hm 10 notes on the transmission of geometry from...

HM 10 Notes

ON THE TRANSMISSION OF GEOMETRY FROM GREEK INTO ARABIC

71

BY WILBUR KNORR STANFORD UNIVERSITY,

STANFORD, CA 94305 USA

In the context of a relatively unfamiliar tradition, one cannot tell with confidence whether a given document is repre- sentative or singular. This is what makes especially interesting the Arabic geometric fragment recently edited by J. P. Hogendijk [1981]. Indeed, one might say, if with a bit of hyperbole, that documents of this kind have revolutionary implications for the study of the transmission of Greek geometry into Arabic.

The text in question is an angle-trisection preserved under the name of *mad (ibn M&Z) ibn ShZkir, a ninth-century patron of mathematical science active in the flourishing circle then at Baghdad. His method and others like it (such as the one received under the name of his colleague, ThZbit ibn Qurra) were familiar to geometers in the 10th century, notably al-Sijzl. As Hogendijk notes, however, the latter was of the opinion that the Arabic scholars originated their methods of angle-trisection, save perhaps for the "ancient method" which is implied in the Archimedean Lemmas [1981, 432 f]. But as comparison reveals at once, the methods of *mad and ThZbit are in fact identical with that preserved by Pappus in Collection, Book IV [1876-1878, 270 ff].

We thus have a clear instance where an Arabic document produced as a translation from the Greek was subsequently per- ceived to be an original contribution due to an Arabic geometer. Is this a bizarre exception or, to the contrary, an example of a type of misunderstanding which might have occurred with some frequency? And if the latter, then with what degree of frequency? Of course, we cannot address this issue until we gain a better grasp of this class of geometric materials, and it may well be that too little now survives to actually permit a firm determination. Nevertheless, we have real grounds for caution. Whenever we come upon a technical treatment securely within the compass of Greek methods and format, we must seri- ously consider whether that treatment was modeled after a prototype in Greek, even when no such Greek document is now known. For instance, certain other trisections described by Hogendijk as "invented by Islamic geometers" [1981, 4191 fall into this category: the method of the BanC MCisZ is merely a variant of the Archimedean neusis, just as their method of cube-duplication but slightly modifies the pseudo-Platonic method reported by

Eutocius; further, the angle-trisection known under the name

Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

72 Notes HM 10

of al-Q*: is reminiscent of another of the methods reported by Pappus. The same uncertainty applies in other areas, as I have discovered in my own studies of the medieval tradition of geometric mechanics. Certainly more of these materials owe their origin to a translating effort than is usually recognized.

Let us look more closely at the two Arabic methods of trisection discussed by Hogendijk, both having, as just said, a provenance in Greek sources. Hogendijk has chosen not to empha- size the similarities between the versions of Ahmad and Pappus, preferring instead to point out their differences, although he does acknowledge that most of these are minor [1981, 420-4231. But if the texts are set side by side in parallel translations, one sees at once that the Arabic is largely in literal agreement with the Greek. A striking sign of this is that the two versions letter their figures in precisely the same way. This degree of conformity is phenomenal, as one soon appreciates from a familiarity with the Greek evidence. In the case of short texts like this one on the angle-trisection, even where the Greek editors presume merely to pass on a received text, they commonly alter minor features; the lettering is especially susceptible to change. One might compare, for instance, the closely similar texts of Archimedes' theorem on the area of the circle presented on the one hand by Pappus [Collection Book V; 1876-1878, I, 312 ff] and on the other hand by Theon [Com- mentary on Ptolemy; 1936, 359 ff], in which the diagrams follow quite different letterings. The same pattern marks the many texts of Hero's method of cube-duplication: the method of proof and, in many cases, the specific wording are the same, yet the diagrams vary with some freedom. Thus, when we come upon an instance where two fairly elaborate diagrams are let- tered exactly the same, as in the case of the angle-trisections by Pappus and Ahmad, we are alerted to a close textual affili- ation. In this regard, the text of Thzbit is a useful check; for it differs from the versions of Pappus and Ahmad both on the diagrams and on several points of technical detail.

What do these parallels signify relative to the Greek tradition? Here, Hogendijk is surely correct in resisting the temptation to hypothesize an Arabic tradition of the Collection; and instead he introduces the possibility of a lost Greek prototype "X" underlying At;mad's text [Hogendijk 1981, 4301. He suggests further that instead of X's being an abridgment based on Pappus' text (A), both A and X might have derived from another text Y. This too seems plausible. But he also proposes that A resulted from major changes by Pappus on his text of Y, while X (and after it, Ahmad's B) retains the essential form Of the earlier text. This ;iew , I maintain, does not conform with the usual procedures of Pappus and other editors in the later Greek tradition. It seems to me far more probable that Pappus' text A represents with greater fidelity than does X (B) the

HM 10 Notes 73

basic form of the hypothesized prototype Y, and that X arose through an editorial abridgment of some derivative of Y. We must of course assume in each case not a single step, but a chain of copies, linking Y to A on the one hand, and Y via X to B on the other.)

The most significant difference between A and B is the absence from the latter of "analyses" held in the former. To fix this point, let us indicate the principal sections of their treatments. A formal ordering of the problem consists of (1) a preliminary lemma, solving the problem of constructing a hyperbola with reference to given lines as asymptotes; (2) the main lemma, solving the problem of inclining a line segment of given length between two given lines as to pass through a given point (this neusis being effected via tonics); and (3) solution of the problem of angle-trisection via this neusis. We may denote by "a" the presence of the analysis, and by I's" that of the synthesis. Then, Pappus' text A consists in order of 24s I 3s, la/s, while Ahmad's text B consists first of the mere statement of 1 followed'by 2s, 3s. (In both versions, 3s is given for the acute case; the right case is then treated speci- ally; and the obtuse case is reduced to the preceding as the sum of an acute and a right angle.) On the one view, then, 2a and la/s would be additions made by Pappus to a source (Y) which was essentially the same as the source X for B. But this type of editorial change is hardly characteristic of Pappus. Far from composing analyses to augment source texts lacking them, Pappus eliminates analyses from texts which had them; in Collection V, for instance, he writes: "we shall write [or, prove] the comparisons of the five [regular] solids having equal surface . . . not via the so-called analytic theory through which some of the ancients effected the proofs of the cited figures, but as prepared by me via the method [lit.: leading] in accordance with synthesis for [its] greater clarity and con- ciseness" [Pappus 1876-1878, I, 410-4121. Again, a text which would dismiss the synthesis as "obvious" [phaneral, after the presentation of the analysis, may receive from Pappus the sketch of the synthesis [1876-1878, I, 144, 146, 154, 282, 2941. In one instance [1876-1878, 284, an alternative method of angle-trisection via hyperbola], the "obvious" synthesis is omitted. We would thus suppose that the brief sketches provided for the other cited passages were supplied by Pappus, and this is supported by their generally similar style. Now, we have from the earlier geometer Diocles an instance of the same sort, where a synthesis is omitted as "obvious" due to the presence of the analysis [Toomer 1976, 861. But at this very point, the later editor Eutocius, who reproduces Diocles' solution in his commentary on Archimedes, steps in and provides the entire, very long and elaborate synthesis [Archimedes 1910- 1915, III, 168-1741. Later in the same work Eutocius presents

74 Notes HM 10

a synthesis of the hyperbola-construction [Archimedes 1910-1915, III, 1761 which is evidently an adaptation of the analysis- synthesis version we have called la/s in Pappus' text A (cf. my article, cited in the bibliographical note at the end). Thus, the commentators Pappus and Eutocius reveal their pref- erence for the synthetic mode of presentation, to the extent of sometimes eliminating analyses or supplying syntheses where absent from the sources. This is precisely how B is related to A: the one lacking the analyses present in the other. I thus find it most appropriate to suppose that the source X of B arose through abridgment of the fuller version, as in A, than that A was an amplification of a text like X.

Texts like B ought to provide valuable insight into the state of the geometric tradition in late antiquity. For instance, in conjunction with C (ThZbit's text, to be discussed below) it reveals that Pappus' version of the angle-trisection was but one of several alternative texts of this same method circulating in the later period. But as to the much earlier provenance of these methods, I think the evidence from Arabic sources is of restricted use; certainly our first recourse here must be to the Greek evidence. Thus, the suggestion that B somehow attests to a pre-Apollonian origin of this method seems to me quite improbable. My principal reason for doubt is the presence of the lemma la/s in Pappus' A. The problem solved here is by its very nature impossible before Apollonius: for it seeks to determine the Apollonian parameters (diameter and latus rectum) of the hyperbola answering to the relation XY = constant; by contrast, earlier writers like Diocles, Archimedes and Menaechmus accept the latter property as practically a defining characteristic of the curve. If, then, Pappus' lemma la/s is an organic part of his text on the angle- trisection, his source for the whole is to be assigned a dating at or after the time of Apollonius. Now, one notes that where Pappus elsewhere introduces a conic answering to a specified locus condition, he is usually quite content to omit explicit reference to the solution of the required locus-problem. For instance, in an alternative solution of the angle-trisection [Collection, Book IV, p. 2841, the analysis reduces the problem to that of a locus specified in terms of the distances from given focus and directrix. Pappus merely says, "B [the required point] thus lies on a [given] hyperbola." He does not even attempt here to specify which hyperbola that would be. Although the solution to this problem does in fact appear much later in the Collection, at the end of Book VII among his lemmas to Euclid's Surface-Loci, one finds at neither place any hint of a cross-reference to the other. Further, among his lemmas to Conies V in Book VII, Pappus gives the hyperbola-construction (la/s) again in a version only trivially different from the one which accompanies the angle-trisection in Book IV.

HM 10 Notes 75

This indicates that Pappus in each case is transcribing a lemma held in his source, not intervening as editor to introduce new materials absent from the source. To propose this as a general fact about Pappus' use of sources would of course demand a careful investigation of the Collection in comparison with other texts from the later tradition. But in view even of the few instances just cited, one would surely incline away from assumptions of Pappus' originality.

I would thus set Pappus' text A (or rather, its proposed prototype Y) sometime after Apollonius. But since the hyperbola- construction supplies a result which is necessary for the application of Apollonius' theory of tonics toward the solution of geometric problems, I would set it not long after Apollonius, and most likely within his generation. The method of trisection via the neusis is doubtless older. It is a variant of the Archimedean method (cf. [Hogendijk 1981, 419 f]) and is likely to have formed the basis of Nicomedes' solution via conchoid. I see no reason to suppose, however, that the effecting of its neusis via tonics (as in 2a/s) derives from a geometer before Apollonius; indeed, the project of substituting neuses for alternative "planar" or "solid" constructions, with which it is naturally affiliated, seems to have been an important re- search interest right at Apollonius' time.

Turning now to ThZbit, his text C of the angle-trisection consists of Is, 2s, 3s. As it differs in substantial respects from A and B, one must decide between viewing it as a form of an improved edition of B or, alternatively, taking it to be an independent translation from yet another Greek variant. For in view of the methodological similarity with the other texts, the possibility of entirely independent composition is firmly excluded. In contrast with Hogendijk [1981, 432; but cf. p. 437 nl, I take C to be not a recension of B, but rather an independent translation of a different text. Since it is not possible here to discuss the issue in detail without access to the actual document (I am preparing a text in the work cited in the note at the end), I will attempt only to offer a general notion of the argument with reference to a few details already noted by Hogendijk. If C is indeed an adaptation of B, we ought to have some signs of actual dependence, as well as indications that C does in fact improve an inadequacy of B. Con- sider ThZbit's phrase, "al-qafc

where he introduces the hyperbola in 2s, al-musamma 'iiburbiili" (SC. "the section called 'hy-

perbola"'); h ow is that in any sense an "improvement" 'over the standard term "al-qaf al-za'id" (SC. "the hyperbolic section") adopted in *ad's rendering? But one can readily see it to be a literal rendering for the Greek phrase hE; tom5 hg kaloumenE hyperbolg, which one finds in ancient authors, including several times in Apollonius' Conies. Now, in ThZbit's text C the standard term qacc z2 'id does in fact appear: four

76 Notes HM 10

times in its opening section 1s. But it happens that this entire section is a verbatim transcription of the text of Conies II, 4 in accordance with the Arabic version made in the ninth century by al-vim@. We must thus take it to be an in- sertion into ThSbit's text, whether by the copyist al-Sijzi or by an earlier editor. But within ThEibit's section 2s, the neusis lemma, we come upon another quotation from the Conies, a statement of II, 12 different from the Arabic of al-Firn$i in its wording , yet more literally faithful to the Greek than al-Himsi is. Doubtless, any number of explanations might be proios;?d; but the most natural one is surely that Th%it found the statement in Greek in his source on the angle-trisection and translated that as literally as he could along with the rest. By contrast, if we compare C against B, we find little agreement in wording, as indeed Hogendijk observes [1981, 4321. For instance, AQmad states the neusis lemma 2s in the form: that the inclining line "falling between GD and BG extended (is) as the given (line) M," where Th%it writes that "what subtends the angle G is equal to the given line I." Note further that ThZbit's lettering is different from qad's. Again, Th%it fails to recognize that his construction in 3s applies only to the acute angle; in this respect his version is markedly infe- rior to that of B and A, where the proper distinction among the three classes of angles is made. On balance, then, these features of C would be quite odd under the assumption that ThZbit is modifying and improving on A$mad's rendering. But such variants are quite common within the Greek tradition of mathematical texts, as mentioned above, so that the discrepancies in C are precisely what we would expect under the view that ThZbit is translating independently a text related to, but different from, that used by Ahmad.

We have, moreover, a circumstantial case for identifying Thait's source: the geometric compilation in three books by Menelaus (second century A.D.). For we know that Th;iibit translated that whole work and that it contained among other things a variant text of Archytas' solution of the cube-duplication. It is thus quite possible that Menelaus adapted solutions of the angle-trisection also (the two problems are, after all, of the same type, as the ancients themselves emphasized) and that it was from this translation of Menelaus by ThZbit that the Arabic copyist excerpted his version of "Thzbit's angle-trisection." The implied misattribution, one may note, is of the same sort which had led others to assign the alternative text of this method to *mad.

These comments merely begin to suggest the intricacies in- herent in the project of tracing out the lines of transmission of geometry from Greek into Arabic. With further investigations of such materials, one may hope to develop in time a credible outline, if not a fully detailed account, of this transmission.

