LOGARITHMS. 



tTie above method made ufe of by different authors, may 

 be feen in the introduftion to Dr. Hutton's Mathematical 

 Tables. It is luirieccflary to obferve, that thefe computa- 

 tions were only required for prime numbers ; for theie being 

 once obtained, the logarithms of all other numbers were 

 found by firaple addition. At prcfcnt, we have only fpokon 

 of logarithms as they are applicable to numerical compu- 

 tations. But they are aifo of very exteniive ufe in the 

 higher geometry, particularly in the doftrine of fluxions, 

 and it will not be amifs, before we quit this part of the fub- 

 jeft. to give an idea of the way in which they have been 

 confidered by writers on the latter fcience. Maclaurin, in 

 his Treatife of Fluxions, has explained the nature and gc- 

 iiefis of logarithms, agreeably to the notion of their firll 

 inventor, lord Napier, delivered in his Mirif. Logar. Canon. 

 Defcript. He thei-e fuppofes logarithn-s, and the quanti- 

 ties to which they correfpond, to be generated by the mo- 

 tion of a point. If this point moves over equal fpaces in 

 equal times, the hne defcinbed by it increafes equally. 



Again, aline decreafcs proportionally when the point that 

 moves over it dofcribes iucli parts in equal times as are always 

 in th.e fame con llant ratio to the lines from which they are 

 fubducled, or to the diftances of that point at the beginning 

 of thofe times, from a given term in the line. In like man- 

 Jier, a line may increafe proportionally, if in equal time 

 the moving point defcribes fpaces proportional to its dif- 

 tances from a certain term, at the beginning of each time. 

 Thus, in the firll cafe, let aChe to ao, ceito co, tU'to Jo, 

 ef tu 



a e d e f g a 



O R 



/ 



eo, fg to fo. alivays in the fame ratio of Q R to Q S ; and 

 fuppofe that the pointy fets out tromo, defcribingar, c J. de, 

 ef,Jg, in equal parts of the time ; and let the i'pace defcribed 

 by p, in any given time, be always in the fame ratio to the 

 d'.llance of p from o, at the beginning of that time, then will 

 the right hneiz? decreat; proportionally ; and the lines a », 

 $ 0, do, e 0, fo, &c. or the diftances of the point p from o, 

 at equal fucceeding intervals of time, are in a contuiued geo- 

 metrical progreflion. 



In like manner, the line oa increafes proportionally, if the 

 point p in equal times defcribes fpaces a c, c d, de, ef, fg, 

 &c. fo that 3 r is to (7 0, c d to c o, de to do, &c. in a con- 

 ilant ratio. 



C D E F 

 A — _B 



d 



f 



If we now fuppofe a point P defcribing the line A B with 

 an uniform motion, equal to that with which p fets out from 

 «, in defcribing the line a o, while the pointy defcribes a line 

 increafing or decreafmg proportionally, the line A P defcribed 

 by P with this uniform motion, in the fame time that a a. by 

 increafing or decreafmg proportionally, becomes equal to 

 op, is the logarithm of op. Thus AC, A D, A E, &c. 

 are the logarithms of o c, o d, oe, &c. refpeftively ; and 

 « a is the quantity whofe logarithm is fuppofed equal to 

 nothing. 



We have here abftrafted from numbers, that the doftrine 

 Vol.. XXI. 



may be the more general ; but it is plain, that if A C, A D, 

 A E, &c. be fuppofed i, 2, 3, Sec. in arithmetic progref- 

 lion ; DC, d, c, &c. will be in geometric progreflion ; 

 and that the logarithm of 9 a, which may be taken for unity, 

 is nothing. 



Lord Napier, in his firft fcheme of logarithms, fuppofes, 

 that while op increafes or decreafcs proportionally, the uni- 

 form motion of the point P, by which ih-.- logarithm of op 

 is generated, is equal to the velocity of p dXa.; that is, at 

 the term of time when the logarithms begin to be generated. 

 Hence logarithms, formed after this model, are called Napier's 

 logarithms, and fometimcs mitural logarithms. 



When the ratio is given, the point /> defcribes the differ, 

 ence of the terms of the ratio in the fame time. When a 

 ratio is. duplicate of another ratio, the pointy defcribes the 

 difference of the terms in a double time. When a ratio is 

 triplicate of another, it defcribes t!ie difference of the terms 

 in a triple time ; and fo on. Alfo, when a ratio is com- 

 pounded of two or more i-atios, the point p defcribes tlie 

 difference of the terms of that ratio, in a time equal to the 

 fum of the times in which it defcribes the differences of thfe 

 terms of the fimple ratios of which it is compounded. And 

 what is here faid of the times of tiie motion of p, when op 

 increafes proportionally, is to be applied to the fpaces de- 

 fcribed by P in tliofe times, with its uniform motion. 



Hence the chief properties of logarithms are deduced. 

 They are the meafures of ratios. The excefs of the lo- 

 garithm of the antecedent above the logarithm of the con- 

 Icquent meafures the ratio of thofe terms. The meafure 

 of the ratio of a greater quantity to a leffer is pofitive, as 

 this ratio compounded with any other ratio increafes it. 

 The ratio of equality, compounded with any other ratio, 

 neither increafes nor diminifhes it ; and its meafure is nothing. 

 The meafure of the ratio of a leffer quantity to a greater is 

 negative, as this ratio compounded witii any other ratio di- 

 miniflies it. The ratio of any quantity A to unity, com- 

 pounded with the ratio of unity to A, produces the ratio 

 of A to A, or the ratio of equality ; and the meafures of 

 thofe two ratios deftroy each other, when added together: 

 fo that when the one is conlidercd as pofitive, the other is to 

 be confidered as negative. 



When op incrcales proportionally, the motion of ^ is per- 

 petually accelerated ; and, on the contrary, when op de- 

 creaies proportionally, the motion of p is perpetually re- 

 tarded. 



If the velocity of the point p be always as the diffanc? 

 op, then will this line mcreafe or decreafe in the manner 

 fuppofed by lord Napier : and the velocity of the point 

 p being the fluxion of the line op, will always vary in the 

 fame ratio as this quantity itfelf. See Maclaurin's Flux, 

 art. 151 — 160. 



The fluxion of any quantity is to the fluxion of its lo- 

 garithm, as the quantity itfelf is to unity. 



Hence the fluxion of the loafarithm of .v will be - 



For .r : I : : A- : — =: the fluxion of the logarithm required. 



When op increafes proportionally, the increments gene- 

 rated in any equal times, are accurately in the fame ratio as 

 the velocities cf p, or tlie fluxions of op, at the beginning, 

 end, x>-c at any fimilar terms of thofe times. 



When op increafes, or decreafes propwtionally, the flux- 

 ions of this line, in all the higher orders, increafe or decreafe 

 in the fame proportion as the line itfelf increafes or decreafcs ; 

 fo that one rule ferves for comparing together thofe of any 

 kind at different terms of time ; and iu this cafe we never 

 J.. 1 arrive 



