And m=,43429, &c. divided by y},0000212879410 20401, Quotes Therefore to the half Sum of the 22, 0043000858809 Logarithms of 100 and 102 — Add the faid Quote 0,0000212879014 And the Sum, viz. 2,0043213737823 is the Logarithm of 101 true to 12 Places of Figures, and obtained by the firft Term of the Series only, whence 'tis eafy 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 20401. This Theorem which we'll call Theorem the third, was firft found out by Dr. Halley, and a notable Inftance of its Ufe given by him in the Philofophical Transactions 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 first made use of, without indefatigable Pains and Labour, if at all; on Account of the great Difficulty that would attend the managing fuch large Numbers. I Our Author's Series for this Purpofe is (Page 363) 7 + &c. the Investigation of which, 4% 24% 360x5 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, becaufe 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 fo ftill; however that be, 'tis certain our Author's converges no fafter than the fe cond Theorem, as I found by the Investigation thereof, which may be as follows. From the foregoing Doctrine the Difference of the Logarithms of z-1 and z+1 is mx2 + ช 2 2 7 27 + 2 2 2 323 + &c. which put equal to y, and the Logarithm of the Ratio of the Arithmetical Mean z, to the Geometrical Mean zz-1 is m x I + + Let A and B be the Logarithms of 2-1 and z+1 +, &c. the Logarithm of %, and if the latter Part 8 of the Series expreffing the faid Logarithm of z be divided by the Series representing the Difference of the Logarithms of z-1 and 2+1 the Quotient will exhi¬ Now because the Dividend is ever equal to the Divifor drawn into the Quotient of the Divifion, it follows I 625 I +&c. is the Logarithm of z. Wherefore +6% I 222 + 424 A+B 2 &c. is the Logarithm Note, I make the Author's 5th Term. 13 to 2520029 Te To illuftrate this Theorem by an Example. 2,0043000858 Add the Difference of the faid Loga-0,0000212875 rithms divided by 4 % equal to And the Sum, viz. 2,0043213733 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 ufing only the firft Term of the Series; whereas if two Terms in each were us'd, perhaps the Difference would have been confiderably greater. Note, This Series of our Author, deduc'd from Theofem the Second, is in effect Dr. Halley's too, but vail'd over by being thrown into a different Form: which however has its Ufe, as being very ready in Practice. Having thus investigated several Theorems, whereby the Tables of Logarithms, according to 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 1 to 1x has been =n I prov'd to be as 1+x − 1 =7×x − {x2+} x3− {x', c. n being any infinite Index whatsoever; whence if L be put for the faid Series, then 1+x-1=L con I n fequently 1+x= 1+L, and 1+x=1+L=1+ n L+in2 L2 + { n3 L3 + —n1 L1, &c. AGAIN, The Logarithm of the Ratio of 1 to 1-x has like I I wife been prov'd to be as. 11-x = × ×+ 1 x2 + { x2 + 1x1, &c. L, wherefore 1-L, and n and i-xi-£= i¬n L+ ¦ m2 L2 24 n + L4, &c. Whence nĹ + in Ĺ+ im L 1+ I x= 'n m2 24 n1 L4, &c. is a general Theorem for finding the Number from the Logarithm given of any Species or Form whatsoever; but in the Application of it to Prac tice we labour under a great Inconveniency, efpecially if the Numbers are large; that is to fay, it converges fo very flow, that it were much to be wifh'd it could be con tracted. However, if L be the Logarithm of the Ratio of a the leffer Term, to b the greater, and either of them are given; then the other will be easily had and expeditiously enough too: For b and a b { x 1 + n L + i n2 Ĺ3 + ¦ n1 Ĺ1‚&« Wherefore it follows by the Help of a Table of Logarithms, that the correfponding Number to any Logarithm may be found to as many Places of Figures as thofe Logarithms confift of; for putting d equal to the Difference between the given Logarithm and the next lefs in the Table, then will the Number fought, viz. Naxi+nd + { n2 d2 + ¦ n3 d3 c. But if d be put equal to the Difference between the given Logarithm, and the next greater, then N bx1 -nd+in2 d2 — ! n3 d3, &c. Both of which Series converge fafter as d is fmaller. But the first three Terms in each may be contracted into two, which is very useful, inafmuch as it faves the Trouble of raising n and d in the third Term to the fecond Power; for letting the first Term remain as it is, the other two are reduc'd to one; thus, make the fecond Term the Numerator of a Fraction, and Unity minus the third Term divided by the Second is the Denominator. Whence Naxi + n d + 1⁄2 n2 d2 = at Number anfwering to the given Logarithm, which tho' it differs a little from the Truth, is fufficient to find the Numbers exact to as many Places as Briggs's Logarithms confifts of, viz. 14, which are the largest Tables extant. Much after the fame Method may the whole Series be contracted, by which Means each alternate Power of d will be exterminated, or which is the fame Thing, every two Terms in the Series will be reduc'd to one, making the whole as short again. To illuftrate thefe Contractions by an Example. Let it be required to find the Number answering to the Logarithm 7,5713740282 in Briggs's Form. From the given Logarithm Subtract the Log. of 37271000 the next nearest the Remainder is equal to d= 7,5713740282 7,5713710453 ,0000029829 And because the Number 37271000 is lefs than the Number fought call it a, which multipliedby,0000029829. and the Product 1,11175 6659, &c. divided by m-d=,434x92, &c. quotes,255,992, which added to 37271000 gives 37271255,992 for the Number fought. Thus, I prefume, the Doctrine of Logarithms has been fufficiently exemplified, whether we confider the Conftruction of them for any given Numbers, or on the contrary the finding of the Numbers from the Logarithms given. But before I conclude I fhall give an Inftance or two of the great Ufe of Logarithms in Arithmetical Calculations, and firft in the purchafing of Annuities. If a be put for any Annuity, p for the present Value, r the Amount of one Pound for one Year at any Rate of Intereft, and t for the Time or Number of Let it be required to find the present Value of an Annuity of 60 1. per Annum to continue 75 Years at the Rate of 4 per Cent. per Annum. Here. |