and then the Principal and this Intereft become a Principal, and so on. It is requir'd to find the Amount of that Sum at the Year's End. Let a be nearly the Intereft of Unity, or of one Pound. Then if one whole Year, or I gives the Intereft a, the indefinitely small Particle of a Year M m, will give the Intereft Mmxa, proportional to Mm; and accordingly, if Unity be expounded by MN, the firft Increment thereof fhall be no Mmxa. This being granted, let a logarithmical Curve be fuppos'd to be defcribed through the Points Nn, whofe Axis is OMQ. Then in this Curve, if the Portion of the Axis MQ expreffes the Time, the Ordinate Qy, will represent the Money proportionally increafing every Moment, to that Time. For if there be taken ml, &c.

Mm, the Ordinates lp, &c. fhall be in a Series of continual Proportionals in the Ratio of MN to mn, that is, they increase in the fame Ratio as the Money doth.

Again, let the right Line NX touch the logarithmical Curve in N, and the Subtangent thereof M X fhall be conftant and invariable, and the small Triangle Non fhall be fimilar to the Triangle X M N. But it has been prov'd, that the Increment no Mm xa=Noxa; and fo no: No:: Noxa: No::d: But as no is to No, fo fhall N M be to MX. Wherefore it fhall be as a is to 1, fo is N M, or 1, to M X Subtangent.

I.

Part of the

Now if the nearly Rate of Interest be Principal, or if a=.05, then will M X=

20.

a

Because of different Forms of Logarithms, the Logarithms of the fame Number, are proportional to the Subtangents of their Curves: If MQ expresses the Time of a whole Year, or Unity, then fhall QY be the Amount of the Money at the Year's End. And to find QY, fay, as M X, or 2 is to 0.4342944, (which Number expounds the Subtangent of the logarithmical Curve expreffing Briggs's Logarithms) fo is one Year or Unity to a Briggian Logarithm, anfwering to the Number QY. This Logarithm will be found 0.0217147, and the Number answering to the fame is 1.05127=QY, whose Increment above

Unity, or the Principal, exceeds the yearly Interest ,05 but a fmall Matter. And fo if the yearly Intereft of 100 Pounds be 5 Pounds, the proportional yearly Intereft, which is added to the Principal 100 at the End of each Particle of the Year, will amount only at the Year's End to 5 Pounds z Shillings and 6 Pence.

20

And if fuch a Rate of Interest be requir'd, that every Moment a Part of it continually proportional to the increafing Principal be added to the Principal, fo that at the Year's End an Increment be produc'd that fhall be any given Part of the Principal; for Example, the Part, fay, as the Logarithm of the Number 1.05 is to 1; that is, as 0.0211893 is to I, fo is the Subtangent o. 432944 to 20. 49, and a then will a=4.0488. For if fuch a Part of the Rate of Intereft .0488 be supposed as answers to a Moment, that is, having the fame Ratio to .0488 as a Moment has to a Year, and it be made as Unity is to that Part of the Rate of Intereft, fo is the Principal to the momentaneous Increment thereof; then will the Money continually increasing in that Manner, be augmented at the Year's End the thereof.

CHAP. VI.

Part

Of the Method by which Mr. Briggs computed his Logarithms, and the Demonftration thereof.

A

Lthough Mr. Briggs has no where describ'd the logarithmical Curve, yet it is very certain that from the Ufe and Contemplation thereof, the Manner and Reafon of his Calculations will appear. In any logarithmical Curve HBD, let there be three Ordinates A B, a b, qs, nearly equal to one another; that is, let their Differences have a very small Ratio to the faid Ordinates; and then the Differences of their Logarithms will be proportional to the Differences of the Ordinates. For fince the Ordinates are nearly equal to one another, they will be very nigh

to

to each other, and fo the Part of the Curve Bs, intercepted by them, will almost coincide with a straight Line; for it is certain, that the Ordinates may be fo near to each other, that the Difference between the Part of the Curve and the right Line fubtending it, may have to that Subtence, a Ratio lefs than any given Ratio. Therefore the Triangles Bcb, Brs, may be taken for Right-lin'd, and will be equiangu

Wherefore, as sr: bc:: Br: Bc:: Ag: Aa, that is, the Exceffes of the Ordinates or Lines above the leaft, fhall be proportional to the Differences of their Logarithms. And from hence appears the Reafon of the Correction of Numbers and Logarithms by Differences and proportional Parts. But if AB be Unity, the Logarithms of Numbers shall be proportional to the Differences of the Numbers.

