Nicole Oresme and His De Proportionibus ProportionumAuthor(s): Edward GrantReviewed work(s):Source: Isis, Vol. 51, No. 3 (Sep., 1960), pp. 293-314Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/226509 .Accessed: 01/10/2012 18:30

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Nicole Oresme and his De proportionibus proportionum

By Edward Grant*

T HE mathematical representation of motion in the ancient and medieval periods was expressed in the language of proportionality. In recent years

considerable attention has been directed to one particular medieval version of proportionality as expounded by Thomas Bradwardine in his Tractatus de proportionibus' of 1328. In this work, Bradwardine related forces, resistances, and velocities in a very special way which gained widespread acceptance amongst scholastics in the fourteenth to sixteenth centuries, and has even been called Bradwardine's "function."2 Research in this area has, in general, em- phasized uniformity of interpretation and application of Bradwardine's "func- tion" - and with good reason - for it was important to show the tremendous impact this had upon medieval scholastic thought as well as to indicate how, in another important area, the scholastics interpreted Aristotle in the light of the many new problems thrust to the fore by inquiring minds grappling with traditional questions.

It is, however, important to recognize that Bradwardine's function gave rise to, and served as a point of departure for, varied mathematical-physical dis- cussions which came to be designated under the heading of proportio propor- tionum - a term which does not appear in Bradwardine's treatise.8

The possible originator of one significant departure may well have been Nicole Oresme, whose interesting ideas appear in his De proportionibus pro- portionum, composed around 136O.4 I should like to show that Oresme's work

* Indiana University. Though considerably augmented and rearranged, the substance of this article was delivered before the joint ses- sion of the History of Science Society and Section L of the American Association for the Advancement of Science on 29 December 1958 in Washington, D.C. The title of that paper was "Euclid, Bradwardine, and Oresme: math- ematical proportionality in the Middle Ages."

1 Edited and translated with analysis by H. Lamar Crosby, Jr., Thomas of Bradwardine His Tractatus de proportionibus (Madison, Wis., 1955).

2 Anneliese Maier, Die Vorlaufer Galileis im 14. Jahrhundert (Roma, 1949), p. 94.

3 The expression appears neither in the text or variant readings of Crosby's modern edi- tion, although Crosby, in at least two places, leaves the impression that Bradwardine used the term. In one place Crosby says "so long as velocities are taken (as they are by Brad- wardine) to vary according to the 'proportion

of proportions'..." (op. cit., p. 31), and then translates "proportiones potentiarum movent- ium ad potentias resistivas,..." as "the pro- portion of the proportions of motive to resistive powers..." (pp. 112, 113). Oresme used the expression in his title and throughout the first four chapters, but he was not the first to do so, since it appears in works by John Dumble- ton, John Buridan, and Albert of Saxony. Latin passages from both Dumbleton and Buri- dan containing the terms proportio proportion- um are given by Marshall Clagett, The Sci- ence of Mechanics in the Middle Ages (Madi- son, 1959), p. 441, n. 39, and p. 442, n. 40. Clagett observes that "it became conventional, at least from the time of Dumbleton and Buri- dan..., to express the Bradwardine formula by saying that velocity follows 'the proportion of proportions"' (p. 441, n. 39).

4 The treatise bears no date and 1360 is a conjecture based on the scanty evidence availa- ble. The argument is too involved for discus-

293

294 EDWARD GRANT

is not only an application and extension of Bradwardine's ideas in directions unthought of by the latter, but that this extension was the outcome of an attempt to furnish a special mathematical foundation for treating the kind of proportionality which Bradwardine had utilized approximately thirty years earlier. As will be seen, the special and unusual use which Oresme made of certain Euclidean definitions and propositions formed the core of the new foundation. Only the most important of Oresme's ideas in the De proportioni- bus can be considered in this article, but the author intends, in the near future, to publish an edition from all the known and available manuscripts accom- panied with a complete analysis of this important treatise.

Before proceeding to Oresme's contributions, it will be essential to explain what is meant by Bradwardine's "function," and this may best be achieved by considering, briefly, the background to its development. It should be noted that I render the Latin term proportio as "proportion" rather than "ratio" even though, in modern usage, the latter term accurately translates proportio whereas the term "proportion" means equality of ratios. But no such distinc- tion existed in medieval proportionality theory and, since ambiguities did not arise, it seemed reasonable to use "proportion" for all cases of proportio.

The entire problem arises from interpretations made by medieval thinkers of some scattered remarks by Aristotle relating motors, mobiles (or things moved), distances, and times,5 as well as Aristotle's statements on the motion of bodies in different media.6 Modern scholars usually represent Aristotle's verbalizations as F/R oc V, where F is the force or mover, R the resistance of a medium or mobile, and V is the velocity. In any case where motion arises it is understood that F must be greater than R. Whether or not this formulation is historically appropriate,7 there is no doubt that this was one of the interpreta- tions placed upon these passages by some medieval scholars who usually con- sidered only those cases where either F is varied and R held constant, or vice versa. It was this interpretation of Aristotle that Bradwardine discussed and refuted as the third of four erroneous opinions regarding proportions of mo- tion.8 This third erroneous opinion was, and remained, the most formidable opposition theory- to Bradwardine's "function," and this explains the consider- able amount of space devoted to its refutation.

Bradwardine directs two arguments against this position, but only the

sion here but the reader will find it in my Ph.D. thesis, The Mathematical Theory of Proportionality of Nicole Oresme (ca. 1320- 1382) (University of Wisconsin, 1957).

5 Physics, Bk. VII, ch. 5, 249b 30-250& 24. 6Physics, Bk. IV, ch. 8, 215a 24-215b 10. 7 Stephen Toulmin asserts that "so far from

formulating any mathematical function or equation relating force and velocity, Aristotle does not even employ any word for 'speed' or 'velocity,' and his word 'dynamis' can only very dubiously be rendered as 'force,'..." "Criticism in the History of Science: Newton on Absolute Space, Time, and Motion, I," Philosophical Review, 1959, 68: 1, n. 1.

8 Bradwardine devotes the whole of Chap-

ter II to the four erroneous opinions. Crosby, op. cit., pp. 87-111 and 32-38. The third opin- ion is found on pp. 95-105.

9 He first characterizes the two arguments: "Ista tamen positio est dupliciter arguenda: primo super insufficientia, secundo super men- dacio consequentis." The first argument then follows: "Est autem insufficiens quia non docet proportionem velocitatum in motibus nisi in quibus est idem motor vel aequalis, seu idem mobile vel aequale. De motibus autem ubi diversantur tam moventia quam mota, penitus nihil dicet." Crosby, op. cit., pp. 96-99. Ac- cording to this erroneous position, if FIR oc V, then F or R must be held constant while the other may vary. The solution by Bradwardine

NICOLE ORESME 295

second,10 and most important, need be treated here. Those who believe that FIR oc V are committed to the absurd position that any given force can move any resistance whatever, and consequently any force is potentially of infinite capacity. Thus if FIR oc V, and F>R, as required, then by continually doubling R we can make R>F. This is shown symbolically by F/nR Mc V/n, where n =

2,4,8,16,32 ... It is evident that when nR>F we still attain some velocity V/n which, for all scholastics, violates the assumption that no motion can arise when a resistance exceeds its mover. In similar fashion, we can represent by F:n/R c V/n the case in which R is held constant and F repeatedly halved so that when R >F:n some velocity, V/n, is still produced.

To avoid this absurdity, Bradwardine proposed his own solution which can be expressed as follows: F1/R, = (F/R),, where n = V1/V and F, R, and V represent the same physical quantities as before."1 Bradwardine remains faithful to the Aristotelian restriction that for motion to arise F must be greater than R. This particular representation of Bradwardine's "function" is an adequate, though anachronistic, rendering of the proportional relations em- bodied in the complex verbal expressions used by Bradwardine and later writers.

We note first that two proportions of force and resistance are related geo- metrically, or exponentially. But the exponent"2 itself is a proportion of velocities expressing an arithmetic relation between the velocities arising from the two proportions of force and resistance. For Bradwardine, to double a velocity arising from some proportion F/R it is necessary to square the pro- portion; to triple a velocity one must cube FIR; and generally to obtain n times any velocity the proportion F/R must be raised to the nth power, so that V1 = nV when Fd/R1 = (F/R)". Conversely, to halve a velocity we take the square root, of FIR; to find Y3 of a velocity arising from a given proportion F/R its cube root must be taken; and generally, to find the nth part of any initial velocity one takes the nth root of F/R so that V1 = V/n when F,R1 - (F/R ) 1/n 13

enables F and R to vary conjointly since the whole proportion varies when it is altered ex- ponentially.

10 substance of the second argument is as follows: "Est autem ista positio ex men- dacio arguenda, quia aliqua potentia motiva localiter potest movere aliquod mobile aliqua tarditate, et potest movere dupla tarditate. Ergo (per istam positionem) potest movere duplum mobile. Et potest movere quadrupla tarditate: igitur quadruplum mobile, et sic in infinitum. Igitur quaelibet potentia motiva localiter esset infinita.

"Similiter autem potest argui de quolibet mobili.... Igitur quodlibet mobile a quolibet motore potest moveri." Crosby, op. cit., p. 98.

11 Bradwardine's verbalization of the above modern formulation is as follows: "Proportio velocitatum in motibus sequitur proportionem potentiarum moventium ad potentias resistivas, et etiam econtrario. Vel sic sub aliis verbis, eadem sententia remanente: Proportiones po-

tentiarum moventium ad potentias resistivas, et velocitates in motibus, eodem ordine propor- tionales existunt, et similiter econtrario. Et hoc de geometrica proportionalitate intelligas." Crosby, op. cit., p. 112.

