DEVELOPMENT OF MATHEMATICAL ANALYSIS 499 



by them arithmetic, their speculative spirit showed little taste for 

 practical calculation, which they called logistic. 



In remote antiquity, the Egyptians and the Chaldeans, and later 

 the Hindus and the Arabs, carried far the science of calculation. 



They were led on by practical needs; logistic preceded arithmetic, 

 as land-surveying and geodesy opened the way to geometry; in the 

 same way trigonometry developed under the influence of the in- 

 creasing needs of astronomy. 



The history of science at its beginnings shows a close relation 

 between pure and applied mathematics; this we shall meet again 

 constantly in the course of this study. 



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 incommensurability 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 progressions must have been 

 considered. 



The procedures of exhaustion which are found in Eudoxus and in 

 Euclid appertain already to the integral calculus, and Archimedes 

 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 me- 

 chanics 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 ex- 

 hausted. 



In plane geometry, two polygons of the same area are either 

 equivalent or equivalent-by-completion, that is to say, can be de- 

 composed into a finite number of triangles congruent in pairs, or 

 may be regarded as differences of polygons susceptible of such a 

 partition. 



It is not the same for the geometry of space, and we have lately 

 learned that stereometry cannot, like planimetry, get on without 

 recourse to procedures of exhaustion or of limit, which require the 

 axiom of continuity or the Archimedes assumption. 



Without insisting further, this hasty glance at antiquity shows 

 how completely then were amalgamated algebra, arithmetic, geo- 

 metry, 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 separated from geometry. Little by little algebra 

 properly so called arrived at independence, with its symbolism and 