If a mean Proportional be found between 1 and 10, or which is the fame thing, if the fquare Root of 10 be extracted, this Root or Number will be in the middle Place between Unity and the Number 10, and the Logarithm thereof fhall be of the Logarithm of 10, and fo will be given. If again, between the Number before found, and Unity, there be found a mean Proportional, which may be done in extracting the fquare Root of the faid Number, this Number, or Root, will be twice nearer to Unity than the former, and its Logarithm will be one half of the Logarithm of that, or one fourth of the Logarithm of 10. And if in this Manner the fquare Root be continually extracted, and the Logarithms bifected, you will at laft get a Number whose Distance from Unity fhall be less than the 10000010507586 Part of the Logarithm of 10. And after Mr. Briggs had made 54 Extractions of the fquare Root, he found the Number 1. 00000 00000 00000 12781 91493 20032 3442, and its Logarithm was 0.00000 00000 00000,055 51 1 1 5 1 2 3 1 25782702. Suppofe this Logarithm to be equal to A q or Br, and let qs be the Number found by extracting the fquare Root then will the Excefs of this Number above Unity, viz. rs, 00000 00000 00000, 12781 91493 20032 34.

Now by Means of thefe Numbers, the Logarithms of all other Numbers may be found in the following Manner: Between the given Number (whofe Loga

rithm

rithm is to be found) and Unity, find fo many mean Proportionals, (as above) till at laft a Number be gotten fo little exceeding Unity, that there be 15 Cyphers next after it, and a like Number of fignificative Figures after thofe. Let this Number be a b, and let the fignificative Figures with the Cyphers prefixed before them, denote the Difference bc. Then fay, As the Difference rs is to the Difference bc, fo is Br a given Logarithm, to B c, or A a, the Logarithm of the Number ab; which therefore is given. And if this Logarithm be continually doubled, the fame Number of Times as there were Extractions of the fquare Root, you will at last have the Logarithm of the Number fought. Alfo by this Way may the Subtangent of the logarithmical Curve be found, viz. in saying, Asrs: Br:: AB, or Unity: AT, the Subtangent, which therefore will be found to be 0.4342944819 03251; by which may be found the Logarithms of other Numbers; to wit, if any Number NM be given afterwards, as alfo its Logarithm, and the Logarithm of another Number fufficiently near to NM be fought, fay, As NM is to the Subtangent X M, fo is no the Diftance of the Numbers to No the Distance of the Logarithms. Now, if NM be Unity A B, the Logarithms will be had by multiplying the fmall Differences be by the conftant Subtangent A T.

By this Way may be found the Logarithms of 2, 3, and 7, and by these the Logarithms of 4, 8, 16, 52, 64, &c. 9, 27, 81, 243, &c. as alfo 7, 49, 343, &c. And if from the Logarithm of 10 be taken the Logarithm of 2, there will remain the Logarithm of 5, fo there will be given the Logarithms of 25, 125, 625, &c.

The Logarithms of Numbers compounded of the aforefaid Numbers, viz. 6, 12, 14, 15, 18, 20, 21, 24, 28, &c. are easily had by adding together the Logarithms of the component Numbers.

But fince it was very tedious and laborious to find the Logarithms of the prime Numbers, and not eafy to compute Logarithms by Interpolation, by firft, fecond and third, &c. Differences, therefore the great Men, Sir Ifaac Newton, Mercator, Gregory, Wallis, and laftly, Dr. Halley, have published infinite converging Series, by which the Logarithms of Numbers

1

Numbers to any Number of Places may be had more expediently and truer Concerning which Series Dr. Halley has written a learned Tract, in the Philofophical Tranfactions, wherein he has demonftrated thofe Series after a new Way, and fhews how to compute the Logarithms by them. But I think it may be more proper here to add a new Series, by Means of which may be found eafily and expeditiously the Logarithms of large Numbers.

-

I

Let z be an odd Number, whofe Logarithm is fought; then fhall the Numbers z I and z be even, and accordingly their Logarithms, and the Difference of the Logarithms will be had, which let be called Therefore, alfo the Logarithm of a Number, which is a Geometrical Mean between Z-1 and I will be given, viz. equal to the half Sum of the Logarithms. Now the Series

ช

汁
1512027 2520029

&c.

fhall be equal to the Logarithm of the Ratio, which the Geometrical Mean between the Numbers z- 1 and +1, has to the Arithmetical Mean, viz. to the Number z.

If the Number exceeds 1000, the firft Term of the Series is fufficient for producing the Logarithm to

42

13 or 14 Places of Figures, and the fecond Term will give the Logarithm to 20 Places of Figures. But if z be greater than 10000, the firft Term will exhibit the Logarithm to 18 Places of Figures; and fo this Series is of great Ufe in filling up the Logarithms of the Chiliads omitted by Briggs. For Example; It is required to find the Logarithm of 20001. The Logarithm of 20000 is the fame as the Logarithm of 2 with the Index 4 prefix'd to it; and the Difference of the Logarithms of 20000 and 20002, is the fame as the Difference of the Logarithms of the Numbers 10000 and 10001, viz. o. 00004 34272 7687. And if this Difference be divided by 4%, or 80004, the Quotient?

A a

4%

fhall