12 No special term is used by Bradwardine or later medieval writers to express an equiv- alent for the exponent. Oresme considers it as a proportion of numbers relating two propor- tions.

1I Oresme, though not Bradwardine, ex- presses all these relations and their converse with the statement that ". . . ex proportione velocitatum arguitur proportio proportionum et iste processus est a-posteriori; quando, vero, ex proportione proportionum arguitur proportio velocitatum tunc proceditur a causa et a priori." Cambridge, Peterhouse 277, Bibliotheca Pep- ysiana 2329, fol. 108r, c. 2. See also pp. 308-9 and note 58. Oresme appears to understand by the phrase proportio proportionum the entire expression F1/R1 = (FIR)", where n is a

296 EDWARD GRANT

It can now be seen why Bradwardine's "function" avoids and remedies the defects inherent in the repudiated theory. Given initially that F>R, it can never happen that R becomes equal to or greater than F. This is obvious, be- cause to repeatedly halve a velocity arising from a given proportion F/R is simply to take (FIR) /,, where n = 2,4,8,16,32,) .. In this way Bradwardine remains faithful to contemporary Aristotelian physics by still maintaining that some relation of force and resistance (where F>R) determines a velocity, but avoids the mathematical difficulties.

It is, however, important to realize that Bradwardine applied his "function" to conclusions involving only the simplest cases where V1 = 2 V and V1 = V/2. This may be partially explicable on the grounds that Bradwardine was arguing against the adherents of the third erroneous opinion, who, like Aristotle, con- sidered instances of doubling and halving velocities. Hence Bradwardine was anxious to show that almost all instances of doubling or halving velocities by the erroneous method would not yield, for identical data, the same results according to his "function." We see this by considering the three cases," Vl>2V, V1 = 2V, V1<2V. If F1 = 2F and R, = R, we find that the rival theories agree in one instance (case a) and disagree in the others (b,c):

Third Erroneous Theory Bradwardine's Function When F/R = 2/1

a) F1/R1 = 2F/R a) F1/R1 = (F/R)2 so that V1 = 2V so that V1 = 2V

When F/R > 2/1 b) Fl/R1 = 2F/R b) F1/R1<(F/R)2

so that V1 = 2V so that V1 < 2V When F/R < 2/1

c) F1/R1 = 2F/R c) F1/R1 >(F/R)2 so that V1 = 2V so that V1 > 2V

A similar arrangement could be made for the three cases V1>V/2, V1 V/2, V1 < V/2.16

The origin of Bradwardine's "function" seems to be the application of geo- metric proportionality, as found in Euclid and works based on Euclid,"6 to proportions of force and resistance which produce velocities. But in the Tractatus de proportionibus, Bradwardine shows no awareness of any further mathematical or physical applications of his function and for this we turn to Nicole Oresme's De proportionibus proportionum.17

proportion of numbers, or an exponent, relating the two proportions. But proportion n, or the exponent n, is also a proportion of velocities V,/V which varies as, or depends on, the "proportion of proportions."

14 The three cases correspond to conclusions 2, 4, and 6 of Chapter III. Crosby, op. cit., p. 112. On some of the difficulties connected with Bradwardine's function see Maier, op. cit., pp. 93-94, n. 23.

15 Conclusions 3, 5, and 7 of Chapter III in Crosby, op. cit., p. 112.

16 See suppositions 2 and 3 and conclusions

1 and 2 in Crosby, op. cit., pp. 76-78. 17 The De proportionibus proportionum has

at least four genuinely known chapters, and may have as many as six. The first three develop the mathematical theory of "propor- tions of proportions" and these I have edited (from manuscript sources) and analyzed in my Ph.D. thesis, The Mathematical Theory of Proportionality of Nicole Oresme. The follow- ing manuscripts were used to establish the edition: (1) Paris, Bibliotheque Nationale, fonds latin, 7371, fols. 269r-278v; (2) Paris, Bibliotheque Nationale, fonds latin, 16621, fols.

NICOLE ORESME 297

By the time Oresme wrote his treatise, Bradwardine's ideas were so well known and influential that it was possible to begin in medias res. Where Brad- wardine had elaborately refuted four "erroneous" theories on proportions of velocities, Oresme considers only the principle opposition theory and mentions only one other without discussion.'8 He also accepts many definitions and ex- planations from other authors, a number of which were discussed in consider- able detail by Bradwardine. For Oresme, the study of "proportions of pro- portions" was valuable not only as a means of more consistently representing proportions of motion, but was a potent tool for understanding the difficulties and secrets of philosophy,1" which meant for Oresme the secrets of the uni- verse. That utterances to this effect were not mere rhetoric is attested by the widespread application of the mathematical conclusions derived in the first three chapters to the physical problems in the latter part of the treatise.

The content and approach of Oresme's De proportionibus differs markedly from Bradwardine's treatise by virtue of the special interpretation given to the term pars which, in turn, determined the way he was to use that term, as well as the terms commensurabdis and multiplex. In what follows, Oresme is

94r-110v; (3) Erfurt, Amplonius Q.385, fols. 67r-82v; (4) Cambridge, Peterhouse 277, Bibli- otheca Pepysiana 2329, fols. 93v-110v; (5) Venice, Marciana, L.VI, 133, fols. 50r-65r. All quotations from the first three chapters are from my edition which shall be cited as MTPO. The fourth chapter applies "propor- tions of proportions" to local motion. All cita- tions from Chapter Four will be from the Cambridge manuscript and will be cited here- after as Pepys 2329. Where necessary, emen- dations, placed in brackets, will be made from the Paris Ed. (q.v. infra).

Although Oresme, in his Prohemium, lists six chapter headings (MTPO, p. 142), the fifth and sixth, which are supposed to treat of "velocities of motion" and the "incommen- surability of the celestial motions" respectively, present many puzzling problems. The fifth and sixth chapters appear as an integral part of the work only in the Venice manuscript and in the two editions cited below. These two chapters also circulated as a separate work bearing the incipit Ad pauca respicientes (or aspicientes), of which a number of manu- scripts are known (see MTPO p. 133). A detailed discussion of these two chapters will be given elsewhere.

There are at least two editions of the De proportionibus both with the title Tractatus proportionum, and both containing six chapters (they are practically identical in actual word order and marginalia):

(1) Questio de modalibus bassani politi ... Tractatus proportionum oren . . . (Venice, 1505). The volume is folio size with seventy- four leaves of which only leaves 2-16 are numbered. Oresme's treatise commences on the first unnumbered leaf, which is 17r, and ends on 26v; (2) Tractatus proportionum

Alberti de Saxonia, Tractatus proportionum Thome brIaduardini, Tractatus proportionum Nicholai horen (Paris n.d.). Sarton, in his Introduction to the History of Science, III (1948), 1496, has dated this around 1510, probably on the basis of information appear- ing on the inside cover of a copy of this edition at Houghton Library, Harvard Uni- versity. References to chapters five and six, or the

Ad pauca respicientes, as the case may be, will be from the Paris edition, to be cited as Paris Ed.

18 "Omnis rationalis opinio de velocitate motuum ponit eam sequi aliquam proportio- nem: hec quidem proportionem excessus po- tentie motoris ad resistentiam seu potentiam rei mote, alia vero proportionem resistentiarum manente eadem potentia vel equali vel pro- portionem potentiarum manente eadem resis- tentia vel equali, tertia proportionem potentie motoris ad resistentiam seu potentiam rei mote quam veram reputo et quam Aristoteles et Averroes tenuerunt" (Proemium, MTPO, p. 141). The first opinion, not mentioned again, seems to correspond to Bradwardine's second "erroneous opinion" (Crosby op. cit., pp. 35, 92-94). The second opinion is Bradwardine's third "erroneous opinion" (Crosby, op. cit., pp. 35-37, 95-105), and Oresme treats it at great length as the first conclusion of Chapter IV (Pepys 2329, fol. 106r c.2- 107r c.1). The third opinion is Bradwardine's "function."

19 "Ut igitur studiosi ad ulteriorem inquisi- tionem excitentur, utile est de proportione proportionum aliqua dicere quorum notitia non solum ad proportiones motuum sed etiam philosophie secreta et ardua negotia prestat inestimabile iuvamentum" (MTPO, p. 141).

298 EDWARD GRANT

dealing solely with geometric proportionality incorporated within the subject which he designates as proportio proportionum. Furthermore, he is concerned only with "proportions of proportions" of greater inequality, i.e., involving only proportions of the type A/B where A >B.2?

The term pars, says Oresme, is taken in two ways-properly and improp- erly.2' A proper part is an aliquot part to which the whole is multiple, as in Il/q where q is any integer. The term partes is simply an aliquot part taken a num- ber of times as in p/q where p> 1 and p and q are integers prime to each other. In this case p/q taken any integral number of times will not equal the whole.

An improper part is non-aliquot and however many times it be taken will not precisely constitute the whole. Oresme says that he will use the term pars in its proper signification.

Thus far we appear to have a mere repetition of customary definitions and distinctions but it soon becomes obvious that Oresme is actually using the term pars in a manner lying wholly outside the main Euclidean tradition, for he applies it exclusively to signify any unit proportion which is the common meas- ure, or base, of all greater proportions related to it in the same geometric series. We may summarize the way Oresme uses the term pars as follows:

If A and B are two proportions, and m and n are integers, then if A >B 1 ) proportion B is a part of proportion A when B . A""

and 2) A is commensurable and multiple to B, since A = Bn; 3) .B is parts of A when B = (A )tm1 where n>m> 1 and m and n

are in their lowest terms so that 4) A is commensurable to B, since they have a common measure

in the unit proportion (A ) /I'. However, A is not multiple to B since A o (B)".

