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Unity, or the Principal, exceeds the yearly Interest ,05 but a small Matter. And so if the yearly In. tereft of 100 Pounds, be 5 Pounds, the proportional
5 yearly Interest, 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 6 Pence.
And if such a Rate of Interest be required, that every Moment a Part of it continually proportional io the increasing Principal be added to the Principal, so that at the Year's End an Increment be produced that shall be any given Part of the Principal; for Example, the zz Part; fay, As the Logarithm of the Number 1.05 is to r; that is, as 0.0211893 is to I; so is the Subtangent 0,4342944 to 5 = 20.49, and then will a = =.0488. For if such a Part of
20.49 the Rate of Intereit .0488 be supposed, as answers to a Moment, that is, having the same Ratio to .0488 as a Moment has to a Year, and it be made, as Unity is to that Part of the Rate of Interest, so 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 76 Part thereof.
CHA P. VI.
Р. Of the Method by which Mr. Briggs com
puted his Logarithms, and the Demonstration thereof. Although Mr. Briggs has no where described the
Logarithmetical Curve, yet it is very certain, that, from the Use and Contemplation thereof, the Manner and Reason of his Calculations will appear. In any Logarithmetical Curve HBD, let there be three Ordinates a B, ab, 9s, nearly equal to one another ; that is, let their Differences have a very small Ratio to the said Ordinates; and then the Differences of their Logarithms will be proportional to the Differences of the Ordinates. For since the Ordinates are nearly equal to one another, they will be very nigh
to each other; and so the Part of the Curve B s, intercepted by them, will almost 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 subtending is, may have to that Subtense à Ratio less 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::A9: Aa; that is, the Excesses of the Ordinates, or Lines above the least, shall be proportional to the Differences of their Logarithms. And from hence appears the Reason 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 same 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 lo 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 said 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 1o. And if in this manner the Square Root be continually extracted, and the Logarithms bifected, you will at last get a Number, whose Distance from Unity shall be less than the TortToOTOOOTTO Part of the Logarithm of 10. And alter 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 o. 00000 oooon 00000 05551 11512 31257 82702. Suppose this Logarithm to be equal to Aq, or Br, and Jet be the Numder found by extracting the Square Root; then will the Excess of this Number above Unity, viz. rs = ,00000 000co ocooo 12781 9149320032 3442.
Now, by means of these Numbers, the Logarithms cf all other Numbers may be found in the following manner : Between the given Number (whole LogaАа2
rithm is to be found) and Unity, find so many bean Proportionals (as above), till at last a Number be gotten so 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 signifi. cative Figures, with the Cyphers prefixed before them, denote the Difference be. Then say, As the Difference rs is to the Difference bc, so is Br a given Logarithm, to Bc or Aa, the Logarithm of the Numberab; 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. Also, by this Way may the Subtangent of the Logarithmetic Curve be found, viz. by saying, 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 NM, be fought, fay, As NM is to the Subtangent XM, rois no, the Distance of the Nurnbers, to No, the Distance of the Logarithms. Now, if N M be Unity = AB, the Logarithms will be had by multiplying the small Differences bc by the constant Subtangent AT.
By this Way may be found the Logarithms of 2, 3o and 7; and by these the Logarithms of 4, 8, 16, 32, 64, & c. 9, 27, 81, 243, &c. as also 7, 49, 343 &C. And if from the Logarithm of so be taken the Logarithm of 2, there will remain the Logarithm of 5; so there will be given the Logarithms of 25, 125, 625, &c.
The Logarithms of Numbers compounded of the aforesaid 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 since it was very tedious and laborious to find the Logarithms of the prime Numbers, and not easy to compute Logarithms by Interpolation, by first, fecond, and third, &c. Differences; therefore the great Men, Sir Isaac Newton, Mercator, Gregory, Wallis, and, lastly, Dr. Halley, have published infinite converging Series, by which the Logarithms of