CHAP. V.

Of the continual Increments of proportional Quantities, and how to find by Logarithms, any Term in a Series of Proportionals, either increafing or decreafing, ... Fig. 3.

I'

F any where in the Axis of the Logarithmetical Curve, there be taken any Number of equal Parts SV, VY, YQ, &c. and at the Points S, V, Y, Q, &c. be raised the Perpendiculars S T, V X, YZ, Q, &c. then from the Nature of the Curve fhall all thefe Perpendiculars be continually proportional; and therefore alfo the continual Increments Xx, Zz, пx, fhall be proportional to their Wholes: For fince ST: VX:: VX: YZ:: YZ: Q", it shall be (by Divifion of Proportion) ST: Xx:: VX: Zz:: Ý Zúz, and (by Compofition of Proportion) VX: Xx: YZ: Zz:: Q. Hence, if Xx be any Part of any right Line ST, then will Zz be the fame Part of the right Line V X, and alfo II the fame Part of the right Line YZ. For Example; if Xx be the Part of ST, then will Zz2 VX, and YZ; or, which comes to the fame, we shall have VX= ST ST, YZ=VX+ VX. Alfo Qu YZ+YZ.

Now make, as ST is to V X, fo is Unity A B to NR; then fhall AN SV; and fo each of the right Lines SV, VY, YQ, &c. fhall be equal to the Logarithm of RN, and A V, the Logarithm of the Term VX fhall be equal to AS+AN Logarithm of ST+Logarithm of NR. Alfo AY, the Logarithm of the Term YZ, fhall be equal to AS+ 2 AN Logarithm ST+2 Logarithm NR, and AQ, the Logarithm of the Term Q fhall be equal to AS+3AN Logarithm ST+3 Logarithm NR. And univerfally, if the Logarithm of the Number NR be multiplied by a Number, expreffing the Distance of any Term from the firft, and the Product be added

tq

to the Logarithm of the firft Term, then will the Logarithm of that Term be had: But if a Series of Proportionals be decreafing, that is, if the Terms diminish in a continual Ratio, and Q be the first Term; then the Logarithm of any other will be had, in multiplying the Logarithm of the Number NR, by a Number that expreffes the Distance of its Term from the first, and fubtracting the Product from the Logarithm of the firft. And if the faid Product be greater than the Logarithm of the firft Term, then the Logarithms muft begin from a Unit in fome Place of Decimal Fractions, as from OP, and then the Logarithm of the Number Qu will be OQ.

Now let LM reprefent any Money, or Sum of Money put out to Intereft, fo that the Intereft thereof be accounted but at the End of every Year, and let Kk be the Gain or Interest thereof at the End of the first Year, then will IK be the Sum of the Interest and Principal. And again, I K becoming the Principal at the End of the firft Year, Hb, which is proportional to IK, or in a conftant Ratio, will be the Gain at the End of the fecond Year; and fo HG, at the End of the fecond Year, will become the Principal; and at the End of the third Year Ff, proportional to GH, will be the Gain. Now let us fuppofe the Principal be augmented every Year Part thereof, fo that IKLM+ LM, GA=IK+ İK, EF GH+ GH, and fo on. And accordingly the Terms L M, IK, GH, EF, &c. continual Proportionals, it is required to find the Amount of the Money at the End of any Number of Years.

=

1

20

Let L M be a Farthing. Because L M is to IK, as I to I+ 209 or as 1 to 1.05. as A B is to NR, then will NR=1.05, whofe Logarithm AN, is o. 021 1893, or more accurately 0.0211892991, it is required to find the Amount of a Farthing put out at compound Intereft, at the End of 600 Years, multiply AN by 600, and the Product will be 12. 7135794, and to this Product add the Logarithm of the Fraction. viz. 97.0177288 (for a Farthing is Part of a Pound) and the Sum 109.7313082 fhall be the Logarithm of the Number fought; and fince the Index 109 exceeds the Index of Unity by 9, there fhall be nine Places of Figures above Unity in the correspondent Num

ber,

ber, and that Number being fought in the Tables, will be found greater than 5386500000, and lefs than 5386600000. And therefore a Farthing put out at Interest upon Intereft, at 5 per Gent. per Annum, at the End of 600 Years will amount to above 5386500000 Pounds; which Sum could hardly be made up by all the Gold and Silver that has been dug out of the Bowels of the Earth from the Beginning of the World to this Time.