Examples are plentiful to show that Oresme is plainly employing the term pars in an exponential, and not arithmetic, sense applying it to both rational and irrational proportions. This can be seen in Chapter I where Oresme im- agines seven possible ways of dividing a rational proportion. For the fifth way he says that any rational proportion can be divided into unequal irra- tional proportions each of which is a part or parts of the whole. He divides 4/1 into (4/1)1/4 (pars) and (4/1)3'4 (partes).22

20 "Sufficit igitur tantummodo investigare proportionem proportionum maioris inequali- tatis per quam haberi potest proportio propor- tionum minoris nec de proportionibus minoris inequalitatis quo ad hoc plura dicam" (MTPO, pp. 148-149). Oresme, like other earlier and contemporary mathematicians, was actually much puzzled by corresponding instances in- volving proportions of lesser inequality and was, therefore, anxious to avoid considering "proportions of proportions" of lesser inequal- ity. Side-stepping this problem was facilitated by the fact that proportions of motion were re- stricted to proportions of greater inequality since the force, F, had to be greater than the resistance, R. The major reason for the diffi- culty is given by A. Maier, op. cit., 91n.

21 Campanus of Novara, in his edition and

commentary on Euclid's Elements, gives essen- tially the same definitions when discussing the term pars (V, Def. 1). Euclidis Megarensis mathematici clarissimi Elementorum geomet- ricorum libri XV (Basilae, per Iohannem Her- vagium, 1546), p. 103. This edition will be cited hereafter as Euc.-Campanus. See also Sir Thomas L. Heath, The Thirteen Books of Euclid's Elements (N.Y., 1956), II, 115.

22 "Quinto per irrationales inequales quarum quelibet sit pars aut partes divise proportionis et quelibet dividi isto modo intelligendo sicut prius, ut quadrupla dividitur in proportionem dyametri ad costam, que est quarta pars eius, et in proportionem quadruple coste ad dyame- trum, que est tres quarte proportionis quad- ruple" (MTPO, p. 156).

NICOLE ORESME 299

In Conclusion 3, Chapter II, Oresme says that "if some quantity should be divided into two unequal quantities, either of which is part or parts of it, those two unequal quantities are related as two numbers in their lowest terms." 3 To aid in understanding this conclusion, Oresme assumes three suppositions. The first2 asserts that every quantity which is a part, or parts, of another may be assigned two numbers, a numerator and a denominator, which are prime to each other. Now this proportion of numbers must be interpreted in an expo- nential sense. For example, if we have some quantity A, and take three of its five equal parts, then A3ls is what Oresme intends, where 3 is the numerator and 5 the denominator of the exponent. The second25 says that if we have some quantity A, and take away some part, or parts, of it the remainder will also be a part or parts of the original quantity. Thus if we take away A3Is we have left A215 because A = A3'15A215. The third holds that any quantity divided in two, where one is a part or parts of the whole, will have the two parts re- lated as their numerators, just as A215 and A3Is are related as a proportion of numbers 2/3, i.e., as the. numerators, or conversely as 3/2. Obviously, Oresme does not mean that A215/A315 = 2/3, but rather that A2'5 is related to A3/5 as two exponential parts to three exponential parts, where each part is Alls. Or, simply that A215 = (A315)2 3, and conversely, that A315 = (A2/5)3/2.

The text of these suppositions given below in the notes reveals that Oresme is here dealing with purely numerical relations between the parts, or unit pro- portions of the quantities, and this explains why he cites Euclid VII, the first of the arithmetic books. That is, he is relating the numerical exponents them- selves where it is understood that each exponent represents one or more unit proportions. But when he refers to Euclid V he appears to be thinking only of the relations of the proportions considered as quantities or magnitudes in the widest signification, that is, embracing both rational and irrational quan- tities or proportions. Since Euclid's fifth book encompasses both rational and irrational magnitudes, citations from that book would be in order. In this way Oresme seems to be preserving a traditional distinction between books V and VII, geometric versus arithmetic treatment of magnitude. But, as we shall see, they are directly related where Oresme makes such statements as "pro- portio istarum proportionum erit sicut proportio istorum numerorum. Numer- orum vero proportionem per arismeticam investigas."27 The following para-

23 "Si aliqua quantitas in duo inequalia di- vidatur quorum quodlibet sit pars eius aut partes illa duo sunt sicut duo numeri minimi" (MTPO, p. 161).

24 "Prima est omnis quantitas que est al- terius pars vel partes duobus numeris signa- retur quorum unus dicitur numerator et alter denominator, ut patet ex commento [sexte septimi] et hii numeri, quidem, sunt contra se primi et in sua proportione minimi sicut dici- mus tres quinte. Unde si sunt minimi sunt primi per 22am septimi, et econverso per 23am eiusdem" (MTPO, pp. 161-162).

25 "Secunda est si ab aliqua quantitate de- matur aliquid quod sit pars aut partes eius

residuum erit similiter pars aut partes eius habens eandem denominatorem cum eo quod a principio demabatur. Verbi gratia, 6 de 10 est tres quinte, et residuum, scilicet 4, est due quinte, et hoc habeteur ex octava septimi" (MTPO, p. 162).

26 "Tertia est omnis quantitas in duo divisa quorum unum sit pars aut partes illa duo par- tialia sunt sicut numeri numeratores eorum. Unde proportio 2/5 ad 3/5 est sicut proportio duorum ad tria, et econverso. Hoc etiam satis habetur ex octava septimi et ex quinto Eu- clidis" (MTPO, p. 162).

2TMTPO, p. 174. Also see note 40 for the full context of the quotation.

300 EDWARD GRANT

graph will show how proportions of quantities on the one hand, and their ex- ponential, or numerical relations on the other, are linked by Oresme via Euclid X, 5 and 6.

Having briefly examined how Oresme uses the term pars, partes, commen- surabilis, and multiplex, it becomes evident that he has departed not only from Bradwardine but from Euclid as well, though whenever he cites Euclid in con- nection with the terms under consideration he does so without any qualifica- tion or comment, thereby implying customary usage.28 By citing the key defi- nitions and propositions from Euclid, and seeing how Oresme utilized them, we shall be in a better position to understand Oresme's approach. The defi- nition of commensurability given in Euclid X, Def. 1, reads: "Quantities are said to be commensurable which have a common quantity numbering them."29 When Oresme employs this definition we must understand him to implicitly substitute the phrase "proportions of quantities" for "quantities" and the term "proportion" for "quantity." It would then read "proportions of quantities [not simply quantities] are said to be commensurable which have a common proportion of quantities [instead of quantity] numbering them." This same substitution must be made implicitly in Euclid X, 5 which reads "any two commensurable quantities have to one another the proportion which a number has to a number."'0 Again substituting the phrase "proportion of quantities" for the term "quantities" this would read "any two commensurable propor- tions of quantities have to one another the proportion which a number has to a number." Indeed this difference can be graphically illustrated when both versions of Euclid X, 5 are symbolized as follows:

Euclid: Given two commensurable quantities, A and B, then A/B = m/n, where m and n are integers.

Oresme: Given two commensurable proportions of quantities, CID and E/F, then CID - (E/P)m/n, where m and n are integers.

Oresme has turned X, 5 into a "proportion of proportions." Euclid X, 6, the converse of X, 5, can be treated similarly. The term commensurabilis, as used in the De proportionibus proportionum, applies only to proportions or terms in a geometric series so that any given proportion, rational or irrational, is commensurable to all proportions in that same series but incommensurable to all those not in that series. This concept of commensurability leads Oresme to seemingly paradoxical expressions such as "an irrational proportion which is commensurable to a rational proportion." But, as we shall see, Oresme properly relates rational and irrational proportions within the special context of this approach.

By consistently interpreting and applying these terms, Oresme distinguishes three tvyes of proportions:31 (1) rational DroDortions: (2) irrational DroDor-

28 It cannot be determined to what extent, if at all, Oresme was aware that he was using Euclid in a "non-Euclidean" way.

29 "Quantitates quibus fuerit una quantitas communis eas numerans, dicentur communi- cantes." Euc.-Campanus, p. 243.

30 "Omnium duarum quantitatum communi- cantium est proportio, tanquam numeri ad numerum." Euc.-Campanus, p. 247.

31 "Sicut inveniebatur in primo capitulo tres sunt modi proportionum. Quedam enim sunt proportiones rationales, alie sunt irrationales habentes denominationes, hoc est rationalibus commensurabiles, et forsitan est tertius modus, scilicet proportiones irrationales que nullam habent denominationem eo quod non sunt com- mensurabiles rationalibus" (MTPO, p. 195).

NICOLE ORESME 301

tions which have denominations, and are therefore commensurable to rational proportions; (3) irrational proportions which have no denominations, and are consequently incommensurable to at least one and possibly more, or even pos- sibly all, rational proportions. Oresme argues for the probable existence of this class in a later section. (See below, pp. 302-304.)

This classification is crucial to the significant portions of the work and it will be worthwhile to examine in some detail just what Oresme may have understood by each of these classes.

