PHILOSOPHICAL TRANSACTIONS. 54^ 



have been furthest carried on by Ludolphus van Ceulen ; Vieta, Hiiygens and 

 Others have given several mechanical. 



The accurate geometrical construction may be had, by which not only an 

 entire circle may be measured, but any section or arch of it also, which is by 

 an exact and ordinate motion; but such notwithstanding as suits with transcen- 

 dental curves, which erroneously are accounted mechanical, though in truth they 

 are as geometrical as those which are commonly so esteemed, though they are 

 not algebraical, nor can be reduced to equations algebraical or of certain degrees, 

 they having degrees proper to themselves, which though they be not algebraical, 

 are yet nevertheless analytical. But these things I cannot in this place explain 

 so fully as they deserve. 



The Analytical Quadrature, or that which is made by accurate calculation, 

 may be again subdivided into three kinds, viz. into the analytical transcendent, 

 the algebraical, and the arithmetical. The analytical transcendent is to be ob- 

 tained, among others, by equations of indefinite degrees, hitherto considered 

 by none. As if X '^ + X be equal to 30, and X be sought, it will be found to 

 be 3, because 3 ^-j- 3 is 27 + 3, or 30 ; which kind of equations for the 

 circle we will give in their proper place. 



The Algebraical is done by vulgar numbers, though irrationally vulgar, or by 

 the roots of common equations, which for the general quadrature of the circle, 

 or its sectors, is indeed impossible. Now there remains the arithmetical qua- 

 drature, which is performed by certain series exhibiting the quantity of the circle 

 exact by a progression of terms, first rational, such as I shall here propound. 



I have found therefore, that if the square of the diameter be put l , the area 

 of the circle will be, J- — 4--!-^ — T4"i---rr + -r3- — -rT + -tV> ^^- viz. the 

 entire square of the diameter being diminished, that it may not be too large, by 

 a third part; and again, because hereby too much is taken away, being aug- 

 mented by one fifth ; and again, because by this last too much is added, di- 

 minished by one seventh, and so onward continually. 



So that the first quantity will be 



too great, viz. 1, but the error will be less than -^; 

 The next too little, viz. 1 — ^, but the error will be less than 3 ; 

 The 3d too much, viz. 1 — i + i^ but, &c. -i-; 

 The 4th too little, viz. 1 — ^ -[- x — ^^ but, &c. i; 

 &c. &c. 



The whole series contains all the approaches together, or the values, both 

 greater than they ought to be, and less than they ought to be. 



So that by continuing the series, the errors may be made less than a fraction 

 giver\, and consequently less than any assignable quantity. Whence it follows, 

 that the whole series must give the true value. And though the sum of the 

 whole series cannot be expressed by one number, and that the series be infinitely 



