Decembeb 23, 1904.] 



SCIENCE. 



859 



We have remained up to this point in 

 the domain which ordinary language calls 

 elementary algebra and arithmetic. 



In fact, from the time that the incom- 

 mensurability of certain magnitudes had 

 been recognized, the infinite had made its 

 appearance, and, from the time of the 

 sophisms of Zeno on the impossibility of 

 motion, the summation of geometric pro- 

 gressions must have been considered. 



The procedures of exhaustion which are 

 found in Eudoxus and in Euclid appertain 

 already to the integral calculus, and Archi- 

 medes calculates definite integrals. 



Mechanics also appeared in his treatise 

 on the quadrature of the parabola, since 

 he first finds the surface of the segment 

 bounded by an arc of a parabola and its 

 chord with the help of the theorem of 

 moments; this is the first example of the 

 relations between mechanics and analysis 

 which since have not ceased developing. 



The infinitesimal method of the Greek 

 geometers for the measure of volumes 

 raised questions whose interest is even to- 

 day not exhausted. 



In plane geometry, two polygons of the 

 same area are either equivalent or equiva- 

 lent-by-completion, that is to say, can be 

 decomposed into a finite number of tri- 

 angles congruent in pairs, or may be re- 

 garded as differences of polygons suscept- 

 ible of such a partition. 



It is not the same for the geometry of 

 space, and we have lately learned that 

 stereometry can not, like planimetry, get 

 on without recourse to procedures of ex- 

 haustion or of limit, which require the 

 axiom of continuity or the Archimedes as- 

 sumption. 



Without insisting further, this hasty 

 glance at antiquity shows how completely 

 then were amalgamated algebra, arithmetic, 

 geometry and the first endeavors at integral 

 calculus and mechanics, to the point of its 



being impossible to recall separately their 

 history. 



In the middle ages and the renaissance, 

 the geometric algebra of the ancients sepa- 

 rated from geometry. Little by little al- 

 gebra properly so called arrived at inde- 

 pendence, with its symbolism and its nota- 

 tion more and more perfected; thus was 

 created this language so admirably clear, 

 which brings about for thought a veritable 

 economy and renders further progress pos- 

 sible. 



This is also the moment when distinct 

 divisions are organized. 



Trigonometry which, in antiquity, had 

 been only an auxiliary of astronomy is 

 developed independently ; toward the same 

 time the logarithm appears, and essential 

 elements are thus put in evidence. 



II. 



In the seventeenth century, the analytic 

 geometry of Descartes, distinct from what 

 I have just called the geometric algebra of 

 the Greeks by the general and systematic 

 ideas which are at its base, and the new- 

 born dynamic were the origin of the great- 

 est progress of analysis. 



When Galileo, starting from the hypoth- 

 esis that the velocity of heavy bodies in 

 their fall is proportional to the time, from 

 this deduced the law of the distances 

 passed over, to afterward verify it by ex- 

 periment, he took up again the road upon 

 which Archimedes had formerly entered 

 and on which would follow after him Cava- 

 lieri, Fermat and others still, even to New- 

 ton and Leibnitz. The integral calculus of 

 the Greek geometers was born again in the 

 kinematic of the great Florentine physicist. 



As to the calculus of derivatives or of 

 differentials, it was founded with precision 

 apropos of the drawing of tangents. 



In reality, the origin of the notion of 

 derivative is in the confused sense of the 

 mobility of things and of the rapidity more 



