O R G A N O N 4(1967) L’ANTIQUITÉ CLASSIQUE ET LES DÉBUTS
DE LA SCIENCE MODERNE
Hans L. W ussing (German Democratic Republic)
EUROPEAN MATHEMATICS DURING THE EVOLUTIONARY PERIOD OF EARLY CAPITALISTIC CONDITIONS
(15th AND 16th CENTURIES)
D uring the 15th and 16th centuries, th e economic developm ent of W estern and C entral Europe was characterized by a stepw ise grow th of commercial capital, principally in consequence of original accum ulation. This process, differentiated both geographically and in time, was followed w ithin the feudal society by th e developm ent of capitalistic m ethods of production: the transition from production by guilds to cottage in d u stry and, la ter on, to th e developm ent of m anufactures.
In this process, the aim of the form ing new class w as to become politically em ancipated by means of developing the economy. The necessity imposed upon the new bourgeoisie of expanding trad e and, la ter on, of im proving th e ir means of production and tran sp o rt tu rn ed th eir in terest in an ever increasing degree to m aking use of n atu ra l science and m athem atics. The revival and fructification of ancient know ledge called the “Renaissance” was, therefore, th e resu lt of changed social conditions and not, by any means, th e ir cause.
It seems w orth investigating in detail, how the m athem atical heritage taken over from antiqu ity has been resum ed un der social conditions of the 15th and 16th centuries, how it was first assim ilated and adopted u n der new aspects, then rejected and again continued in an im proved m anner. Speaking m ore generally how, in th e period of an altered orientation of objective social conditions offered to m athem atics, th e m athem atical knowledge, accum ulated under en tirely d ifferent social conditions, was being “worked over.”
Evidently the sh ort tim e granted to speakers suffices barely for p u ttin g fo rth very few essential moments on this subject. We therefore m ust concentrate, as far as the period of early European capitalism is concerned, upon those social domains of m athem atical orientation from w hich impelling forces tow ards reorientation have issued. These w ere:
9 0 H. L. Wussing
1. The stepw ise and extrao rd in ary increase in m o n e y c i r c u l a t i o n led to a m ultitude of m athem atical problem s: book-keeping, w idening of the com plem ent of num bers, conversion of wide varieties of currencies and of different units of m easure and weight, rates of interest and compound interest.
2. N a v i g a t i o n required new knowledge for high-sea sailing and naval construction work; hence dem ands w ere raised for nautical and astronom ic data, th a t is, for spherical geom etry and the com putation of centres of gravity of vessels. In addition there w ere to be solved hydraulic problem s of inland w aterw ays involving the construction of canals, sluizes and riv er regulation.
3. The emergence and developm ent of g u n n e r y made arm ourers and ordnance men face a num ber of new ballistic problems; these problem s had to be solved w ith highest exertion in view of the fact, th a t each shot was extrem ely expensive and th a t therefore sighst had to be taken very carefully. From these requirem ents heavy demands
resulted as to geom etry and trigonom etry.
4. A s t r o n o m y , w ith its close relations to research on scientific cosmology and to the replacem ent of the geocentric by the heliocentric universe, to astrology and to the com putation of the almanacs, brought, combined w ith the dem ands made by m ilitary and civil surveying, a m ultitude of incentives tow ards trigonom etry. As a resu lt of this, the necessity arose of im proving and sim plifying trigonom etric calculations.
5. A r c h i t e c t u r e was confronted w ith difficult problems, especially in m ilitary building, such as defilading a fortress, th a t is, now to locate it including its bulw arks, ram parts, com ers, etc. w ith due consideration of land forms, so th a t none of its parts, should be exposed to shelling by th e besieging enem y’s guns. Moreover, because of the increasing piercing pow er of shells, it became necessary to construct fortifications a t greater depths, to build casemates. Hence the need of picturing three-dim ensional objects in a plane which, in turn , developed fundam entals of descriptive geometry.
