362 PHILOSOPHICAL TRANSACTIONS. [anNO 1747. 



XXIII. Let A =+ 9' - g" And a' = - a 



B = — q'" + q" + Aq' b' = — B — aa' ■ 



c = + 9" — q"' + Bq' — Aq" c' = — c — ba' — ab' 



D = — q^" + q''"' + cq' — Bq"' -\- Aq". d' = — D — ca' — bb' — Ac'. 



&c. &c. 



Tangent t = a + Aa^?—^ + Ba'r— " + caV— ' + va?r~^, &c. 

 Or = 1 + AN + bn'^ + CN* + dn* + en', &c. X a. 



Co-tangent t" = a-V^ + ace + s'a^r-^ + c'aV-* + D'a^r-^, &c. 



Or = rr + Aa' + b'no^ + c'n^a'^ + n'm^a^kc. X -. 



XXIV. Also let a = + q' And ^=-{-q" " 



y = + yv _ «9'" + (iq' y'=+ q"' - «9''+ (i'q" 



S = _ 9'" + a^v _ p^,/ + y^' J' = _ ^v//^. ^^'v_ p'^'v + ^y^ 



&c. &c. 



Secant/ = r -j- aaV- ' + P^'r-" + yaV"' + <ra»r-% &c. 



Or = 1 + aN + |3n' -l-yN^ + in", &c. X r. 



Co-secanty"=: c-V* + da + |3'aV ^ + y'a'r-* + ^a'r-«, &c. 



Or =1 rr -\- aaa + (3Naa + y'-a'^aa + S'n^aa, &c X -• where n = '^. 



a rr 



XXV. Putting/;^ = 4-; p"= ^p;p"'= ^p"; /'= i/)"'; Jb'= -^p'"; &c. n = Ji. 



Then arc a = 1 + i-p'n + ip^N' + -f/>" V + i/j'-'N^ &c. X j. • 

 Or =s + -t/^'an + -yi^BN + -fft^'CN + ip'^DN, &c. 



Or = 5 + ^; AN + i;] BN -I- I7J CN + ^7^ DN, &c. whcrc A, B, c, &c. 



stand for the respective preceding terms. 



XXVI. If V be the versed sine of an arc a, diameter being d,M =: -, k = V~ 



Then arc a = 1 + ji m -j- J^ am + ^7 bm + sis cm, 8cc. X k; a,b, c, &c. are as 

 before. 



XXVII. And putting n = - , a= t, b = an, c = bn, d = on, &c. 

 Then arc a = « — ^an + jbn — ^cn -f- -i^oN — ^-v en, &c. 



Or = 1 — iN + ^n' — ^N* + -^N* — TVN*r&c. X t. 



XXVIII. Also, if c be the chord of an arc (a) ; and n = j-.. 



Then arc a = c + rij an + j^j bn + |^' cn -|- ^l dn, &c. where a, b, c, &c. 

 stand for the respective preceding terms. 



Extract of a Letter, dated at Rome, Jug. 5, 1747, from Mr. Hoare, a young 

 Statuary, pursuing his Studies there, to his Brother Mr. Hoare, an eminent 