(1) Rational proportions are always immediately denominated by some num- ber or numbers. If, for example, we have a proportion of commensurable quantities, A/B, then A/B = n, where n is some integer or ratio of in- tegers.82

(2) Irrational proportions which, though incapable of immediate denomina- tion by any number or numbers, can be mediately denominated by some number. This is possible in all cases where the irrational proportion is a part or parts of some rational proportion, and this is equivalent to as- serting the commensurability of the irrational and rational proportions.88

If, for example, (A/B)p/q is an irrational proportion, where A/B is a rational proportion of quantities, and p and q are integers where p<q, then Oresme would maintain that the irrational proportion, (A/B) /q, is immediately de- nominated by the rational proportion A/B. But we have already seen in (1) that every rational proportion of quantities is immediately denominated by some number n, so that A/B = n. Hence n may be said to mediately denomi- nate the given irrational proportion by way of the rational proportion A/B. That is, A/B mediates between the irrational proportion, (A/B)I'/, on the one hand, and the number n on the other. It is evident from all this that every irrational proportion of the form (A/B)pq/ is immediately denominated by the same rational proportion, namely A/B, and therefore by the same mediate number. In such cases, the only way to distinguish one irrational proportion from another is by specifying the proportion of numbers, or the exponent, p/q, which denotes what part or parts the irrational proportion is of the rational.34

82 "Omnis proportio rationalis immediate de- nominatur ab aliquo numero, aut cum fractione aut fractionibus aut sine fractione..." (MTPO, p. 151).

88 "Proportio, vero, irrationalis dicitur me- diate denominari ab aliquo numero quando ipsa est pars aliquota aut partes alicuius propor- tionis rationalis, aut quando est commensura- bilis alicui rationali, quod est idem, sicut pro- portio dyametri ad costam est medietas duple proportionis" (MTPO, p. 151). Oresnim's ex- ample asserts that the proportion of the diag- onal of a square to the side (which is taken as 1) of the square is (2/1)1/2 (medietas duple proportionis).

8"In Chapter Four, Oresme expresses the ideas in this paragraph as follows: ".. . omnis irrationalis proportio cuius denominatio scita est denominatur a proportione rationali. Aut,

ergo, denominatur a maiori rationali quam ipsa irrationalis, aut a minori. Si a maiori tunc illa irrationalis dicitur esse pars illius rationalis sicut '/2, %4, vel Y4, et cetera; aut est partes illius sicut Y3, vel Y4, et cetera. Et est unus numerus numerator, et alter denominator harum partium..." Pepys 2329, fol. 109r, c.l.

By citing similar passages from Bradwar- dine's Tractatus de proportionibus and Albert of Saxony's Tractatus proportionum it will be evident in what way Oresme's concept of de- nominating an irrational proportion is different.

Bradwardine: "Proportio est duarum quan- titatum eiusdem generis unius ad alteram habi- tudo. Et haec est duplex; nam rationalis, et in primo gradu proportionalitatis, est illa quae im- mediate denominatur ab aliquo numero: sicut proportio dupla, et tripla, et sic de aliis. Se- cundum vero gradum illa tenet quae irration-

302 EDWARD GRANT

(3) Irrational proportions which have no numerical denomination. Such a proportion is not an aliquot part or parts of the rational proportion which is supposed to denominate it. Thus if A/B is a double proportion, namely 2/1, then an irrational proportion of this type cannot be expressed in the form (2/1 )pM, where p and q are integers and p<q.

"Therefore, [such a proportion] will be some proportion which will be part35 of a double proportion but will not be half of a double [(2/1)1/2], and so forth; but it will be incommensurable to a double proportion and consequently, in- commensurable to any proportion commensurable to a double proportion.... And, by the same reason, some irrational proportion might be incommensura- ble to a double proportion, and also to a triple [(3/1)], and incommensurable to any proportion commensurable to a double and triple.... And there might be some irrational which would be incommensurable to any rational propor- tion. The reason for this seems to be that if there is some proportion which is incommensurable to two rationals, and some to three, and so forth, then there might be some proportion incommensurable to any rational whatever, though this does not follow from the form of arguing.... However, I do not know how to demonstrate this; but if the opposite should be true it is in- demonstrable and unknown.""6

In this passage, Oresme certainly seems to have arrived at the concept of an irrational exponent by pushing to the limit his use of the term pars and commensurabilis. Since all irrational proportions of the form (2/1)I'"l, where n is any integer, are commensurable to the rational proportion 2/1, Oresme

alis vocatur, quae non immediate denominatur ab aliquo numero, sed mediate tantum (quia immediate denominatur ab aliqua proportione, quae immediate denominatur a numero: sicut medietas duplae proportionis, quae est propor- tio diametri ad costam.. ." Crosby, op. cit., p. 66.

Albert of Saxony: "Proportio irrationalis est duarum quantitatum incommensurabilium adinvicem habitudo. Vel sic proportio irration- alis est que non potest immediate denominari ab aliquo numero sed immediate denominatur ab aliqua proportione que immediate denomi- natur ab aliquo numero sicut proportio que medietas duple nominatur qualis est proportio dyametri quadrati ad costam eiusdem." Trac- tatus proportionum, All c.l.

Though all agree that an irrational propor- tion is immediately denominated by a rational proportion, and mediately by some number, Oresme introduces the notion of part and com- mensurability which enables him, quite logi- cally, to take the next step and conceive of an irrational proportion which has no denomi- nation, that is, which is not a part or parts of some rational proportion and is, therefore, in- commensurable to it. (See below, under the third class of proportions.)

We note further that Oresme also denomi- nates irrational proportions by smaller ra-

tionals but these will not be considered since they are introduced only once in the De pro- portionibus.

35 Part, in this context, must be understood as improper. See above, p. 298. From a modern standpoint the term "improper part" would only be applicable to irrational proportions which could only be expressed with irrational exponents. But an irrational proportion such as medietas duple, (2/1)1/2, is a proper part of 2/1. The distinction depends upon whether the exponent is rational or irrational.

38 "Igitur erit aliqua proportio que erit pars duple et tamen non erit medietas; duple, nec tertia pars, nec quarta, nec due tertie, et cetera, sed erit incommensurabilis duple et, per conse- quens, cuicumque commensurabili ipsi duple.... Et iterum, pari ratione, aliqua poterit esse in- commensurabilis duple, et etiam triple, et cuili- bet commensurabili alicui istarum.... Et sic, forte, poterit, esse aliqua irrationalis que sit incommensurabilis cuilibet rationali. Nunc vi- detur ratio si aliqua est incommensurabilis duabus, et aliqua tribus, et sic ultra quoniam sit aliqua que sit incommensurabilis cuilibet licet non sequatur ex forma argumendi ... Istud, tamen, nescio demonstrare sed si op- positum sit verum est indemonstrabile et ig- notum" (MTPO, p. 152).

NICOLE ORESME 303

asks whether there may be some irrational proportion which is not any part whatever of 2/1. The proportion (2/1)1 V2 seems to represent what Oresme had in mind, since raising it to any integral nth power will never make it equal to 2/1. Hence it is not a part or parts of 2/1, and the two proportions can- not be related as a number to a number. Oresme, of course, was incapable of expressing this with the limited mathematical tools and language at hand, and was, therefore, reduced to negative statements about this category of propor- tions. This becomes more obvious when it is realized that Oresme, utilizing the concept of part, was able to make statements about, and even to manipu- late, irrational proportions which could be designated as exponential parts of some rational proportion. The next step was to ask whether there might be other irrational proportions which are not parts of rational proportions. As we shall see in the next paragraph, Oresme answered this in the affirmative, but having done so, he came by and large to a dead-end since the concept of part was no longer applicable and some new concept, not forthcoming, was necessary to make any further progress. Lack of adequate terminology, sym- bols, and rules prevented any effective handling of irrational exponents. How- ever, Oresme does say that where such proportions are involved it is enough if one can approximate to the unknowable proportion by determining, suffi- ciently closely, one greater and one lesser knowable proportion, which serve to locate the unknowable proportion."7

Though Oresme seems to have arrived at the concept of an irrational ex- ponent from a persistent logical application of the terms pars and commen- surabilis, what reason did he have for believing that any such proportions might really exist? Here Oresme draws upon an earlier supposition which we might call his "principle of mathematical plenitude." This appears as his fifth sup- position of Chapter I: "Another supposition, and let this be the fifth, is that any proportion is as a continuous quantity in this respect, that it is divisible into infinity just like a continuous quantity-that is, it is divisible into two equal [parts], into three, four, and so forth; it is also divisible into any what- ever unequal parts; it is divisible into commensurable parts, and similarly into parts which are incommensurable to it, and so forth in any whatever other way....".8. Indeed, Oresme cites this very supposition as the basis for his belief in the existence of irrational proportions which have no rational denomina- tions.89 Hence, his thought pattern seems to be as follows: any proportion is like a continuous quantity and therefore divisible in any conceivable mathe- matically logical manner; now one conceivable way is to divide a rational pro-

37 "Verumtamen de qualibet proportione nobis data poterimus investigare per secundam conclusionem [of Chapter 4] utrum ipsa sit maior vel minor tali proportioni irrationali incognobili et innominabili ttandem] poterimus invenire duas proportiones satis propinquas ad quas proportio ignota se habebit ita quod erit minore maior et maiore minor et hoc debet sufficere." Pepys 2329, fol. 108v, c.l.

38 "Alia est, et sit quinta, quod quelibet pro- portio est sicut quantitas continua in hoc, quod in infinitum est divisibilis sicut quantitas con- tinua et in 2 equalia, et in 3, et in 4, et cetera,

et per inequalia quomodolibet, et in partes com- mensurabiles. et similiter in partes sibi invicem incommensurabiles, et cetera et quolibet alio modo...." (MTPO, p. 151).

39 "Dico, ergo, quod non apparet verum quod omnis proportio irrationalis sit commensura- bilis alicui rationali, et ratio est quia omnis proportio est sicut quantitas continua quo ad divisionem ut patet per ultimam suppositionem. Ergo potest dividi in duo quorum quodlibet est incommensurabile toti . . ." (MTPO, p. 152). This passage immediately precedes the one in note 36.

304 EDWARD GRANT

portion into smaller proportions one of which is irrational and no part, or parts, of the given rational, and hence incommensurable to it; therefore such a con- ceivable proportion, logically possible, must correspond to some real category of proportions even though we cannot express a single instance of it.

