RECTIFICATION. 



14 19 ^ 



L ) X v' »■) &c. And confequently z = y + 



4f'° 16 c"/ 



I' v^ / b- i'\ y' / 1" t* , i' \ 



14c' 



By the very fame way of proceeding, the arc of an tOipfis 

 may be found, the equations of the two curves differing in 

 nothing but their figns. 



IV. To reaify the Jpiral of Archimedes. The value of / 



(AT./f. 13.) being denoted by —^J-—^ (fee Tan- 

 gent of the Spiral, &c.) we have z ( = 7") = ■> 



^ ^ ILjLi ; which fluxion being the fame as that exprefl'- 



b 

 jng the arc of the common parabola (Prob. II.), by inferting 



in the cxpreffion i = — x (a' + 4/)^, b for \ a, its 



fluent will, therefore, be truly rcprefented by the meafure 





— - + -i— 

 i.3 2 



= (,-^. 



2.3 2 



T 



.4.5 



.4.5 



7 



+ 



4.6.7 



+ 



5/ 



— + 



_3 

 2 



_ 3 

 2.4.6 



/' 

 9 



X V * 



4- &C.) X >> 



( 



1 — r s' 



+ 



+ 



3 



+ &c.) X t 

 r') 



+ &c.) 



X r 



s 2 • 3 .f 2 . 4. 5 r 



where ;• is retained in the latter for fake of analogy. 



It is obvious, therefore, that the arc may be computed 

 by any of thefe, in terms either of the fine or verfed fine, 

 tangent or fecant, and confequently alfo in terms of the co- 

 fine, co-tangent, co-fecant, &c. 



Thus, in the firfl, taking x = — , which h the verfed 



fine of 60^, we have 

 I 



arc 60'^ = ( I -I- 



-f 



of the faid arc, or by 



hy v/(^' f/) 



+ i * X hyp. log 



y '^ ^ S — ■^ ' , the value there found, by making the 

 b 



propofed fubftitution. 



V. To reaify the involute of a circle, whofe nature is fuch, 

 that the part P R [fg. 14.) of the tangent intercepted by 

 the point of contafi; and the perpendicular C P, is every 

 where equal to the radius C O of the generating circle. 



In this cafe i t— ^— j = '—-, we obtain e = 

 - which, correfted by making _)■ = a = A C, becomes 



cp- 



2 a 



ylz 



2 



AR. 



_ _i / C P' \ 



— I ^ - J, the true meafure of the required arc 



VI. To find the length of a circular arc. — This may be ex- 

 prefled either in terms of the fine, cofine, verfed fine, or 

 any other trigonometrical line, as follows. Firft, 



Let the verfed fine = .v, the fine =; y, radius = r, and arc 

 = 2, then, by the property of the circle, y^ ^^ 2 r x — .v- 



^ r^ + f 



or, putting tangent = / 



and fecant = s, gives - - y'^ = '- p^ j-' 



as are readily deduced from the known properties of the 

 circle. 



Nov? , by means of thefe values of y'', or of 2 r .v — x", 

 and the general equation i =; ^/ (i^ + })> we readily 

 draw the following values of 2;, "viz. 



(2. 



.V) V (' 



f) 



r' + t' ^ (x' - r) 



the fluents of which can only be found in feries, which 

 iu-e as follows ; making radius r ■=. i, viz. 



2.3 2'.3.4 



-^-^^ 



J.; 5 

 •4 



6.'/ 2 



In the fecond, afTuming y ■= — z= fin. 30°, we have 



arc 30° = ( I -I 



I 



2'. 3 2' 



.4. 



+ 



3 -5 



-) X -• 



5 2'. 4. 6. 7' 2 

 In the third, alfuming ; = i = tang. 45°, we obtain 



/'III 



arc 45° =( I -1 1- — 



3 5 7 9 II 

 In the fourth, afluming ;• = 2 = fee. 60°, we get 



-— -f&c.).r 



arc 60° = ( 



2 — 



'- + ''- 



+ 



3(2^- i^) 



-1- &c.).r 



2 • 2*. 3 2°. 4. 5 



Then multiplying the numbers obtained from thefe feries 

 by the number of times that the arc is contained m the whole 

 circumference, will give the circumference required. 



But no one of thefe feries is fufficiently convergent for 

 afcertaining the circumference of the circle to a great degree 

 of accuracy, and therefore other methods have been con- 

 trived, in order to produce feries better calculated for this 

 purpofe, of which that of Machin has been the moft popular ; 

 though it does not appear that he employed it in his cele- 

 brated quadrature or reftification, in which he found the 

 circumference to one hundred places of figures. 



In order to render thefe feries more converging, it is ob- 

 vious that lefs arcs muft be aflumed, and the difficulty con- 

 fifts only in finding the tangents (for example, ufing that 

 feries) of a fmall arc, which may be expreffed in numbers 

 that are tolerably manageable in the general feries. 



For this purpofe Machin, knowing the tangent of 45° to 

 be I, and that the tangent of an arc being known, any 

 multiple of it is readily found, confidered, that if there 

 were aflumed fome fmall fimple number for the tangent of an 

 arc, and then the tangent of the double arc were continually 

 taken, until a tangent be found nearly equal to i, the tan- 

 gent of 45° : by taking the tangent of this fmall difference 

 between 45° and the multiple arc, there would be had two 

 very fmall tangents, the one of the firft arc, and the other 

 of this difference. Then computing the arc to thefe tan- 

 gents, whether the meafure of them in degrees, &c. were 

 known or not, the whole arc of 45^ would become known ; 

 •viz. by multiplying the firft by the aflumed multiple, and 

 adding the laft arc to the produft, if the tangent of the 

 multiple arc were lefs than i , or the arc itfelf lefs than 45° ; 

 but fubtrading it if greater. 



Having 



