550 PHILOSOPHICAL TRANSACTIONS. [aNNO 1682. 



continued, yet because it consists of one regular method of progression, the 

 whole may sufficiently enough be conceived by the mind. For since the circle 

 is not commensurate to the square, it cannot be expressed by one number, but 

 it must necessarily be exhibited in rational numbers by a series, in the same 

 manner as the diagonal of a square ; and the section made according to extreme 

 and mean proportion, which some have called the divine section, and many 

 other quantities which are irrational. And indeed if Van Ceulen could have 

 given a rule by which his numbers 314159 &c. could have been continued in 

 infinitum, he would have given us the arithmetical quadrature exact in whole 

 numbers, which we have here done in fractions. 



Now lest some less knowing in these matters might think that a series con- 

 sisting of infinite terms cannot possibly be equal to a circle, which is a finite 

 quantity, they must know that there are many series whose terms are infinite in 

 number, whose whole sum is equal to finite quantities. As, to give an easy 

 example, the series from 1 decreasing in subduple geometrical proportion, 

 ■A- -f 4- -f i -f- tV + 3-V 4- -bVj &c. in infinitum, the whole of which notwith- 

 standing makes no more than 1. 



There are several things relating to this quadrature, which might be taken 

 notice of. But I have not now the leisure to prosecute it. However I must 

 not omit one, viz. that the terms of this our series -f, -f , -f, -f, -f, &c. are in a 

 continued harmonical progression. And a series made by skipping, as -f, -i-, i, 

 ,j^j _>_^ &c. is also of harmonical progression. And -i-, i, -^V, -^, -j^, &c. is 

 also a series of harmonical proportionals. Wherefore since the circle = j. -f- x -|- 

 + i tV + tV + &c- — i — T — -rV — tV — -tVj &c. by subtracting the latter 

 partial series from the former partial series, the circle will be the difference of two 

 series in harmonial progression. And because the sum of any number of terms 

 in harmonial progression, how many soever, may by some compendium be ob- 

 tained; hence compendious approaches maybe deduced, if one would in this 

 our series take out the terms affected with the sign — by adding the two next 



into one, + i — -i- and + ^ — ^ and -f -l — -^, and -|- -jV — tV. and + -rV 

 — _i_ , and so onward, he will have a new series for the magnitude of the circle, 

 namely f (that is i - i) + VV (that is -i- - i) + iV (that is i - -l-), &c. 

 wherefore the square inscribed being i, the area of the circle shall be J- -j- -313- -f 



■5V "T" -rW "T TWi *^^' 



But the numbers 3,35,99,1 95,323, &c. by skipping, are taken out of the series 

 of square numbers, 4, 9, 16, 25, &c. diminished by a unit, and so made the series 

 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 36o, 

 399, &c. out of the members of which series every fourth after the first, is a 

 number of this our series. But I have found the sum of this infinite series 

 ■i- + i + iV + -^ + tV + ^r + ^ + -sV + W. &c. to be 4. And by culling 

 out by single skipping, as -i- + -^V + -aV + -sV + w^ ^c. the sum of this infinite 



