am ateur artist, musician, even a stint in "
ways helpful, because it can over whelm both the beginner and the sea soned researcher with so much infor mation that mental paralysis results from the overdose." A very large focus of the book is his fascination with Thomas Jefferson. Yes, the Thomas Jefferson, third Pres ident of the United States, also amateur scientist and mathematician, who "said that if he had not gotten involved in politics, he would have devoted his life to these things." (p. ix) Perhaps he should have. If I recall correctly, Albert Schweitzer, student and visitor of wild animals, was also originally attracted to mathematics as a career. And-al though I was glad to learn that Jeffer son asked whether the mathematically
way out" or too revealing of their inner selves), and general insecurity and stress-then the sciences and the arts will, like everything else, corrupt rather than purify morals. Davis has little to say about women. For exan1ple, p. 80: "There is a saying an10ng mathematicians that mathe matical talent is often passed down from father to son-in-law, and there are numerous cases to back this up" . . . What about father to daughter-in-law, mother to son-in-law, and so on? But also: What of the daughter through whom it's passed? What was/is her role in this legacy? Davis says that he has learned a lot from his own father-in-law, and I am curious about his wife. It seems she
. he manages to t u rn small events
i nto s h i n i n g gems that reflect truth . "Show Biz" (where the likes of Alexan der Kerensky, Carol Channing, and Imogene Coca's husband Bob Burton showed mild interest in the calculus book he would read on the set). His artist hat has qualified him to make in teresting speculations on computer art, capsulated in the chapter title "A NEW Esthetic?" Also pp. 187-188: "What emotions are stirred up in me? Vener ation, awe, fear, love, hate, surprise? Awe, sometimes. Occasionally amusement." " . . . While I attain a feeling of elation . . . there is little in the images that operates for me in a didactic way, outside of mathematics itself. This may be contrasted to the religious art of the Renaissance. . . . The computer Virgin has not yet appeared to provide peace of mind, comfort, and salvation in a horror-ridden world. . . . " He concludes, "At the technological level, awareness of progress comes fairly easily. With re gard to moral progress, one wonders." Yes, to what extent does computer art depict the human condition? And-a question of my own-to what extent is computer art really art and not craft? (Perhaps to the same extent that craft is art.) Davis has things to say about computers in general, not only com puter art. For example, p. 99: "The amount of computerized database in formation is tremendous. It is not al-
detern1ined motions of the sun and planets would ever, due to these same mathematical laws, come to an end via, perhaps, a crash, and that this made Jefferson a kind of forerunner of "the stability of dynamical systems" (p. 134)-I personally am more fascinated by Schweitzer than by Jefferson. Davis devotes considerable space toward the end of his book to describ ing the mathematics of Jefferson's day and how mathematics has changed since then. He also involves Jefferson in his discussion of the age-old topic, "Have the sciences and the arts puri fied morals?" Basically both he and Jef ferson believe that, despite many ex amples to the contrary, they can. My own feeling is that the answer to that question depends on how the sciences and art are taught (especially to young children), disseminated, and generally treated within society. I've written much on the subject of education, and I repeat here: If individuals (including scientists, artists, and laypersons) are taught and conditioned to associate art or science with negative aspects of life-such as too much discipline (other than self-discipline), hypocrisy, negative self-inmge, fear of diminishing talent, other fears (for mathematicians, of discovering something "trivial," and for artists, of doing something "just too
"
has also written books, and only one sentence is devoted to this. In the Ac knowledgements (p. 353) he con cludes, "My wife, Hadassah F. Davis, has been the absolute without-whom not of this book, and I have often said of her and continue to say, "Thy word is a lan1p unto my feet.' " I'd be inter ested in seeing at least one of her words. Not only women get dismissive jibes. On p. 169 Davis tells us that he once wrote a spoof, "that a certain Bernstein, who was naturally left handed, felt aggrieved because auto mobile traffic in the United States proceeded on the right side of the road. . . . " Now is that fun, or is it mak ing light of minority and disability is sues? Davis himself "detected a slight conservative tinge to it." This tinge did n't seem to bother him, however; rather he simply published the spoof in the National Review. The article was in tended to emphasize the abuses of the class-action suit; I feel that, while there might be some isolated cases of such abuse, class-action suits are important and serious weapons in our society, one of the few that many people have. His article regrettably helped shape public opinion in the wrong direction. (The mathematical "orientations" he mentions do not justify this.)