Then, by extending the cases, one can conceive of irrational proportions in- commensurable to every rational proportion. A deductive proof of this was out of the question for Oresme because of the inductive form of the argument. All that could be done was to show, on the basis of his "principle of mathe- matical plenitude" (supposition five), that there is some irrational proportion which is no part whatever of some larger given rational proportion. But from that point his argument was inductive since the fifth supposition applies only to the division of some one given proportion. In order to extend the conclu- sion to two, or more, rational proportions we find Oresme resorting to the expression "pari ratione" (note 36) which he recognized as having only rea- sonable persuasiveness-and not logical force. For if such an irrational pro- portion is incommensurable to 2/1 and, therefore, to every rational proportion of the form (2/1)P where p is any integer, or improper fraction, then pari ratione it is reasonable to suppose there is such a proportion incommensurable to (2/1)P and (3/1)P, and so on inductively. Hence it is only reasonable to suppose one or more such irrational proportions which are incommensurable to every rational proportion-but this "non sequatur ex forma argumendi" (note 36).

We have now seen the three types of proportions which Oresme would dis- tinguish when considering proportions independently. But Oresme is inter- ested in "proportions of proportions" and therefore his task is to investigate the different ways in which any two proportions may be related. Any two proportions must be drawn either from any one of the three independent cate- gories or from any two of them. Oresme devotes most of his attention to the former, that is to "proportions of proportions" involving two proportions from only one category at a time, and we shall discuss only those.

All "proportions of proportions" are either rational or irrational. A "pro- portion of proportions" is said to be rational when the two proportions are commensurable, and relatable as a proportion of numbers, that is related ex- ponentially by integral or fractional exponents; or, as Oresme would also ex- press it, when the smaller proportion is a part or parts of the greater. But a "proportion of proportions" is irrational when the two proportions are incom- mensurable and hence unrelatable as a number to a number, or stated in positive terms, when they are relatable only by an irrational exponent, though Oresme, of course, never expresses it in this modern fashion. As already noted, the concept of an irrational exponent, though always expressed negatively, is a logical outgrowth of the special approach involving the notions of part and commensurable as confined to a "proportion of proportions."

In the class of rational "proportions of proportions" the three major types are the following:

1) A rational proportion which is commensurable to another rational pro- portion. An example is 8/1 and 2/1 which are related as a number to

NICOLE ORESME 305

a number, namely 3 to 1, since 8/1 = (2/1) 1/. 40 It is obvious that when Oresme relates two rational proportions he is not concerned with their immediate denominations, but rather in what he calls their "meeting or participating in means," or, as we would say, their common base.4'

2) An irrational proportion denominated by, and therefore commensurable to, some rational proportion, which is commensurable to another irra- tional proportion also denominated by, and commensurable to, some rational proportion. In other words, Oresme is relating two proportions from the second category. An example42 is (4/1)'1/ and (2/1) 1/2 which are related as a sexquitertia proportion, that is as 4/3, since (4/1)1/3 [(2/1) 1/2]4/3

3) An irrational proportion having no rational denomination which is com- mensurable to some other irrational proportion which also lacks any rational denomination.43 Oresme is here relating two proportions from the third category. In modern terms each of these proportions has an irrational exponent. We have already noted that Oresme had no means of expressing such proportions and therefore we take the liberty of offer- ing one in his behalf in which (8/1) V2 and (2/1) V2 are related as 3 to 1 because (8/1)V2 = [ (2/ 1 )V2]3/1.

Each of the above types of rational "proportions of proportions" has a

40 Oresme says: "Capiatur, igitur, numerus mediorum inter numeros proportionis minoris et addatur unitas, et consimiliter numerus mediorum inter numeros maioris et addatur unitas. Dico quod proportio istarum propor- tionum erit sicut proportio istorum numer- orum. Numerorum vero proportionem per aris- meticam investiges."

"... sint proportio octupla et dupla. Quia igitur inter numeros maioris, qui sunt 8 et 1, sunt duo numeri medii, scilicet 4 et 2, secundum proportionem minorem, huic numero mediorum addas unitatem et sunt tres. Dico quod maior continet ter minorem igitur est tripla ad eam..." (MTPO, pp. 174-175).

41 "Adverte quod propter brevitatem lo- quendi voco proportiones in mediis convenire seu participare quando inter primos numeros maioris est numerus medius seu numeri se- cundum proportionem minorem aut secundum aliquam aliam proportionem secundum quam inter primos numeros minoris sit etiam nu- meros seu numeri medii eodem modo quo dice- batur in probatione none secundi capituli" (MTPO, p. 183). Thus, in the example above, 8/1 contains the number 2, as a mean which is secundum proportionem mintorem, namely 2/1. But should 32/1 and 8/1 be the given proportions we see that 32/1 does not contain 8 as a mean number secundum proportionem minorem, that is 8/1. However, both 32/1 and 8/1 contain 2 as a mean number which is the denomination of a double proportion. There- fore both 32/1 and 8/1 "meet in the mean" 2,

and are related as a proportion of numbers 5/3 which is to be understood as (2/1)5 = [(2/1)315/3 or simply 32/1 = (8/1)5/3. The expression "in mediis convenire seu partici- pare" refers to some exponential base of the given proportions.

42 "Cum, enim, queris de tertia parte quad- ruple et de medietate duple et proportio quad- ruple sit dupla ad duplam, sicut patebit post, capias unum numerum, habentem tertiam, qui sit duplus ad aliquem alium numerum habentem medietatem vel duplam partem. Deinde accipe tertiam partem maioris et medietatem minoris et qualis erit proportio unius istarum partium ad alteram talis erit proportio proportionum predictarum et ita poterit in aliis operari.

"Verbi gratia, 12 est unus numerus habens tertiam duplus ad 6 qui habet medietatem. Est igitur 12 loco proportionis quadruple, et 6 loco duple. Qualis est itaque proportio 4, que est tertia pars 12, ad 3, que est medietas 6, talis erit proportio tertie partis quadruple ad medie- tatem duple, scilicet proportio sexquitertia et eodem modo in aliis est agendum" (MTPO, p. 155). If we have two commensurable pro- portions (A)1/P and (B)1/q, where (A)1/P > (B)1/q, A = (B)r and pJ q, and r are inte- gers, then Oresme's procedure is, in effect, to transform (B)l/q to (B)Pq/q and (A)1/P to (A)r(Pq)IP so that Al/P = (Bl/q)rq/P. The two proportions will then be related as the numbers rq to p.

43 See note 52.

306 EDWARD GRANT

counterpart in the class of irrational "proportions of proportions." All of the corresponding types given below involve irrational exponents which Oresme was compelled to express negatively by explaining that the smaller proportion is neither a part nor parts of the greater proportion and consequently they are incommensurable and unrelatable as a number to a number. Thus the con- cept of part as an aliquot part in the exponential sense must be utilized, al- though negatively, to get at the notion of irrational "proportions of propor- tions."

1) A rational proportion which is incommensurable to another rational pro- portion. For example," 9/1 and 2/1 since 9/1 s (2/1)plq where p/q is a proportion of integers and p>q. The same holds for any two rational proportions where A/B p (C/D)pIq where A/B, C/D, and p/q are ra- tional proportions.

2) An irrational proportion denominated by, and therefore commensurable to, some rational proportion which is incommensurable to, another ir- rational proportion also denominated by, and commensurable to, some rational proportion. Oresme is here relating two proportions from the second category of independent proportions (see p. 301). An example4" is (3/1) 1/4 and (2/1 )1/2 since (3/1) 1/4 [(2/1) 1/2]plq where p/q is a proportion of integers, or the exponent.

3) An irrational proportion having no rational denomination which is in- commensurable to some other irrational proportion also lacking any rational denomination." These two proportions are from the third cate- gory of independent proportions. Once again, Oresme can offer no ex- amples for this type but one fitting the specifications would be (8/1) V2

and (5/1) V2 which are unrelatable as a number to a number, that is they are unrelatable by any rational exponent p/q since (8/1 )V2 ;

[ (5/1) 'V2 ]plq.

The reader of the De proportionibus proportionum, upon arriving almost to the end of Chapter III, the last of the strictly mathematical chapters, is met by an unexpected burst of enthusiasm which serves as introduction to the tenth conclusion. Oresme informs the reader that the more deeply one reflects on this conclusion and its consequences, the more will one come to admire it.47

In that conclusion, Oresme shows the high degree of probability that any two proposed unknown proportions are incommensurable because if many un- known proportions are taken it would be very probable that some one of them, taken at random, would be incommensurable to some other one of them taken

44it ... sint nonacupla et dupla. Cum igitur

inter numeros maioris, qui sunt 9 et 1, sit medium secundum proportionem triplam, scili- cet 3, et non secundum proportionem minorem propositam, scilicet secundum duplam, dico quod maior non est multiplex ad minorem" (MTPO, p. 173).

45 i. si queratur de proportione inter me- dietatem duple proportionis et quartam partem triple dico quod si proportio dupla et tripla sint incommensurabiles, sicut est rei veritas et infra

patebit, similiter et quelibet partes aliquote earum sunt incommensurabiles" (MTPO, p. 154).

48 See note 52. 47 "Sed finaliter pono unam aliam conclu-

sionem que videtur sequi ex precedentibus cuius fructus non modicus, per dei gratiam, in sequentibus apparebit et tanto amplius admira- beris quanto circa eam et ea que ex ipsa se- quuntur profundius cogitabis" (MTPO, pp. 194-195).

NICOLE ORESME 307

at random."8 In other words, Oresme is saying that it is probable that any two such proportions would form an irrational, rather than a rational "proportion of proportions." The basic cases which he considers are identical to the three categories of rational and irrational "proportions of proportions" already dis- cussed.

