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deduced from the Number whose Logarithm is fought, the swifter does the Series converge.
ite Here b= =1::21=2=3- 3eie= }, and ce s
The OPERATION is as follows:
55590 T1118 445
50 me!! =
18 Briggs's Logarithm of i=
1170091259 To which and the Logariihm of 2= 301029993 The Suin is the Logarithin of 3=
?:?52 A vain, to find the Logarithm of 4, becaul: 2 X =4 there'ore the Logarithm of 2 ad led to it cli, or muluplied by 2, the Product 0,602059986 is the Logajithm of 4.
To find the Logarithm of 5, becaufe 1 = 5, therefore from the Logarithm of 10 1,coccccoco fubtract the Logari.hın ot 2
,301029993 There remains the Logarithm of 5= And because 2 *3=6; therefcre
To find the Logari.hm of 6, To the Logarithm of 3
2477121252 Add the Logarithm of 2
,301029993 The Sum will be the Logarithm of 6= 2778151245 Which being known, the Logarithm ot 7, the next prime Number, may be easily found by the Theorem; for because 6x1=7, therefore to the Lorarithm of 6 add the Logarithm of 1, and the Sum will be the Logarithm of 7.
E X A M P L E.
ite Here b = =? :'( ='sand ee = rs.
=,066814536 me = ,366814535 me3 395352
131784 mes = 2339
468 me? =
me? = Briggs's Logarithm of
,C66946789 To which add the Log. of 7
,778151245 The Sum is the Log. of 7=
,845098034 Again, because 4 X2=8; therefore To the Logarithm of 4
,60205998 Add the Logarithm of 2
,30102999 The Sum is the Logarithm of 8 ,90308997
And because 3x3=9; therefore To the Logarithm of 3
947712125 Add the Logarithm of 3
»47712125 The Sum is the Logarithm of 9= ,95424250 And the Logarithm of 10 having been determined to be 1,00000000, we have therefore obtained the 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 forın 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 composed of the Sum of the Logarithms of 100 and 101, because 100
ite x 100 = 101, in which Cafe b= zóti whence, in making of Logarithms according to the preceding Method, it may be cbserved, that the Sun and Difference of the Numerator and Denominator of the Fraction whose Logarithm is fought, is ever equal to the Numerator and Denominator of the Fraction represented by e; that is, the Sum of the Denominator, and the Difference, which is always Unity, is the Numerator; confequently, the Logarithm of any prime Number may be readily had by the Theorem, having the Logarithm either next above or below given.
Tho' if the Logarithms next above and below that Prime are both given, then its Logarithm will be obtained fomewhat easier. For half the Difference of the Ratios which constitutes the first Theorem, viz. dd d4
dB m X +
&c. is the Logarithm 255 454 656 of 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-sake, may be called 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 derived from the fame Ratios as Theorem the first. For the Difference of the Terms between a b and 1ss, or a + ab + bb, is La a
ab + 4bb = amb = dd = 1, and the Sum of the Terms ab and ss being put = y, therefore (since y in this Cale = 5, and d=1) it follows, that
2 , &c. is the Logarithm
+ у 3y"
7y7 of the Ratio of a b to's: Whence X-t + +
&c is the Logag
3y3 rithm of the Ratio of v ab to s, which converges exceeding quick, and is of excellent Use for finding the Logarithms 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 Logarithm of the prime Number 101; then a = 100, and b= 102; whence y=20401; put=m=,4342944819, &c. Then
Andm = .43429, Es divided by } ,00002.12879017
y= 20406, quotes Therefore to the half Sum of the
Logarithms of 100 and 1025}2,0043000858810 Add the said Quote
0,0000212879017 And the Sum, viz.
2,0043213737827 is the Logarithm of 101 true to 12 Places of Figures, and obtained by the first Term of the Series only; whence it is easy to perceive what a vast Advantage the second Term would have, were it put in Practice, since m is to be divided by 3 multiplied into the Cube of 20401.
This Theorem, which we'll call Theorem the third, was firft 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 357)
7 ух to
&c. the Investigation of which 47' 2423' 360zs as he was pleased to conceal, induced me to inquire 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 published, that it converged quicker than any Theorem then made public, and in all Probability does so still. However that be, 'tis certain our Author's converges no fafter than the second Theorem, as I found by the Investigation thereof, which may be as follows:
From the foregoing Doctrine, the Difference of the Logarithms of zur and zti is 2
&c. which put equal to y, and the Logarithm of the Ratio of the Arithmetical Mean z, to the Geometrical Mean v zz-I is m x + +
8. per Theo22Z 626 82
&c. 47' 2423 ' 36ozs"
Now, because the Dividend is ever equal to the Divi-
& c. is equal to
s, &c. is the Lo-