VOL. XXX.] PHILOSOPHICAL TRANSACTIOXS. 303 



Hence by the 2d irrational formula, v = y/l^2^^ + 2d^ _ IM^i^P ^ 



xaciiivc uy , V 5,18419* ' 5,18419 5,18419 



0,01 5 16 ; which gives y = z + t; = 2,315l6, which is true to six places. If 

 you desire it more exact than to the extent of the tables of logarithms, 

 taking z = 2, 3 1 5 1 6 f or the next supposition, the calculation must be repeated 

 by computing zz + 1 K* to a sufficient number of places ; which must be done 

 by the binomial series, or by making a logarithm on purpose, true to as many 

 places as are necessary. 



Exam. 2. — For another example, let it be required to find the number whose 

 logarithm is 0,29, supposing we had no other table of logarithms but Mr. 

 Sharp's, of 200 logarithms to a great many places. This amounts to the re- 

 solving this equation Li/ = 0,29, ^^ ^j/ — Oj^Q = O. Hence therefore we have 

 X = Ix — 0,29, i' = -, (a being the modulus belonging to the table we use, 



viz. 0,43429448 1 9 &c.) ^ = ^, i = ^, i = ::i|^, &c. In this case, be- 



z z z 



cause cV- has a negative sign, changing the signs of all the co-efficients, the 

 canon for v will b e found in the 4th case, which in the irrational form 



e-ives V =. - — \/ —, -\- —. -zirnv — -——rrv &c. = z — 



° X ^ x'^ ' X 2.3a: 2.3.4X- 



> / 2 , 2/.Z — 0,58 ,, 7" , 2f'' 2i'* , 2t)^ T *u- .. -j r. 



\/ 2^-j X 2+-5 Th-^" T3 ^^' ^^ ^"'S ^^se, to avoid often 



CI oZ T'S O4 



dividing by z, it will be most convenient to compute -, which is got from this 



,. V , . /' ] 2I.Z — 0,58 , 2t)' 2v* , 2t)^ J mu 



equation - = 1 - \/ ] -f — ^ f. _ _ _ + _ &c. The nearest 



logarithm, in the tables proposed, to the proposed logarithm 0,29, is 

 0,2900346114, its number being 1,93. Therefore for the first supposition 

 taking z = 1,95, we have x ( = /.z — 0,29 = 0,290034dll4 — 0,29) = 



0,0000346.14, and 5£!^-- = 2^2222|£|i = 0,000.5939139, and 1 + 



2l z 58 



— ^ — = 1,00015939139. Whence for the hrst approximation we have 



- = 1 — ^1,00015939139^ — 0,00007969247, and i; = — 0,00015540032, 

 and 7/ = z -i- V = 1,94984459968. Which is true to eleven places, and 

 may easily be corrected by the terins -— &c. which I leave to the readers 

 curiosity. 



Being upon the subject of approximations, it may not be amiss to set down 

 here two approximations I have formerly hit upon. The one is a series of 

 terms for expressing the root of any quadratic equation : and the other is a 

 particular method of approximating in the invention of logarithms, which has 



