1

belter, followed by Sherwin in his Mathematical Tables, published at London in 1705; wherein are the Logarithms from 1 to 101000, confifting of feven Places of Figures. To which are fubjoined the Differences, and proportional Parts, by means of which, may be found easily the Logarithms of Numbers to 10000000; obferving, at the fame Time, that thefe Logarithms confift only of feven Places of Figures. Here are alfo the Sines, Tangents, and Secants, with their Logarithm, and Differences for every Degree and Minute of the Quadrant, with fome other Tables of Ufe in practical Mathematics.

OF

OF THE

Nature and Arithmetic

OF

LOGARITHMS.

CHA P. I.

Of the ORIGIN and NATURE of
LOGARITHM S.

A

S in Geometry the Magnitudes of Lines are often defined by Numbers; fo, likewife, on the other Hand, it is fometimes expedient to expound Numbers by Lines, viz. by affuming fome Line which may reprefent Unity; the Double thereof, the Number 2; the Triple, 3; the one Half, the Fraction; and fo on. And thus the Genefis and Properties of fome certain Numbers are better conceived, and more clearly confidered, than can be done by abftract Numbers.

Hence, if any Line a* be drawn into itself, the Quan- Fig. 1. tity a2, produced thereby, is not to be taken as one of two Dimenfions, or as a Geometrical Square, whose Side is the Line a, but as a Line that is a third Proportional to fome Line taken for Unity, and the Line a. So, likewife, if a be multiplied by a, the Product a3 will not be a Quantity of three Dimenfions, or a Geometrical Cube, but a Line that is the fourth Term in a Geometrical Progreffion, whofe first Term is 1, and fecond a; for the Terms 1, a, a2, a3, aa, a3, ao, a7, &c. are in the continual Ratio of 1 to a. And the Indices affixed to the Terms fhew the Place or Distance that every Term is from Unity. For Ex

ample,

ample, as is in the fifth Place from Unity, a in the fixth, or fix Times more diftant from Unity than a, or a', which immediately follows Unity.

If, between the Terms 1 and a, there be put a mean Proportional, which is ✔a, 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 fora. And if a mean Proportional be put between a and a2, the Index thereof will be 1, or ; for its Diftance will be fefquialteral of the Diftance 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 Diftant from Unity only by a third Part of the Distance of a from Unity;

Hence

and fo the Cube Root may be expreffed by a. 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 afcending towards

a3, aa, a3, &c. are all in the fame Geometrical Progreffion. And fince the Distance of a from Unity is towards the Right Hand, and pofitive or + 1, the Distance equal to that on the contrary Side, viz. the

I

Distance of the Term-, will be negative, or-I,

a

I

which fhall be the Index of the Term-, for which

a

2

may be written a . So likewife in the Terms a the Index-2 fhews, that the Term ftands in the second Place from Unity towards the Left Hand, and the Ex

I

preffions a2 and -are of the fame Value. Alfo a 3

that the Terms belonging to them go from Unity the

6

contrary

contrary Way to that by which the Terms, whose Indices are pofitive, do. Thefe Things premised,

If on the Line AN, both Ways indefinitely extended, be taken A C, CE, EG, GI, IL, on the Right Hand; and also Aг, III, &c. on the Left; all equal to one another; and if, at the Points II, г, A, C,¡E, G, I, L, be, erected to the Right Line A N, the Perpendiculars П, гa, AB, CD, EF, GH, IK, LM, which let be continually proportional, and reprefent Numbers, whereof A B is Unity: The Lines A C, AE, AG, AI, AL,-Aг,-AП, refpectively exprefs the Diftances of the Numbers from Unity, or the Place and Order that every Number obtains in the Series of Geometrical Proportionals, according as it is diftant from Unity. So fince AG is triple of the Right Line A C, the Number G H fhall be in the third Place from Unity, if CD be in the firft: So likewife fhall L M be in the fifth Place, fince AL=5A C. If the Extremities of the Proportionals, 2, 4, B, D, F, H, K, M, be joined by Right Lines, the Figure zn L M will become a Polygon confifting of more or lefs Sides, according as there are more or lefs Terms in the Progreffion.

If the Parts AC, CE, EG, GI, I L, be bifected in the Points c, e, g, i, 4, and there be again raised the Perpendiculars cd, ef, gb, ik, lm, which are mean Proportionals between AB, CD; CD, EF; EF, GH;GH,IK; IK, L M; then there will arife a new Series of Proportionals, whofe Terms, beginning from that which immediately follows Unity, are double of thofe in the firft Series, and the Differences of the Terms are become lefs, and approach nearer to a Ratio of Equality than before. Likewife in this new Series, the Right Lines A L, A C, exprefs the Distances of the Terms LM, CD, from Unity; viz. fince AL is ten Times greater than A c, L M fhall be the tenth Term of the Series from Unity: And because A e is three Times greater than A c, ef will be the third Term of the Series, if cd be the firft; and there fhall be two mean Proportionals between A B and ef; and between AB and L M there will be nine mean Proportionals.

And if the Extremities of the faid Lines, viz. B, d, D, f, F, h, H, &c. be joined by Right Lines, there will be a new Polygon made, confifting of more, but fhorter Sides than the last.

If,

If, again, the Diftances A c, cC, Ce, e E, &c. be fuppofed to be bifected, and mean Proportionals between every two of the Terms be conceived to be put at thofe middle Distances; then there will arife another Series of Proportionals, containing double the Number of Terms from Unity than the former does ; but the Difference of the Terms will be lefs; 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 leffer, because of the Diminution of the Dif tances 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 A L be five Times greater than A C, and CD be the fourth Term of the Series from Unity, then L M will be the twentieth Term from Unity.

If in this manner mean Proportionals be continually placed between every two Terms, the Number of Terms at laft will be made fo 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 fo leffened, as to become less than any given Right Line; and confequently the Polygon will be changed into a curve-lined Figure; for any curveJined Figure may be conceived as a Polygon, whofe Sides are infinitely fmall, and infinite in Number.

A Curve defcribed after this manner is called Logarithmetical; in which, if Numbers be reprefented by Right Lines ftanding ar Right Angles to the Axis AN, the Portion of the Axis intercepted between any Number and Unity fhews the Place or Order, which that Number obtains in the Series of Geometrical Proportionals, diftant 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 fhall be the two hundredth Term of the Series from Unity; and let the Number of Terms from A B to L M be fuppofed what it will, then the Number of Terms from A B to CD will be one fifth Part of that Number,

The