VOLUME 24. NUMBER 2. 2002
71
Single sentences, "teasers," seem to pepper his book, sentences that I wish were paragraphs. For example, p. 317: "Mathematics often promulgated a spirit and a view of the world that I found unacceptable." p. 291: "I believe that the mathematical spirit both solves problems and creates other problems." p. 280: "The mathematical spirit and mathematical applications are not always benign." And especially in a more personal vein, p. 174: "I al ways considered myself a bad teacher. . . . " In general, I'd like to know more about his vulnerabilities and how he deals with them. Maybe it's a man/ woman thing: if I were to write my own mathematical autobiography, that au-
expository, philosophical, fictional, at about one every three or four years. These books have won me numerous awards and have reached people in many strange places . . . including a high-security prison in Colorado . . . . " There are certainly passages where he goes out of his way to be modest. I have quoted some; here is another, p. 1 10, perhaps more honest than modest: "I naturally inherited a bit of the snob bism described above. I would give my self a 3 [out of 10) in that regard. " "But how could an autobiography possibly avoid sounding arrogant?" asked a mathematician friend of mine when I voiced these thoughts. My an swer, after taking time to reflect, would
he given up anything (as most have) to "get where he is"? A chapter that caught my fancy is "What Are the Dreams?" (p. 206). This is shorthand for "What are the motiva tions, the drives, the questions, per haps the political causes that could in fluence a young person into becoming a mathematician (or philosopher, physicist, or student of some other expression of human intelligence)'?" Davis wrote to experts in various dis ciplines and received answers. For ex ample, from his father-in-law Louis Finkelstein the theologian: "Can we de velop a civilization in which the most pious people are also the most moral?" From Sir Alfred Ayer the philosopher:
Single sentences , "teasers , " seem to pepper h i s book, sentences that I wish were parag raph s . tobiography would be more on the hu man side, and more about the dark side. I both identified with and resisted his passage on p. 109: "I've felt great wonder in the statements of mathe matics, and yet, I have experienced the waning and the disappearance of that wonder as I've succeeded in under standing what lay behind the state ments. Rationalism and wonder are in conflict." Yes, I have felt that polarity, but wonder stays in the lead. First, an swers almost always point to, or al ready are, further questions. Second, a mathematical proof doesn't always convey why a theorem is true and thus doesn't explain away the mystery. Third, a "math-poem" of mine begins, "You can draw pretty pictures WITH OUT Cartesian geometry. I You don't need x-square plus y-square to draw a circle . . . I But," begins the next stanza, "it wouldn't be as pretty. Cu rves look prettier with equations running along side them . . . " The conclusion: "Beauty isn't as pretty without truth." Under standing (and rationalism) can often enhance wonder. In writing about his life, Davis is not always modest. Take p. 156: "It (The Lore of Large Numbers) was my first book and I could hardly have guessed . . . that over the next four decades I would be producing books, technical, 72
THE MATHEMATICAL INTELLIGENCER
be, "By getting in touch with that arro gance, and sharing that, his whole re ality, with the reader." Another question that I asked as I read the book is something I had asked before reading it: Just how attractive is the academic life? For that matter how attractive is the intellectual life? After all, both involve risks and in securities-graduate student blues, tenure-track (or non-tenure-track) woes, reviews or "mere" passing com ments from colleagues; it's sometimes hard not to associate in one's mind the discipline itself with anxieties. Then, too, when academic or intellectual peo ple interact with one another, the un derlying, if vague, assumption often seems to be that everything NOT aca demic or intellectual is somehow laughable. (It is, perhaps, this ethno centrism that provides much of the hu mor in Davis's book.) How does this af fect communication among academic or intellectual people? Are rationalism and human communication in conflict? Davis seems to like his life; still, do I detect a few undertones? It is the omissions that stand out. I've already remarked on the peripheral and tem porary nature of many of his profes sional friendships; and what about non-professional friendships? We are not told how Davis places his work in the context of his life. Has he or hasn't
