PHYSICAL SCIENCE. 



IX 



He entered into the society of Jesuits, was sent 

 to the university of Pisa, where he acquired his 

 geometrical knowledge, and was afterwards 

 professor of astronomy in the university of 

 Bologna. 



Cavalleri preceded in his geometry of indi- 

 visibles, on the following principle : areas may 

 be considered as made up of an infinite number 

 of parallel lines ; solids, of an infinite number 

 of parallel planes ; and lines, of an infinite 

 number of points. Thus, the cubature of a solid 

 was reduced to the summation of a series of 

 planes; and the quadrature of a curve, to the 

 'summation of a series of ordinates. Now the 

 rule for summing an infinite series of terms in 

 arithmetical progression, had been long known, 

 and the application of it to find the area of a 

 triangle, according to the method of indivisibles, 

 was a matter of no difficulty. The next step 

 was, supposing a series of lines in arithmetical 

 progression, and squares to be described on each 

 of them, to find what ratio the sum of all these 

 squares bears to the greatest square, taken as 

 often as there are terms in the progression. 

 Cavalleri showed that when the number of 

 terms is infinitely great, the first of these sums 

 is just one third of the second ; this evidently 

 led to the cubature of many solids. Proceeding 

 one step further, he sought for the sum of the 

 cubes of the same lines, and found it to be one- 

 fourth of the greatest, taken as often as there 

 are terms ; and continuing his investigations, he 

 was able to assign the sum of the nth power of 

 a series in arithmetical progression, supposing 

 always, the difference of the terms to be infinitely 

 small, and the number of terms to be infinitely 

 great The number of curious results obtained 

 from these investigations was prodigious. It 

 may be considered as constituting as great a 

 step over the calculus of exhaustions, as the 

 i ntegral calculus, was over the geometry of in- 

 divisibles. 



The next general step, (for we are under the 

 necessity of passing over many individual dis- 

 coveries of great importance), in the extension 

 of mathematics, was the arithmetic of infinites of 

 Dr Wallis, first published in the year 1655. 

 Wallis was a native of Kent, where he was born 

 at Ashford, in the year 1G16. He was appoint- 

 ed Savillian professor of mathematics at Oxford, 

 in 16-M); which place he occupied till the year 

 1703, when he died. 



The origin of the arithmetic of infinites, was 

 an attempt by Wallis to discover the quadrature 

 of the circle, after he became acquainted with 

 the method of indivisibles of Cavalleri. He 

 began by observing, that if we have a series of 

 numbers, arithmetically proportional (the natural 

 numbers for example) beginning with 0, and 



proceeding regularly on, the sum of this series, 

 is equal to half the sum of the last term, repeated 

 as many times as there are terms in the series. 

 Thus, 



0+1 1 . 0+1+2 _ 3 _ ! 0+1+2+3 6 _ 1 

 1 + 1 2' 2+2+2 6 ~ 2' 3+3+3+3" 12 ~ 2' 



From this it follows, that a triangle is half a 

 parallelogram, on the same, or equal basis, and 

 between the same parallels. For a triangle may 

 be considered as composed of an infinite number 

 of lines, beginning at a point or 0, and increas- 

 ing arithmetically, the greatest of which is the 

 base ; while a parallelogram consists of an infinite 

 number of lines of equal length, and all equal 

 to the base. From this analogy he deduced the 

 quadrature of a variety of figures. 



He next shows that the sum of an infinite 

 series of numbers, beginning with 0, and pro- 

 ceeding as the squares of the natural numbers, 

 that is, the series 0, 1, 4, 9, 16, 25, 36, 49, &c., 

 is to the last term, repeated as often as there are 

 terms in the series, as 1 to 3. From this he 

 shows that the compliment of a semi-parabola 

 is to the parallelogram contained between the 

 corresponding absciss and ordinate, as 1 to 3 ; 

 and deduces from the same analogy, many other 

 of the quadratures of curvilinear spaces. 



He then shows that the sum of an infinite 

 series of quantities, increasing from 0, as the 

 cubes of the natural numbers, or the series 0, 

 1, 8, 27, 64, 125, 216, &c., is to the sum of the 

 last term, repeated as often as there are terms in 

 the series, as 1 to 4. This analogy furnishes the 

 quadrature of many figures. 



But it would be too tedious to attempt an 

 analysis of the whole of this curious book, which 

 may be considered as furnishing the first germ 

 of the integral calculus. There are however a 

 few other particulars which it would be improper 

 to omit. He was led by analogy to express the 

 denominators of fractions by means of negative 

 powers. The numbers 



ar 3 f 2 x 1 a? for D &r 



or, x~, x , x , v or i), ^, ^ <xu, 



are in continued geometrical progression. This 

 led him to express them in this way, r*, ar 2 , a: 1 , 

 ar, x~\ x~ z , or- 3 . Simple as this improvement 

 may appear, it has led to consequences of the 

 greatest importance. It put him in possession 

 of the measure of every space, the elements of 

 which are reciprocally as any power of the 

 abscissa. A prodigious number of new dis- 

 coveries was the necessary consequence of these 

 new views. 



Most of the discoveries made by other mathe- 

 maticians immediately after the publication of 

 the Arithmetic of Infinites, were little else than 

 developments of the views of Wallis. Neil 

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