ne 468 = ,868588963 = ,066814536 me =,066814535 me = 395352 131784 mes 2339 me? 14 2 Briggs's Logarithm of} ,066946789 To which add the Log. of 6 2778151245 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 990308997 And because 3x3=9; therefore To the Logarithm of 3 947712125 Add the Logarithm of 3 947712125 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 Lo. garithms of the first ten Numbers. After the same manner the whole Table may bé constructed ; and as the prime Numbers increase, lo 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 composed of the Sum of the Logarithms of 100 and 10%, because 100 Ite Х Toi: zóti whence, in making of Logarithms according to the preceding Method, it may becbserved, that the Sune 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; consequently, the Logarithm of any prime Number may be readily had by the Theorem, having the Logarithm either next above or below given. Thom + + 838' Tho' if the Logarithms next above and below that Prime are both given, then its Logarithm will be obtained somewhat easier. For half the Difference of the Ratios which constitutes the first Theorem, viz. dd de do ds m x + &c. is the Logarithm 255 454 656 of the Ratio of the arithmetical Mean to the geome trical Mean, which being added to the half Sum of the Logarithms, next above and below the Prime sought, will give the Logarithm of that prime Number, which for Distinction-fake, may be called Theorem the seconds 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 same Ratios as Theorem the first. For the Difference of the Terms between a b and Iss, or a 2 + ab +4bb, is ça a ab + bb=1a-4b= 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 &c. is the Logarithm у 3y3 575 of the Ratio of a b to ss: Whence + + &o. is the Logag 393 ' rithm of the Ratio of v ab tos, 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. I 2 2 2 n 7y7 1 Х + n 5ys E X A M P L E. Let it be required to find the Logarithm of the prime Number 101; then a = 100, and b= 102; whence y=20401; put =m= 24342944819, &c. Then the Series will stand thus, it & C. y 373 m m m + 5y5 And And m=434297&s, divided by } ,0000212879017 y=20406, quotes Therefore to the half Sum of the Logarithms of 100 and 102=3 2,0043000858810 Add the faid Quote 0,00002 12879017 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, fince m. is to be divided by 3 multiplied into the Cube of 204CI. 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 firft 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) 2 ух + 36025 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 faster 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 2-1 and z+ı is to + &c. which put equal 323 525 + to y, and the Logarithm of the Ratio of the Arithmetical Mean Z, to the Geometrical Mean v. 2Z.--I is mx + + ác. per Theo+ : 626 474 I 2 2 2 2 m x Z 727 I 2%Z rem 22Z I I then is A+B + 2 rem the second; for z=; s; whence dd_1 Let A and B be the Logarithms of z-1 and 2 + I re, spectively; then is +mx + Esco 222 the Logarithm of z; and if the latter Part of the Series expressing the said Logarithm of z be divided by the Series representing the Difference of the Logarithms of Z-1 and 2+1, the Quotient will exhibit the Series required, uz. 7 42 2473 360% L&C. 1 I 2 1&. I + + Now, because the Dividend is ever equal to the Divifor drawn into the Quotient of the Division; it follows, that y X + &c. is equal to 42 2423 m X + + &c. I + + 1 &C. 2 бz6 7. Esc. is the Lo4% 2470 36oz garithm of z. E.I. to be Note, I make the Author's sth Term 13 25200% 1903 226800z9 To illustrate this. Theorem by an Example: Let it be required to find the Logarithm of 101. To the half Sum of the Logarithms of 100 and 102= 2,0043000159 Add the Difference of the said Lo 0,00002 garithms divided by 42 equal to And the Sum, viz. 2,0043213735 is the Logarithm of 101 true to 9 Places of Figures : Whence it appears, that our Author's Series falls short of Dr. Halley's, in finding the Logarithm of the prime Number 101, three Places of Figures, by using only the first Terms of the Series ; whereas, if two Terms in each were used, perhaps the Difference would have been considerably greater. Note, This Series of our Author, deduced from Theorem the second, is in Effect Dr. Halley's too, but dirguis'd by being thrown into a different Form ; which, however, has its Use, as being very ready in Practice. Having thus investigated several Theorems, whereby Tables of Logarithms, of any Form, may be constructed; it remains to shew how, from the Logarithm given, to find what Ratio it expresses, The Logarithm of the Ratio of i to 1+* has been proved to be as i+x" -I=*****+ ***-***, &c. n being any infinite Index whatsoever; whence, if L be put for the said Series, then 1 +*"-I=L; con Itt sequently I +x"=1+L, and I +x=1 + La = It n L + n2 L? + ở n3 L3 + 74 n4 L4, 3C. A GAIN: The Logarithm of the Ratio of 1 to 1 — * has likewise been proved to be as I X*+* +73+ 4*, &= L; wherefore I + 1 |