"(1) To find a secure foundation for our claims to knowledge. (2) To agree upon a criterion for deciding what there is . . . . " And from himself: "(1) The Dream of the Universal Language. . . . (2) The Dream of Indubitability. (3) The Dream of Infinity. (4) The Dream of Description and Oracularity." And, my own favorite, from Josephine Hardin the writer, "the dream of vul nerability conquered." It seemed an interesting and easy ex ercise to list my own "dreams". Some would be more personal, more poetic, and as such deeper than the dreams Davis has in mind. For one thing, "dreams" in Davis's sense of motivation and drive can be psychologically based. I admit that (subconsciously and later consciously) I learned the axiomatic method in order to prove that my par ents, not I, were wrong. And consider p. 178 (in another chapter entirely): "The famous art critic Kenneth Clark conjectured that the impulse for math ematics [at least some mathematics) with its lines and curves came from men's fascination with the female body." (However, I would put Clark's conjecture in more "unisex" terms.) I searched through my "math-poems." According to them, I do math for a va riety of reasons: (1) in order for things to not be simple, nor finished; (2) to pick at things-points, lines, wiry little
x's and y's; yes, math as mannerism; (3) "to love that for which/there is no space"; (4) to have the pleasure of writ ing not in cursive; and (5) "to rescue the insides" (for example, irrationals and transcendental numbers, or re mote theorems and lemmas like some one buried alive on some distant planet). "What are the dreams?" is the title of a book Davis confesses he consid ered but eventually abandoned. Maybe we should revive the project! Speaking of abandoning, Davis ends his book in perhaps the only way one could-by dubbing this ending an abandoning. Perhaps I may do the same with this review. Department of Mathematics and Computer Science Drexel University Philadelphia, PA USA
1 9 1 04
Trigonometry by I.
M.
Gelfand and Mark Saul
BOSTON, BASEL. BERLIN: BIRKHAUSER, 200 1 , x + 229
pp., 1 85 illustrations US $1 9.95 ISBM 0-81 76-39 1 4-4, 3-7643-39 1 4-4 )paperback)
REVIEWED BY EDWARD J. BARBEAU
T
he ancient discipline of trigonometry has many faces. The astronomer's need to make exact calculations gave birth to spherical trigonometry in the work of the Greek mathematician Hipparchus in the second century B.C.E. The subject was systematized by Ptolemy in his Almagest. By the fifth century, Indian mathematicians had produced sine tables, and their contri butions were later consolidated and extended by Arab mathematicians. From the fifteenth century, the subject developed beyond its astronomical ap plications to become an important part of the mathematics of Western Europe. Plane trigonometry was developed and used not only in surveying, cartogra phy, and navigation but also for pure mathematical purposes like solving cu bic equations. All of this still does not account for the subject's prominence in the modem cmriculum. Since the
latter part of the seventeenth century, The book under review is one in the it has been clear how central the Gelfand School Outreach Program se trigonometric functions are in the ries, that includes also Algebra, Func study of analysis and physics. The sub tions and Graphs, and The Method of ject has a richness and elegance that Coordinates, all co-authored by Pro make it well worth the trouble of con fessor Gelfand. Its authors, on the whole, have selected wisely and cre veying to the young. Trigonometry is a bridge between ated a book that gives a gentle and geometry and analysis. Although geom clear introduction to high-school stu etry has much to say about structural dents. One of its goals is to prepare "for relationships, it has few tools designed a course in calculus by directing . . . at to provide quantitative infom1ation. tention away from a particular value of There are criteria for congruence of tri a function to a study of the function as angles, but how can we determine all an object in itself." Written in an easy the measurements of a triangle from a style, it proceeds through generally detem1ining set of data? Larger sides accessible examples and exercises to of triangles rest opposite larger angles, cover the fundamentals of the sub but how are the ratios of the sides re ject-the definition of the trigonomet lated to the measures of the angles? ric ratios, basic identities, the sine and The power of trigonometry rests on cosine laws, Hero's area formula, the observations that each acute angle extending the definition of the ratios