6. The practice of s t r u c t u r a l a n d d e s c r i p t i v e a r t in volved a v ariety of m athem atical factors. P rom inent buildings, statues and paintings, if m eant to comply w ith the revided ideals of antique beauty, had to be designed in accordance w ith canonical rules, th a t is, th e ir individual p arts had to show definite proportions, like those called the “golden c u t” (dev goldene Schnitt). Consideration of linear perspective on paintings, an essential accomplishm ent of the a rt during the Re naissance, led to the determ ination of the vanishing point and the vanishing line.
It was w ithin these domains th a t the demands of social life made on m athem atics during the early capitalistic period in Europe were concentrated; and w ithin this range the m athem atical heritage of ancient
European Mathem atics in the 15th and 16th Centuries 9 1
times had to be assim ilated, if m athem atics of the ancient slave-holding society had already p rep ared the corresponding m aterial. The an tiq u ity became outdistanced or new m athem atical disciplines developed, w herever the new evolution required new form s of thinking th a t had not advanced to m atu rity in the tim e of the antique.
Beginning w ith the 11th century, w hile still in contact w ith the Islamic W orld and the B yzantine Em pire, p arts of antique m athem atics had reached Europe and had become p a rt of the quadrivium ta u g h t at the universities of feudal Europe. These studies referred m ainly to the content of Euclid’s planim etrical works, to elem entary num erals, to w hat was called planisphere courses, and to P tolem y’s Almagest, as w ell as to m athem atical data from P lato and A ristotle. P ractical m athem atics w ere lim ited to some details of surveying, to com putus, and to figuring by means of the abacus). It was not until the 13th century th a t an independent resum ption and in terp retation of traditional knowledge began, for instance by Leonardo Fibonacci from Pisa (ca. 1180— 1250). And w hile in th e 13th century th e re was taken up the translation of some of the m asterpieces of antique m athem atics—such as the works of Euclid by Johannes Cam panus (ca. 1260), of A ristotle, Proclus, Archimedes, Heron and Ptolem y by W ilhelm von M oerbeke (d. 1286)— initially these achievem ents w ere practically futile, because commonly they w ere beyond comprehension, despite the fact th a t m athem atics occupied a consolidated position w ithin the stru ctu re of Scholastics.
Rem arkable in th e transition from the late Scholastics to th e Re naissance is th e sharp break th a t occurred not so much as to time, b u t ra th e r in the shift in the social domain. During a process lasting for entire generations this domain, from w hich was supposed to originate the impulse tow ards the fu tu re evolution of m athem atics, moved aw ay from th e universities to practice. There was a wide field of operation for m athem atics in practical life; in this m anner social evolution cleared the road to the critical perusal of th e more advanced p a rts of ancient m athem atics. It w as only w hen m athem atics, derived from practical application, had reached some sort of m aturity, th a t it became responsive to Diophantes and to the complex p a rts of th e w ork of Apollonius, A rchim edes and other scientists. The great advancem ent of m athem atics during the end of the 16th and the beginning of the 17th centuries resulted thus from th e m asterpieces of ancient m athem atics being joined to a new conception of m athem atics p u t in relief during the previous period.
This step-by-step reception and recast of the ancient m athem atical heritage m ight be illu strated by a m ultitude of examples, proving it to be distinctly typical; however, for the sake of brevity, only a few facts shall be called to mind here.
92 H. L. M ussing
Tow ards th e end of the 16th century, the altercation between th e abacists and the algorithm ics came to a close w ith a victory of the latter, the disciples of calculating by the use of Indian-A rabic num bers. However, it became necessary to develop in an appropriate w ay the m ethods of w ritten calculation—a task to be accomplished w ithout recourse to antiquity. The m ultitude of m athem atical textbooks published in the 16th century—beginning w ith W idmann and Adam Ries up to the Cossist w ritings—picture faith fu lly this slow and difficult procedure. And w ith M. S tifel’s excellent Arithm etica integra, w ritten in 1544, a new stage in the appropriation of an tiqu ity was reached. Using the Cossist method, Stifel succeeded in making available Euclid’s difficult Volume X of the Elem ents, the p a rt dealing w ith irrationals of type j / a _|_ j/fj“ a fte r the ancient method of geom etrical algebra. In this m anner, im plicitly inspired by Euclid Stifel m anaged to surpass Euclid and to
n / ~ study irrationals of the type i / a + j/5.
