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79914, &c. and its Reciprocal, viz. 0, 43429, 44819, 03251, 82765, 11289, &c. which by the Way, is the Subtangent of the Curve'expressing Briggs's Logarithms ; from the double of which the said Logarithms may be had directly. For because
= 0,43429449, &c. :-=,868588 9638, &c. which put =m, and then the Logarithm eten
me' me? of b=
me} me +
Let it be required to find Briggs's Logarithm of 2.
2:e and ee
Whence Briggs's Logarithm of 2 is ®,30102999.
Let it be requir'd to find Briggs's Logarithm of 3 ; now because the Logarithm of 3 is equal to the Logarithm of 2 plus the Logarithm of 1 2 (for 2 x ! 2= 3) therefore find the Logarithm of 1 1 and add it to the Logarithm of 2 already found, the Sum will be the Logarithm of 3, which is better than finding the Logarithm of 3 by the Theorem directly, inasmuch as it will not converge fo fast as the Logarithm of 1 \; for the smaller the Fraction represented by e, which is
deduc'd from the No. whose Logarithm is sought, the swifter does the Series Converge.
e-+1 Here b =
szet2 = 3-36''c=s
The OPERATION is as follows.
me 7 me
Again, to find the Logarithm of 4, because 2 X 2 =4, therefore the Logarithm of 2 added to itself, or multiplied by 2 the Product 0,60205998 is the Logarithm of 4.
To find the Logarithm of 5, because I = 5, therefore from the Logarithm of 19 1,000000000 Subtract the Logarithm of 2
93.01029990 There remains the Logarithm of 5 =
,6989700! And because 2 x3=6. Therefore
To find the Logarithm of 6 To the Logarithm of 3
,477121246 Add the Logarithm of 2
7301029990 The Sum will be the Logarithm of 65,778151236 Which being known, the
Logarithm of 7, the next Prime Number, may be easily found by the Theorem ; for because 6 x = 7, therefore to the Logarithm of 6 add the Logarithm of ; and the Sum will be the Logarithm of 7.
7 me' =
Again, because 4x2= 8. Therefore "To the Logarithm of 4
,60205998 Add the Logarithm of 2
230102999 The Sum is the Logarithm of 8 90308997 And because 3x3 = 9.
Therefore To the Logarithm of 3
2477121246 Add the Logarithm of 3
2477121246 The Sum is the Logarithm of 9 ,954242492 And the Logarithm of 10 having been determined to be 1, 0000000, we have therefore obtained th¢ Logarithms of the first ten Numbers.
After the same manner the whole Table may be constructed, and as the prime Numbers increase, so fewer Terms of the Theorem are required to form their Logarithms; for in the common Tables which extend but to seven Places, the first Term is sufficient to produce the Logarithm of 101, which is compos'd of the Sum of the Logarithms of 100 and 1oó, because 100 x = 101, in which cafe b= noy, whence in making of Logarithms according to the preceding Method, it may be observ'd that the Sum and Difference of the Numerator and Denominator of the Fraction whose Logarithm is fought, is ever equal to the Numérator and Denominator of the Fraction
represented by e; that is, the Sum is the Denominator, and the Difference which is always Unity, is the Numerator; consequently the Logarithm of any Prime Number may be readily had by the Theorem, having the Logarithm cither next above or below given.
Tho', if the Logarithms next above and below that Prime are both given, then its Logarithm will be obtained something easier, For half the Difference of the 4
Ratios which constitute the ift Theorem, viz. (n =)
d: db d +
&c. is the Logarithm of
696 8,8? the Ratio of the arithmetical Mean to the geometrical Mean, which being added to the half Sum of the Logarithms next above and below the Prime fought, will give the Logarithm of that Prime Number, which for Distinction's Sake may be calld Theorem the second, and is of good Dispatch, as will appear hereafter by an Example.
But the best for this purpose is the following one, which is likewise deriv'd from the fame Ratios as Theosem the first. For the Difference of the Terms between a b and Ass or faa + ab + 4 bb, is 1 an -4abtbb=iamb= dd, 1, and the Sum of the Terms a b and ss being put = y, therefore (since y in this Case =s and d=1) it follows that
+2+3+2, &c. is the Logarithm of the у
y y5 ?
į 79 which converges exceeding quick and is of excellent Use for finding the Logarithm of Prime Numbers, having the Logarithms of the Numbers next above and below given, as in Theorem the Second,
Let it be required to find the Logarithms of the Prime Number 101, then a = 100 and b=101, whence y=20401, put-Em 24342944819, &c. then
m the Series will stand thus to + +
Esc. 5 y 7 y.
And m= 243429, &c. divided by y},0000212879410
0,0000212879014 And the Sum, viz. 2,00432!3737823 is the Logarithm of roi true to 12 Places of Figures, and obtained by the first Term of the Series only, whence 'tis easy to perceive what a vast Advantage the fecond Term would have were it put in Practice, since mis to be divided by 3 multiplied into the Cube of 20401.
This Theorem which we'll call Theorem the third, was first found out by Dr. Halley, and a notable Instance of its Use given by him in the Philosophical Transactions for making the Logarithm of 23 to 32 Places, by five Divisions performed with small Divisors; which could not be obtained according to the Methods first made use of, without indefatigable Pains and Labour, if at all; on Account of the great Difficulty that would attend the managing such large Numbers.
Our Author's Series for this purpose is (Page 363) yx+
&c. the Investigation of which, 4 Z 242 360 % as he was pleas'd to conceal, induc'd me to enquire into it, as well to know the Truth of the Series, as to know whether this or that had the Advantage, because Dr. Halley informs us when his was first publish'd that it converged quicker than any Theorem then made publick, and in all Probability does so still; however that be, 'tis certain our Author's converges no faster than the lecond Theorem, as I found by the Investigation thereof,
follows. From the foregoing Doctrine the Difference of the Logarithms of z~1 and z+r is mx++ +
&c. which put equal to y, and the Logarithm of the Ratio of the Arithmetical Mean Z, to the Geometrical Mean zz-r is mx +
2 Z Z