deduced from the Number whofe Logarithm is fought, the fwifter does the Series converge. Again, to find the Logarithm of 4, becaul: 2 × 2=4, therefore the Logarithm of 2 added to it ell, or muluplied by 2, the Product 0,602059986 is the Logaithm of 4. To find the Logarithm of 5, becaufe 10=5, therefore from the Logarithm of 10 fubtract the Logari.hm of 2 There remains the Logarithm of 5= And because 2×3=6; therefore To the Logarithm of 3 Add the Logarithm of 2 The Sum will be the Logarithm of 6= 1,000000000 $301029993 698970007 477121252 ,301029993 778151245 Which being known, the Logarithm of 7, the next prime Number, may be eafily found by the Theorem; for because 6 ×2=7, therefore to the Logarithm of 6 add the Logarithm of 7, and the Sum will be the Logarithm of 7. ,95424250 The Sum is the Logarithm of 9 = 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 fame manner the whole Table may be conftructed; 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 feven Places, the firft Term is fufficient to produce the Logarithm of 101, which is compofed of the Sum of the Logarithms of 100 and. x 100 = 101, in which Cafe b=1+ Ie because 100 10 ΤΟΥ = r; whence, in making of Logarithms according to the preceding Method, it may be cbferved, that the Sun and Difference of the Numerator and Denominator of the Fraction whofe Logarithm is fought, is ever equal to the Numerator and Denominator of the Fraction reprefented 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 Tho' if the Logarithms next above and below that Prime are both given, then its Logarithm will be obtained fomewhat eafier. For half the Difference of the Ratios which conftitutes the firft Theorem, viz. c. is the Logarithm dd d4 m X + + + 454 656 255 858 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 Diftinction-fake, may be called Theorem the fecond, and is of good Difpatch, as will appear hereafter by an Example. But the beft for this Purpofe is the following one, which is likewife derived from the fame Ratios as Theorem the firft. For the Difference of the Terms between a b and ss, or 1a a + 1⁄2 a b + 1 b b, is 1 a a - 1/2 ab + 1 bb = 1 a — b2 = dd = 1, and the Sum of the Terms ab and 4 ss being put =y, therefore (fince y in this Cases, and d=1) it follows, that I n 2 X- + 2 3y3 2 2 + + c. is the Logarithm 5ys 7y7 of the Ratio of a b to 'ss: Whence X + n y + rithm of the Ratio of + &c. is the Loga7y7 ab tos, which converges exceeding quick, and is of excellent Ufe for finding the Logarithms of prime Numbers, having the Logarithms of the Numbers next above and below given, as in Theorem the second. EXAMPLE. Let it be required to find the Logarithm of the prime Number 101; then a 100, and 102; whence y=20401; put=m=,4342944819, &c. Then And m=,43429, &c. divided by},0000212879017 y= 20406, quotes Therefore to the half Sum of the Logarithms of 100 and 102 'Add the faid Quote And the Sum, viz. 2,0043213737827 is the Logarithm of 101 true to 12 Places of Figures, and obtained by the firft Term of the Series only; whence it is easy to perceive what a vaft Advantage the fecond 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 Inftance of its Ufe given by him in the Philofophical Tranfactions for making the Logarithm of 23 to 32 Places, by five Divifions performed with fmall Divifors; which could not be obtained according to the Methods firft made ufe of, without indefatigable Pains and Labour, if at all ; on account of the great Difficulty that would attend the managing fuch large Numbers. Our Author's Series for this Purpose is (Page 357) I 7 + + &c. the Investigation of which 4% 24%3 360zs as he was pleafed 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 fo ftill. However that be, 'tis certain our Author's converges no fafter than the fecond Theorem, as I found by the Inveftigation thereof, which may be as follows: From the foregoing Doctrine, the Difference of the Logarithms of 2-1 and 2+1 is Z 323 525 + 727 c. which put equal to y, and the Logarithm of the Ratio of the Arithmetical Mean z, to the Geometrical Mean . per Theo rem d d I rem the fecond; for zs; whence 22 1 255 2ZZ Let A and B be the Logarithms of z-I and z + I re A+B I I I fpectively; then is +m x + + &c. the Series required, viz. + I 7 + as appears by the following Operation : &c. 323525 &c. Now, because the Dividend is ever equal to the Divifor drawn into the Quotient of the Divifion; it follows, that y X + I &c. is equal to 360zs |