By far the greatest amount of space is devoted to the first category, namely "proportions of proportions" involving two rational proportions."9 His method"0 is to take 100 rational proportions from 2/1 to 101/1, and by relating them two at a time show that we can have 4,950 possible "proportions of propor- tions." Of this total, only 25 are rational "proportions of proportions" con- stituted out of the following geometric series: (2/1)", where n = 1,2,3,4,5,6, thus producing 15 possible rational "proportions of proportions"; (3/1)" where n = 1,2,3,4, and yields 6 rational "proportions of proportions"; and one "pro- portion of proportions" from each of the following four proportions: (5/11)', (6/1)", (7/1)", and (10/1)" where in each case n = 1,2. The remaining 4,925 "proportions of proportions" are irrational. The ratio of irrational to rational "proportions of proportions" is thus 197 to 1. As more and more rational pro- portions are taken, the ratio of irrational to rational "proportions of propor- tions" becomes greater and greater. Therefore, says Oresme, if one were asked whether or not any two unknown rational proportions formed an irrational "proportion of proportions," the answer, based on probability, ought to be in the affirmative.5' The same argument, without specific examples, is applied by

48 "Propositis duabus proportionibus ignotis verisimile est eas incommensurabiles esse quod si multe proponantur ignote verisimilimum est aliquam alicui incommensurabilem fore" (MTPO, p. 195).

49 "Sint, igitur, due proportiones ignote. Aut igitur utraque est rationalis, scilicet de primo modo, et tunc arguitur sic: quibuscumque et quotlibet proportionibus rationalibus secundum unum ordinem denominationum aut secundum plures ordines multo pauciores sunt que sunt invicem commensurabiles et que sunt incom- mensurabiles multo plures. Igitur duabus earum ignotis propositis verisimile est eas incommen- surabiles esse" (MTPO, p. 195).

50 "Sumantur enim secundum ordinem su- arum denominationum 100 proportiones in genere multiplici sicut dupla, tripla, quadrupla, quintupla, et cetera, usque ad lOlam et sint sicut 100 termini ad invicem comparati. Tunc inter huius terminos seu proportiones compa- rando quemlibet cuilibet sunt 4950 proportiones que sunt proportiones proportionum et illarum 25 sunt rationales et non plures, et omnes alie sunt irrationales sicut postea declarabo. Et si plures proportiones rationales tamquam ter- mini sumerentur sicut 200 vel 300 et acciper- entur proportiones eorum adhuc esset propor- tio irrationalium ad rationales multo maior" (MTPO, pp. 195-196). The actual detailed working out of this part of the tenth conclu- sion, which is summarized above, is found in a conclusio practica which follows upon the

tenth conclusion (MTPO, pp. 199-201). 51Oresme illustrates this probability argu-

ment by analogy with cube and perfect num- bers. The more numbers one takes the greater does the ratio of non-cube numbers to cube numbers become. Hence if you are asked whether an unknown number is a cube or not it is safer to reply in the negative since this is more probable. This reasoning is applied by Oresme to rational and irrational "proportions of proportions." The text of the passage is as follows: "Videmus enim in numeris quod quibuscumque seu quotlibet per ordinem ac- ceptis numerus perfectorum seu cubicorum multo minor est numero aliorum et quanto plures capiuntur tanto maior est proportio non cubicorum ad cubicos, aut non perfectorum ad perfectos. Ideo si sit aliquis numerus de quo penitus ignoretur, quis est aut quantus, et utrum sit magnus vel parvus, ... erit veri- simile est quod talis numerus sit non cubicus, sicut est in ludis si peteretur de numero ab- scondito, utrum sit cubicus vel non, tutius est respondere quod non, cum hoc probabilius et verisimilius videatur.

"Modo sicut est de numeris quantum ad hoc, ita est de proportionibus proportionum ration- alium, sicut prius est ostensum, quia irration- ales sunt aliis multo plures ad sensum prius dictum" (MTPO, pp. 195-196).

The important question, from a modern standpoint, is whether Oresme intended to apply this probability argument only to finite

308 EDWARD GRANT

Oresme to "proportions of proportions" in the second and third categories52 already discussed, as well as to proportions taken from any two of the three categories of proportions.53

Why should Oresme have been so enthusiastic about this conclusion? The answer to this question is intelligible only when it is realized that almost all of the mathematical conclusions concerning "proportions of proportions" were meant to be applied to physical problems.5" This is hardly surprising since its originator, Bradwardine, formulated it in direct response to a mathematical- physical problem. The significance of this conclusion for Oresme is made evi- dent in Chapter IV of the De proportionibus where he applies it to physical problems. After applying "proportions of proportions" to problems of local motion, Oresme utilizes the first supposition55 of Chapter IV, which is essen- tially Bradwardine's "function," and assumes that a proportion of velocities varies as a "proportion of proportions." This supposition is also the basis for Oresme's assertion that "proportions of proportions" of forces and resistances

numbers of proportions, or to any number of proportions taken to infinity. In the former case, Oresme is correct, provided the terms in the sequence of proportions are properly cho- sen. Obviously it would not do to select any geometric series since they would form only rational "proportions of proportions." If, how- ever, the series of finite proportions is properly chosen, the ratio of irrational to rational "pro- portions of proportions" increases as the num- ber of proportions in the series increases. But if Oresme meant to extend the sequence of proportions to infinity, there would be as many rational as irrational "proportions of propor- tions" since a one-to-one correspondence could be established. If, for example, he meant to take the series 2/1, 3/1, 4/1, 5/1, etc., to in- finity, then for every irrational "proportion of proportions" formed from the sequence there can be a corresponding rational "proportion of proportions" between any two proportions in any subclass formed from any geometric series, say (2/1)", where n is the sequence of natural numbers. Since Oresme was not a fourteenth century Georg Cantor, it is highly unlikely that he was aware of this and if he avoided this pitfall it must have been for other reasons.

The difficulty lies in trying to determine what Oresme means by such expressions as "quibuscumque et quolibet proportionibus ra- tionalibus secundum unum ordinem denomina- tionum" (see note 49). Does he mean some definite number of proportions which may be increased by some finite amount, though not indefinitely? Or, does he mean a series of proportions taken to infinity? It should be noted in this connection that Oresme never uses the expression ad infinitum, et cetera, or et sic ultra, though these and equivalent ex- pressions are frequently used in Chapter I where sequences of proportions are clearly in- tended to be taken to infinity.

52 "Et si forte proportiones proposite essent de secundo modo, scilicet irrationales habentes denominationes et rationalibus commensura- biles, probabitur hoc idem" (MTPO, p. 197). A demonstration follows: "Et si forte essent de tertio modo proportionum, si sint alique tales ita quod nullam haberent denominationem adhuc estimandum esset, et verisimile est, quod ita sit de illis sicut de aliis quantum ad hoc, scilicet quod inter proportiones illarum pro- portionum rationales sunt rariores quam irra- tionales, ideo verisimile esset proportiones propositas incommensurabiles esse" (MTPO, pp. 197-198).

53 "Et si forte una esset de uno modo, et alia de alio, tunc arguitur sic: in quolibet modo per se sumpto proportiones commen- surabiles inter se sunt rariores aliis igitur similiter erit in totale multitudine proportion- um" (MTPO, p. 198).

54 In Chapter IV, after applying some math- ematical conclusions from Chapter III to pro- portions of motion, Oresme says: "... et ita iuxta quamlibet conclusionem de tertio capitulo una vel plures conclusiones de velocitatibus poterunt demonstrari quas, ut brevis transeam usque ad decimam eiusdem capituli iuxta quam eligitur talis conclusio propositis duabus veloci- tatibus quarum proportio sit ignota verisimile est earum proportionem irrationalem esse et illas velocitates incommensurabiles fore..." (Pepys 2329, fol. 109v, c. 2-110r, c. 1).

55 "Prima sit hec: velocitas sequitur pro- portionem [potentie motoris] ad mobile vel ad resistentiam eius. Unde proportio unius veloci- tatis ad alteram est sicut proportio propor- tionis potentie unius motoris ad suum mobile ad proportionem proportionis alterius motoris ad suum mobile. Ista suppositio patet per Aris- totelem secundo celi et per Commentatorem ibidem et quarto et septimo physicorum" (Pepys 2329, fol. 105v, c. 1-2).

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can be derived from a given proportion of velocities. For example, Oresme considers the case56 in which F1/R1 - F/R . R/R1 where F1- F and F,I/R1 (F/R) Vl/v. If R/R1 and V1V are known then the proportions F1/R1 and F/R can be determined. Similarly if R1 = R and F1/RR = F1/F . F/R, the two proportions FI/R1 and F/R can be determined if both F1/F and V1/V are known. If, however, we are not given the proportion of resistances or forces,;" but know V1/V and either Fl/R1 or F/R, then the other proportion can be de- termined so that all the proportions will be known in the formulation Fl/R1 - (FIR) vi/v. Indeed, Oresme notes in an important, though brief, passage58 that deriving a "proportion of proportions" of forces and resistances from a pro- portion of velocities is working a posteriori from effect to cause, and that deter- mining a proportion of velocities from a "proportion of proportions" is working from cause to effect and is a priori. He also notes at the commencement of that passage that a proportion of velocities can be derived from a proportion of distances or times, and generally from any such quantities. This passage serves to link the kinematic treatment of velocity with the dynamic, and incorporates both within the general subject of "proportions of proportions." This means that it is proper to think in terms of the formulation F/R, = (F/R)81 8 since V1/V = S1/S when T = T. We can also substitute T1/T, as exponent, since V1/V= T1/T when S1 S.