HM 10 Notes 77

The principal difficulty is that neither tradition is yet well understood. The Arabic documents have been edited only piece- meal, while most of what has been written about the Greek tradition of later antiquity is unhelpful. I perceive that in both instances the interpretive effort is obstructed by concern over the question of the originality of these writers. To be sure, there is nothing original here as far as basic technique is concerned; but what of the details of their application, as in variant constructions or reformulations of proofs? Although T. L. Heath admits that Pappus does not "pretend (...) to great originality," he nevertheless concludes that "Pappus stands out as an accomplished and versatile mathematician, a worthy repre- sentative of the classical Greek geometry" [Heath 1921, II, 3581. This statement inevitably assigns to Pappus ultimate responsi- bility for the actual texts assembled in the Collection. To the contrary, when we fully appreciate that the writers in these later traditions are editors and archivists, not "original" mathematicians in any sense of the term meaningful to us today, then we shall know the better how to exploit their testimony.

Bibliographical note: A preliminary version of the sub- stance of these remarks, as they pertain in particular to ThZbit's trisection, was presented to a session of the Collo- quium on the History of Mathematics of the Courant Institute (NYU) in November 1979. Thanks to David King (NYU), I learned of J. P. Hogendijk's presentation, "On the trisection of an angle . . . in medieval Islamic geometry" (University of Utrecht, Dept. of Math. preprint No. 113, March 1979); although the author does not present actual texts or translations of the items in his well-documented survey, he does present for comparison the diagrams from Pappus (A) and Ahmad (B), and from this I inferred an unusually close textual*agreement as well. Subsequently, Hogendijk edited the text from Ahmad, in the article which has occasioned the present note. I have included the texts from AQmad and Thait, together with related texts in Arabic and Greek, as a portion of Part II of a study of the ancient tradition of geometric problems, forthcoming with Birkhxuser. A discussion of the Greek versions of the hyperbola-construction has appeared [Knorr 19821. A preliminary version of findings on the medieval mechanical tradition was read by me before a session of the History of Science Society during its conference in December 1979. An expanded treatment is currently in press as a monograph of the AnnaZi issued by the Istituto e Museo di Storia della Scienza in Florence.

REFERENCES

Archimedes 1910-1915. Opera omnia, J. L. Heiberg, ed., 2nd ed., 3 vols. Leipzig (repr. Stuttgart, 1972).

78 Notes HM 10

Eutocius 1915. Commentarii in Archimedem: see Archimedes 1910-1915, Vol. III.

Heath, T. L. 1921. A history of Greek mathematics, 2 ~01s. Oxford.

Hogendijk, J. P. 1981. How trisections of the angle were transmitted from Greek to Islamic geometry. Historia Mathematics 8, 417-438.

Knorr, W. R. 1982. The hyperbola-construction in the Conies, Book II. Centaurus 25, 253-291.

Pappus 1876-1878. Collectionis quae supersunt, F. Hultsch, ed., 3 vols. Berlin (repr. Amsterdam, 1965).

Theon of Alexandria 1936. Commentaires . . . sur l'Almageste, A. Rome, ed. Studi e Testi, 72.

Toomer, G. J. 1976. Diocles: On burning mirrors. Berlin/ Heidelberg/New York: Springer-Verlag.

EXPECTATION AND THE EARLY PROBABILISTS

BY EDDIE SHOESMITH UNIVERSITY COLLEGE AT BUCKINGHAM,

BUCKINGHAM, ENGLAND

In an earlier issue of this journal, Daston [1980] discussed the role of the "expectation" concept in early probability theory. With respect to late-17th-century work, she stressed the seminal influence of Huygens' De ratiociniis in aleae ludo [1657]. No explicit indication was given of the contribution of Pascal and Fermat to the development of expectation-based concepts and methods, though in a later paper concerned mainly with the 18th and 19th centuries, Daston sug.- gested that "Pascal, Fermat, and Huygens . . . took the legal notion of equal expectation, rather than probability per se, as their point of departure" [Daston 1981, 2881. I should like to complement Daston's interpretation of the 1650s by pointing out and briefly commenting on three recent accounts [Hacking 1975, Chap. 11; Freudenthal 1980; Edwards 19821 which throw some light on the contributions made by Huygens, Pascal and Fermat to the emerging concept of "mathematical expectation."

In Chapter 11 of The Emergence of Probability, Hacking [1975] discussed in detail Huygens' use of the "expectation" concept in De ratiociniis in aleae ludo. Huygens' approach to deriving and proving rules of chance was squarely based on the notion of a fair bet or fair lottery, as also noted by Daston [1980, 2391. Hacking [1975, 951 pointed out further that


HISTORIA MATHEMATICA ll(1984). 417-423

Changing Canons of Mathematical and Physical intelligibility in the Later 17th Century

MICHAELS.MAHONEY

Princeton University. Princeton, New Jersey 08544

Learning to use the new calculus in the late 17th century meant looking at quantities and configurations, and the relationships among them, in fundamentally new ways. In part, as Leibniz argued implicitly in his articles, the new concepts lay along lines established by Vi&e, Fermat, Descartes, and other “analysts” in their development of algebraic geometry and the theory of equations. But in part too, those concepts drew intuitive support from the new mechanics that they were being used to explicate and that was rapidly becoming the primary area of their application. So it was that the world machine that emerged from the Scientific Revolution could be both mechanically intelligible and mathematically transcendental. 0 1984 Academic Press. Inc.

A la fin du Dix-septieme sitcle, apprendre a employer le nouveau “calcul” impliquait de regarder sous un nouveau jour les quantites et les configurations ainsi que leurs relations. Comme le soutenait implicitement Leibniz dam ses articles. ces nouveaux concepts repre- naient en partie les idles etablies par Vitte, Fermat, Descartes et d’autres “analystes” dans leur recherche en geometric algebrique et en theorie des equations. Neanmoins, ces concepts retiraient aussi une partie de leurjustification intuitive des nouvelles mCcaniques dont its Ctaient des outils explicatifs et qui devenaient rapidement leur domaine priviligie d’apph- cation. De la sorte, le monde mtcanique qui emergera de la Revolution scientifique pourra @tre a la fois mecaniquement intelligible et mathematiquement transcendant. 0 1984 Academic

Press. Inc.

Das Erlernen des Gebrauchs des neuen Kalkulus am Ende des 17. Jahrhunderts be- deutete. GroRen und Konfigurationen und die zwischen diesen bestehenden Zusam- menhange auf fundamental neue Weise sehen zu lernen. Wie Leibniz in seinen Beitrlgen implizit zeigte, lagen die neuen Begriffe zum Teil auf der Linie, die durch die von Vi&e, Fermat, Descartes und anderen “Analytikern” vollzogene Entwicklung der algebraischen Geometrie und Gleichungstheorie vorgezeichnet war. Teilweise jedoch bildete die neue Mechanik die intuitive Basis fur die neuen Konzepte. insofern diese in der Mechanik entwickelt wurden und dort such in erster Linie zur Anwendung kamen. Daher konnte die Weltmaschine, wie sie aus der Wissenschaftlichen Revolution hervorging, zugleich me- chanisch einsehbar und mathematisch transzendent sein. 0 I984 Academic Press. Inc.

During the 17th century the world became a machine, and mechanics became the mathematical science of motion. The two developments proceeded in tandem, driven, one generally supposes, by people’s desire for intelligibility greater than that offered by the traditional world picture. One understood nature better because it was now mechanical, and mechanics better because it was now mathematical. But what about mathematics itself? It too underwent radical change over the course of the century, not only in its theories and techniques but also in how mathematicians understood their subject. The Ancients would hardly have recog-

417 03 15-0860184 $3 .oO

Copyright 0 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.

418 MICHAEL S. MAHONEY HM 11

nized geometry in the book Descartes published by that name, and what he thought incomprehensible in the 1630s formed part of standard mathematical prac- tice in the 1690s.

Only by taking account of the altered canons of intelligibility in mathematics can one understand how the world machine of the late 17th century could be comprehensible and yet transcendental. By the 1690s the mechanics of nature consisted of relations describable only in terms of logarithms and exponential functions, of sines, cosines, and tangents. Planets moved by an area law on curves that were not algebraically quadrable or rectifiable. Pendulums swung in periods only approximately measurable by trigonometric functions, and even then only by assuming a value for rr. Mathematical and physical space held a host of new curves-caustics, isochrone, brachistochrone, tractrix, catenary, and so on- many of them expressible only by means of differential equations.

Yet, differential equations themselves belonged to a new conceptual realm. Algebraic in appearance, they symbolized mathematical operations at the upper and lower boundaries of finite quantity. For many thinkers, those boundaries marked the limits of understanding. Moving along them or across them required a point of reference by which the mind could check the course of its operations. In some cases, that point of reference was nothing other than the new mechanics that was leaning on the new mathematics for conceptual support. So proponents claimed on the one hand that infinitesimal analysis yielded mechanical insights otherwise unattainable and on the other that the concepts of the calculus were rooted in kinematical experience. In the developmental pattern now familiar as “bootstrapping,” mechanics and mathematics helped one another to intelligibility. They did so in part by conspiring to change the canons of intelligibility by which understanding of them was measured. Mathematicians and mechanicians at the end of the century did not understand their subjects better than had their predecessors of the early 1600s but rather differently. The new analysts gauged their understanding by new standards, among them heuristic efficacy. Fruitful use of a notion conveyed its own sort of intelligibility.

What follows is an attempt to chart the course by which mathematicians made the transition to new forms of understanding. The brief compass of this essay permits no more than the identification of several crucial points at which practi- tioners veered into new directions of thinking. They took their start in the realm of algebra.

In 1638, to the consternation of admirers in Paris, Descartes announced his intention to turn from mathematics to other fields of inquiry. In response to the protests conveyed by Mersenne, he explained:

I am obliged to M. Desargues for the concern he graciously shows for me by averring his disappointment in my no longer wishing to study mathematics. But I have resolved to leave only abstract mathematics, that is to say, the investigation of questions that serve only to exercise the mind. And I do so to the end of having that much more leisure to cultivate another sort of mathematics, which sets itself as questions the explication of natural phenom- ena. For, if he will please consider what I have written on salt, on snow, on the rainbow, and so on, he will know well that my whole physics is nothing other than mathematics. [l]

HM 11 CANONS OF INTELLIGIBILITY 419

In retrospect, given such goals, Descartes left off where he should have begun. For the mathematics in which he had been engaged that spring and to which he would be recalled willy-nilly over the next years is the mathematics to which physics was eventually reduced. But Descartes did not see it that way. Such questions as determining the tangent and area of the cycloid (and of other “special” curves) or of finding a curve given a defining property of its tangent seemed to him merely curiosite’s, especially since in many cases neither the curves nor the methods to handle them fitted his notion of what was mathematical.

Mathematical for Descartes was what could be grasped in a single intuition or could be reduced in clearly understood steps to such immediate apprehension. In geometry the straight line was immediately intelligible; in arithmetic, the unit and the combinatory acts of addition and subtraction. From these one derived the notion of ratio (or rapport) and equality of ratio (or proportion), and thence the concepts of multiplication and division, raising to a power and finding mean proportionals. By building on these elements, he argued at the beginning of his GPom&rie, one could give clear meaning to all finite algebraic expressions and clearly relate those expressions to plane curves, thus supporting one’s intuition of the simple figures and providing a vehicle for understanding the complicated ones La.

For Descartes, then, mathematical intelligibility came to rest in closed algebraic expressions. What they did not suffice to express did not participate in the clarity of mathematical knowledge and hence was not mathematics. So it was that, even though Descartes easily determined the area of the cycloid by two different methods-the Archimedean method of exhaustion and a method akin to Cavalieri’s- and went on to find the tangent to the curve by a technique akin to determining its center and radius of curvature at the given point, he could discern nothing of mathematical import in such exercises. He could not express his techniques algebraically. Why people like Roberval, claiming to be interested in mathematics, made such a fuss over such problems was something Descartes could not understand.

Roberval, in fact, shared Descartes’ view of what was properly mathematical, if not of what was mathematically interesting. In a letter to Torricelli written something after 1645, he contrasted his method of infinites to Fermat’s use of analysis in finding centers of gravity, where by “analysis” he meant “algebra.” Fermat’s approach seemed to Roberval “most abstruse, most subtle, and most elegant”; his own, perhaps simpler and more universal [3].

But the standard was clear. It shines forth in James Gregory’s Vera quadruturu hyperbolue et circuli of 1667 and even in Christiaan Huygens’ strenuous criticism of that work. A “true quadrature,” i.e., a truly mathematical quadrature, is an analytic quadrature, A problem is analytic if it can be expressed and investigated algebraically, where “algebra” now denotes the theory of equations. Given the foundations of intelligibility in the basic operations of algebra, talking about curves and what could or could not be known about them came down to talking about equations and how they could be transformed. Three features of this atti- tude in the 17th century are worth noting here.

420 MICHAEL S. MAHONEY HM II

First, for reasons that I have set out in detail elsewhere, both Fermat and Descartes, and then their successors for some time afterward, thought their methods of tangents and maxima and minima to rest in the theory of equations and hence to be fully analytic. But they were analytic only insofar as they were applied to analytic curves. Other curves presented special challenges, and many of the methods of tangents published and touted in midcentury aimed at handling the wider range of curves in a uniform manner. The title of Leibniz’ first published account of the calculus in 1684 described it in terms of this problem, and it was probably for that reason that many initially failed to appreciate the profound novelty of his approach [Leibniz 1684/1859]. His next few articles were aimed at showing that his calculus of differentials was not just another extension of the method of maxima and minima.

Second, one of the major themes of both Viete’s and Descartes’s theory of equations was the reduction of quadratic, cubic, and quartic equations to forms equivalent to the determination of mean proportionals between given extremes. It confirmed the understanding of equations as basically compound proportions. Equations of the fifth degree and higher resisted such a general reduction to the pure form. By the 1670s the question of whether it was in fact possible seems to have grown in urgency [4]. The range of opinions-and in the absence of a proof or a disproof there could be only opinions-suggests what people then thought to be at stake. Cartesians argued, to take a phrase of the time, that the fault lay in the artisan rather than the art. For them equations were intelligible because, as compound rapports or relations, they were reducible to the simple intuitible rapport that is ratio. To surrender that notion was to surrender intelligibility. Others, by contrast, were readier to accept that the rules of the art might change at the fifth degree. They had come to find algebraic expressions intelligible in themselves, or at least no less intelligible for not being explicitly resolvable into simpler forms. Equations expressed relations. “A quantitative relation,” wrote Leibniz, “is a way of finding one quantity by means of another,” and, as will become clear below, an equation of whatever degree constituted a modus inueniendi [Leibniz ca. 16901. He said this in Latin, using relatio. When he talked this way in French, relatio became rapport. (One can trace a shift from the narrow to the extended sense of rapport over the course of Malebranche’s later career.)

