LOGARITHMS. 



tkmals, whose terms beginning from that 

 which immediately follows unity, are dou- 

 ble of those in the first series, and the 

 difference of the terms are become less, 

 and approach nearer to a ratio of equality 

 than before. Likewise, in this new se- 

 ries, the right lines A L, A c, express the 

 distances of the terms L At, c d, from uni- 

 ty ; viz. since A L is ten times greater 

 than A c, L M shall be the tenth term of 

 the series from unity ; and because A e 

 is three times greater than A c, e/ will be 

 the third term of the series [fed be the 

 first, and there shall be two mean pro- 

 portionals between A B and ef\ and be- 

 tween A B and L AI there will be nine 

 mean proportionals. And if the extre- 

 mities of the lines B rf, D/, F h, &c. be 

 joined by right lines, there will be a new 

 polygon made, consisting of more but 

 shorter sides than the last. 



If, in this manner, mean proportionals 

 be continually placed between every two 

 terms, the number of terms at last will 

 be made -so great, as also the number of 

 the sides of the polygon, as to be greater 

 than any given number, or to be infinite; 

 and every side of the polygon so lessened, 

 as to become less than any given right 

 line ; and consequently the polygon will 

 be changed into a curve lined figure ; for 

 any curve-lined figure may be conceived 

 as a polygon, whose sides are infinitely 

 small and infinite in number. A curve 

 described after this manner, is called lo- 

 garithmical. 



It is manifest from this description of 

 the logarithmic curve, that all numbers at 

 equal 'distances are continually propor- 

 tional. It is also plain, that if there be 

 four numbers, A B, C D, I K, L M> such 

 that the distance between the first and 

 second be equal to the distance between 

 the third a;,d the fourth ; let the distance 

 from the second to the third be what it 

 will, these numbers will be proporti6nal. 

 For because the distances A C, I L, are 

 equal, A B shall be to the increment D s, 

 as I K is to the increment M T. Where- 

 fore, by composition, A B : D C : : IK : 

 M L. And, contrarywise, if four num- 

 bers be proportional, the distance be- 

 tween the first and second shall be equal 

 to the distance between the third and 

 fourth. 



The distance between any two num- 

 bers is called the logarithm of the ratio 

 of those numbers; and, indeed, doth not 

 measure the ratio itself, but the number 

 of terms in a given series of geometrical 

 proportionals, proceeding from one num- 

 ber to another, and defines the number 



of equal ratios by the composition where- 

 of the ratio of numbers are known. 



LOGARITHMS, are the indexes or ex- 

 ponents (mostly whole numbers and de- 

 cimal fractions, consisting of seven places 

 of figures at least) of the powers or roots 

 (chiefly broken) of a given number ; yet 

 such indexes or exponents, that the seve- 

 ral powers or roots they express, are the 

 natural numbers 1, 2, 3, 4, 5> &c. to 10 

 or 100000, &c. (as if the given number be 

 10, and its index be assumed 1.0000000, 

 then the 0.0000000 root of 10, which is 1, 

 will be the logarithm of 1; the 0.301036 root 

 of 10, which is 2, will be the logarithm of 

 2 ; the 0.477121 root of 10, which is 3, will 

 be the logarithm of 3; the 0.612060 root 

 of 10, the logarithm of 4; the 1.041393 

 power of 10, the logarithm of 11 ; the 

 1 079181 power of 10, the logarithm of 

 12, &c.) being chiefly contrived for ease 

 and expedition in performing of arithme- 

 tical operations in large numbers, and in 

 trigonometrical calculations; but they have 

 likewise been found of extensive service 

 in the higher geometry, particularly in 

 the method of fluxions. They are gene- 

 rally founded on this consideration, that 

 if there be any row of geometrical pro- 

 portional numbers, as 1, 2, 4, 8, 16, 32, 

 64, 128, 256, &c. or 1, 10, 100, 1000, 

 10000, &c. And as many arithmetical 

 progressional numbers adapted to them, 

 or set over them, beginning with 0. 



. 5 0, 1, 2, 3, 4, 5, 6, 7, &c. 

 tnus l 1, 2, 4, 8, 16, 32, 64, 128, &c. 

 1, 2, 3, 4, &c. 

 10, 100, 1000, 10000, &c 



Then will the sum of any two of these 

 arithmetical progressionals, added toge- 

 ther, be that arithmetical progressional 

 which answers to, or stands over the ge- 

 ometrical progressional, which is the pro- 

 duct of the two geometrical progression- 

 als, over which the two assumed arithme- 

 tical progressionals stand : again, if those 

 arithmetical progressionals be subtracted 

 from each other, the remainder will be 

 the arithmetical progressional standing 

 over that geometrical progressional, 

 which is the quotient of the division of the 

 two geometrical progressionals belong, 

 ing to the two first assumed arithmetical 

 progressionals ; and the double, triple, 

 &c. of any one of the arithmetical pro- 

 gressionals will be the arithmetical pro. 

 gressional standing over the square, cube, 

 &c. of that geometrical progression which 

 the assumed arithmetical progressional 

 stands over, as well as the one-half, one- 

 third, &c. of that arithmetical progres- 