corresponds to a class of similar right beyond acute angles, radian measure, triangles and so can be characterized addition and multiple-angle fommlae, by certain ratios. Because each trian sun1-product conversion formulae, gle can be decomposed into right tri expression of the trigonometric func angles, and other figures can be ana tions in tem1s ofthe tangent of the half lyzed through triangles, trigonometric angle, and inverse trigonometric func ratios apply in many geometric set tions. There are some nice excursions tings; trigonometric formulae and ta on the way-a dissection proof of the bles are handy and efficient. The peri Pythagorean theorem, Ptolemy's theo odicity of the trigonometric functions rem on cyclic quadrilaterals and its and their connection with the simple relation to the addition formulae, a differential equation y" + y = 0 en dimensional-analysis appreciation of sure that they intervene in many areas Hero's theorem, the role of trigono of applied mathematics, and their metric identities in approximating 1T close affinity to polynomials and ex and summing trigonometric series, and ponential functions makes still further a preview of the role of lim11 0 relevance. (sin h)lh in finding tangents to and the All of this presents strategic ques area under an arch of the sine curve. tions for the writer of a primer in The teaching experience of the authors trigonometry. How can a treatise of shines through in their anticipation of reasonable size capture the flavor of student confusion or oversights and the subject and prepare the reader for their use of carefully modulated and later developments in analysis? How probing questions. However, their will should the functions be introduced ingness to introduce an idea and then through right triangles or a moving return to it later made me regret the point on a tmit circle? How much em lack of an index. phasis should be placed on surveyor The authors take the reader by the problems, where the context is imme hand, pausing to pose small questions, diate and easily understood? To what providing a commentary on the un extent should complex numbers or po folding mathematics, and occasionally lar coordinates intervene? How exten offering some advice on how to ap sive should be the coverage of fomm proach the subject. On page 46, the lae and technique? How should reader is advised, "From the identities periodicity be treated? How can one we have, we can derive many more. capture the significance of this deep But there is no need for anxiety. We subject? Inevitably choices have to be will not have an identity crisis. If you made, and topics must be left out. forget all these identities, they are eas_,
VOLUME 24, NUMBER 2 . 2002
73
ily available from the fundamental identities. . . . " On page 1 1g, they sell the reader on the advantages of using radians, despite the appearance of the nonrational 7r and its lack of a compact nun1erical representation, and urge, "Don't let this slight inconvenience stop you from using radian measure." This is the sort of book that a student can work through on her own, or that a teacher can take as a text and be sure that the important material is covered. However, I kept getting the feeling that the authors were sometimes too reluctant to push the reader. Without adding much heft to the book, there was more that might have been done in adding content and providing chal lenge in the exercises. Although stu-
completely convincing; a little more argument could have been given to nail it down. One might look at a triangle ilBC with the lengths of CA and CB fixed, CB :s CA, CA horizontal and B a vari able point that moves on a circle of ra dius a and center C. As the angle BCA increases, B moves from a position B1 on CA to a position B2 on .4C produced. Let Bo be a possible position of B. The circle with center A that passes through B0 contains the arc BoB1 in its interior and the arc BoB2 in its exterior. Thus, if LBCA < LB0CA, then B, on the arc BoB 1 is closer to A than B0. Sim ilarly, if LBCA > LB0CA, then BA > BoA . On page 72, it is pointed out that one
the circumference of a unit circle? This would underscore that the trigonomet ric functions, like polynomials and ex ponential functions, are tools of analy sis defined for real variables. However, the reader is quickly made aware of the key fact that sin x - x for small values of x measured in radians. In Chapter 7, the authors wish to ex tend the range of applicability of trigonometric identities, such as sin(a + {3) = sin a cos {3 + cos a sin {3, from positive acute angles to any angle whatsoever. At this point, they have ac cess to such facts as sin
( (} + ; )
=
cos e.