The question thus became: What constitutes a “way of finding,” a modus inueniendi? One can solve finite equations-or some of them-but is an infinite series a way of finding? Is a trigonometric relation a relation in Leibniz’ sense? Does a differential or an integral express a relation?

This is the third point. Leibniz, taking algebraic equations as themselves basically intelligible, moved to extend the notion of what constituted an equation, or rather a quantitative relation. For example, in a 1686 article titled “On a Hidden Geometry and on the Analysis of Indivisibles and Infinites,” in which he explained the deeper meaning of the method published in 1684, he said

I prefer moreover to set out dx and similar [expressions], rather than letters for them. because this dx is a sort of modification of x itself. Thus with its help, when it is necessary, only the

HM 11 CANONS OF INTELLIGIBILITY 421

letter x with its powers and differentials enters the calculation. and the transcendent relation between x and some other [quantity] is expressed.

Here Leibniz offered as examples “algebraic” expressions of the versine x of arc a, a = Jdx:m, x and of the cycloid, y = m + Jdx:m, the latter of which

perfectly expresses the relation between the ordinate y and the abscissa x; from it all the properties of the cycloid can be demonstrated. In this way analytic calculus is extended [pro~otus] to those lines that have hitherto been excluded for no other cause than that they were not believed to be subject to it. [Leibniz 168611859. 2311

This passage requires considerable exegesis, more than the current occasion allows. To begin with, what are transcendent relations? They are those that “are not plane or solid or sursolid or of any determinate degree, but rather transcend every algebraic equation.” In short, they are what Descartes called “non-mathematical” relations. For Leibniz, by contrast, they can be made mathematical by considering dx to be the result of some operation on x of a sort comparable to the basic combinatory operations. Leibniz went into no further detail in the article, but later on in an unpublished treatise he tried at some length to show that the rapports among differentials can be reduced to those among finite quantities and hence that differential equations are finite equations at heart [5].

The goal of the effort and the language in which it is couched are more important than its ultimate futility. Accustomed to understanding complex mathematical relations and intricate geometric configurations by means of their equations, Leibniz sought to include new relations and configurations by introducing new sorts of equations into analysis. For him the scope of mathematics was set by the methods of solving ordinary and differential equations, which thereby became themselves the proper and central object of mathematical understanding. In this, followers of Descartes saw a triumph of technique over intelligibility.

Not only mathematics was at stake. As mentioned at the outset, the new curves and techniques of solution Leibniz was bringing under the aegis of analysis per- tained largely to physical problems. As he put it in concluding his 1694 article, “Considerations sur la difference qu’il y a entre l’analyse ordinaire et le nouveau calcul des transcendantes,”

Finally. our method is properly that part of general mathematics that treats of the infinite, and that is why it is so necessary for applying mathematics to physics; because the character of the infinite Author ordinarily enters into the operations of nature. [LM.S.V.308].

At heart, nature itself was transcendent in Leibniz’ sense, and a mathematics of nature would have to be similarly so.

An example from Leibniz’ early mentor, Huygens, and then one from Leibniz himself show where this led. In deriving the period of a simple pendulum in 1659, Huygens began with a geometric picture of the bob’s circular arc over a short swing and then superimposed on it first the parabolic gradient of its speed as a function of distance fallen and then an auxiliary curve, the quadrature of which would yield him his answer (for details of the derivation see Mahoney [ 1980]). But

422 MICHAEL S. MAHONEY HM II

to carry out the quadrature-no, carry out the transformation of the curve’s equations into a form Huygens could square-he had to replace the bob’s trajectory by a parabolic arc quite close to it [6]. Asking himself then for what trajectory the substitution would be exact rather than approximate, he recognized a recently derived property of the cycloid 171. In fact, with the cycloid in place he no Ionger needed to assume that the arc be small. It could be of any size. The cycloid is the tautochrone. Since it is also its own evolute, cycloidal leaves constrain a pendulum to follow a tautochronic path.

Some years later, in 1675, while studying the rectification of the cycloid, Huy- gens recognized that the arc length measured from the vertex of an inverted cycloid was proportional to that segment of the tangent that would correspond to the accelerative force on a body moving along the cycloid at the point of tangency. That is, in an inverted cycloid the distance from the lowest point is proportional to the gravitational force moving a body along it. But, he reasoned, the cycloid is a tautochrone; hence, that relation is itself the tautochronic relation. Moreover, the relation holds for springs. Therefore, springs are tautochrones. Indeed, he could think of a host of mechanisms expressing the relation. His notes of the early 1690s are filled with them. All of them must be tautochrones. As one reads through his studies on this subject, it seems clear that from 1675 the physical pendulum that had once embodied his and his predecessors’ understanding of what we now call harmonic motion was replaced, not by another physical instance, but by a mathematical relation. Huygens himself expressed it in words with reference to a diagram. Leibniz’ new calculus then provided the symbolic form: ddS + kSdt’ = 0. That was the tautochrone; it was an equation, not a device.

Something of the sort may underlie Leibniz’ 1689 effort to place the vortex theory on a mathematical footing, “Tentamen de motuum coelestium causis” [Acta eruditorum (1689); LMS.VI.144-1611. One reads his definition of “harmonic oscillation” as that in which the linear speed of the circulating body is inversely proportional to its distance from the center of circulation, and one wonders what fluid Leibniz thought he was describing. It may be that in his mind at the time the fluid in question was precisely the juid he was describing. In the order of things, first the tautochrone, then the tautochronic mechanism; first harmonic circulation, then a harmonically circulating fluid. Again, the way he is thinking, that is, the grounds of his understanding and what he takes to be the grounds of his readers’ understanding, is more important historically than whether he was correct in thinking that way.

These examples from the work of Leibniz can be multiplied many times over by means of material from the writings of the Bernoullis, of Varignon, and of similar Continental proponents of the new calculus and its application to mechanics. As important, perhaps, are the examples from the writings of contemporaries, especially Cartesians, whose criticisms of the calculus seem strangely off the mark until one realizes what lies behind them, namely, an inability to “see” what is going on or an unwillingness to accept heuristic success as a substitute for clear understanding. Discourse among the “new” analysts rested on new canons of

HM 11 CANONS OFINTELLIGIBILITY 423

mathematical and physical intelligibility. That people of intellectual standing at the time had difficulty accepting them shows how new they were.

NOTES 1. Letter from Descartes to Mersenne, 27 July 1638. in Descartes [Vol. II, p. 2681, hereafter referred

to in the form AT.II.268. 2. For a discussion of Descartes’ treatment of curves. see Molland [1976] and Bos [1981].

3. Letter from Roberval to Torricelli [post 16451, Memoires de I’Acudemie Royale des Sciences, 1666-1699 (hereafter MARS), Vol. X. 440-478; at 449.

4. Cf. Leibniz’ correspondence with Oldenburg and Collins. and the subsequent literature of the 1780s and 1790s.

5. See his manuscript work, “Cum prodiisset atque increbuisset Analysis mea infinitesimalis . .” [post 17011, in Gerhardt [1846], translated by Child [1920] and discussed by Bos [1974]. 56-64.

6. The circle was the osculant of the parabola. 7. He had been working on the cycloid at the time, stimulated by the controversy provoked by

Pascal’s challenge of 1654.

REFERENCES Bos, H. J. M. 1974. Differentials, higher-order differentials, and the derivative in the Leibnizian

calculus. Archive for History of Exact Sciences 14, l-90. - 1981. The representation of curves in Descartes’ GComPtrie. Archive for History of E.wr.t

Sciences 24,295-338.

et al. 1980. Studies on Christiann Huygens. Lisse: Swets & Zeitlinger. Child, J. M. 1920. Early mathematical manuscripts ofleibniz. Chicago: Open Court. Descartes, Rend. 1897/1913. Oeuvres de Descartes, Ch. Adam and P. Tannery. eds. Paris: L. Cerf. Gerhardt, C. I. 1846. Historia et origo calculi differentialis a G. G. Leibnitzii conscripta. Hannover:

Hahn. (Trans. in Child [1920].) Leibniz. G. W. 1684/1859. Nova methodus pro maximis et minimis, itemque tangentibus, quae net

fractas, net irrationales quantitates moratur & singulare pro illis calculi genus. Acta enrditorum 1684, 467-473; reprinted in Leibnizens Mathematische Schriften (Leipzig, 1859). C. 1. Gerhardt, ed., Vol. V, pp. 220-226 [hereafter cited in the form LMS.V.220-2261.

_ 168611859. De geometria recondita et analysi indivisibilium atque infinitorum. Acttr entdi- torum, 292-300; LMS.V.226-233.

ca. 169011859. Dynamica de potentia et legibus naturae corporetre, Sect. 1, Cap. 1; LMS.VI.295.

Mahoney, M. S. 1980. Christiaan Huygens: The determination of time and of longitude at sea. In Studies on Christiaan Huygens, H. J. M. Bos et al., eds. Lisse: Swets & Zeitlinger, pp. 234-270.

Molland, A. G. 1976. Shifting the foundations: Descartes’ transformation of ancient geometry. Historia Mathematics 3, 21-49.

HISTORIA MATHEMATICA 14 (1987), 3 I l-324

The Calculus of the Trigonometric Functions

VICTOR J. KATZ

Department of Mathematics, University of the District of Columbia. 4200 Connecticut Ave. N. W., Washington, D.C. 20008

The trigonometric functions entered “analysis” when Isaac Newton derived the power series for the sine in his De Analysi of 1669. On the other hand, no textbook until 1748 dealt with the calculus of these functions. That is, in none of the dozen or so calculus texts written in England and the continent during the first half of the 18th century was there a treatment of the derivative and integral of the sine or cosine or any discussion of the periodicity or addition properties of these functions. This contrasts sharply with what occurred in the case of the exponential and logarithmic functions. We attempt here to explain why the trigonometric functions did not enter calculus until about 1739. In that year, however, Leonhard Euler invented this calculus. He was led to this invention by the need for the trigonometric functions as solutions of linear differential equations. In addition, his discovery of a general method for solving linear differential equations with constant coefficients was influenced by his knowledge that these functions must provide part of that solution. o 19x7 Academic

Press. Inc.

Les fonctions trigonometriques sont entrees dans I’analyse lorsque Isaac Newton a ob- tenu une serie de puissances pour le sinus dans son De Anafysi de 1669. Par contre, aucun manuel jusqu’a 1748 n’a portt sur le calcul de ces fonctions. C’est-a-dire que, dans aucun des douzaines d’ecrits portant sur Ie calculs publies en Angleterre ou sur le continent pendant la premiere moitie du XVIII’ sitcle, il n’y avait pas d’etude de la dtrivte et de I’integrale du sinus ou du cosinus, ni d’examen des prop&t& de ptriodicitt ou d’addition de ces fonctions. Cela contraste fortement avec ce qui est arrive pour les fonctions exponen- tielle et logarithme. Nous essayons d’expliquer ici pourquoi les fonctions trigonometriques ne sont entrees dans I’analyse qu’aux environs de 1739. D’ailleurs, en cette an&e, Leonhard Euler a inventt ce calcul. II a ete conduit a cette decouverte par la necessitt d’utiliser les fonctions trigonomttriques comme solutions des equations differentielles lineaires. En outre, sa decouverte d’une methode g&&ale de resolution des equations differentielles lineaires a coefficients constants a Ctt influencee par sa connaissance que ces fonctions doivent foumir une partie de Cette sohrtion. 0 1987 Academic Press, h.

Die trigonometrische Funktionen traten in die “Analysis” ein, als Isaac Newton die Potenzreihe der Sinusfunktion in seiner Arbeit De Analysi von 1669 herleitete. Andererseits betrachtete kein Lehrbuch vor 1748 den Kalktil dieser Funktionen. Das heisst, man findet weder eine Behandlung der Ableitung und des Integrals vom Sinus oder Cosinus noch eine Behandlung der Periodizitlits- oder Additionseigenschaften dieser Funktionen in irgen- deinem Lehrbuch tiber Differential- und Integralrechnung aus der ersten Htifte des achtzehnten Jahrhunderts. Hierin liegt ein deutlicher Gegensatz zum Falle der Exponen- tialfunktion und der logarithmischen Funktionen. lm vorliegenden Aufsatz versuchen wir zu erkhiren, weshalb die trigonometrischen Funktionen bis urn 1739 rechnerisch nicht behan- delt wurden. In diesem Jahr erfand Leonhard Euler den betreffenden Kalkiil. Er wurde zu dieser Ertindung durch den Bedarf an trigonometrischen Funktionen als Liisungen linearer

311

0315-0860187 $3.00 Copyright 0 1987 by Academic Press. Inc.

All rights of reproduction in any form reserved.

312 VICTOR J. KATZ HM 14

Differentialgleichungen gefiihrt. Zusltzlich beeinfluRte sein Wissen darum, daR diese Funktionen einen Teil der Liisung linearer Differentialgleichungen mit konstanten Koeffi- zienten liefern miissen, seine Entdeckung eines allgemeinen LGsungsverfahrens fiir solche Gleichungen. 0 1987 Academic Press, Inc.

AMS 1980 subject classifications: OlASO, 34-03, 26-03. KEY WORDS: Trigonometric functions, Leonhard Euler, linear differential equations, Johann Ber-

noulli, exponential function.

The trigonometric functions entered “analysis” with Isaac Newton. It is well known that in De Analysi [ 16691, Newton derived the power series for the sine by inverting the power series for the arcsine; the latter he had derived from the binomial theorem and geometrical considerations. Within the next decade, these series showed up in various places, including the correspondence between New- ton and Leibniz in 1676. Leibniz noted in particular that the sine series could be derived from the cosine series by term-by-term integration since “the sum of the sines of the complement to the arc . . . is equal to the right sine multiplied by the radius, as is known to geometers” [Turnbull 1960, 651. Nevertheless, as we will see, the calculus of the trigonometric functions did not come into existence until 1739. That is, until that date there was no sense of the sine and cosine being expressed, like the algebraic functions, as formulas involving letters and numbers, whose relationship to other such formulas could be studied using the developing techniques of the calculus. Since such was not the case for the other large class of what we call the transcendental functions, the exponential and logarithmic functions, this 70-year gap calls out for explanation. Not only will we attempt that explanation here, but we will also see why and how the trigonometric functions finally did enter calculus.