Also in th e middle of the 16 th century and in a sim ilar m anner the antique doctrine on equations was m astered. The transition to the algebraic solution of quadratic, cubic and biquadratic equations was accomplished by a group of Italian m athem aticians: Tartaglia, A. M. Fior, and several others, who abandoned the antique m ethod of applying plane geometry. And it was C ardano who, in 1545 in his A rs magna, gave a sum m ary of antique achievem ents considered from the viewpoint of the new treatm en t of equations.
Quite as typical as the above progress was the tu rn in the methodical transform ation of th e antique, based, as far as substance is concerned, on ancient tradition, th a t is seen in the elaboration of algebra by Fr. Vieta (1540— 1603). V ieta’s Logistica speciosa, his splendid contribution, re presents an attem p t of reconstructing the proper m ethod suggested by D iophantes whose doctrine had m eanwhile become commonly acceptable. However, Vieta achieved much more than expanding the disposition of algebra during the antique; he attain ed a degree of perfection in num erical methods, unknow n in the antique.
These few exam ples m ight easily be supplem ented by m entioning the discovery of logarithm s, by pointing out the interrelation between the sexagesimal and the decimal system, by indicating the history of tradition in compiling and im proving astronom ical tables, by calling attention to th e transform ation of the ancient trigonom etry of secants into a sine geometry, accompanied by the evolution of the num erical methods involved, or by the changes arrived at in the evaluation of the doctrine of conical sections. In the present paper it was m erely intended to exhibit the most fundam ental principles.
In general outlines, the evolution of m athem atics attained in the 15th and 16th centuries m ay be summed up as follows: as the result
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of the demands made by social life, m athem atics advanced in th ree principal directions: im provem ent of the calculation technique, algebra- ization of methods of com putation, and developm ent of trigonom etry into a homogeneous system.
W ith the rise of early capitalism in Europe m athem atics attained, compared w ith the antique, a fundam entally novel social position. M athem atics had ceased to be m erely a science in P lato ’s sense, and both directly and indirectly—also in view of its ties w ith n a tu ra l sciences—th e possibilities of m athem atics in consideration of production had found recognition. In certain fields, th e m athem atical heritage taken over from the antique—particu larly geom etry and trigonom etry— constituted sound m athem atical foundations, w hereas w ith regard to algebraic evolution it stim ulated incentives. On th e other hand, the new social requirem ents dem anded th e command of m ultifarious num erical methods, a domain in w hich ancient tradition was by no means p rep ared to satisfy the needs set off by the Renaissance. U ndoubtedly, the emphasis p u t on num erical methods by the Islam itic m athem aticians has consider ably aided the acquisition of m athem atical knowledge in antiq u ity as well as supported the developm ent of European m athem atics during the Renaissance.
As far as necessary, traditional knowledge was absorbed in the new perspectives envisaged by social requirem ents. It was only later, in the 17th century, th a t other suggestive topics initiated in antique m athem atics w ere resum ed and fu rth e r developed, such as the fructification of the difficult p arts of A rchim edes’ teachings dealing w ith researches on problem s of q u ad ratu re and rectification. Also m uch later, in th e 17th century, w hen m ethods suggested by Apollonius and studies of conics w ere resumed, elem ents of the geom etry of co-ordinates w ere taken over from the antique and expanded. The new social prospects opened to m athem atics and constantly im proved during the Renaissance cleared the road for these p a rts of the antique m athem atics to effective operation in the fu tu re and to the release of essential im pulses for the evolution of m odern m athem atics of variable quantities.
