OF THE NATURE and ARITHMETICK OF LOGARITHMS. CH A P. I.“ Of the ORIGIN and NATURE of LOGARITHMS. A S in Geometry, the Magnitudes of Lines are pound Numbers by Lines, viz. by assuming some Line which may represent Unity, and the Double thereof; the Number 2, the Triple 3, the one half, tie Fraction, and so on. And thus the Genesis and Properties of fome certain Numbers are better conceived, and more clearly considered, than can be done by abstract Numbers. Hence, if any Line a * be drawn into itself, the Quan-* Fig. 1. tity a' produced thereby, is not to be taken as one of two Dimensions, or as a geometrical Square, whose Side is the Line a, but as a Line that is a third Pro, portional to some Line taken for Unity and the Line a. So likewise, if a' be multiplied by a, the Product a', will not be a Quantity of three Dimensions, or a Geometrical Cube, but a Line that is the fourth Term in a Geometrical Progression, whose first Term is 1, and second a; for the Terms 1, a, a, a', a', a', a', &c. are in the continual Ratio of 1 to a. And the Indices" affixed to the Terms, Ahew the Place or Y 3 Dir an Distance that every Term is from Unity. For Example, as is in the fifth Place from Unity, qó is the fixth, or fix times more diftant from Unity than an or a', which immediately follows Unity. If between the Terms i and a, there be put a mean Proportional which is d, the Index of this will be for its Distance from Unity will be one half of the Distance of a from Unity; and fo a { may be written ✓ a. And if a mean proportional be put between a and a, the Index thereof will be į or }, for its Distance will be fefquialteral of the Distance of a from Unity. If there be two mean Proportionals put between i and a; the first of them is the Cube Root of a, whose Index must be , for that Term is distant from Unity only by a third Part of the Distance of a from Unity; and fo the Cube Root must be expressed by a', Hence, the Index of Unity is o, for Unity is not distant from itself. The fame Series of Quantities, geometrically proportional, may be both ways continued, as well descending towards the Left Hand, as ascending towards the Right ; for the Terms i, a, a, a, at, a, a, a, a, a, a, Edc. are all in the fame Geometrical Progression. And since the Distance of a from Unity is towards the Right Hand, and positive or +1, the Distance equal to that on the contrary Side, viz. the Distance of the Term.will be Negative or — 1, i 1 which shall be the Index of the Term L for which 2 and a2 may be written a!. So likewise in the Term a The Index -- 2 thews that that Term stands in the second Place from Unity towards the Left Hand, and the Terms are of the same Value. Also 973 is the fame as 1. For these negative Indices shew that the Terms belonging to them, go from Unity the contrary Way to that by which the Terms whose Indices are positive, do. These Things premised. If If on the Line AN, both Ways indefinitely extended, be taken, AC, CE, EG, GI, IL, on the right Hand. And also Ar, ru, &c. on the left, all equal to one another. And if at the Points, 11, T, A, C, E, G, I, L, be erected to the right Line AN, the Perpendiculars I , r4, A B, CD, EF, GH, IK, LM, which let be continually proportional, and represent Numbers, whereof A B is Unity. The Lines A C, AE, AG, AI, AL-A, - AI, respectively express the Distances of the Numbers from Unity, or the Place and Order that every Number obtains in the Series of Geometrical Proportionals, according as it is distant from Unity. So since AG is triple of the right Line A C, the Number GH shall be in the third Place from Unity, if. CD be in the first: So likewise fhall LM be in the fifth Place, since ALE 5 AC. If the Extremities of the Proportionals, 2, 0, B, D, F, H, K, M, be joined by right Lines, the Figure TIL M will become a Polygon confifting of more or lefs Sides, according as there are more or lefs Terms in the Progression. If the Parts AC, CE, EG, GI, IL, be bisected in the Points c, 6, 8, i, l, and there be again raised the Perpendiculars cd, ef, gh, ik, lm, which are mean Proportionals between AB, CD; CD, EF;EF GH; GHIK; IK, LM; 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 are become lefs, and approach nearer to a Ratio of Equality than before. Likewise in this new Series, the Right Lines A L, AC, express the Diftances of the Terms L M, CD, from Unity, viz. Since AL is ten times greater than Ac, L M shall be the tenth Term of the Series from Unity: And because Ae 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 A B and ef, and between A B, and LM, there will be nine mean Proportionals. And if the Extremities of the Lines Bd Df Fh H, &c. be joined by right Lines, there will be a new Polygon made, consisting of more, but shorter Sides than the last. If, again the Distances Ac, cC, Cè, e E, &c. be supposed to be bisected, and mean Proportionals, between every two of the Terms, be conceived to be put at those middle Distances; then there will arise another Series of Proportionals, containing double the Number of Terms from Unity than the former does ; but the Differences of the Terms will be less, and if the Extremities of the Terms be joined, the Number of the Sides of the Polygon will be augmented according to the Number of Terms; and the sides thereof will be lesser, because of Ahe Diminution of the Distances of the Terms from each other. Now in this new Series, the Distances AL, AC, &c. will determine the Orders or Places of the Terms; viz. if AL be five times greater than A C, and CD be the fourth Term of the Series from Unity, then LM will be the twentieth Term front Unity. If in this manner mean Proportionals be continually placed between every two Terms, the Number of Terms at laft will be made so great, as allo 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 fo lefsened, as to become less than any given right Line ; and consequently the Polygon will be changed into a curve-lind Figure; for any curve-lin'd 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; in which, if Numbers be represented by right Lines standing at right Angles to the Axis AN, the Portion of the Axis intercepted between any Number and Unity, Mews the Place or Order that that Number obtains in the Series of Geometrical Proportionals, distant from each other by equal Intervals. For Example, if AL be five times greater than AC, and there are a thousand Terms in continual Proportion from Unity to L M; then will there be two Hundred Terms of the fame Series from Unity to CD, or CD shall be the two hundredth Term of the Series from Unity; and let the Number of Terms from A B to L M be supposed what it will, then the Number of Terms from AB to CD, will be one fifth Part of that Number, TIe The Logarithmical Curve may also be conceived to be described by two Motions, one of which is equable, and the other accelerated, or retarded, according to a given Ratio. For Example, if the right Line AB, moves uniformly along the Line AN, fo that the End thereof describes equal suces in equal Times; and, in the mean Time, the said Line AB fo encreases, that the Increments thereof, generated in equal Times, be proportional to the whole encreasing Line; that is, if A B, in going forward to cd, be encreafed by the Increment od, and in an equal Time when it is come to CD, the Increment thereof is Dp, and Dp to dc is as do is to AB, that is, if the Increments generated in equal Times are always proportional to the Wholes ; or, if the Line AB moving the contrary Way, diminishes in a constant Ratio, so that while it goes thro' the equal Spaces, the Decrements AB - PA PA, nę, are Proportionals to AB, ra. Then the End of the Line encreasing or decreasing in the said Manner, describes the Logarithmical Curve : For fince AB:do :: dc:DD::DC:f9, it shall be (by Composition of Ratio) as AB:dc::dc:DC:: DC:f e, and so on. X X Х By these two Motions, viz. the one equable, and It is manifest from this Description of the Logarithmick 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 Dirtance between the first and second, bę equal to the Distance between the third and the fourth: Let the Distance from the second to the third be what it will, thefe Numbers will be proportional. For because the Distances AC, IL, are equal, AB shall be to the Increment Ds, as IK is to the Increment MT. Wherefore ( by Composition) AB:ĐC::IK:ML. And contrariwise, if four Numbers be proportional, the Distance between the firft and the second, shall be |