BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 279 



analysis, his Italian contemporary, Cavalieri (1598-1647) was 

 rendering a similar service to the integral calculus in developing 

 his theory of indivisibles. 



The problem of measuring the length of a curve or the area of 

 a figure having a curved boundary, or the volume of a solid bounded 

 by a curved surface goes back indeed to comparatively ancient 

 Greek times. Most notable in this direction was the work of 

 Archimedes. Kepler, attempting to resolve astronomical difficulties 

 by the hypothesis of elliptical orbits, is confronted at once with 

 the problem of determining the circumference of an ellipse. He 

 gives the approximation TT (a + 6) where a and b are the semi- 

 axes. This is close if a and b are nearly equal, as in most of the 

 planetary orbits. Interesting himself in current methods of 

 measuring the capacity of casks, he published in 1615 his Nova 

 Stereometric, Doliorum Vinariorum, in which he determines the 

 volumes of many solids bounded by surfaces of revolution. The 

 Greek method had in case of the circle, etc., depended on an 

 "exhaustion" process of inscribing and circumscribing polygons 

 differing less and less from the curve both in boundary and in 

 area. Kepler however divided his solid into sections, determined 

 the area of a section and then sought the sum. He lacked an 

 adequate system of coordinates, a clearly defined conception of a 

 limit, and an effective method of summation. In view of the 

 intrinsic difficulty of this important problem, however, the extent 

 of his success is remarkable. 



He also sought to determine the most economical proportions 

 for casks, etc., expressing his view of the underlying mathematical 

 theory by the theorem " In points where the transition from a less 

 to the greatest and again to a less takes place, the difference is 

 always to a certain degree imperceptible." 



Cavalieri, in 1635, adopted the form of statement that a line 

 consists of an infinite number of points, a surface of an infinity of 

 lines, a solid of an infinity of surfaces, but later revised this on the 

 basis of the assumption "that any magnitude may be divided 

 into an infinite number of small quantities which can be made to 

 bear any required ratios one to the other." On this basis, open 



