MX. For let there be fuppofed an infinite Number of mean Proportional Terms between AB, CD, or MN, QY, in the Ratio of AB to ab, or MN to mn; and fince A B MN, then will ab = mn, as. alfo bc= =no. And because the Number of propor-, tional Terms in each Figure are equal, they do divide the Lines A C, MQ, into equal Numbers of Parts, the first of which are A a, Mm, and fo the said Parts fhall be proportional to their Wholes; that is, it will be as Aa: Mm:: AC: MQ. And becaule the Triangles TA B, Bcb, are fimilar (for the Part of the Curve Bb nearly coincides with the Portion of the Tangent), as alfo the Triangles XMN, Non, we have Aa, or Be: bc:: TA: AB. Alfo, as no, or be: No:: MN, or AB: MX. Where (by Equality of Proportion) it will be, Be: No: TA: MX:: Aa: Mm:: AC: MQ; which was to be demonftrated. If AT be called a, fince AB: AT:: be: Be, then will Be axbc. AB For Hence, if the Logarithm of a Number extremely near Unity, or but a fmall Matter exceeding it, be given, then will the Subtangent of the Logarithmetick Curve be had. For the Excefs be is to the Logarithm Bc, as Unity A B is to the Sub angent A T. Or even if there are any two Numbers nearly equal, their Difference fhall be to the Difference of the Logarithms, as one of the Numbers is to the Subtangent. Example, if the Increment be be ,00000 0000000001 02255 31945 60259, and Be or A a the Logarithm of the Number ab be ,00000 00000 0000044408 92098 30062. Now if a fourth Proportional be found to the said two Numbers and Unity, viz. 434294481903251, 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 Interest thereof be accounted every Moment of Time, viz. fo that at the End of the first Moment of Time, or indefinitely small 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 second Moment of Time, and . and then the Principal and this Intereft become a Principal, and fo on; it is required 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 Mm will give the Interest Mm+a, proportional to Mm; and, accordingly, if Unity be expounded by MN, the first Increment thereof shall be no Mmx a. This being granted, let a Logarithmetic Curve be fuppofed to be defcribed through the Points Nn, whofe Axis is OMQ. Then, in this Curve, if the Proportion of the Axis MQ expreffes the Time, the Ordinate QY will reprefent the Money proportionally increafing every Moment, to that Time. For if there be taken ml, &c. Mm, the Ordinates lp, &c. fhall be 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 N X touch the Logarithmetical Curve in N, and the Subtangent thereof MX fhall be conftant and invariable, and the small Triangle Non fhall be fimilar to the Triangle XMN. But it has been proved, that the Increment no Mm xa Noa; and fono: No: : Noxa: N 0::a: But as no is to No, fo fháll MN be to MX. Wherefore it shall be, as a is to 1, fo is M N, or I, to MX== Subtangent. I. 20 Now if the yearly Rate of Interest be Part of the Principal, or if a .05, then will MX x ÷ = 20. 209 Because in different Forms of Logarithms, the Logarithms of the fame Number are proportional to the Subtangents of their Curves: If MQ expreffes the Time of a whole Year, or Unity, then shall QY be the Amount of the Money at the Year's End. And to find QY, fay, As MX, or is to 0.4342944 (which Number expounds the Subtangent of the Logarithmetic 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 anfwering to the fame is 1.05127 =QY, whose Increment ab ve Unity, A a Unity, or the Principal, exceeds the yearly Intereft ,05 but a small 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 2 Shillings and 64 Pence. And if fuch a Rate of Interest be required, 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 produced that shall 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 0,4342944 to 20.49, and 23 I then will a=.0488. For if fuch a Part of 20.49 the Rate of Intereit .0488 be fuppofed, 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 Part thereof. CHAP. VI. Of the Method by which Mr. Briggs computed his Logarithms, and the DemonAtration thereof. Although Mr. Briggs has no where defcribed the Logarithmetical Curve, yet it is very certain, that, from the Ufe and Contemplation thereof, the Manner and Reafon of his Calculations will appear. In any Logarithmetical Curve HBD, let there be three Ordinates A B, ab, 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 almoft coincide with a strait 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 Subtense a Ratio lefs then any given Ratio. Therefore the Triangles Bcb, Brs, may be taken for Right-lined, and will be equiangular. Wherefore, as sr: bc:: Br: Bc:: Aq: 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 fhall 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 Square 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 shall 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 by extracting the Square 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 Square Root be continually extracted, and the Logarithms bifected, you will at laft get a Number, whose Distance from Unity fhall be lefs than the 9000000000000 Part of the Logarithm of 10. And after Mr. Briggs had made 54 Extractions of the Square Root, he found the Number 1.00000 00000 0000012781 91493 20032 3442; and its Logarithm was 0.00000 0000 00000 05551 11512 31257 827c2. Suppofe this Logarithm to be equal to A 9, or Br, and let qs be the Numder found by extracting the Square Root; then will the Excefs of this Number above Unity, viz. rs,coooo oooco oco00 12781 9149320032 3442. Now, by means of these Numbers, the Logarithms of all other Numbers may be found in the following manner: Between the given Number (whofe Loga 1 rithm is to be found) and Unity, find fo many mean Proportionals (as above), till at last a Number be gotten fo little exceeding Unity, that there be 15 Cyphers next after it, and a like Number of fignificative Figures after those. Let this Number be ab, and let the fignificative Figures, with the Cyphers prefixed before them, denote the Difference be. Then fay, As the Difference rs is to the Difference bc, fo is Br a given Logarithm, to Bc, or Aa, 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 Square Root, you will at laft have the Logarithm of the Number fought. Alfo, by this Way may the Subtangent of the Logarithmetic Curve be found, viz. by faying, As rs: Br:: A B, or Unity: AT, the Subtangent, which therefore will be found to be 0.434294482903251; by which may be found the Logarithms of other Numbers; to wit, if any Number N M be given afterwards, as also its Logarithm, and the Logarithm of another Number, fufficiently near to N M, be fought, fay, As NM is to the Subtangent X M, fo is no, the Distance of the Numbers, to No, the Distance of the Logarithms. Now, if N M be Unity AB, the Logarithms will be had by multiplying the small Differences be by the conftant Subtangent A T. By this Way may be found the Logarithms of 2, 3 and 7; and by thefe the Logarithms of 4, 8, 16, 32, 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 ealy to compute Logarithms by Interpolation, by first, 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 |