If on the line AN both ways indefinitely extended, be taken AC, CE, EG, GI, IL, on the right hand; and also AG, gP, Plate &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 AN, the perpendiculars PS, gd, AB, CD, EF, GH, IK, LM, which let be continually proportional, and represent numbers, viz. AB, 1; CD, 10; EF, 100, &c. then shall we have two progressions of lines, arithmetical and geometrical: for the lines AC, AE, AG, &c. are in arithmetical progression, or as 1, 2, 3, 4, 5, &c. and so represent the logarithms to which the geometrical lines AB, CD, EF, &c. do correspond. For since AG is triple of the first line AC, the number GH shall be in the third place from unity, if CD be in the first: so likewise shall LM be in the fifth place, since AL = 5 AC. If the extremities of the proportionals S, d, B, D, F, &c. be joined by right lines, the figures SBML will become a polygon, consisting of more or less sides, according as there are more or less terms in the progression.

If the parts AC, CE, EG, &c. be bifected in the points, c, e, g, i, l, and there be again raised the perpendiculars cd, ef, gh, ik, lm, which are mean proportionals between AB, CD; CD, EF, &c. then there will arise a new series of proportionals, whose terms, beginning from that which immediately follows unity, are double of those in the first series, and the difference of the terms is become less, and approach nearer to a ratio of equality, than before. Likewise, in this new series, the right lines AL, Ac, express the distances of the terms LM, cd, from unity, viz. since AL is ten times greater than Ac, LM shall be the tenth term of the series from unity: and because Ac is three times greater than Ac, ef will be the third term of the series if cd be the first; and there shall be two mean proportionals between AB and ef; and between AB and LM there will be nine mean proportionals. And if the extremities of the lines Bd, Df, Fh, &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 continual- Logarithms 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 logarithmical.

It is manifest from this description of the logarithmic curve, that all numbers at equal distances are continually proportional. It is also plain, that if there be four numbers, AB, CD, IK, LM, such that the distance between the first and second be equal to the distance between the third and the fourth, let the distance from the second to the third be what it will, these numbers will be proportional. For because the distances AC, IL, are equal, AB shall be to the increment DT, as IK is to the increment MT. Wherefore, by composition, \( \frac{AB}{DC} = \frac{IK}{ML} \). And, contrariwise, if four numbers be proportional, the distance between the first and second shall be equal to the distance between the third and fourth.

The distance between any two numbers, 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 proportions, proceeding from one number to another, and defines the number of equal ratios by the composition whereof the ratio of numbers is known.