They complain that "checking the for-
The teac h i n g experience of the authors shi nes t h roug h . dents are likely to see only the analyt ical side of trigonometry, it might have been worth providing a few triangula tion problems more substantive than fmding the height of a tree by measur ing its shadow (p. 56). The introductory chapter draws attention to a sinusoidal curve for the hour of sunrise during the course of the year, with a promise that, in a later chapter, the reader will be shown how trigonometry "allows us to describe it mathematically." The ques tion is indeed revisited (p. 203), but the reader is simply told that "if we graph the number of hours of daylight in each day, we get a sinusoidal curve." This is an appropriately challenging question of mathematical modeling, and it might have been worth a few exercises to guide the student to a reasonable set of assumptions that lead to a formula for the number of daylight hours. In some places, there is the question of how much explanation needs to be provided. For example, on page 13, the authors describe how the side of length c opposite the angle C in a triangle ABC gets longer as the angle C increases, given that the lengths, a and b, of the remaining sides are fixed. They justify this by the plausible argument that if the triangle is hinged at C, one can see that the side opposite C gets smaller or larger as one closes down or opens up the hinge. This is insightful, but not
74
THE MATHEMATICAL INTELLIGE�JCER
consequence of the sine law is that the greatest side lies opposite the greatest angle. This is true, but a subtlety is glossed over when one of the angles of the triangle is obtuse, for the sine of the angle decreases when the angle ex ceeds goo. However, once we know that supplementary angles have the same sine, we can note that if a > goo > {3 > y are the angles of a trian gle, then {3 + y is acute and sin a
=
sin({3
+ y) > sin {3 > sin y
by the monotonicity of the sine func tion for acute angles. So, indeed, larger angles have larger sines, and so are op posite longer sides. A tricky issue for anyone teaching trigonometry is how to make the tran sition from degrees to radians and from trigonometric ratios to trigonometric functions. The choice here (on page 103) is to define the radian measure of an angle as the ratio of the arc it cuts off to the radius of any circle whose center is the vertex of the angle. The main reason I can see for this is that the authors want to emphasize that the ra dian measure is dimensionless, but it is not clear what students are to make of this point. On the face of it, measuring angles in radians or degrees is just a matter of choosing a unit. Wouldn't it be better right off the bat to define an gle in terms of the length of arc cut off
mula for sin( a + {3) for general angles becomes very tedious. You can try it for other angles, reducing each sine or cosine to a function of a positive acute angle. But pack a lunch, because such a procedure takes a long time." This, they say, can be avoided by invoking the Principle of Analytic Continua tion, which says that "any identity in volving rational trigonometric func tions that is true for positive acute angles is true for any angle at all." This is a pretty big black box to put in front of high-school students; to me it seems hardly better than the authors simply exerting their authority. Justi fication does not seem to cost more than an induction argument using steps like this:
(
sin a
+ {3 + = =
;)
cos
= a
cos(a + {3) cos {3 - sin a sin {3
;) + sin a cos ( {3 + ; }
(
cos a sin {3 +
On pages 33-35, before the discus sion of angle-sum and double-angle identities, the authors raise the ques tion of what is the maximum of sin a + cos a and sin a cos a. I agree with their decision to have the students explore these and make some conjectures. It is
only much later, however, that they dis
two sinusoidal functions is again sinu
Complex numbers are not men
pose of the question. But the tools are
soidal. It is tempting at this point to
tioned at all in the book Should they
already at hand. Just note that
look at the special case of
2 - (sin a + cos a) 2 = 2(sin2 a + cos2 a) - (sin a = (sin a - cos a)2 2: 0,
+ cos a)2
and (sin a
+
cos a) 2
= 1 + 2 sin
a cos a
sin
k 1.r +
= 2 sin
have been? Many students leave sec
sin k2x
(tul[ + lv2)x) (t cos 2
(k t
- k-2)x
)
ondary school with no acquaintance of these beyond their incidental occur rence in the quadratic formula. The de sire in many quarters to incorporate
and relate it to the concept of "beats"
calculus into the secondary curriculum
when two sounds with very close fre
has preempted time that could have
quencies are sounded together, as might
been used to provide students with
will lead to an expeditious answer. Stu
occur in the tuning of a violin. It can be
a broader experience
dents could be invited to tum to the
seen that the sum
more general question of maximizing
sinusoidal, being now amplitude-modu
a
lated, but it may still be periodic.