First, however, we want to review what was known about the sine and cosine in the last quarter of the 17th century and then briefly discuss their rare appearance in the first calculus textbooks of the early 18th century. Given that the sine and cosine are the most familiar examples of periodic functions, one might expect that they would make an appearance whenever there was any discussion of a periodic physical phenomenon. In fact, they did, but in ways so geometrical that there was no development of the analytic ideas. For example, in 1678 Hooke’s law appeared in print in the published version of his Cutlerian lecture [Hooke 16781. In an effort to describe the motion of a weight on a stretched spring based on his law, Hooke drew a rather complex diagram and showed that the velocity of this weight is as certain ordinates in a circle; these ordinates may be thought of as the sines of the arcs cut off. He also drew a curve which represented the time for the weight to be in any given location; this curve is in fact an arccosine curve. Hooke, however, does not use these trigonometric terms; he is content with the geometry of the situation.

A few years later, a more explicit result appears in Newton’s Principia as Proposition XXXVIII, Theorem XII:

HM 14 TRIGONOMETRIC FUNCTIONS 313

Supposing that the centripetal force is proportional to the altitude or distance of places from the centre, I say, that the times and velocities of falling bodies, and the spaces which they describe, are respectively proportional to the arcs, and the right and versed sines of the arcs. [Newton 16871

It may be, as Truesdell says in referring to this theorem, that “for Newton, simple harmonic motion was a familiar and completely mastered concept” [Trues- dell 19601. The theorem, however, occurs in the section entitled “Concerning the Rectilinear Ascent and Descent of Bodies” and Newton makes no reference there to motions repeating themselves. He simply describes the motion of bodies moving on certain curves subject to various types of forces. Newton’s proof of the theorem involves taking the limit of a body moving on an arc of an ellipse which is an affine transform of a circle, as the ellipse is squeezed onto its diameter. But his diagram to the theorem shows only one quadrant of a circle; the right sine of the statement is, as is usual for that time, simply a line from the circle’s diameter to its circumference. To a modern reader, the calculus of the sine and cosine is only a hair’s breadth away from Newton’s discussion; but Newton himself says no more about it and there is little reference to this idea in any other work over the next 30 years.

One might also expect the sine and the cosine to appear as the solution to a simple differential equation, in particular as the solution to y” = -ky. Again, one does find, in effect, this equation. Leibniz in [1693] derives from his differential method the infinitesimal relation between the arc and its sine in a circle of radius a: a2dy2 = a2dx2 + x2dy2 (Fig. l), assuming dy is constant. Leibniz takes the differential of this equation to get 2a2dxd2x + 2xdxdy2 = 0 or a2d2x + xdy2 = 0. We would write this equation as d2xldy2 = -x/a=, the standard differential equation for x = sin (y/a). Leibniz does in. fact derive this solution, by his method of undetermined coefficients, and writes it as a power series. Again, we wonder why Leibniz did not go further and discuss the properties of this series. But neither he nor Johann Bernoulli, who discussed the same differential equation and power series in a paper of the following year, moved any closer to the calculus of these functions.

(a’-x2)dy2= a2dx2

a2dy2=a2dx%2dy2

FIGURE 1


A physical problem leading to a differential equation which could also have produced the calculus of the sine function was the vibrating string problem first attacked by Brook Taylor [ 17131. Taylor in fact needed to find the fluent of c?/ w; he showed that this expression was the fluxion of the circular arc whose sine is y with radius c. And since Taylor was most interested in the periodic time of the motion of the string, he did not write this in terms of the sine itself. In fact, as we will see, it was quite common in later works to deal with what we call the arcsine function rather than the sine.

In 1696 there appeared the first textbook on the new calculus of Newton and Leibniz, the Analyse des infiniment petits by the Marquis de I’Hospital [1696]. It was followed in France by the Analyse demontre’e of Charles Reyneau [ 17081 and in England by a number of texts including works by George Cheyne [1703], Charles Hayes [ 17041, Humphrey Ditton [ 17061, John Craig [ 17181, Edmund Stone [ 17301, James Hodgson [1736], John Muller [1736], and Thomas Simpson [ 1737). What do we find in these texts on the calculus of the trigonometric functions? Essentially, we find nothing. As we already noted, this is in contrast to the situation with exponential and logarithmic functions. For even though these functions are not treated as inverses of one another, their derivatives and integrals are dealt with. In most of these texts we find some sort of derivation of the basic result that the derivative (differential, fluxion) of the logarithm of a quantity is the derivative (differential, fluxion) of the quantity divided by the quantity itself. We also usually find an extensive treatment of the derivatives of expressions of the type ax where the exponent x is a variable and where a is either a constant or a variable. In fact, most of the authors deal with even more complicated exponential expressions.

What does appear about trigonometric functions? If there is anything at all, it is only a discussion of the relationship between the sine or tangent and the arc; this treatment is then carried out in the manner of Newton’s original work via power series. The only one of the authors cited who has anything more is Thomas Simpson. In the course of solving a problem dealing with spherical triangles, he proves geometrically the result that “the Fluxion of any circular arch is to the Fluxion of its Sine, as Radius to the Cosine” [Simpson 1737, 1791. This proof uses, in effect, the differential triangle of Leibniz, and its similarity with the triangle whose hypotenuse is the radius of the circle and whose legs are the sine and cosine of the intercepted arc (Fig. 2). We can of course translate the theorem into the standard calculus result that the derivative of the sine is the cosine.

The cited proof was not, however, original to Simpson. It appeared some 1S years earlier when the manuscripts of Roger Cotes were published 6 years after his untimely death at the age of 34. Cotes proved the result at the beginning of a tract On the Estimation of Errors in which he analyzed the errors which occurred in astronomical observations. The particular lemma was stated by Cotes as “the small variation of any arc of a circle is to the small variation of the sine of that arc, as the radius to the sine of the complement” [Gowing 19831. The “small variations” can be considered as fluxions or as differentials; in any case Cotes uses


FIGURE 2

essentially the same diagram as Simpson did later and gives the same proof. (The same result with the same proof is also found in a paper of Euler’s colleague at St. Petersburg, F. C. Maier [1727].) Cotes followed this lemma with two others in which he proved by similar methods results equivalent to the theorems that the derivative of the tangent is the square of the secant and the derivative of the secant is the product of the secant and the tangent.

Cotes’ manuscripts contain other work related to the calculus of the trigonometric functions. As part of his Logometria he calculates the integrals of the tangent and secant functions. Here I use the word function in the modern sense, because Cotes in fact sketches several periods of both of these functions and shows how these curves are derived from the geometric definitions.

As a final example of Cotes’ use of the trigonometric functions in calculus, we may cite his extensive table of integrals. Cotes notes in effect that for many integrands involving sums or differences of powers, a change of sign in the inte- grand changes the integral (or fluent) from a logarithm to an inverse trigonometric function. In fact, Cotes uses the same notation for both, stating that one or the other is meant depending on the sign of a certain quantity. In other words, the inverse trigonometric functions, at least, had the same standing in Cotes’ mind as the logarithmic functions. He could deal with one as easily as with the other. This “equivalence” of the two types of functions led Cotes to a result close to Euler’s famous expression of ei8 = cos 8 + i sin 8. Cotes’ result was stated, though, in terms of logarithms.

Unfortunately for the development of the calculus of the trigonometric functions, Cotes died before he could formulate his results in a systematic way. Some of the English textbook writers referred briefly to Cotes’ work, but no one devel- oped it much further.

Why do the sine and cosine not appear in these texts? We will give two explanations, both of which will be justified by the work of the man who unquestionably was responsible for the ultimate introduction of these functions into calculus, Leonhard Euler. First of all, though sine tables existed in abundance, the sine was not considered as a “function,” even to the extent that logarithms or exponentials were. It was thought of geometrically as a certain line in a circle of a given radius;


one did not, in general, draw a graph of such a function so there was no question of finding tangent lines or areas. That is not to say that graphs of the sine function did not occur at all. There are several early examples of the appearance of one or more arches of the sine in the mid-17th-century works of Roberval and Wallis [Boyer 19471. But the idea of the sine as a “function” did not enter the mathematical mainstream at this time.

As another indication of this, we cite the unpublished manuscript of Euler’s Culculus Diffeerentialis, which Yushkevich 119831 believes to have been written about 1727 for use as a text in the university at St. Petersburg. In this text, Euler gave a definition of function similar to the one he was later to give in his Intro&c- tio in analysin injinitorum; he then proceeded to subdivide all functions into two classes, algebraic and transcendental. But this latter class, he noted, consists solely of the exponential and logarithmic functions. Euler gave a complete treatment of the differential calculus of these latter functions, but he did not mention the trigonometric functions at all.

A second reason for their nonappearance may be related to a statement of Edmund Stone’s in a somewhat different context. Stone, commenting on the absence even of exponential functions from the work of L’Hospital and also from his own work, noted,

As the illustrious author [L’Hospital] has omitted the exponential calculus, or manner of finding the fluxions of exponential quantities, such as xX = a, xx = yr, etc. where the index’s of the variable quantities are also variable, thinking, as I suppose, this branch of doctrine to be of very little or no use, so I have been silent in this matter also; which it is much better to be, than take up the reader’s time in learning what is only mere speculation. [Stone 1730, Intro- duction]

That is, trigonometric functions may have been avoided because no one saw any reasonable use for them as yet. As a possible confirmation of this reason, we will again cite Euler. In the abstract to a paper he published in 1754 which dealt extensively with the calculus of the sine and cosine, Euler claimed this invention for himself. As he, or the journal’s editor, wrote,

In addition to the logarithmic and exponential quantities there occurs in analysis a very important type of transcendental quantity, namely the sine, cosine and tangent of angles, whose use is certainly most frequent. Therefore this type rightly merits, or rather demands, that a special calculus be given, whose invention in so far as the special signs and rules are comprised, the celebrated author of this dissertation is able rightly to claim all for himself, and of which he gave examples in his Introduction to Analysis and Institutions of the Differ- ential Calculus. Numerous examples stand out in his work on the motion of the moon and on the perturbations of the motion of Saturn and Jupiter, in which this type of calculus is frequently used in the investigations, and without the help of which these were scarcely able to be performed. Therefore, for this new calculus, which is called the calculus of sines, not only does Euler present the first principles and reveal the highest uses in various parts of mathematics through the most impressive examples, but also he continues to enrich it by new inventions, which the present article splendidly demonstrates. [Euler 1754, 5431

One could thus say that by 1754 there were definitely uses for these functions; in 1727 Euler himself saw none.


We have already mentioned on several occasions Euler’s famous text Zntroduc- tie in anatysin in.nitorum 11748~1. In this text Euler provides a complete treatment of what we may call the precalculus of the trigonometric functions. That is, he defines them numerically, not as lines in a circle, discusses their various properties including addition formulas and periodicity, and develops their power series. He also draws the graphs of some of these functions. Since this text was not designed to be a calculus text, there is no mention of derivatives. This lack was only temporary, however. Euler was already writing his differential calculus text, which appeared 7 years later [Euler 17551. In this text, of course, there was a derivation, using power series, of the standard rules for the derivatives of the trigonometric functions. As we shall see, Euler was in full command of these rules much earlier. The question remains, then, since Euler claimed the invention of the calculus of trigonometric functions, what were the circumstances under which this invention took place, or, more specifically, how did these functions enter calculus?

A consideration of Euler’s papers before 1740 provides an answer. The trigonometric functions entered calculus via the study of differential equations. Not only did this study give the sine and cosine the status of “function” in our sense, and give them an equal status with the exponential and logarithmic functions, but it also provided the necessary uses for these functions. The study of differential equations was not just the cause of the sine and cosine functions entering calculus, however. It was Euler’s knowledge of these functions which led him, I believe, to the development of the standard method of solving linear differential equations with constant coefficients. The remainder of this paper will be devoted to convinc- ing the reader of the truth of these assertions.

In some of Euler’s earliest papers, in the late 172Os, Euler needs to integrate equations of the form dt = dxlw As was the custom of the time, he gave as the solution that t was the arc whose sine was x with radius a. In fact, Euler probably learned this solution from his teacher Johann Bernoulli. Bernoulli himself, in a paper dealing with vibrations [ 17281, studied an object which moves subject to a force proportional to its distance from a given point. He in effect set up the equation d2yldt2 = a - y, where a is a constant, and solved it using two integrations. We will present the essence of his solution, which is similar to one worked out a decade earlier by Jakob Hermann. We first set u = dyldt. From duldt = a - y, we then derive the result that duldy = (a - y)/u or udu = (a - y)dy. An integration of this equation leads to u* = 2uy - y? or u = m or finally dyldt = w. Bernoulli, however, does not put the equation in that form; he rewrites it as dt = dylw and proceeds to integrate once more to solve for t by use of the arcsine. That is, he, like Taylor earlier, is interested in the time of the motion, not the motion itself. Bernoulli does get a sine curve out of this problem, but it is simply the shape of a string; he does not deal with sinusoidal motion. Over the next several years, Euler dealt with similar integrals in the same way as Bernoulli, by solving for the arcsine.

On the other hand, during this same time Euler was certainly familiar with the


periodicity properties of the sine function. In a well-known paper [ 1735b] he used the fact that the power series for sin S, s - s3/3! + ss/5! - . . . had nonzero roots at +7~, +2~, . . . to “factor” 1 - s2/3! + s4/5! - . . . as (1 - s2/7r2) (1 - sz/47r2) . . . and derive his result that 1 + l/2* + 1/32 + . . . = ~~/6. Similarly, in another paper [1736], in which he integrated a differential equation of the above type to get the arc of a circle whose cosine is y/a, he stated the various values of the arc at which y would be 0 or a.

Nevertheless, Euler did not always recognize the sine function as the solution of a differential equation. Daniel Bernoulli wrote to Euler on May 4, 1735, to discuss a problem on the vibrations of an elastic band. As part of this problem he had to solve the differential equation k4 (d4yldx4) = y. He wrote, “This matter is very slippery. . . . The logarithm satisfies the equation . . . but no such logarithm is general enough for the present business” [Truesdell 1960, 1661. By “logarithm,” Bernoulli of course meant the exponential function. Euler dealt with the same problem and the same equation in a paper later that year and was also unable to find a complete finite solution. He was, however, able to solve it using power series. But since he incorporated the initial conditions into that solution, he did not recognize that there was a sine or cosine hidden in the series he finally obtained [Euler 1735al.

