RECTIFICATION. 



Hnviiig thus laid down his plan of operation, by a few 

 trials he k-U upon a number well fuited to his purpofe, viz. 



knowing the tangent of — of 45°, or 11° 15', to be very 

 nearly — , radius being i, ho allumed for his firft arc that 



whofe tangent is — ; then fince tan. 2 a 



2 tan. n 



I — tan.' , 



120 



he had — for the tanpent of his double arc, and -^-^ for 

 12 '' 119 



the double of this, or of four times the firll, which being 

 a little greater than 45^, was well adapted to his views ; for 

 by a known trigonometrical property, tan. (3 — 45) = 



; that is, the tangent of the finall arc, which is 



tan. a + I ° 



120 



equal to the excefsof his multiple above 45°, was — — 



— I 



120 

 119 



+ I 



=39 



He had, therefore, two arcs to compute, the one 



having for its tangent — , and the other ; and then four 



5 239 



times the firft of thefe arcs minus the latter, would evi- 

 dently give the exaft arc of 45°, and both thefe numbers 

 being fuch as to converge very well in the general feries, the 

 difficulty attending the ufual approximation was avoided. 

 Other approximations, however, have fince been difcovcrcd, 

 which, if not more rapid, their invelligation is, at lead, 

 more fimple, of which, perhaps, that of Euler's is the moft 

 deferving of notice. This celebrated geometer obferves, 

 that every arc whofe tangent is commenfurable with the ra- 

 dius, as, for inftance, 45'^, may be divided into two arcs, of 

 which the tangents, though much fmaller, are ftill com- 

 menfurable with the radius ; for fince 



. ,^ tan. a -j- tan. i> 



tan. [a + b) =: ■ -•, 



I — tan. a tan. b 



we have alfo 



tan. a = 



tan. (a + i) — tan. i 



tan, 



[a + b) . tan. 6 + 1 ^ 



I I 



3'^ riF 



I I 



— ; + - &c. 



•3' 9-3' 



3 3-3' 5-3' ?• 



In both which feries, the terms diminifh much more ra- 

 pidly than in the original feries, and may therefore be com. 

 puted with tolerable eafe. 



But it is evident that we may proceed farther in this ap- 

 proximation, by dividing each of thefe into other two arcs, 

 by which means the convergency will obvioufly be much 

 more rapid ; and though, generally fpeaking, for every fub- 

 divifion we double the number of our feries, yet the degree 

 of convergency is fo much the greater as amply to compen- 

 fate for the additional number of feries. Bcfides, we may 

 always fubdivide our greater arc, fo that one of its fubdivifions 

 fliall be the fame as the fmaller arc, in which cafe we do not 



increafe the number of feries. Thus ; arc to tan. — = arc 



2 



therefore arc to tan. 1 = 2 



I I 



to tan. 1- arc to tan. — 



3 7 



arc to tan. 1- arc to tan. 



3 



I 2 



= arc to tan. f- arc to tan. — 



7 II 



1 



Again 



therefore arc to tan 



arc to tan. — 

 3 



3 arc to tan. — -)- 2 arc to tan. 



and fo on to 



2 



+ 2 arc to tan. — •, 

 7 II 



any extent required, wliich might in courfe be purflied fo 

 far as to render the operation as fimple and as little laborious 

 as can be expefted in fuch kind of computations. Even 

 with thefe already mentioned, the circumference of the circle 

 might undoubtedly be computed to 200 places of decimals, 

 with Icfs labour than it colt Vieta to carry them to 10 places, 

 or Romanus to 15. 



The reader will obferve, that this approximation differs 

 from Machin's in nothing except the fimplicity and gene- 

 rality of the invcltigation ; for if we make the fucceffive 

 lubdivifion of the greater arc, fo as always to include in it 

 the fmaller one, we (hall find in our refults the identical 

 formula of Machin. 



Let us repeat our former expreflion 



tan. i 



tan. (a + i) 

 tan. a = . ^ — 



tan. a 



R 



Let tan. {a + b) = =-, tan. a = 

 R' R/-Ti 



tan. b 

 R> 



, tan.*= -,and 



where it is obvious that if tan. (a + b) and b be rational, 



tan. a will alfo be rational ; thus, if tan. {a i- i) = tan. 45° ^e (hall have generally — 



I, and tan. b = — , we have tan. 

 2 



-, and we fhall 



evidently have a fimilar refult, whatever rational fraiftion we 

 adume for tan. b. 



We (hall find, therefore, by the feries which gives the arc 

 in terms of the tangent, each of thefe arcs, the fum of which 

 will evidently be the meafure of the whole arc fought ; 

 whether the arcs themfelves, which belong to thefe tangents, 

 be rational or irrational, with relpedl to the whole arc of 

 which they foi'm the parts. Thefe tangents, fubllituted in 



t\)e general feries abore, give arc to tan. ( — J = 



- &c. 



Rr + Ti 



Let now tan. [a + b) = l, anfwering to arc 45^, fo that 

 R := I, and T := l ; atfume alfo r = I, then, by con- 

 (tantly fubftituting, in the general exprefTion, the values 

 found for R', R", R"', &c. and T', T", T'", &c. for R and 

 T refpeftively, we fhall have 



R' / - I 



T' 



R'^ 



"-£11 



TTJi" 



R^ _ ___^ 



T'" ~ /• -t- 4/' — 6/- - 4/ -(- I 



R' _ ." — 5/-* — l ot' ■+ IP/' -i- 5/ — I 



aw 