Oresme's next step is to show that the subject of "proportions of propor- tions" is applicable to celestial motions. After his long discussion of the con- ditions, pertaining to local motion, by which a proportion of force to resistance can be determined if the proportion of velocities is known (see note 58), Oresme describes how in celestial motion it is possible to determine an unknown pro- portion of force and resistance,59 say, F1/R1. It is only necessary to know two

56 Pepys 2329, fol. 107v, c. 1-2. 57 Pepys 2329, fol. 108r, c. 2-108v, c. 1. 68 "Sciendum quod proportio velocitatum ar-

guitur et scitur ex proportione temporum et spatiorum pertransitorum vel acquisitorum vel aliquorum talium ut patet 6 et 7 physicorum. Et ex proportione velocitatum arguitur pro- portio proportionum et iste processus est a posteriori; quando, vero, ex proportione pro- portionum arguitur proportio velocitatum tunc proceditur a causa et a priori" (Pepys 2329, fol. 108r, c. 2). Oresme admits he cannot mathematically derive the proportion which produces some velocity if that proportion has an unknowable denomination, and concedes that, even for proportions which have know- able denominations, he does not have a general rule for finding them from known proportions of velocities-though he believes he can always determine whether such an unknown propor- tion is greater or smaller than some known proportion of force and resistance (Pepys 2329, fol. 108v, c. 1).

Oresme lists certain other obvious necessary mathematical conditions (Pepys 2329, fol. 108v, c. 2) to enable one to derive proportions of force and resistance from known proportions

of velocities, and then mentions the necessary physical conditions for achieving this. All the specific physical conditions retduce to the single assertion that every force under consideration must be wholly and uniformly applied, withont interruption, to a mobile or resistance which is moved uniquely by that force. The passage reads: "Ubi potentia non potest a resistentia separari nec diversis motibus applicari; nec idem mobile pluribus motibus coaptari" (Pepys 2329, fol. 108v, c. 2).

59 Immediately following the last passage quoted in the preceding note, we read: "Sic igitur si alicuius velocitatis circulationis pro- portio cognoscatur per doctrinam preceden- t[em] ita ut possit dici hec velocitas est a pro- portione dupla vel a proportione tripla et cetera, et sciatur proportio velocitatis motus alicuius orbis ad illam velocitatem, quod [per] astrologiam sciri potest ex proportione quanti- tatum [motuum] vel circulorum descriptorum et ex proportione temporum in [quibus] revol- vuntur, ex istis duobus, scilicet ex [notitia pro- portionis a qua venit velocitas demonstrata et] notitia proportionis velocitatis orbis ad velocitatem datam, poterit comprehendi pro- portio intelligencie moventis ad orbem que,

310 EDWARD GRANT

things: (1) the proportion F/R which produces some velocity V, and (2) the proportion which some other velocity bears to V, i.e., V1/V. The proportion of celestial velocities can be arrived at, just as in local motion, from knowledge of a proportion of distances traversed (when T1 = T) which are circles or parts of circles in this case, or from knowledge of the proportions of the times it takes two celestial bodies to traverse equal distances. Knowing the proportion of velocities, one can then arrive at the proportion between the moving intelli- gence and the celestial sphere it moves considered as a resistance. Oresme ex- plains, however, that it is improper to speak of proportions of force and resist- ance, except by way of analogy or similitude because, strictly speaking, the immaterial intelligences move the celestial spheres effortlessly through a me- dium which offers no resistance.

If we know the proportion of the distances traversed in equal times, or if we know the proportion of the times it takes two celestial bodies to traverse equal distances, we can determine the proportion of velocities. Then a pos- teriori the "proportion of proportions" of mover to moved (which must not be called proportions of force to resistance) varies as the proportion of veloci- ties. Should one of the proportions of mover to moved be known, the other can be determined-just as in local motion.

We have now seen that Oresme has linked "proportions of proportions" of forces and resistances with proportions of velocities as cause to effect and pro- portions of velocities with "proportions of proportions" as effect to cause. But in addition, proportions of velocities can be derived from, and are related as, proportions of distances (when T1 = T) and times (when S1 = S) (see note 58). Most of this is reiterated when Oresme connects all this with the proba- bility and commensurability arguments which we have already examined.6"

Since any given unknown "proportion of proportions" would probably be irrational, the same conclusion and reasoning may be applied to any propor- tion of velocities, times, distances (or any quantities relatable as times or distances). This is so because all these proportions of different kinds of quan- tities are relatable as "proportions of proportions" of forces and resistances. For this reason Oresme feels justified in asserting the following general con- clusion: "Assuming that the proportion between any two things acquirable by continuous motion is unknown, it is probable that they are incommensurable; and if more such proportions are proposed it is more probable that some one

quidem, proportio non debet vocari proportio virtutis ad resistentiam nisi secundum simili- tudinem sicut puto quia intelligencia movet [sola] voluntate et nulla alia virtute sive conata et difficultate, et celum non resistit ei sicut credo fuisse de mente Aristoteles et Averyoes" (Pepys 2329, fol. 108v, c. 2-109r, c. 1).

6? The entire passage reads: "quia sepe dic- tum est per primam [suppositionem] quod ita est de proportione velocitatum sicut de propor- tione proportionum. Sed proposita una propor- tione proportionum [ignota] verisimile est eam incommensurabilem esse et illas proportiones incommensurabiles fore, quod si plures propor- tiones proportionum proponantur verisimilli-

mum est aliquam esse irrationalem quia inter proportiones proportionum rariores sunt ra- tionales sicut inter numeros sunt numeri cubici rariores sicut in ista conclusione 10 terti capi- tuli dicebatur. Igitur de proportionibus veloci- tatum consimiliter dicendum est, scilicet quod propositis duabus velocitatibus et cetera, quod est propositum. Cumque proportio quantitatum sit sicut proportio velocitatum quibus ille quan- titates [pertranseuntur] in eodem tempore vel [equalibus] temporibus; et proportio temporum sicut velocitatum quibus contingeret illis tem- poribus equalia [pertransiri], et econverso, ut patet in sexto phisicorum" (Pepys 2329, fol. lIOr, c. 1).

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of them would be incommensurable to some other one of them. And this. same thing can. be affirmed of two times, and any two continuous quantities what- ever.!"6 To show what he means Oresme says that if V1/V is unkn'own (which means that. Fd/R, and F/R are also unknown), it is probable that SJ/S is in- commensurable or. irrational when Ti = T. Similarly T1/T is probably incom- mensurable when S1 = S.62

As a special case of this general conclusion Oresme enunciates the following special conclusion: "it is probable that any two proposed motions of celestial bodies are incommensurable, and it is most probable that some motion of the heavens is incommensurable to some motion of another sphere. And if the op- posite should be true it could not be. known, and this seems especially true because harmony comes from incommensurable motions, as I shall discuss afterward."63

It was this special case in particular and sometimes the general conclusion as well, which served as important weapons in Oresme's attacks against astro- logical prediction. Derived ultimately, as we have seen, from the mathemati- cal arguments advanced in Conclusion 10, Chapter III, we find one or the other, or both, cited by Oresme in a number of works64 some of which were attacks against those astrologers who claimed it was possible to make precise astrological predictions from exact celestial data. If astrology depends upon

61 "Propositis [duobus] quibuscumque ac- quisibilibus per continuum motum quorum pro- portio sit ignota verisimile est [illa] esse in- commensurabilia. Et si plura proponantur verisimillius est aliquid alicui incommensura- bile fore, et de duobus temporibus contingit hoc [idem] affirmare, et de quantitatibus continuis quibuscumque" (Pepys 2329, fol. l1Or, c. 1).

62 "Verbi gratia sint duo motus [inequales] quorum proportio sit ignota qui durent per equale tempus, dico quod verisimille est quod quantitates [pertransite] sint incommensura- biles et quelibet alia per motum huius acquisita vel acquisibilia. Et si sint duo motus in- equales in duratione quorum proportio sit ig- nota et quibus equalia acquirerentur, verisimille est quod huius tempora sint incommensurabilia et [sic] de pluribus temporibus ut prius est dicendum. Igitur verisimille est quod dies et annus solaris sint [incommensurabilia] tem- pora, quod, si fuerit, impossible est invenire [veram anni] quantitatem ut si annus duret per [aliquot] dies et per unam partem diei [incommensurabilem] diei, et de aliis consimi- liter est dicendum" (Pepys 2329, fol. lOr, c. 1- liOr, c. 2).

63 "Ex predictis etiam sequitur [ista] con- clusio: propositis duobus motibus corporum celestium verisimile est illos esse incommen- surabiles atque [verisimilimum] est quod ali- quis motus celi sit alicui motui alterius orbis incommensurabilis. Et oppositum huius si foret verum non tamen sciri posset et hoc videtur maxime verum quia ex motibus incommensura- bilibus provenit armonia, ut postea declarabo" (Pepys 2329, fol. 110r, c. 2).

64 In a later work, De commensurabilitate vel incommensurabilitate motuum celi, Oresme expressly cites (Vat. lat. 4082, fol. 104r, c. 1) the De proportionibus proportionum in another connection and seems to refer to it again in connection with an argument for believing in the incommensurability of the heavenly mo- tions. Both the general and special conclusions are cited to show that the difficulties stemming from the arguments of those who believe they can foretell the future are avoided if it can be shown that the heavenly motions* are incom- mensurable. "Magis igitur tponenda- est in- commensuratio motuum celestium ex qua. hec inconvenientia non sequitur. Que quidemx in- commensurabilitas adhuc aliter ostenditur quo- niam sicut alibi probatum est [quibuslibet] ignotis magnitudinibus .demonstratis verisimi- lits est istas esse incommensurabiles quam commensurabiles sicut quantumque ignota mul- titudine proposita magis verisimile est quod sit non perfectus numerus quam perfectus, igitur de proportione quorumlibet duorum motuum nobis ignota verisimilius et probabilius est ipsam esse irrationalem quam rationalem" (Vat. lat. 4082, fol. 108v).