The sine and cosine do appear in other papers of Euler in the late 1730s as well as in some of his correspondence. But if the papers deal with calculus at all, these functions appear only in the contexts discussed above. Outside of calculus, we can find references to various trigonometric formulas, especially to those dealing with multiple angles. It is only in 1739 that Euler is able to put all of his knowledge of these functions together.

On March 30, 1739, Euler presented the paper De nouo genere oscillationurn [1739a] to the Academy of Sciences at St. Petersburg. The paper dealt with the motion of what we would call a sinusoidically driven harmonic oscillator; that is, Euler considered the motion of an object in which the force acting was composed of two separate parts, one proportional to the distance, the other one varying sinusoidally with the time. Euler noted in a letter to Johann Bernoulli on May 5 that “there appear . . . motions so diverse and astonishing that one is unable altogether to foresee until the calculation is finished” [Enestrom 1905, 331. Note that Euler wrote about the motions; these now become central rather than merely the period. What calculations did Euler perform to derive these motions? He began by deriving three simultaneous differential equations which the four given variables S, t, y, u had to satisfy. Here t is time measured along the arc of a circle of radius a while y is the sine of that arc. In addition, s represents the position of the object while u/a represents the square of the velocity. The equations are

du =

Euler’s aim was to use these equations to eliminate y and u, thus getting a single


equation relating s and t which he can solve. In particular, he solves the second equation, a very familiar one, in a new way. Not only does he get the usual solution t = a arcsin (y/a) but also, for the first time, he writes this in the inverse form as y = a sin (t/a). As we already mentioned, Euler is now dealing with the motion; so t becomes the independent variable. Using this result as well as the solution u = uds2/dt2 of the third equation, he easily derives the desired differential equation

udt? t 2ud2s + F + - g sin ; = 0

relating s and t. He now wanted to find a solution to this in finite terms. It is not necessary here to discuss his entire solution. But there are a few major

points of interest. First of all, he solves the special case where “b = a,” that is, 2gd2s + dt* sin(tlu) = 0. To do this, he obviously needs to know the differentials of the sine and cosine; so he writes them down: diff. sin(tlu) = (dtlu) cos(tlu) and diff. cos(tlu) = -(dtlu) sin(tlu). (There was no necessity to derive these; as we have noted, the results themselves had in one form or another been known for years.) After solving the equation in the form s = (u2/2g) sin(tla), he proceeds to analyze the periodicity properties of this solution. Second, in the process of solving the general case he first deletes the sin(tlu) term and considers the equation 2ud2s + sdt2/b = 0. Note that this is similar to the equation of Bernoulli already mentioned. This time, however, Euler solves it by integrating twice to get first 2ubds2 + s2dt2 = C2dt2 or dt = -V%&dslm and then, by what he usually calls a quadrature of the circle, the result t = X6%$ arccos(slC), or finally, as before, s = C cos(tlV?&). Again, the trigonometric functions appear explicitly. Finally, Euler solves the general case by postulating a solution of the form s = u cos(tlV%6) where u is a new variable. He proceeds to substitute that expression into the equation and solve for u. This, of course, involves being thoroughly familiar with the calculus manipulations of the sine and cosine. After much of this type of manipulation, he demonstrates the “motions so diverse and astonishing” about which he wrote to Bernoulli.

We note that in a paper which appeared only 45 pages later in the same volume of the Commenturii of the St. Petersburg Academy [1739b], Euler again uses the differentials of the sine and cosine. Already, he is becoming “fluent” in their use.

We have now seen how the calculus of the sine and cosine appeared as part of the process of solving an interesting differential equation. But there is more. In the same letter of May 5 to Johann Bernoulli, Euler mentions that he also solved, in finite terms, the equation a3d3y = ydx3. He writes, “though it appears difficult to integrate, needing a triple integration and requiring the quadrature of the circle and hyperbola, it may be reduced to a finite equation; the equation of the integral is

y = b&cl + ce-.T/2~~ sin <f + x>fl 2u


. . . where 6, c, f are arbitrary constants arising from the triple integration” [Enestrom 1905, 311. Euler does not state explicitly how he found this solution. But one can hazard a guess based on his statement that it required the quadrature of the circle and the hyperbola and on some of his earlier methods of solving differential equations. Namely, Euler was certainly aware that y = &’ was a solution. As far back as 1728 he used the fact that for that relation, dy = ( I ln)ydx; differentiating twice more would give that function as a solution to Euler’s third order equation. Of course, using the exponential function as a solution meant using the “quadrature of the hyperbola”; Euler was aware of the relationships between the logarithm and both the hyperbola and the exponential function. Once Euler had one solution, he could use it to reduce the order of the equation. That is, he could use e-xla as an “integrating factor.” He had dealt with such factors in [Euler 1735~1; later he wrote a detailed study of their use in reducing the order of equations [1750]. In this particular case, if we multiply the original form u3d3y - ydx3 by e-xia and assume it is the differential of c”‘“(Ad’y + Bdydx + Cydx?), it is not difficult to show that a new solution of the original equation must also satisfy the second-order equation a’d’y + adydx + ydx’ = 0, or, in derivative notation, a2(d2yldx2) + a(dyldx) + y = 0.

How would Euler solve this second-degree equation? Using the technique of multiplication mentioned in De nouo genere oscillationurn, he could easily guess as a solution y = uea. If we differentiate that twice and put it into the equation, we are able to eliminate the term in dudx by setting a! = - 112a. Now using y = ue-“2u, we reduce the equation to a2d2u + $u dx * = 0. The latter equation is similar to several that we have already seen Euler solve by two integrations and the quadrature of the circle. The result in this case is u = C sin((x + f)fi/2a) as desired. The complete solution to the third-order equation then follows immediately. (We note that Euler used this method explicitly in [Euler 17431.)

In any case, since Euler has now used the sine and the exponential function together in a solution of a differential equation, it is clear that the former now has equal status with the latter insofar as the calculus is concerned; that is, the sine, and of course the other trigonometric functions as well, have now entered calculus. But we will go further. It is the introduction of these functions into the calculus which gave Euler the impetus to find the general solution to linear differential equations with constant coefficients, some special cases of which he had already solved in early 1739.

On September 15, 1739, Euler wrote to Johann Bernoulli giving this solution. Several months later, he wrote again noting that “after treating this problem in many ways, I happened on my solution entirely unexpectedly; before that I had no suspicion that the solution of algebraic equations had so much importance in this matter” [Enestrom 1905, 461. What is this “unexpected” solution? It is the standard method in use today. We replace the given differential equation

& d’y d3y y+az+bz+cB+. . .=O


by the algebraic equation

1 + up + bp2 + cp3 + . . . = 0.

We factor the polynomial on the left into its real linear and quadratic factors. For each linear factor I - crp we take as a solution y = Ce x’a while for each irreducible quadratic factor 1 + crp + pp2 we take as solution

Euler gave as his example of this process the solution to the equation

k4d4y o Y-x=.

Since the algebraic equation 1 - k4p4 factors as (I - kp) (1 + kp) (I + kzp2), Euler finds the complete solution

y = Cemxik + Dexik + E sin i + F cos i.

In this letter, Euler did not consider the case of multiple factors to the polynomial. How did Euler happen upon his general method of solution for this class of

differential equations? Euler does not tell us. But it seems that the examples dealt with in the March paper and the May letter must have been influential. It was by dealing with these examples that Euler first learned that the trigonometric functions were necessary parts of that general solution. He surely had known for years that exponential functions had to be involved. In fact, Johann Bernoulli noted in his December 9 response to Euler’s letter that he had found such solutions at least 20 years earlier by assuming that y = e x/p was a solution and solving the resulting equation

I?+ i I+;+++;+. . . =o 1

for p. But Bernoulli had only dealt with a single real solution. For example, for Euler’s case of

k4d4y Y-F=0

Bernoulli solved p 4 - k4 = 0 as only p = k. He then stated that “the logarithm, whose subtangent = k, satisfies the proposed equation” [Enestrom 1905,411, that is, what we call the exponential function eX lk. Bernoulli was not able to deal with complex solutions to the algebraic equation, or even with more than a single real solution. It is this major advance that Euler was able to take.

Once Euler knew that the trigonometric functions were in fact part of the complete solution, it was only necessary to see how the parameters of these


functions could be found algebraically. And if, in fact, Euler found the solution to the third-order equation by a method similar to what I have proposed, he would have noted that the second-order equation to which the original reduced was in some sense a factor of the original. Euler’s genius in dealing with formal identities would then have led him to writing down the characteristic polynomial to the differential equation and exploring the factors. It would have been clear that it was the irreducible quadratic factors which led to the trigonometric solutions.

Over the next couple of years, Johann Bernoulli debated with Euler about the validity of this method of solution in a series of letters. In essence, he did not understand how complex roots of the characteristic polynomial could lead to solutions involving the “real quadrature of the circle.” Euler finally showed him in 1740 that in fact 2 cos x and e& + e-“” were equal. Daniel Bernoulli, on the other hand, accepted Euler’s solution and in an article showed that the power series solution agreed with Euler’s for the equation d4y = ydx4/f4 [Bernoulli 17411. In any case, beginning in the early 1740s Euler was able to use the calculus of the trigonometric functions with ease; it appears in several of his papers, including a paper in which he published his method for solving linear differential equations with constant coefficients and further explained the relationship between trigonometric and complex exponential solutions [1743]. In that paper, Euler also han- dled the case of multiple factors of the characteristic polynomial.

It is well that Euler had invented this calculus when he did, for in the mid-1740s he became involved in several major areas of investigation in which the trigonometric functions were to play a crucial role. First of all, he needed the explicit solution of the same fourth-order differential equation in the first appendix of his work on the calculus of variations [ 17441. Second, Euler introduced trigonometric series in a prize paper he wrote for the Paris Academy in 1748 on the question of the inequalities in the movements of Saturn and Jupiter [1748a]. It is worthy of mention that in this paper Euler did not yet expect his readers to be familiar with the calculus of the trigonometric functions. In the early pages of that work, Euler discussed this calculus and showed how important it was to the understanding of the topic at hand. Third, at the same time Euler also introduced trigonometric series and the necessary calculus into his initial debates with d’Alembert on the question of the vibrating string [1748b].

As we noted earlier, Euler finally published this calculus in textbook form in his Znstitutiones calculi differentialis [ 17551. His methods immediately drove out the earlier geometric methods of dealing with the sine and cosine. Practically without exception, the calculus texts published in the second half of the 18th century all adopted Euler’s calculus for their treatment of the trigonometric functions. And, of course, we have continued to use Euler’s invention to the present day.

REFERENCES Bernoulli, D. 1741. De vibrationibus et sono laminarium elasticarium. Commentarii Academiae Scien-

tiarum Petropolitanae 13, 105-120.


Bernoulli, J. 1728. Meditationes de chordis vibrantibus. Commentarii Academiae Scientiarum Petro- politanae 3, 13-25; Opera Omnia 3, 198-210.

Boyer, C. 1947. History of the derivative and integral of the sine. Mathematics Teacher 40,267-275.

Cannon, J., L Dostrovsky, S. 1981. The evolution of dynamics-Vibration theory from 1687 to 1742. New York: Springer-Verlag.

Cheyne, G. 1703. Fluxionum methodus inversa sive quantitatum juentium. London: Mathews. Craig, J. 1718. De calculofluentium libri duo. London: Pearson. Ditton, H. 1706. Institution ofjkxions. London: Botham. Enestrijm, G. 1905. Der Briefwechsel zwischen Leonhard Euler and Johann 1 Bernoulli. Bibliotheca

Mathematics (3) 6, 16-87. Euler, L. 1735a. De minimis oscillationibus corporum tam rigidorum quam Aexibihum methodus nova

et facilis. Commentarii Academiae Scientiarum Petropolitanae 7, 99-122; Opera Omnia (2) 10, 17-34. Here and in all the other references to Euler, we will give the location of the work in the Opera Omnia, in the process of publication in 4 series since 1911.

- 1735b. De summis serierum reciprocarum. Commentarii Academiae Scientiarum Petropoli- tanae 7, 123-134; Opera Omnia (1) 14, 73-86.

- 1735~. De infinitis curvis. Commentarii Academiae Scientiarum Petropolitanae 7, 174-189; Opera Omnia (1) 22, 36-56.

- 1736. De oscillationibus tili flexihs quotcunque pondusculis onusti. Commentarii Academiae Scientiarum Petropolitanae 8, 30-47; Opera Omnia (2) 10, 35-49.

- 1739a. De novo genere oscillationum. Commentarii Academiae Scientiarum Petropolitanne 11, 128-149; Opera Omnia (2) 10, 78-97.

- 1739b. Methodus facilis computandi angulorum sinus ac tangentes tam naturales quam artifi- ciales. Commentarii Academiae Scientiarum Petropolitanae 11, 194-230; Opera Omnia (1) 14, 364-406.

- 1743. De integratione aequationum differentialium altiorum graduum. Miscellanea Berolinensia 7, 193-242; Opera Omnia (1) 22, 108-149.

- 1744. Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes, Lausanne: Marcus-Michaelis Bousquet; Opera Omnia (1) 24.

1748a. Recherches sur la question des inegalites du mouvement de Saturne et de Jupiter, Paris; Opera Omnia (2) 25, 45-157.

- 1748b. Sur la vibration des cordes. Histoire de l’dcade’mie Royale des Sciences et Belles-

Lettres de Berlin 4, 69-85; Opera Omnia (2) 10, 63-77. - 1748~. Zntroductio in analysin infinitorum, Lausanne; Opera Omnia (l), 8-9. - 1750. Methodus aequationes differentiales altiorum graduum integrandi ulterius promota. Novi

Commentarii Academiae Scientiarum Petropolitanae 3, 3-35; Opera Omnia (1) 22, 181-213. - 1754. Subsidium calculi sinuum. Novi Commentarii Academiae Scientiarum Petropolitanae 5,

164-204; Opera Omnia (1) 14, 542-584. - 1755. Znstitutiones calculi differentialis, St. Petersburg; Opera Omnia (I) 10. Gowing, R. 1983. Roger Cotes-Natural philosopher. Cambridge: Cambridge Univ. Press. Hayes, Ch. 1704. A treatise offluxions. London: Midwinter. Hodgson, J. 1736. The doctrine offluxions. London: Wood. Hooke, R. 1678. Lectures de potentia restitutiva or of Spring: Explaining the power of springing

bodies. Reprinted in Early science in Oxford, R. T. Gunther, Ed., Vol. 8, pp. 331-356. Oxford: Oxford Univ. Press, 1968.

