,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 firft ten Numbers. After the fame manner the whole Table may be conftructed; and as the prime Numbers increase, fo 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 10%, because 100 X 100 101, in which Cafe b Ite Ie ΙΟΙ 100 Tor; whence, in making of Logarithms according to the preceding Method, it may be cbferved, that the Sum 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 somewhat eafier. For half the Difference of the Ratios which conftitutes the first Theorem, viz. do + &c. is the Logarithm m X dd + d+ 255 454 +656 858 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 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 Purpose is the following one, which is likewife derived from the fame Ratios as Theorem the first. For the Difference of the Terms between a b and ss, ora a + ab + 1 bb, is a a aa 1⁄2 6+ • 1⁄2 ab + 1 bb = 1 a — 1 b2= 1 dd=1, and the Sum of the Terms a band ss being put = y, therefore (fince y in this Cases, and di) it follows, that 2 + + n y 3y3 515 of the Ratio of a b to n y 3y3 I 2 535 &. is the Loga rithm of the Ratio of 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 b 102; whence y=20401; put=m=,4342944819, &c. Then And m =,43429, &c. divided by 2,0000212879017 20406, quotes Therefore to the half Sum of the 22,0043000858810 Logarithms of 100 and 102. Add the faid Quote 0,0000212879017 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 eafy to perceive what a vast Advantage the fecond Term would have, were it put in Practice, fince m is 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 Ufe given by him in the Philofophical Tranfactions for making the Logarithm of 23 to 32 Places, by five Divifions performed with small 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 Purpofe is (Page 357) I 7 42 2423 360zs I &c. the Investigation of which 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 Investigation thereof, which may be as follows: From the foregoing Doctrine, the Difference of the Logarithms of 2-1 and z+1 is 2 &c. which put equal + 7222 to y, and the Logarithm of the Ratio of the Arithme tical Mean z, to the Geometrical Mean ✔zz-1 is rem the fecond; for zs; whence dd 255 222 Let A and B be the Logarithms of z-I and %+ I re spectively; then is A+B 2 +mx Ι + + &ci 222 424 6z6 the Logarithm of z; and if the latter Part of the Series expreffing the said Logarithm of z be divided by the Series representing the Difference of the Logarithms of z-I and 2+1, the Quotient will exhibit Τ I 7 the Series required, viz. —— + + &€€ 42 2423 36025 as appears by the following Operation: I &c. 222+42+ ' 6z° + I 7 + &c. &'c. I + Now, because the Dividend is ever equal to the Divifor drawn into the Quotient of the Divifion; it follows, Note, I make the Author's 5th Term to be 1903 22680029 To illuftrate this Theorem by an Example: 13 25200% Let it be required to find the Logarithm of 101. Add the Difference of the faid Logarithms divided by 42 equal to 2,0043000159 } 0,0000212876 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 fhort 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 confiderably greater. Note, This Series of our Author, deduced from Theorem the fecond, is in Effect Dr. Halley's too, but difguis'd by being thrown into a different Form; which, however, has its Ufe, as being very ready in Practice. Having thus investigated several Theorems, whereby Tables of Logarithms, of any Form, may be conftructed; it remains to fhew how, from the Logarithm given, to find what Ratio it expreffes. The Logarithm of the Ratio of i to 1+x has been I 4 4 proved to be as 1+x”—1 = - 1 x2 + 2x3 — &c. n being any infinite Index whatsoever; whence, if L be put for the faid Series, then 1+x^— 1 = L; con fequently += I + L, and 1 + x = 1 + L® I + n L + = n2 L2 + 1⁄2 n3 L3 + 24 n+ L+, &c. AGAIN: I The Logarithm of the Ratio of 1 to I likewife been proved to be as I - 1 - × ×+ 1/2 x2 + 1 x2 + = x2, &c. L; wherefore |