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Unity, or the Principal, exceeds the yearly Interest ,05 but a small Matter. And so if the yearly Intereft of 100 Pounds be 5 Pounds, the proportional 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 requir’d, that every Moment a Part of it continually proportional to the increasing Principal be added to the Principal, so that at the Year's End an Increment be produc'd that shall be any given Part of the Principal ; for Example, the <a Part, say, as the Logarithm of the Number 1.05 is to ; that is, as 0.0211893 is to 1, fo is the Subtangent 0.432944 to 5= 20.49, and then will a=iu ty=.0488. For if such a Part of the Rate of Interest .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 ó Part thereof.
CH A P. VI.
Of the Method by which Mr. Briggs computed his Logarithms, and the Demonstra
tion thereof. A
Lthough Mr. Briggs has no where describ'd the logarithmical Curve, yet it is very certain that
from the Use and Contemplation thereof, the Manner and Reason 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 said 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 each other, and so the part of the Curve B s, intercepted by them, will almost coincide with a straight Line; for it is certain, that the Ordinates may be so 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 less than any given Ratio. Therefore the Triangles Bob, Brs, may be taken for Right-lin'd, 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 į of the Logarithm of 10, and so 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 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 10. And if in this Manner the square Root be continually extracted, and the Logarithms bisected, you will at last get a Number whofe Distance from Unity shall be less than the T0510610 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 co000 12781 91493 20032 3442, and its Logarithm was 0.00000 00000 00000,0555111512 31257 82702. Suppose this Logarithm to be equal to Aq or Br, and let ys be the Number found by extracting the square Root ; then will the Excess of this Number above Unity, viz. rs=,00000 00000 00000, 12781 91493 20032 34.
Now by Means of these Numbers, the Logarithms of all other Numbers may be found in the following Manner : Between the given Number (whole Loga
rithm is to be found) and Unity, find so many mean 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 significative Figures with the Cyphers prefixed before them, denote the Difference bc. Then fay, As the Difference rs is to the Difference b c, so is Bra given Logarithm, to Bc, or A a, the Logarithm of the Number ab; which therefore is given. And if this Logarithm be continually doubled, the same Number of Times as there were Extractions of the square Root, you will at last have the Logarithm of the Number fought. Also by this Way may the Subtangent of the logarithmical Curve be found, viz. in faying, A srs:Br::AB, or Unity: A T, 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 also its Logarithm, and the Logarithm of another Number fufficiently near to NM be fought, fay, As NM is to the Subtangent XM, so is no the Distance of the Numbers to N o the Distance of the Logarithms. Now, if NM be Unity=AB, the Logarithms will be had by multiplying the small Differences b c by the constant Subtangent AT.
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 also 7, 49, 343, &c. And if from the Logarithm of 1o be taken the Logarithm of 2, there will remain the Logarithm of 5, so there will be given the Logarithms of 25, 125, 625, E86.
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, second 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
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 Transactions, wherein he has demonstrated those Series after a new Way, and shews 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 easily and expeditiously the Logarithms of large Numbers.
Let z be an odd Number, whose Logarithm is fought; then shall the Numbers % i and z tibe even, and accordingly their Logarithms, and the Difference of the Logarithms will be had, which let be called y: - Therefore, also the Logarithm of a Number, which is a Geometrical Mean between 2- Land zti will be given, viz. equal to the half Sum of the Logarithms. Now the Series
&c. 42 247
1512027 shall be equal to the Logarithm of the Ratio, which the Geometrical Mean between the Numbers Z-I and 2+1, has to the Arithmetical Mean, viz. to the Number z.
If the Number exceeds 1000, the first Term of the Series is sufficient for producing the Logarithm to
42 13 or 14 Places of Figures, and the second Term will give the Logarithm to 20 Places of Figures. But if z be greater than 10000, the first Term will exhibit the Logarithm to 18 Places of Figures; and fo this Series is of great Use 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 same 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 same 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
0. 00000 00005 42813 And if the Logarithm of the '4. 30105 17093 02416 Geometrical Mean be added
4. 3010517098 45231 to the Quotient, the Sum will be the Logarithm of 20001. Wherefore it is manifelt, that to have the Logarithm to 14 Places of Figures, there is no Neceflity of continuing out the Quotient beyond fix Places of Figures. But if you have a Mind to have the Logarithm to 10 Places of Figures only, as they are in Vlag's Table, the two first Figures of the Quotient are enough. And if the Logarithms of the Numbers above 20000 are to be found by this way, the Labour of doing them will mostly confift in setting down the Numbers.
Note, This Series is eafily deduced from that found
out by Dr. Halley; and those who have a Mind to be inform’d more in this Matter, let them consult his above-nam's Treatise.