the Differences and proportional Parts, by Means of which may be found eafily the Logarithms of Numbers to 10000000, obferving at the fame Time that thefe Logarithms confift only of 7 Places of Figures: Here are alfo the Sines, Tangents, and Secants, with the Logarithms and Dif ferences for every Degree and Minute of the Quadrant, with fome other Tables of Ufe in practical Mathematicks.

OF THE

NATURE and ARITHMETICK

OF

LOGARITHMS.

CHA P. I.

Of the ORIGIN and NATURE of
LOGARITHMS.

A

S in Geometry, the Magnitudes of Lines are often defined by Numbers; fo likewise on the other hand, it is fometimes expedient to expound Numbers by Lines, viz. by affuming fome Line which may represent Unity, and the Double thereof; the Number 2, the Triple 3, the one half, the Fraction, and so 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. 1a tity a 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 a, will not be a Quantity of three Dimensions, or a Geometrical Cube, but a Line that is the fourth Term in a Geometrical Progreffion, whofe firft Term is 1, and fecond a; for the Terms 1, a, a2, a', a', a3, a', &c. are in the continual Ratio of 1 to a. And the Indices affixed to the Terms, fhew the Place or

а,

Y 3

T

Dif

Distance that every Term is from Unity. For Example, as is in the fifth Place from Unity, a is the fixth, or fix times more diftant from Unity than a, or a', which immediately follows Unity.

If between the Terms 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 ✔a. And if a mean proportional be put between a and a, the Index thereof will be or, for its Distance will be fefquialteral of the Distance of a from Unity.

If there be two mean Proportionals put between 1 and a; the firft 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 Diftance of a from Unity; and fo the Cube Root muft be expreffed by as. Hence, the Index of Unity is o, for Unity is not diftant from itself.

The fame Series of Quantities, geometrically proportional, may be both ways continued, as well defcending towards the Left Hand, as afcending towards the Right; for the Terms

I I

I i 1, a, a2, a3, at, a2, a, a, a', a, a, &c. are all in the fame Geometrical Pro greffion, And fince the Diftance of a from Unity is towards the Right Hand, and pofitive or+1, the Distance equal to that on the contrary Side, viz. the Distance of the Term will be Negative or — 1,

a

which fhall be the Index of the Term for which

a,

may be written a So likewife in the Term a

The Index 2 fhews that that Term ftands in the fecond Place from Unity towards the Left Hand, and the Terms a andare of the fame Value. Also

q is the fame as

a2

fhew that the Terms belonging to them, go from Unity the contrary Way to that by which the Terms whofe Indices are pofitive, do. Thefe Things premised.

If

"

If on the Line AN, both Ways indefinitely extended, be taken, AC, CE, EG, GI, IL, on the right Hand. And alfo Ar, rn, &c. on the left, all equal to one another. And if at the Points, I, I, A, C, E, G, I, L, be erected to the right Line A N, the Perpendiculars I, г4, A B, C D, E F, GH, IK, LM, which let be continually proportional, and represent Numbers, whereof A B is Unity. The Lines AC, AE, AG, AI, AL-Aг,-Aг, refpectively exprefs 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 diftant from Unity. So fince A G is triple of the right Line A C, the Number GH fhall be in the third Place from Unity, if CD be in the firft: So likewife fhall LM be in the fifth Place, fince AL 5 AC. If the Extremities of the Proportionals, 2, 4, B, D, F, H, K, M, be joined by right Lines, the Figure LM 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, IL, be bifected ́in the Points c, e, g, i, l, and there be again raised the Perpendiculars cd, ef, gb, ik, lm, which are mean Proportionals between AB, CD; CD, EF; EF GH; GĤ,IK; IK, LM; then there will arife a new Series of Proportionals, whofe Terms beginning from that which immediately follows Unity, are double of those in the firft Series, and the Difference of the Terms are become lefs, and approach nearer to a Ratio of Equality than before. Likewife in this new Series, the Right Lines AL, AC, exprefs the Diftances of the Terms L M, CD, from Unity, viz. Since AL is ten times greater than A c, LM fhall be the tenth Term of the Series from Unity: And becaufe A is three times greater than Ac, ef will be the third Term of the Series, if cd be the firft; and there fhall be two mean Proportionals between AB and ef, and between A B, and LM, there will be nine mean Próportionals.

And if the Extremities of the Lines Bd Df Fh H, &c. be joined by right Lines, there will be a new Polygon made, confifting of more, but fhorter Sides than the laft.

If, again the Distances Ac, C, Cè, eE, &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 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 leffer, because of the 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 AC, and CD be the fourth Term of the Series from Unity, then LM 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-lin'd Figure; for any curve-lin❜d Figure may be conceived as a Polygon, whofe Sides are infinitely fmall and infinite in Number.

A Curve described after this Manner, is called Logarithmical; in which, if Numbers be reprefented by right Lines standing at right Angles to the Axis AN, the Portion of the Axis intercepted between any Number and Unity, fhews the Place or Order that 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 LM; 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 AB to CD, will be one fifth Part of that Number.

The