Leibniz, G. W. 1693. Supplementum geometriae practicae sese ad problemata transcendentia ex- tendens, ope novae methodi generalissimae per series infmitas. Acta Eruditorum, pp. 178-180; Mathematische Schriften, C. I. Gerhardt, Ed., Vol. 5, pp. 285-288. Hahe: Schmidt, 1858.


L’Hospital, G. F. 1696. Analyse des infiniment petits. Paris: L’Imprimerie Royale. Maier, F. C. 1727. De planetarium stationibus. Commentarii Academiae Scientiarum Petropolilanae

2, 82-90. Muller, J. 1736. A mathematical freatise. London: Gardner. Newton, I. 1669. De analysi per aequationes numero terminorum infinitas. In The mathematical

papers of Isaac Newron, D. Whiteside, Ed., Vol. 2, pp. 206-247. Cambridge: Cambridge Univ. Press, 1969.

- 1687. The mathematicalprinciples of naturalphilosophy. Translated into English by A. Motte, 1729. Reprinted in 1968 by Dawsons of Pall Mall, London.

Reyneau, C. 1708. Analyse de’montre’e. Paris: Quillau. Simpson, T. 1737. A new treatise on jluxions. London: Gardner. Stone, E. 1730. The method ofjuxions, both direcr and inverse. London: Innys. Taylor, B. 1713. De motu nervi tensi. Philosophical Transactions of the Royal Society ofLondon 28,

26-32. The same material is also found in Methodus Incrementorum Directa et Inversa, London: Innys, 1715.

Truesdell, C. 1960. The rational mechanics of flexible or elastic bodies. In Leonhard Euler Opera Omnia (2) 11, Pt. 2.

Turnbull, H. W., Ed. 1960. The Correspondence of Isaac Newron, Vol. 2. Cambridge: Cambridge Univ. Press.

Yushkevich, A. P. 1983. L. Euler’s unpublished manuscript Calculus Differentialis. In Leonhard Euler 1707-1783, Beitriige zu Leben und Werk. Basel: Birkhauser.

HISTORIA MATHEMATICA 16 (1989), 69-85

Sharaf al-Din al-Tiisi on the Number of Positive Roots of Cubic Equations

JAN P. HOGENDLIK

Department of Mathematics, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands

In the second part of his Algebra, Sharaf al-Din al-T&i (l2th-century) correctly determines the number of positive roots of cubic equations in terms of the coefficients. R. Rashed has recently published an edition of the Algebra [al-Tiisi 19851, and he has discussed al- Tiisi’s work in connection with 17th century and more recent mathematical methods (see also [Rashed 19741). In this paper we summarize and analyze the work of al-Tiisi using ancient and medieval mathematical methods. We show that al-Tiisi probably found his results by means of manipulations of squares and rectangles on the basis of Book II of Euclid’s Elements. We also discuss al-Tiisi’s geometrical proof of an algorithm for the numerical approximation of the smallest positive root of x3 + c = ax2. We argue that al-Tiisi discovered some of the fundamental ideas in his Algebra when he was searching for geomet- Ikd proofs of such algorithms. 0 1989 Academic Press, Inc.

Dans la seconde par-tie de son Algebre, Sharaf al-Din al-Tusi (XII’ sitcle), a correctement determine le nombre de racines d’une equation du troisitme degre en fonction de ses coefficients. R. Rashed a rtcemment publie une edition de cette Algebre [al-Ttisi 19851 et a CtudiC l’ouvrage d’al-Tiisi en se servant des methodes mathematiques du XVII’ siecle et de methodes encore plus recentes (voir aussi [Rashed 19741). Dans cet article, nous resumons et analysons l’ouvrage d’al-Tiisi en utilisant les methodes mathematiques connues dans 1’Antiquite et au Moyen-Age. Nous montrons qu’al-Tiisi a probablement trouve les resultats auxquels il est parvenu par des operations effect&es sur des car& et des rectangles, operations basees sur le Livre II des klements d’Euclide. Nous Ctudions Cgalement la demonstration geometrique d’un algorithme utilise par al-Tiisi pour calculer par approximation la valeur numerique de la plus petite racine positive de l’tquation x3 + c = ax2. Nous essayons de montrer qu’al-Tiisi a trouve certaines des idees fondamentales de son AlgPbre alors qu’il tentait de trouver des demonstrations geometriques a de tels algorithmes. o 1989 Academic Press, Inc.

69 03 15-0860/89 $3 .OO


JAN P. HOGENDIJK HM 16

AMS 1980 subject classifications: OlA30. KEY WORDS: historyofalgebra, cubicequations, Islamicmathematics, Sharafal-Din al-JXsi,geometri-

cal algebra, numerical approximation.

1. INTRODUCTION

The recent edition of the Algebra of Sharaf al-Din al-Tiisi (12th century, not to be confused with Nagir al-Din), which was published by R. Rashed in [al-Tiisi 19851, is an important contribution to the history of Arabic mathematics. Until recently the mathematician and poet ‘Umar al-Khayyam (ca. 1048-l 131) was supposed to have given the most advanced medieval treatment of cubic equations. Thanks to Rashed’s publications [al-Ttisi 19851 and [Rashed 19741 we now know that al-?&i went considerably further.

The publication [al-Ttisi 19851 contains an edition of the Arabic text with a literal French translation, a transcription of al-Ttisi’s reasoning in modern notation, and a discussion of most of the text in terms of modern algebra and analysis. Rashed conveniently divided the very long text of the Algebra into two parts, consisting of 116 and 127 pages of Arabic text, and printed in two volumes of [al- Ttisi 19851; these volumes will henceforth be denoted as [Tl] and [T2]. We will be concerned with the second part of the Algebra, on cubic equations that do not for all positive choices of the coefficients have a positive root. This second part consists mainly of a sequence of very long proofs in Euclidean style. The proofs are correct, but as Rashed points out, they do not necessarily reflect the way in which al-Ttisi found his results. In the introduction in [Tl, xviii-xxxi], Rashed relates al-TM’s discussion of the cubic equation f(x) = c to a method of P. de Fermat (1601- 1665) for the determination of maxima and minima of a cubic curve y = f(x). Act or d ing to Rashed, the concept of the derivative of a function or of a polynomial is also implicit in al-Ttisi’s work (see also [Rashed 1974, 272-273,290] = [Rashed 1984, 175-176, 1931, and for some further consequences [Rashed 1984, 3 121, reprinted from [Rashed 19781).

HM 16 AL-TuSI ON POSITIVE ROOTS 71

As far as is known, cubic curves were never drawn by medieval mathematicians, and the method of Fermat and the derivative are not mentioned explicitly in any known medieval Arabic text. Thus the question arises of whether al-Ttisi’s methods and motivation can also be explained in terms of standard ancient and medieval mathematics. In this paper I propose such an alternative explanation.

Section 2 of this paper is a concise analysis of the second part [T2] of al-TM’s Algebra, by means of methods and concepts attested to elsewhere in the Greek and Islamic tradition. Section 3 is about al-Ttisi’s motivation. The appendices contain notes to the Arabic text and the French translation in [T2], for the reader who wishes to compare this paper with the original text. I conclude the present section with a brief summary in modern notation of the results that al-Ttisi proves in the Algebra.

The Algebra is a detailed treatment of linear, quadratic, and cubic equations in one unknown. Because the mathematicians in the Islamic tradition only recognized positive coefficients and roots, they had to distinguish 18 different types of cubic equations. Al-Khayyam had already shown that the five types without a constant term can be reduced to quadratic equations, and for each of the remaining 13 types he had given a geometrical construction of a root by means of two intersecting conic sections, or by means of one conic section intersecting a circle [al-Khayyam 198 11.

Eight of these thirteen types have for all (positive) choices of the coefficients a (positive) root. In the first part of the Algebra [Tl], al-Ttisi renders al-Khayyam’s geometrical constructions for these eight types, and he describes a numerical procedure (essentially the Ruffini-Horner scheme, see [Luckey 19481) for approximating the root.

The second part of the Algebra [T2] is entirely devoted to the five remaining types of cubic equations, namely

x3 + c = ax2 (0

x3 + c = bx (2)

with a, b, c > 0.

x3 + ax2 + c= bx (3)

x3 + bx + c = ax2 (4)

x3 + c = ax2 + bx (5)

Al-Khayyam pointed out that the number of roots of these equations depends on the number of intersections of the two conic sections used in the construction. He does not give the precise relation between the number of intersections and the coefficients of the cubic equation (cf. [al-Khayyam 1981, 711). For a given choice of the coefficients one could of course draw the conic sections on a piece of paper and determine the number of intersections empirically. Al-Khayyam does not mention this procedure, perhaps because it cannot be completely accurate. However, al-Ttisi succeeded in determining the exact relationship between the

72 JAN P. HOGENDIJK HM 16

number of roots and the coefficients of the equation. Neither al-Khayyam nor al- Tiisi was able to determine the roots themselves in terms of the coefficients; there is no evidence whatsoever that the algebraic solution of the cubic equation was known before the Italian Renaissance.

Al-Ttisi treats the five equations in the order (1) [T2, l-181; (2) [T2, 19-341; (3) [T2,34-481; (4) [T2,49-701; and (5), case a = fi [T2,70-761, case a > fi [T2, 76-1041, case a < fi [T2, 104-1271. For each of the types (l)-(4), and for each of the three cases of (5), the treatment is structured as follows (for detailed references to the text, see note [ 11). For sake of brevity I write the equations (l)-(5) as f(x) = c.

A. First al-Tusi defines a quantity m in a way that depends on the type of equation: (1) m = ($)a, (2) m = X@Bj, (3) m2 + (@am = b/3, (4) m2 + (b/3) = #am (here m is the largest of the two positive roots), and (5) m2 = @am + 6/3. (In ail five cases we havef’(m) = 0, but in my opinion al-Ttisi did not know the concept of a derivative.) He then proves f(x) <f(m) for all (positive) x # m. Thus if c > f(m), f(x) = c has no root and if c = f(m) there is exactly one root x = m.

B. He then supposes c <f(m), and he considers the equation

y3 + py2 = d, (6)

with d = f(m) - c for all types and p depending on the type of equation, as follows: (1) p = a, (2) p = 3m, (3) p = 3m + a, (4), (5) p = 3m - a with m defined as above; it can be shown that p > 0 always. The (unique positive) root yl of (6) had already been constructed geometrically in [Tl, 56-571 by means of a parabola and a hyperbola, and an algorithm for the computation of yI had been described in [Tl, 58-661. Al-Tusi proves that x1 = m + yl is a root of f(x) = c. Thus the existence of at least one root xl > m is guaranteed (by the geometrical construction of yl), and in part F it will turn out that there is no other root x > m. The root xl can be computed from m and yl.

C. For types (4) and (5) al-Tusi provides an upper bound of x1 in terms of a and b.

Dl. For type (1) only, al-Ttisi geometrically constructs a segment of length 4 such that

q2 + da - XI) = da - XI), (7)

where x1 > m is the unique positive root of (1) constructed in B. He shows that x2=a- xi + q is another root of (1) with x2 < m. He also proves that if z2 = m - x2, then z = z2 is a root of

z3 + d = pz2, (8)

with d = flm) - c = (4/27)a3 - c, p = a as above. He then explains an algorithm for the computation of x2 from (l), assuming that

c 5 (&)a3 (see below for more details). If c > (&)a3 we have d < (&)a3; in this case he first computes 22 = @a - x2 by the same algorithm applied to (8).

D2. For types (2)-(5), al-Tiisi considers the auxiliary equation

HM 16 AL-‘j’oS1 ON POSITIVE ROOTS 73

z3 + d = pz2,

with p and d as in (6); this equation is of type (1). Let 22 be the smallest (positive) root and x2 = m - 22. He then proves that x2 is a root offlx) = c. The root x2 can be computed from z2 and m.

E. For types (4) and (5), al-Ttisi discusses positive lower bounds for x2 in terms of a and b if such bounds exist.

F. Al-Ttisi proves separately that if x1 > m is a root offlx) = c, y1 = x1 - m is a root of (6).

G. He proves similarly that if x2 < m is a root offlx) = c, z2 = m - x2 is a root of (9).

H. He finishes the discussion of most types with a summary or a numerical example.

Thus al-T&i determines the number of solutions directly from the coefficients, and he shows that al-Khayyam’s separate geometrical constructions for (l)-(5) are superfluous, because they can all be reduced to the geometrical construction for (6) in [Tl , 56-571. Therefore [T2] does not contain conic sections at all. Al- Tiisi does not mention the fact that the equation x3 + bx = ax2 + c can have two or three positive roots for suitable positive coefficients a, b, c (compare [Tl , 107- 1161).

2. ANALYSIS OF THE SECOND PART OF AL-TGSI’S ALGEBRA

In the Algebra al-Tiisi uses similar reasoning in many different situations, and his solutions of Eqs. (l)-(5) are to a large extent analogous. This makes it possible to render the essentials of the 127 pages of Arabic text in [T2] in a concise way. The purpose of the following presentation is to make al-Ttisi’s ideas easily accessible to the reader, and to explain his ideas in the context of ancient and medieval mathematics. The presentation is very close in spirit to the text of the Algebra, although I do not follow the order of the arguments in the text, labeled A-H in the preceding section. I rather intend to give a plausible reconstruction of how al-Ttisi found his results. In ancient terminology one could say that al-Tiisi’s Algebra is a synthesis and my reconstruction is a corresponding analysis. The text of the Algebra contains several indications of al-Tusi’s original line of thought (see the parts labeled F and G in Section 1, and also, for example, [T2,36,39,57]), and my reconstruction is consistent with these indications.

For sake of brevity and clarity I use some modern notation in the transcription of ancient and medieval concepts. I indicate the algebraical “cube,” “square,” and “root” as x3, x2, and x (or y3, y2, y, z3, z2, z), and I transcribe equations such as “a cube plus a number equals squares plus roots” as x3 + c = ax2 + bx; here a and b stand for the “number of squares” and the “number of roots,” respectively.