Let Q expound any Sum of Money due to fome Perfon at the End of a full Year. Now it is certain, that if the Debtor fhould pay down present the whole Sum of Money, he would lofe the yearly Ufury or Interest that his Money would gain him; and fo a leffer Sum, being put out to Intereft, will at the End of one Year together with the Intereft thereof, be equal to the Sum of Money Q. Now this prefent Sum of Money, which together with the Interest thereof, is equal to the Sum of Money Q", is called the prefent Worth of the Money Q, Let A N be the Logarithm of the Ratio which the Principal has to the Sum of the Principal and Intereft, that is, if the Principal be twenty times the yearly Intereft, let AN be the Logarithm of the Number 1 or 1.05, and take QY equal to AN; then will AY be the Logarithm of the present Worth of the Money Q. For it is manifeft, that the Money Y Z put out to Intereft, will at the End of one Year amount to the Money Q", and so to have the Logarithm of the prefent Worth thereof, or Y Z, the Logarithm AN, must be taken from the Logarithm A Q, and there will remain the Logarithm AY of the prefent Worth, or YZ. But if the Sum Q be not due till the End of two Years, then the Logarithm 2 AN must be subtracted from the Logarithm AQ, and there will remain AV, the Logarithm of the prefent Worth, or of the Sum that must be paid down present for the Money Qu due at the End of two Years. For it is manifeft, that the Money VX being put out to Interest, will, at the End of two Years, amount to the Sum of Money Q". By the fame Reafon, if the Sum Q, be not due until the End of three Years, the Logarithm 3 AN must be subtracted from the Logarithm of Q", and the Remainder AS fhall be the Logarithm of the Number ST, or

ST

ST fhall be the prefent Worth of the Sum Qn due at the three Years End. And univerfally, if the Logarithm AN be multiplied by the Number of Years, at the End of which the Sum Q is due, and the Number produced be taken from the Logarithm AQ, then will the Logarithm of the prefent Worth of the Sum Qu be had. And from hence it is manifeft, if 5386500000 Pounds be due to fome Society at the End of 600 Years, then would the present Worth of that vaft Sum of Money be fcarcely a Farthing.

If the proportional right Lines HG, EF, AB, CD, Fig. 4. are Ordinates to the Axis of the logarithmical Curve, and if their Ends FH, DB, be joined by right Lines, which, produced, meet the Axis in the Points P and K, then the right Lines GP, AK, will be always equal. For fince GH: EF::AB: CD it will be as GH: FS::AB: DR. But because of the equiangular Triangles PGH, Hs F, as alfo KAB, BRD, we have PG: HS:: (GH: Fs::AB: DR::) KA: BR. And fince the Confequents Hs, BR, are equal, the Antecedents PG, KA, fhall be alfo equal. W. W. D.

If the right Lines CD, EF, equally accede to AB, GH, fo that the Point D at laft may coincide with B, and the Point F with H, then the right Lines DBK, FHP, which did cut the Curve before, will be changed into the Tangents BT, HV. And the right Lines AT, GV, will be always equal to each other; that is, the Portion of the Axis A T, or GV, intercepted between the Ordinate and the Tangent, which is called the Subtangent, will every where be a conftant and given Length. And this is one of the chief Properties of the logarithmical Curve; for the different Species or Forms of thofe Curves are determined by the Subtangents.

The Logarithms or the Distances from Unity of the fame Number, in two logarithmical Curves of different Species, will be proportional to the Subtangents of their Curves. For let HBD, SNY, Fig. 4, 5. be Curves, whofe Subtangents are AT, MX, and let AB MN- Unity; alfo DC-QY, then shall AC the Logarithm of the Number CD, in the logarithmical Curve HD be to MQ, the Logarithm of the Number QY, (or of the faid CD,) in the Curve SY, as the Subtangent AT is to the Subtangent

MX.

.

MX. For let there be fuppofed an infinite Number of mean proportional Terms between AB, CD, or NM, QY, in the Ratio of A B to ab, or MN to mn; and fince AB MN, then will abmn, as alfo be no. And because the Number of proportional Terms in each Figure are equal, they do divide the Lines AC, MQ, into equal Numbers of Parts, the first of which Aa, M m, and fo the said Parts fhall be proportional to their Wholes; that is, it, will be as Aa: Mm:: AC: MQ. And because the Triangles TAB, Bcb, are fimilar, (for the Part of the Curve B b nearly coincides with the Portion of the Tangent,) as alfo the Triangles X MN, Non, we have A a, or Bc:bc::TA:AB.

Alfo as no, or bc: No:: MN, or A B: M X.

Where (by Equality of Proportion) it will be Bc: No ::TA: MX::Aa: Mm:: AC: MQ; which was to be demonftrated. If AT be called. a, fince AB: AT:: axbc bc: Bc, then will Bc

སྐ

AB

Hence, if the Logarithm of a Number extreamly near Unity, or but a small matter exceeding it, be given, then will the Subtangent of the logarithmica! Curve be had. For the Excefs be is to the Logarithm Bc, as Unity AB is to the Subtangent AT. Or even if there are any two Numbers nearly equal, their Difference fhall be to the Difference of their Logarithms, as one of the Numbers is to the Subtangent. For Example, if the Increment b c be ,00000 00000 00001 02255 31945 60259, and Bc or A a the Logarithm of the Number ab be ‚00000 00000 00000 44408 92098 50062. Now if a fourth Proportional be found to the faid two Numbers and Unity, viz. 43429 4481903251, this Number will give the Length of the Subtangent AT, which is the Subtangent of the Curve expreffing Briggs's Logarithms.

If a Sum of Money be put out to Intereft on this Condition, that a proportional Part of the yearly Rate of Intereft thereof be accounted every Moment of Time, viz. fo, that at the End of the firft Moment of Time, or indefinitely fmall Particle of a Year, the Intereft gotten thereby be proportional to that Time; which being added to the Principal, again begets Intereft at the End of the fecond Moment of Time,

and