It seems reasonable to suppose that we have here a reference to the De proportionibus, since the phrase "sicut alibi probatum est" explicitly refers to (1) the demonstrations concerning the probability that any given unknown mag- nitudes are incommensurable, and (2) to the example concerning perfect and non-perfect numbers. The assertion of (1) is made in Chapter IV of the De proportionibus but based, as we have seen, on Conclusion 10, Chapter III

312 EDWARD GRANT

precise predictions of conjunctions, oppositions, quadratures, entries of planets into the signs of the zodiac, and so forth, then Oresme believes he has the means of destroying the very basis for such predictions. In his attack two main arguments are discernible, the first physical, the second mathematical.

First, Oresme assumes that we can never know any proportion of quanti- ties pertaining to celestial motions."5 Indeed, our senses are so defective that we are incapable of knowing proportions of quantities very proximate to us. Though we are, by the very nature of things, doomed to ignorance con- cerning actual proportions of velocities between celestial bodies, we can try considering whether the actual, but unknown, proportions of motions are com- mensurable or not. Conclusion 10 of Chapter III supplies the mathematical answer for Oresme. Even if we could know those celestial motions, the proba- bility is great that any two such proportions of velocities, times, or distances, would be incommensurable because these proportions of quantities are as pro- portions of proportions, and as more terms are taken there are correspond- ingly more irrational than rational proportions of proportions. Precise celestial positions are therefore quite unpredictable and the very best astronomical data could not remedy this inherent mathematical indeterminancy.

Why was Oresme so determined an opponent of astrology? He no doubt opposed it on general religious grounds, but his dislike for it must have im- measurably intensified because of the strong fascination it held for the King of France, Charles V, friend and benefactor of Oresme. Oresme had been in close association with Charles long before his formal ascension to the French throne in 1364, and had had opportunity to note the considerable influence of

of the same work; the citation of (2) is an example from the De proportionibus (see note 51).

The De commensurabilitate, in turn, was cited by Oresme in his Le Livre du Ciel et du Monde (edited by A. D. Menut and A. J. Denomy, in Mediaeval Studies, 1941, 3: 252- 255), as the basis for the belief that the heav- enly motions are probably incommensurable. The De commensurabilitate is also cited in Oresme's Le Livre de Divinacion (G. W. Coopland (ed. and tr.), Nicole Oresme and the Astrologers, A Study of his Livre de Divina- cions [Cambridge, Mass., 1952], p. 54), solely in connection with the incommensurability of the heavenly motions.

The general conclusion, qualified in connec- tion with a preceding logical argument by the terms "possible est" and "dubium," is found in the Ad pauca respicientes (see note 17) as a supposition applied to the celestial motions: "Propositis multis quantitatibus quarum pro- portio est ignota, possible est, et dubium, et verisimile est aliquam alicui incommensura- bilem esse" (Paris Ed., E,1, c. 2).

The same conclusions may underlie a state- ment by Oresme in his Contra judiciarios astronomos in regard to astrological predic- tion where he says ". . . tamen hoc astrologi nequeunt prescire, tum quia proporciones sunt

inscibiles, ut alibi demonstravi, tum quia . . ." (Edited by Hubert Pruckner, Studien zu den Astrologitschen Schrif ten des Heinrich vo)I Langenstein [Leipzig, 1933], p. 235).

In his Questiones de sphera, Oresme ex- plicitly cites the general conclusion as a suppo- sition: "Hoc posito pono istam suppositionem quod ista tempora, scilicet A et B, sint incom- mensurabilia et hoc est verisimile, et hoc probatur quia quibuscumque temporibus vel quantitatis duabus demonstratis verisimile est illa esse incommensurabilia, et quod eorum proportio sit irrationalis sicut patet in libro de proportionibus" (Florence, Bibl. Riccardiana, Ms. 117, fol. 134v). It would hardly be sur- prising to find these same conclusions utilized in still other of Oresme's works.

65 "Non quelibet proportio omnium quanti- tatum pertinentium motibus celestibus cor- porum est cognita, deinde proportio circulorum aut magnitudinum pertransitarum temporibus equalibus aut cuiuslibet distantie, et cetera, non est cognita et hoc est satis notum intelli- genti. Hoc enim proprie sciri non potest de quantitatibus prope nos stantibus propter de- fectum sensuum" (Paris Ed., E,,, c. 2). Though this passage appears in the Ad pauca respi- cietttes, which may not be an integral part of the De proportionibus (note 17) it is still an attitude basic to Oresme's thought.

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the court astrologers.66 His vigorous attacks may well have been prompted by a fear that important decisions of state might be influenced by vain astro- logical predictions.67 The mathematical argument of Conclusion 10, Chapter III, as applied to celestial motions, may have been, in Oresme's mind, the great antidote required to counteract the insidious poisons of astrological delusion. Oresme had formulated an intellectual mathematical argument with fond hopes of appealing to a king well known for his intellectual interests, so that with one mighty blow the king might be brought back to reason and made to real- ize that he had been duped into believing in the possibility of accurate astro- logical prediction, which was now shown to be inherently impossible.

But alas, for all his prodigious and praiseworthy effort, Oresme, judging from Charles' unswerving allegiance to his astrologers, failed dismally in his reform attempt.

The mathematical arguments in the De proportionibus proportionum formed only one weapon in Oresme's overall efforts against certain aspects of astrol- ogy, though judging from the frequent use made of them by Oresme in later works, they were a favorite weapon. It should not, however, be thought that Oresme turned away from astronomy as a science simply because he believed it inexact. In his Livre de Divinacion, a later attack against astrology, he di- vides astrology into six parts, of which the first is essentially what we would call astronomy. He describes it as "speculative and mathematical, a very noble and excellent science and set forth in the books very subtly, and this part can be adequately known but it cannot be known precisely and with punctual ex- actness..... 18 And one might add that Oresme, like so many medieval thinkers, believed firmly in the physical influence of the celestial bodies on human ac- tivities.69 But he was wholly skeptical that human beings could foretell future events from celestial motions which were, very probably, mathematically in- commensurable.

With the completion of this summary of Oresme's De proportionibus some brief remarks seem in order concerning its relation to another-and better known-mathematical treatise, namely his Algorismus proportionum.70 In the Algorismus Oresme considers exponential relations for both rational and ir- rational proportions but in the class of irrational proportions restricts himself only to those which can be denominated by rational proportions and thus,

66 On this point see the following: Charles Jourdain, "Nicolas Oresme et les astrologues de la cour de Charles V," Revue de questions historiques, Vol. 18 (1875); L. Thorndike, A History of Magic and Experimental Science, III (N.Y., 1934), 585-589; A. D. Menut and A. J. Denomy (eds.), "Maistre Nicole Oresme, Le Livre du ciel et du monde, Text and Com- mentary," in Mediaeval Studies, 1943, 5: 240- 241.

67 Though Oresme does not explicitly men- tion Charles V, his Livre de divinacions prob- ably had Charles in mind when, after de- nouncing efforts to foretell the future by astrology and other occult arts, he remarks "that such things are most dangerous to those

of high estate, such as princes and lords to whom appertains the government of the com- monwealth." G. W. Coopland (ed. and tr.), Nicole Oresme and the Astrologers, p. 51.

68 Ibid., p. 54. 69 Ibid., pp. 54-57; Thomndike, op. cit., p. 417. 70 An edition of this work was published by

M. Curtze, Der Algorismus proportionum Zum ersten Male nach der Lesart der Hand- schrift R.402 des K. Gymnasium zu Thorn herausgegeben (Berlin, 1868). The date of composition is unknown though a terminus ante quem of 1361 is fairly certain. The order of composition of the Algorismus and De pro- portionibus is likewise unknown. For a full discussion see MTPO, ch. 7.

314 EDWARD GRANT

apparently, reserved discussion of irrational proportions with no rational de- nominations for his De proportionibus. Generally, the Algorismus may be said to provide a series of rules and procedures-devoid of any theoretical founda- tion-for manipulating proportions exponentially. For each rule there are concrete examples and in the latter portion of the work these rules are applied to geometrical and physical problems. The De proportionibus, in sharp con- trast to the Algorismus, furnishes a solid theoretical basis for the exponential treatment of proportions and is thoroughly grounded in Eucidean proportion- ality theory.

The possible significance of Oresme's De proportionibus proportionum may well go beyond the exponential treatment of proportions, which was sufficiently significant so that we find some scholastic works discussing the very same problems in the sixteenth century.71 Its importance may go beyond its signifi- cance as a source for mathematical arguments against astrology. One would like to know how, if at all, Oresme's special discussion of mathematical proba- bility was related to probabilistic philosophical thought in the fourteenth cen- tury and later. Were such arguments new or commonplace? Might the idea for such probability discussions have arisen from games of chance of which Oresme makes brief mention in the De proportionibus?72 Is this the first dis- cussion involving the concept of an irrational exponent, and if so, was it of any consequence in the main stream of the history of mathematics? These are only some of the questions and problems which arise from a significant, but hitherto largely neglected work, by one of the most inquiring and pene- trating minds in medieval science.

71 The author hopes to consider, in a sep- arate article, the influence of Oresme's De proportionibus on later authors.

72". . . sicut est in ludis si peteretur de

numero abscondito, utrum sit cubicus vel non tutius est respondere quod non cum hoc proba- bilius et verisimilius videatur" (MTPO, p. 196).