LOGARITHMIC curve. If on the line A N (Plate 'III. Miscel. fig. 12) both ways in definitely extended, be taken A C, C E, E G, G I L, on the right hand. And also A g, g P, &c. on the left, all equal to one another. And, if at the points P, g. A, C, E, G, I, L, be erected to the right line, A N, the perpendiculars P S, g d, A C D, E G H, I K, L M, which let be continually proportional, and repre sent numbers, viz. A B, 1, C I), 10, E F, 100, &c. then shall we have two progres sions of lines, arithmetical and geometri-e cal : for the lines A C, A E, A G, &c. are in arithmetical progression, or as 1, 2, 3, 4, 5, &c. and so represent the logarithms to which the geometrical lines A C D, E F, &c. do correspond. For since A G is triple of the right line A C, the number G II shall be in the third place from unity, if C D be in the first: so, likewise, shall L M be in the fifth place, since A L = 5 A C. If the extremities of the propor tionals S d, B, D, F, &c. be joined by right lines, the figure SBML will become a polygon, consisting of more or less sides, according as there is more or less terms in the progression.
If' the parts A C, C E, E G, &c. be bisected in the points c, e, g,i, 1, and there be again raised the perpendiculars c d, g b, i k, I in, which are mean proportion als between A13, C D; C D, E F, &c. then there will arise a new series of propor tionals, 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 al, c d, from uni ty; viz. since A L is ten times greater than A c, 1- M shall be the tenth term of the series from unity ; and because A e is three times greater than A c, e f will be the third term of the series if c d be the first, and there shall be two mean pro portionals between A B and ef ; and be tween A 11 and L AI there will be nine mean proportionals. And if the extre.
mities of the lines B d, D f, 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 garitlimical.
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, IK, L M, such that the distance between the first and second be equal to the distance between the third ly.c1 the fourth ; let the distance from the second to the third be what it will, these numbers will be proportional. 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, AB:DC I K 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.