sin
x+b
x.
cos
There are a couple of missed op
is no longer strictly
It is pointed out in Chapter
functions,
inequalities,
with algebra, and,
indeed,
trigonometry, so that they can embark on the calculus with a better array of
5 that
technical and conceptual tools. I see fa
portunities that would not have re
the area under an arch of the sine func
miliarity with complex numbers as pati
quired much extra space but would
tion is 2, but the argument is deferred
of this tool box.
have enhanced the book For example,
to a fmal appendix because of the need
Still, it may well be argued that com
8, it
plex numbers are justifiably omitted
might be worth pointing out that the
area under an arch of the graph of y = 2 sin x can be calculated in a much more
from this book There is enough con
elementary way, by exploiting reflec
Moivre's theorem, roots of unity, poly
Chapter
8 treats the graphs of the
trigonometric functions and eventually discusses linear combinations of sinu soidal functions, using the example '\
..!. sin k.r
tive
.::... k
to adumbrate the Fourier series repre sentation. There is a careful explana tion of how to transform the graph of y
= sin x
to
a sin k(.r - {3),
obtain
graphs
to consider a limit. In Chapter
of y
=
and how the sine and
cosine curves are related by shifts. The reader is led to a realization that a lin ear combination of sinusoidal func tions with the same frequency is also sinusoidal with that frequency, before raising the question of combining func tions with different frequencies. In the part of Chapter 8 dealing with linear combinations of sinusoidal func tions, there is a very short section treat ing the question whether the sum of
symmetry
(not
otherwise
tent-absolute value, polar representa tion,
geometrical
applications,
de
dis
nomial equations, use for trigonomet
cussed). Restrict attention to the half
ric results-to warrant a separate com
arch, for which 0 :S x :S 1Tiz. Because 2 2 cos (x - 1Tf2) = sin x, the graph of 2 cos x is the reflection of that of sin2 x
panion volume. The present text has struck a good balance. It is written for a wide range of students, and it ought
about the line x = trf4. Because sin2 x = 1 - cos2 x, the graph of cos2 x is the 2 reflection of that of sin x about the 1 line y = /z. Accordingly, the graph of 2 sin x is carried to itself by the prod
and be purchased by students who
to be negotiable by any who are going on to study mathematics or science at the tertiary level. It should be on the bookshelf in every secondary school,
uct of these two reflections, a rotation
want a succinct introduction to this im
of
portant subject.
equal to the area between the graph
University of Toronto
180° about the point ( 7T/4, 1/z). Thus for 0 :S x :S 7Th the area between the graph of sin2 x and the line y = 0 is and the line y
= 1; both these areas are
equal to 7Th Thus, the area under a
complete arch is
7Tfz.
