QUADRATURE. 



lias been done on this fubje<3;, by way of approximation. 

 Wc have already obfcrvcd that Archimedes was the firft. who 

 gave an approximation of tlie ratio of the diameter to tlie 

 circumference of a circle, placing it between the limits I to 

 34", and I to 3)4; and it is faid that Apollonius and Philo 

 found more accurate approximations, wliich, however, iiave 

 not been tranfmitted to us. 



Towards the year 1585, Melius, combatting the falfe 

 quadrature of Simon Duch^ne, gave the ratio of 113 to 

 355, which is very exaft, being only T-ij-o-o-tnnru i" cxcefs. 

 Victa found a ftill nearer approximation, carrying it to ten 

 decimals, whereas the former is true only to fix places. He 

 alfo gave a kind of ferics, the infinite fum of which was equal 

 to the entire circle. 



Adrianus Romanus carried tho approximation to feventeen 

 figures, and Ludolph van Ceulen to thirty-fix ; which he 

 publiflied in his work "DeCirculoet Adfcriptis ;" and of 

 which SnelliuspublinicdaLatintranflation in 1619. He after- 

 wards verified Van Ceulen's approximation by fome theorems 

 of his own invention, which greatly facilitated the compu- 

 tation, and whicli he publiflied in 1 621, under the title of 

 •' Willebrordi Snellii Cyclometricus de Circuli Dimenfione, 

 &c." 



Defcartes found a geometrical conftruftion from which it 

 was eafy to draw an exprcflion in the form of a feries ; and 

 Hnygens afterwards difcovered fome curious theorems con- 

 nefted with this fubjeft, but did not advance the approxima- 

 tion, though he made fome ufeful rules for approximating 

 towards the length of the circular arc. 



One of the moil curious difcoveries connefted with this 

 fubjeft, which had yet been publiflied, was that given by 

 Wallis in his " Arithmetica Infinitorum," in 1655: where 

 he fhews that the ratio of a circle to the fquare of its dia- 

 meter, is truly exprefled by the infinite fraftion 



3'. 5^ f. 9'. ii\ &c. 



2. 4-. 6'. 8'. io\ 12'. &c. 



'If we limit ourfelves, as we muft do to, a finite number of 

 terms, we mall have a ratio alternately too great and too fmall, 

 according as we take an even or an odd number of terms ; 



thus, - is too great ; ^-^ too fmall : ' too great ; and 



2 ° 2.4 2.4.4 



too fmall, and fo on 1 but each of thefe is a nearer 



3-3-5-5 

 2.4.4.6 



approximation than the preceding ones. But in order to 

 approach ftill nearer in both cafes, the author propofed to 

 multiply the whole producl by the fquare root of a binomial, 

 ysiz. unity, plus unity divided by the lalt figure, with which 

 the feries terminated either m the numerator or denominator ; 

 in which cafe, the produtt will be a much nearer approxima- 

 tion ; it will be too great if we ufe the lait figure of the 

 numerator, and too fmall if the laft; of the denominator. 

 Thus we fliall have for the ratio fought, alternately in excefs 

 and defeft. 



lows. The circle ilfelf being 1, the fquare of its diameter 

 isexpreffed by the infinite continued fraftion. 



I 



I + 



2 -(- &c. 

 is obvious. 



See 



In excefs. 

 3-3-5-5 



2.4.4.6 



3-3J^5-7i7 

 2.4.4.6.6.8 

 &c. 



v/(l X 



r) 



In defeft, 



3-3-5-5 



2.4.4.5 



3-3-5-5-7-7 

 2.4.4.6.6.8 

 &c. 



y (i X ^) 

 ^(i X 



.) 



Prior to the above feries of Dr. Wallis, however, fome 

 thing of an equivalent expreffion, though given under a dif- 

 ferent form, was difcovered by lord Brounker, which is as fol- 



7 



of wliicli the law of the denominators 

 Circle. 



Such was the progrcfs which mathematicians had made 

 towards the folution of this interefting problem prior to the 

 invention of fluxions, which, by reducing the quadrature of 

 all curves to one general principle, again revived the hopes of 

 fuccefs with regard to the circle, notwithftanding fome pre- 

 tended demonilrations of its impolTibility ; and its quadra- 

 ture was accordingly again attempted with the greateft 

 eagernefs. The quadrature of a fpace, and the redlification 

 of a curve, were now reduced to that of finding the fluent of 

 a given fluxion but ftill the problem was found to be inca- 

 pable of a general folution in infinite terms. The fluxion of 

 a given fluent was found to be always aflignable, but the 

 converfe propofition, vix. of finding the fluent of a given 

 fluxion, could only be effedled in particular cafes ; and amongft 

 the exceptions, to the great regret and difappointment of 

 geometricians, was included the cafe of the circle with regard 

 to every form of fluxion under which it could be obtained. 

 Some exceedingly near approximations have, however, fince 

 been made towards the true ratio of the diameter to the 

 circumference of the circle, but thefe belonging rather to 

 the article Rectification than to Quadrature, we fliall 

 enter again upon the fubjeifl under the former term, and 

 fliall occupy the remainder of the prefent article on the 

 quadrature of curves m general. 



On the Quadrature of Curves by Fluxions. In order to ex- 

 hibit more diftindlly and at large the ufe of fluxions, ac- 

 cording to the modern method of notation, in finding the 

 areas of curves, we fliall premife the two following cafes. 



Cafe I. — Let A R C ( Plate XIII. Analyfis,fg. 10. ) be a 

 curve of any kind, whofe ordinates R ^, C B, are perpendi- 

 cular to an axis A B. Imagine a right line ^ R ^, perpen- 

 dicular to A B, to move parallel to itfelf from A towards 

 B ; and let its velocity, or the fluxion of the abfciffe A b, 

 in any propofed fituation of that line, be denoted by b d% 

 then will the retlangle b n exprefs the fluxion of the gene- 

 rated area A ^ R, which (if A ^ = .v, and ^ R = j>) will be 

 =:= jy .V : whence, by fubftituting for y or x (according to 

 the equation of the curve,) and taking the fluent, the area 

 itfelf will become known. 



Cafe 2. — Let ARM {fg. II.) be any curve whofe ordi- 

 nates C R, C R, are all referred to a point or centre ; and 

 conceive a right line C R H to revolve about the given cen- 

 tre C, and a point R to move along the faid line, fo as to 

 defcribe the curve live ARM. If this point were to move 

 from Q, without changing its direftion or velocity, it would 

 proceed along the tangent Q S (inftead of the curve), and 

 defcribe areas Q J- C, Q S C, about the centre C, propor- 

 tional to the times in which they were defcribed ; becaufe, 

 having the fame altitude C P, they arc as the bafes Q^r and 

 O S. Confequently, if R S be taken to denote the value 

 of k the fluxion of the curve line A R, the correfponding 



fluxion 