The second part of the Algebra contains very little of what we would call algebra, i.e., direct manipulation of algebraic equations (for an exception see [T2, S-91). Al-Ttisi immediately casts his equations in a geometrical form, and he works with the resulting geometrical expressions. Thus in the case of x3 + c = ax2


+ bx, he chooses on a straight line three segments BE = X, BC = a, and BA = -\Tb (the square root is necessary for reasons of homogeneity). Then c can be inter- preted as “the excess of BC times the square of BE and the square of AB times BE over the cube of BE.” I will transcribe this as c = BC . 3 2 + AB2. BE - BE3. I denote the points in the geometrical figures as much as possible in the way of the French translation in [T2].

Turning to al-Ttisi’s ideas, first consider Eq. (S), that is x3 + c = ax* + bx, to which al-Ttisi devotes the last 58 pages of Arabic text [T2, 70-1271.

Fix segments BC = a and AB = a, as in Fig. 1. Al-Ttisi discusses the three cases a = a, a > d, and a < fi separately. I omit the relatively easy case a = a [T2,70-761. First suppose a < d [T2, 104-1271. Al-Tusi is interested in the . relationship between x and c. Let x = BE as in Fig. 1. Al-Ttisi sometimes uses a technical term baqiya c;liZ’ BE (“the remainder for side BE”) [2] for the quantity BC l BE* + AB* * BE - BE3, and I therefore feel entitled to write this quantity as f(BE). Then (5) can be written as f(BE) = c.

Al-Tusi interprets AB* and BE* as real squares ABlYa!, EBKE, and the difference AB* - BE2 as a “gnomon” (Arabic: ‘alam) A&K&E as in Fig. 1, in the manner of Book II of Euclid’s Elements (see [Heath 1956 I, 370-3721). I write the squares and the gnomon as [Bcx], [BE], and [ECY], respectively. Then

f(BE) = BC . [BE] + BE . [ECY]. (10)

If D is a point between E and C, then similarly

f(BD) = BC . [B6] + BD . [&XI. (11)

Al-Tusi investigates the difference between f(BD) and f(BE), but he does not use zero or negative quantities. For the sake of brevity I will use the minus sign in the modern way; thus I use “a - b = c - d” to shorten expressions like “if a > b thenc>danda-b=c-d;ifa=bthenc=d;ifa<bthenc<dandb-a=d - c.”

B C FI I -

FIG. 1. BC = a < fi = BA.

HM 16 AL-TnSj ON POSITIVE ROOTS 75

Al-Tusi simplifies f(BD) - f(BE) by decomposing all squares and gnomons as far as possible. We have [BE] = [B6] + [Ss] and [&xl = [&I + [ECY]. Therefore by (10) and (11)

f(BD) - f(BE) = (BC - [BS] + BD . [Sa]) - (BC a [BE] + BE l [~a]) = BD . [&TX] - (BC . [&I + BE . [~a]) = BD . [&I - (BC . [&I + DE . [~a]) = CD a [&I - DE s [EC&

Al-Tusi calls CD - [&I the characteristic (khassa) off(BD) and DE . [~a] the characteristic off(BE) (see the index in [T2, 1591). Note that both characteristics depend on D and E.

Using [as] = DE . (BD + BE) we obtain

f(BD) - f(BE) = DE . (CD . (BD + BE) - [~a]). (12)

Similarly, if F is between C and D

f(BF) - f(BD) = FD . (CF . (BF + BD) - [&Y]). (13)

We now try to find D such thatf(BD) is maximal. Then by (12) and (13) D must be a point such that for all E between D and A

CD . (BD + BE) > [~a] (14)

and for all F between D and C

CF . (BF + BD) < [&Y]. (15)

Since CD * (BD + EB) > 2CD * DB, and [6a] > [~a], (14) is true if 2CD l DB 1 [h]. Since CF l (BF + BD) < CD . (BD + BD), (15) is true if 2CD . BD I [aa]. Therefore, if D is such that

2CD . BD = [h], (16)

then f(BD) is maximal. (Al-Tiisi shows that for D defined by (16) and for all relevant points P not between C and A also f(BP) < f(BD).)

Putting m = BD, (16) can be reduced to

m* = (3)m - BC + ($)AB*. (17)

Al-Trisi defines m algebraically by (17) and he then derives (16). The rest of his argument is based exclusively on (16) and the ideas of the present analysis.

I now investigate the possible relationships between al-Tiisi’s definition of D and the derivative. We havef’(m) = 3m2 - 2ma - b = 2m(m - a) - (b - m*) = 2 CD . DB - [6ar] = 0 (cf. (16)). However, for x = BE,f’(x) = 2CE . BE - [ECY], but this quantity does not occur in al-Ttisi’s argument. This means that al-Tusi does not find m by computing the derivative f’ and by putting f’(x) equal to zero. Therefore the concept of derivative is not implicit here.

To return to al-TM’s ideas, it is now clear that the original Eq. (5) has no solution if c > f(m) and one solution, namely x = m, if c = f(m).


NOW let c <f(m), write x1 = BE, and put y1 = DE, then y1 = x1 - m. We have CD * (BD + BE) - [ECY] = CD l DE + [&I, and therefore by (12) f(m) - c = f(BD) - f(BE) = DE * (CD * DE + [&I) = yl((m - a)yl + y1(2m + y,)) = yT(3m - a + yl). Therefore y = y1 is the (unique positive) root of y3 + y2(3m - a) = f(m) - c, that is (6).

Similarly, if we let x2 = BF, and put 22 = FD, then 22 = m - x2 andf(m) - c = FD l ([&I - CF * (BF + BD)) = 22 * (CD * FD + [$a]) = &3m - a - .Q), and therefore z = z2 is the unique positive root of z3 + f(m) - c = z2(3m - a), that is (9), such that z < m.

These are the essential ideas in the solution of (5). The parts labeled A, B, D2, F, and G in Section 1 are lengthy elaborations of these ideas (see [T2, 104-1271).

The preceding reasoning answers the question: for which c does a root x exist? Al-Ttisi also studies the similar question: For which x does c > 0 exist; i.e., what x can be roots of an equation of type (5) for fixed a and b? Such x should satisfy c = f(x) > 0, that is to say x2 > ax + b. The further details (in parts C and E in Section 1) are mathematically trivial.

This concludes the discussion of the case a < b, so suppose BC = a > fi = AB, as in [T2, 76-1041, and let the notation be as in Fig. 2. Then

f(BD) = BC . BD* + AB* . BD - BD3 = BC . [B6] - BD a [cd], (18)

and by a similar reasoning as above

f(BD) - f(BE) = DE . ([cd] - EC a (BE + BD)) (19)

and

f(BF) - f(BD) = FD . ([a$] - DC . (BD + BF)). (20)

We now wish to find D such that f(BD) is maximal. Then for all E between D and C

[cd] > EC . (BE + BD) (21)

and for all F between D and A

[a+] < DC l (BD + BF). (22)

J BGAFDEC 0

O C Y

“4

%

-

E

FIG. 2. BC = a > V’& = AB.

HM 16 AL-TuSj ON POSITIVE ROOTS 77

First consider (21). The term [aa] does not depend on E. We now determine the maximum of EC a (BE + BD) for E a variable point between C and D. If we choose J on DB extended such that BJ = BD, then EC . (BE + BD) = EC * JE.

Suppose that the midpoint of segment JC lies between J and D. Then by Euclid, Elements II : 6 [Heath 1956 I, 3851 EC. JE < DC. JD = DC * 2DB. Therefore (21) holds for all E between C and D if

2DC . DB 5 [&I. (23)

Note that if 2DC . DB I [&I, then 2DC . DB < [B6], so that DC < BD/2, hence the midpoint of JC is in fact between D and B.

At first sight the analysis of (22) seems more complicated, because both terms increase monotonically ifF tends to D. The difficulty disappears if we guess (with (23) in mind) that D should also be defined by (16), that is, 2DC . DB = [&I, and if we then consider the differences [c&l - [a+] = [@I = (BD + BF) * FD and 2DC - DB - DC l (BD -I- BF) = DC. FD.

By (16), DC < BD < BD + BF, so that DC * FD < [@I, and (22) follows. Thus if D is defined by (16), f(BD) is maximal. Everything else is the same as in

the case a < V’%. This concludes my analysis of al-Tusi’s solution of Eq. (5). Al-Ttisi treats Eq. (2), that is x3 + c = bx, and (3), that is x3 + ax* + c = bx, in

the same way as x3 + c = ax2 + bx, case a < fi. For (2), C coincides with B in Fig. 1, and in (3), C is chosen on AB extended such that IBCl = a.

The treatment of Eq. (4), that is x3 + bx + c = ax2, resembles that of x3 + c = ax2 + bx, case a > fi. For (4), al-Tiisi draws a segment BA = fi perpendicular to BC (Fig. 3).

In (19) and (20) one obtains instead of gnomons [&I and [a+] quantities AB2 + BD2 and AB2 + BF2, respectively (which al-Tusi interprets geometrically as the squares of hypotenuses of right-angled triangles). Thus f(BD) > f(BE) and f(BD) > f(BF) are seen to be equivalent to

AB2 + BD2 > EC . (BE + BD) (24)

A P

J I3 IfD EC

-&

FIG. 3. x3 + bx + c = uxz. AB = ti. BC = u.


and

respectively.

AB* + BF* < DC . (BD + BF), (25)

The inequalities (24) and (25) can be investigated in similar ways as (21) and (22), leading to the result that f(BD) is maximal if D is such that AB* + BD* = 2DC s DB.

The equation (I), x3 + c = ax*, is treated in the same way as the case a > I6 of x3 + c = ax* + bx, with A coinciding with B in Fig. 2. For (l), al-Ttisi derives the quadratic equation q* + q(a - x1) = xl(a - x1), where x1 > m and x2 < m are the two positive roots and x2 = a - x1 + q (see Section 1, part Dl) in the following manner. Referring to Fig. 2, put x1 = BE, x2 = BF, a = BC. From

c = ad - xf = BE* . CE = ax: - xi = BF* * CF

we get, subtracting from BE* . CF,

BE* a EF = CF. [.$I,

hence

BE* = CF. (BE + BF) (26)

hence

BF . (BE + BF - CB) = BE. CE. (27)

In order to cast (27) in a nice geometrical form, al-Ttisi defines G on BC such that BG = CE. Then (27) can be written as

BF . GF = BE. CE. WV

If B, E, and C (and hence G) are known, the construction of F is a standard Euclidean problem: to apply to BG a rectangle, equal in area to BE . CE, and exceeding by a square (GF*). Or, in other words, GF* + BG . GF = BE . CE (this is the equation used in [T2, 71). The fact that al-Ttisi uses GF and not BF (in (27)) as the unknown shows that his method is basically geometrical.

The preceding summary contains the essence of the second part of the Algebra, with the exception of trivialities and the Ruffini-Horner process (see the next section). Al-Tiisi discusses each equation in such an elaborate way that his Alge- bra resembles the Cutting-off of a Ratio of Apollonius of Perga. Unlike Apollo- nius, al-Tiisi sometimes makes his proofs more complicated than necessary by introducing useless proportions. Suter also noted complications of this kind in another text of al-Ttisi [Suter 1907-19081. My analysis does not take account of such complicating factors.

3. AL-TUSI’S INITIAL MOTIVATION

In the preceding section we have seen that certain identities for a cubic polyno- mialf, such asf(BD) -f(BE) = DE l (CD . (BD + BE) - [~a]) (that is (12)), play a

HM 16 AL-Y@SI ON POSITIVE ROOTS 79

cardinal role in the reasoning of al-TM. Clearly al-Ttisi discovered many of the results in the Algebra, such as (16) and (17), after he had found identities such as (12). Thus one wonders for what reasons al-J&i initially studied (12).

A possible reason may have been his search for geometrical proofs of numerical algorithms for the approximation of roots of cubic equations. A proof of this kind appears in [T2, 15-181, in connection with the approximation of the smallest positive root of Eq. (l), that is x3 + c = ax*.

The algorithm is essentially the method of Ruffini-Homer (see [Luckey 19481). This method was used for the computation of cube roots before the middle of the third century A.D. in China [Wang and Needham 1955; Vogel 1968, 41-42, 113- 1191 and in the 10th century A.D. in the Islamic world [Ktishyar 1965,26-28, lOO- 1041. The extraction of cube roots was apparently well known in the time of al- Ttisi, who does not even bother to explain the details [Tl , 241. The generalization to arbitrary cubic equations is straightforward (see [Luckey 1948, 220-221, 229- 2301) and may have been used in the early I lth century A.D. by al-Biriini for the computation of the roots of x3 + 1 = 3x and x3 = 1 + 3x [Schoy 1927, 19, 211. In the first part of the Algebra, al-Ttisi describes the generalized algorithm for all cubic equations of the form x3 + t-ax + sbx = c with r and s equal to - 1, 0, or 1, not both zero. In these cases al-Ttisi adds numerical examples and a verbal explanation of why the algorithm is correct. It seems that he felt more uncertain about (l), that is x3 + c = ax*, perhaps because a (positive) root does not always exist. This may have prompted him to develop the geometrical proof in [T2, 13-151, which will now be rendered in modern notation.

Suppose x0 is the smallest positive root of (1). (We assume c 5 (4/27)a3, so that x0 exists.) Let x0 = nl . lOk + n2 * IOk-* + . . . be the decimal expression, with n1 # 0. We can estimate k using x0 = m (see [T2, 151 and [l] below). We then find by trial and error xl = n1 * 10“ as the maximal number X = n . 10k such that n is an integer and aX* 5 X3 + c. We then compute the following quantities:

a’ = a - x1, a” = a’ - x1, al = a” - xl

6’ = xla’, bl = b’ + xla”,

cl = c - x,b’

(note that x0 = xi + y with y(bl + y(al - y)) = cl). We now find by trial and error yl = n2 * lok-’ as the maximum number Y = n .

lok-i such that n is an integer and Y(bl + Y(al - Y)) I cl. We then compute

ai = al - YI, a; = ai - yl, a2 = a;’ - yl

bi = bl + ylai, b2 = bi + yla’i,

c2 = cl - ylh

(note that y = yi + z with z(b2 + z (a2 - z)) = Q) and so on. With each step we find one further decimal of the root; the successive approximations of x0 are xl, x1 + ~1, etc.