Department of Mathematics Toronto, Ontario
M5S 3G3 Canada
e-mail:
[email protected] .edu
VOLUME 24, NUMBER 2. 2002
75
l$@ii•i§u6hi£11@1§
[email protected] i,i,i§
M i c hael K l e b e r a n d Ravi Vaki l , E d i to rs
Cross-Num ber Puzzle Robert Haas
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so
Decimal points are omitted. Decimals are truncated, not rounded. Answers to clues marked ''reversed" are written right-to-left or bottom-to-top.
elegant, suprising, or appealing that one has an urge to pass them on. Contributors are most welcome.
a
b
c
d
k
e
g
h
n
0
u
X
a
Please send all submissions to the
'
h'
Mathematical Entertainments Editor,
Ravi Vakil, Stanford University,
Department of Mathematics. Bldg. 380, Stanford , CA 94305-21 25, USA
Earlier versions of this puzzle appeared in
e-mail: vakil@math. stanford .edu
been received for reproduction.
76
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
The Journal of Irreproducible Results, from which permission has
ACROSS
a. Circle constant i. Mid-April headache in the U.S. j. 116 + 1/12 + 1/24 + 1148 + . . . k. Subtracting it from its reversed dig its gives a multiple of 5; adding it, a multiple of 7. l. a'/1081 - 2£/289 = -857 (see a') 0. 4! - 3! + 2! - 1! + 0! p. Downing prime's number in U. K. [reversed] q. (Sides in Gauss's new polygon)2 r. Plane t. Tchaikovsky overture - 200 u. Liquid butter in India (A = 1, B = 2, . . . ) w. (Total gifts in "12 Days of Christmas") + 1 [reversed] x. (Arab base + computer base) x (Babylonian base - 1)
Vv' 1464 1
y. z.
a'. d'. e'. f'. h'.
(Demuth or Beethoven number) X 10 [reversed] 3a'/1081 - 5£/289 1012 (see f) Adult (Race, or tuning note) X 10 Hastings - 60 Natural constant =
DOWN
a. \/ 10 b. 10 in binary c.
I � �1
n. 385 r. 117 [reversed] s. Product of two triangular non square numbers that's triangular non-square B t. A = H4 (A =F B integers) v. log (6.4339) w. 2(32) y. Futuristic film [reversed] z. Good vision [reversed] b ' . Toll-free [reversed] c ' . Emergency g'. Days around the world [reversed]
-
d . Bad room i n 1984 e. (10 010 001 1 10 0 1 1 )2 f. Fissionable U g. Sexy shape [reversed] h. cos sin- 1 cos sin- 1 (.53881) m. Bond
1 081 Carver Road Cleveland Heights, OH 441 1 2 USA
The solution will appear in Volume 24, Number 3.
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VOLUME 24, NUMBER 2. 2002
77
41£i,I.M9.h.I§i
R o b i n W i l son
Navigational Instruments
here are T
many stamps featuring
navigational instruments, ranging
from
armillary spheres
and astrolabes
to sextants and octants.
A rm illa ry sphaes usually con sisted of metal circles representing the
ST. C H R I STO PHER N£VJS·ANGUI LLA
Armillary sphere
Mariner's astrolabe
main circles of the universe, and were used to measure celestial
coordinates. Other instruments were used by navigators to measure the altitude of a heavenly body, such as the sun or pole star, in order to determine latitude at
seas. The astrolabe reached its matu rity during the Islarnk period and con sists of a brass disc suspended by a ring, fixed or held in the hand. On the front are calculating devices for mea suring the heavenly bodies. The back has a circular scale on the rim and a rotating bar. To measure altitude, the observer views the object along the bar and reads the altitude from the scale. For navigational purposes a more ba sic mariner's ast1'0labe was later de veloped.
Used in Europe from the thilteenth
century, the qu.admnt has the shape of a quwter-cirde (90°); the se.rtant and octant similarly correspond to a sixth (60°) and an eighth (45°) of a circle. To measure altitude, the observer looks along the top edge of the instrument and the position of a movable rod on the circular lim gives the required
Islamic astrolabe
Quadrant
reading.
Around 1300 the mathematician and astronomer Levi ben Gerson invented the Jacob 's staff or cross-staff, for mea suring the angular separation between
two
celestial bodies. Although widely had a m