igO ' PHILOSOPHICAI- TRANSACTIONS. [aNNO 1771. 



versed sine, to the like scale, as before, of the complement of the right ascen- 

 sion, of the point of the ecliptic on the meridian. And thus may be found the 

 fittest interval of time for the 3 calculations of the parallaxes, &c. I have above 

 proposed in general an hour; but in great eclipses it would be best to assume 

 this interval something greater, and in small eclipses less. 



XLVll. Of Logarithms, by the late Wm. Jones, Esq., F. R. S. Communicated 

 by John Robertson, Lib. R. S, p. 455. 



The following paper on the nature and construction of logarithms, was com- 

 muniaited to Mr. R. many years before, by that eminent mathematician the late 

 Wm. Jones, Esq. The familiar manner in which he explains, their nature, and 

 the great art with which he obtains the modes of computation, not being ex- 

 ceeded, if equalled, by any writer on this subject, may claim a place in thePhilos. 

 Trans. 



j . Any number may be expressed by some single power of the same radical 

 number. For every number whatever is placed somewhere in a scale of the se- 

 veral powers of some radical number r, whose indices arem — l, m — 2, m — 3, 

 &c. where not only the numbers r", r"-', r" ~ », &c. are expressed; but also 

 any intermediate number x is represented by r, with a proper index z. The 

 index z is called the logarithm of the number x. 



1. Hence, to find the logarithm z of any number x, is only to find what 

 power of the radical number r, in that scale, is equal to the number x; or to 

 find the index z of the power, in the equation x = r". 



3. The properties of logarithms are the same with the indices of powers; that 

 is, the sum or difference of the logarithms of two numbers, is the logarithm of 

 the product or quotient of those numbers. And therefore, n times the loga- 

 rithm of any number, is the logarithm of the nth power of that number. 



4. The relation of any number x, and its logarithm z, being given; to find 

 the relation of their least synchronal variation x and z. Put 1 -j- w for r, the 



radical number of any scale, and q = j-r;^- 



Let a = 9 + i9' -I- W + ^9*> &c.: /= K 



Then/f = xz shows the relation required. For x = r* = (l + r)". 



Now, let X and z flow so, that x becomes x -\- x, at the same time as z shall 

 become z -\- z. 



Then x -{- x = {I + ny + =" = {1 + ny X {I + ny = x X {1 + zq -\- -^zq-" 

 + 4-%' + W &c.) 



Therefore x = xz X (9 + W + if + -rg* &c.) = xza = xz X j. Con- 

 sequently /r ^ xz. 



5. If 1 -\-n = r = 10, as in the common logarithms of Briggs's form. 