80 JAN P. HOGENDIJK

B E ID A

HM 16

FIGURE 4

Al-Ttisi proves the correctness of this procedure in a somewhat obscure passage [T2, U-181, which we paraphrase as follows (Fig. 4). The algebraical notation a, b, c, xi, yi, z and the symbols K, K1 are mine. Let AB = a, BD = x0, BE = xi,ED=y.Thenc=ax$-xi= DA.BD2=DA*BE2+DA.(BD2-BE2)=EA. BE2 - ED a BE2 + DA * (2 BE . ED + ED2). Therefore

cl = c - (a - x,)x; = DA . BD2 - EA . BE2 = ED . K

with

K = DA . (2BE + ED) - BE2 (29) =2(DA+ED).BE-2ED*BE+DA*ED-BE2 = 2 EA a BE + DA . ED - BE2 - 2ED . BE = EA a BE + (EA s BE - BE2) + DA . ED - 2BE. ED = EA . BE + (EA - BE) . BE + (EA - BE - BE - ED) . ED. (30)

Thus cl = y * K with K = bl + y(al - y) as desired. Similarly, let y1 = EZ, z = ID, x2 = x1 + y1 = BZ. The text is very concise, but the

underlying line of thought seems to be as follows ([T2, 17 line 20-18 line 61: We have in the algorithm c2 = cl - y1 * (bl + yl(al - yi)), or geometrically c2 =

cl - EZ . K1 with K1 = EA . BE + (EA - BE) . BE + (EA - BE - BE - EZ) . EZ (cf. (30)). Hence, as above, c2 = cl - EZ . (IA . (2BE + EZ) - BE2) (cf. (29)). Thus c2 = cl + EZ * BE2 - IA . (2EZ. BE + EZ2), as stated in the text. Therefore c2 = cl +EA.BE2-ZA.BZ2=c - IA * BZ2 = c - &a - x2). It is also easily verified that b2 = x2(a - x2) + xz(a - 2x2) and a2 = a - 3x2.

We can now apply the proof of (30) to a2, b2, c2, x2, z instead of al, bl , cl, x1, y. It follows that

c2 = z(b2 + z(a2 - z))

as desired. Differences such as DA . BD2 - EA l BE2 play an important role in this proof

(cf. (29) and (30), or [T2, 16 line 5-17 line 19 (Arabic), 16 line 3-17 line 21 (French)]). Hence it is conceivable that al-Ttisi first studied the differencesf(BD) - f(BE) while he was searching for this proof, and possibly for similar proofs for Eqs. (2)-(5). In the beginning he may not have known that the roots of (2)-(5) can be found by solving (1) and x3 + ax2 = c. Anyhow, it would be natural for al-Ttisi to begin with (l), because the necessary and sufficient condition c 5 (4/27)a3 for the existence of a root was known in his time. This condition had been derived geometrically by Archimedes, and it had been stated algebraically in the 10th century (see [Woepcke 1851,96-1031 = [Woepcke 1986 I, 168-1751). Note that it was important for al-Ttisi, who did not work with negative numbers, that the

HM 16 AL-TUSI ON POSITIVE ROOTS 81

quantities al = a - 3x1, a2 = a - 3x2, etc., in the algorithm are all positive. This is only true if x o I ($)a. For x0 > (Qa, one can use Fig. 4 for BD = (@a, BE = x0, ED = y to obtain y3 + [(&)a3 - c] = ay2 using methods which are even simpler than the proof of (30). Because y 5 (+)a one can now use the algorithm to compute y. Hence al-Ttisi may well have discovered the substitution y = (@a - x (22 = m - x2 in the notation of Section 2) in connection with his investigation of the proof of the algorithm for Eq. (1).

In conclusion, it seems to me that the Algebra of al-Ttisi can be explained as the result of a project that started with a more modest aim, namely the search for geometrical proofs of algorithms for approximating the roots of cubic equations. I believe that I have shown that al-Tiisi’s motivation and ideas can be explained without the assumption that he drew cubic curves and determined their local maxima and minima by means of the method of P. de Fermat. And as we have seen in Section 2, there is no evidence that al-Tfisi used the derivative. The absence of traces of these concepts does not detract from the intrinsic value of al- Tosi’s work. On the contrary, al-Ttisi’s ingenuity appears very clearly when one realizes that he used only traditional ancient and medieval mathematical methods.

4. NOTES TO THE TEXT AND TRANSLATION OF THE ALGEBRA

The following notes are intended for the reader who wishes to study the original text or the translation of the second part of al-Tiisi’s Algebra, which has been analyzed in Sections 2 and 3 of this paper. I wish to stress here that the edition and translation in [T2] are in my opinion very good, and that my notes on details do not imply a qualification of this general judgment. This section contains notes to the Arabic text, followed by corresponding notes to the translation (not all notes to the text entail a change in the translation). A notation such as 98:2 refers to line 2 of page 98 of the Arabic text or the translation. In the transcription of the Arabic text I conform to the conventions in [T2]; thus letters denoting points in the geometrical figures are transcribed according to the system used in [T2] (therefore jim = C, ztiy = G, {a’ = I), and angular brackets contain editorial additions to the Arabic text in the manuscripts. I also put the French translation of these words in angular brackets, even though such brackets do not appear in the translation.

Notes to the Arabic Text

1. 15: 11 delete (murabba’). 2. 16:16 and 16:17 for BE read (murabba’) BE. 3. 17:22-18:l (AI, wa-darabna EZ fi): In view of the singular mablagh on 18:l

one should add here something like (Al, wa-naqasnti al-mablagh min al-‘adad, wa- darabnti El fi).

4. 30:5 for illa m&Ian read wa-ill; malan, and 30:7 for illti kacban read wa-ills kacban, as in the mss. (cf. the apparatus); illa functions as the minus sign. Com- pare 69: l-2 (wa-illa amwalan), 102:21-103: 1 (wa-illa ka’ban).

5. 35: 11 delete (murabba’).


6. 38: 11 for EM read CM. The reading in the footnote to 38: 12 is preferable to the text in 38:12. The mathematical context requires that ka-dhalika in 38:13 be emended, for example to wa-dhalika.

7. 40:6 delete (ma’him). 8. 40: 17 for wa-(huwa) mithl di’f read wa-di’f, the word mithl in the manuscript

should be deleted from the text and put in the apparatus, because it is a scribal error.

9. 46: 1, 2 for BC read MC. 10. 49: 16 delete (wa-qutruha AB); the words ‘ala AB indicate that AB is the

diameter. 11. 5 1: 11-12 for fa-la yu’radu . . . Ii-1-istihala read: fa-la ya’ridu . . . . al-istihala

(al-istihala and li-1-istihala are indistinguishable in the London manuscript). Delete the footnote to 5 1: 11.

12. On p. 64 interchange ya and sad in the figure. 13. 67:3 for (fi BE) read (fi EG). 14. 73: 16 if DA is emended to BA, the additions (wa-huwa musawin li-murabba’

AB) and (DK wa-huwa) can be omitted. 15. 74:16 the emendation must be incorrect, because the quantity in question

does not in fact have a (positive) lower bound, as al-j&i proves in the subsequent passage (75: l-5). Perhaps li-bayan should be emended not to Ii-1-bayan (lahu), but to laysa lahu (the final ruin in the manuscript being a trace of lahu).

16. 77:5 for BG read AB as in the mss. (see the apparatus). 17. 78:7 delete (wa-huwa), and for wa-huwa read huwa. 18. 79: 10 note that [ma’a] is evidently a trace of (murabba’) in 79: 11. 19.84:21 fa-darb: the fa- makes no sense here, and the text is much clearer if we

emend wa-(huwa) darb; this takes care of the difficulty mentioned in the footnote to 85:l. In 85:3 delete (huwa) and for bi-muka’ ‘ab read mukac ‘ab.

20. 85: 10 emend CO to DJ, delete (madruban fi OM), for li-kawn read lakin as in the mss. (see the apparatus). Note CM = DJ.

21.94:5-6 delete (BE . . . Ii-#I’), instead of the footnotes to 94:6 and 94:6-9 put: 94:6-9 BD ‘ala muka’ ‘abihi . . . dil’: naqisa L.

22. 98:3 al-awwal: there is no need for this emendation, read al-th%ni as in the mss. (see the apparatus).

23. 98:17 delete (wa I-ashya’ wa I-mal) (the gnomon is AE(EB + BA)), 98:19 delete (ziyada).

24. 108:12 for DC read GC. 25. 109: 1 for BD read BG. 26. 116: 11 delete (DB maca BE), 116: 12 for DC read EC as in the mss. (see the

apparatus), 116: 15 for DC read EM (cf. the apparatus), 116: 16 for DC read EC as in the mss. (see the apparatus). In 116: 14-16 note DC + EK = DC + EM - MK = EM, because KM = DC (cf. 110: 12). The emendation (alladhi) in 116: 13 is mathematically correct, but (wa-murabba’ DE fi EK) is perhaps more plausible from a paleographical point of view.

HM 16 AL-‘J’uSi ON POSITIVE ROOTS 83

Notes to the French Translation 1. For 15: 12 et il peut-15: 15 infe’rieur I suggest the following alternative transla-

tion et il peut convenir qu’il n’ait pas d’ecart (pour le premier chiffre), et que l’ecart ait seulement lieu pour les autres chiffres cherchees (du quotient). Le premier chiffre de ce quotient (par AB) sera done le (premier chiffre) exact (du quotient par AD) ou un nombre voisin qui lui est inferieur. The following il refers to the premier chiffre de ce quotient (par AB). Footnote 47 on page 15 and footnote 1 on p. xix of the commentary are misleading. In 15:16 for le car& du nombre read le nombre.

2. 16:22 and 16:23 for BE read le carve de BE. On p. 17 footnote 50 read petit for grand; thus the translation in 17:24 is correct. 3. 17128 for (AZ, et multiplie’ AZ) read (AZ, et soustrait ce produit du nombre et

que nous ayons multiplie EZ par). 18: 1 for ces produits read ce produit. 5. 35:15 delete (le carre). 6. 38:15-16 for (par EM) read (par CM). For 38: 17 le reste sera done (la

difference du) premier (solide) et du deuxieme. Aussi puisque read: le premier reste sera done plus grand que l’autre (reste). Car puisque.

7. 4019 for le nombre des cart-e’s est (connu) read est le nombre des car&s. 8. The translation 40:23 corresponds to the text as I have corrected it. 9. 46:3 for BC read MC (MC in 46: 1 is correct). 10. 49:21 for nous . . . et read nous construisons sur AB un demi-cercle de

centre G, et. 11. 51: 17 for le probleme read tel probleme, the reference is to the quadratic

equation in 51: 16. Footnote 59 is misleading. 12. On p. 64 interchange J and U in Fig. 59. 13. 67:5 for BE read EG. 14. 73: 20-22 for le car& de DA . . . serait read le car& de BA ou (un quantite)

plus grand que lui par trois fois AB ou (un quantite) plus grand que lui, serait. 15. 74: 19 the translation is based on an emendation which must be incorrect,

because the nombre cherchee does not in fact have a positive limite en petitesse, as is proved subsequently in 75: l-7.

16. 77~5 delete alors, 7716 for BC seraient read AB sont. 19. 84123 for par CD. Le produit read par CD, c’est-d-dire le produit. 20. 85: 11 for par CO, (multiplie par OM,) du fait que read par DJ, mais. Note

CM = DJ. 21. 94:6-7 delete (BE . . . c&e). 94:8 delete ce qui. 22. 9814 for premier read deuxieme. 23. 98:24 delete (les chases et le carre) (the gnomon is AE(EB + BA)), 98:27 for

qui reste de l’augmentation du cube read qui reste du cube. 24. 108:16 for DC read GC. 25. 109:2 for BD read BG. 26. 116: 13 delete (DB plus BE), 116: 14 for DC read EC, 116: 17 for Mais read

Done, 116: 18 for DC read EM, 116: 19 for DC read EC.


ACKNOWLEDGMENTS I am grateful to Professors H. J. M. Bos (Utrecht), E. S. Kennedy (Princeton), F. Oort (Utrecht),

and especially to Professor J. L. Berggren (Vancouver) for their very helpful comments on an earlier version of this paper. Dr. Kh. Jaouiche (Paris) kindly translated the summary of this paper into French and Arabic.

NOTES

1. The following references are to the Arabic text of [T2]. The French translation has the same pagination as the Arabic text, but the line numbers may be different. A notation such as 3:8 refers to line 8 of page 3. (1) A 1: l-5:9, B 5: lo-6:18, Dl and G 7: l-8:2 and 10:6-18:22, F 8:3-10:5. (2) A 19: l- 23:14, B 27:1-28:20, D2 23:15-26:13, F 29:1-30:16, G 31:1-32:4, H 32%34:2. (3) A 34:3-40:4, B 40:5-41:22, D2 42: l-43: 19, F 44: l-45: 10, G 45: 1 l-46: 14, H 47: l-48:20. (4) A 49:1-58:4, B 58:4-60:7, D2 60:8-62:2, C and E 63:1-66:7, F 66:8-67:19, G 68:1-69:9, H 69:10-70:13. (5) case a = a, A 70: 17-72:9, B 72:9-73:11, C 73: 12-73: 19, D2 73:20-74: 15, E 74:16-75:6, F 75:7-75: 15, G 75:16-76:3, H 76:4-76: 15. (5) case a > a, A 76: 16-84:4, B 84:5-89:8, C 89:9-90: 12, D2 90:13-95:15, F 95:16-99: 13, G99:14-103:16, H 103:17-104:15. (5) case a < a, A 104:16-llO:lO, B llO:lO-114:12, C 114:13- 115:16, D2 116:1-119:18, F 119:19-123:11, G 123:12-126:21, H 127:l -127:17.

2. Compare [T2,41 lines 11, 15-20; 43 lines 11, 15; 65 line 61; al-Ttisi also uses variant expressions such as “the remainder which is together with BE” (al-baqiya alladhi maCa BE) on [T2, 52 line 121.

REFERENCES

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Aleppo: Institute for the History of Arabic Science. Ktishyar ibn Labban. 1965. Principles of Hindu reckoning (Kitiibfi- U;til I;iiscib al-Z$nd), translated

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Paris: Les belles lettres. (The two volumes are indicated as Tl and T2 in this paper.) Vogel, K. 1968. Chin Chang Suan Shu. Neun Biicher arithmetischer Technik. Braunschweig: Vieweg

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Wang, L., & Needham, J. 1955. Homer’s method in Chinese mathematics: Its origin in the root- extraction procedures of the Han-Dynasty. T’oung Puo 43, 345-401.

Woepcke, F. 1851. L’algPbre d’Omar Alkhuyyumi. Paris: Duprat (reprinted in [Woepcke 1986 I]). 1986. Contributions ci l’ktude des muthkmutiques et ustronomie urubo-islumiques. 2 ~01s.

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