L. 10 = L. 4 2 L. 2, L. 100 2 L. 10, L. 8 = 3 L. 2, L. 2401 L. 3403+L.2400, L. 24 = L. 8+ L. 3, L. 11 =L+L. 9, L.2400 L.100+ L.24, L. 6 = L. 2 + L. 3. Thus we have got the logarithms of 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. And this is upon the whole, perhaps the beft method of computing logarithms that can be taken. There have been indeed fome methods difcovered by Dr. Halley, and other mathematicians, for computing the logarithms of the ratios of prime numbers to the next adjacent even numbers, that are ftill fhorter than the application of the foregoing ferics. But thofe methods are lefs fimple and easy to understand and apply than these feries; and the computation of logarithms by these feries, when the terms of them decrease by the powers of 10, or of fome greater number, is fo very fhort and easy (as we have seen in the foregoing computations of the logarithms of the ratios of 10 to 9, 11 to 10, 81 to 80, 121 to 120, &c,) that it is not worth while to feek for any fhorter methods of computing them. And this method of computing logarithms is very nearly the fame with that of Sir Ifaac Newton in his fecond letter to Mr. Oldenburg, dated October 1676, as will be seen in the following article. Of Sir Ifaac Newton's Methods. The excellent Sir I. Newton greatly improved the quadrature of the hyperbolical-afymptotic spaces by infinite feries, derived from the general quadrature of curves by his method of fluxions; or rather indeed he invented that method himself, and the conftruction of logarithms derived from it, in the year 1665 or 1666, before the publication of either Mercator's or Gregory's books, as appears by his letter to Mr. Oldenburg dated Oct. 24, 1676, printed in pa. 634 et feq. vol. 3 of Wailis's works, and elsewhere. The quadrature of the hyperbola, thence tranflated is to this effect. Let dFDD be an hyperbola, whofe center is C, vertex F, and interpofed fquare CAFE E 1. In CA take AB and Ab on each fide = 7% or o.1: And, erecting the perpendiculars BD, bd; half the fum of the spaces AD and Ad will be = 0.1 +0.001 +0.00001 +0.0000001 &c. 3 100000*0 7 СЪАВ 1.0000000000000 0.0050000000000 The fum of thefe 0.1053605156577 is Ad. 250000000 and the differ. 0.0953101798043 isAD. 1666666 In like manner putting AB and Ab 12500 each 100 Ad 0.1003353477310 0.0050251679267 0.2, there is obtained 0.2731435513142, and 0.1823215567939. 1.2 I 2 Having thus the hyperbolic logarithms of the four decimal numbers 0.8, 0.9, 1.1, and 1.2; and fince x2, and 0.8 and 0.9 are lefs than unity; add their logarithms to double the logarithm of 1.2, and you will have 0.6931471805597 the hyperbolic logarithm of 2. 2 X 2 X 2 To the triple of this add the logarithm of 0.8, because 0.8 10, and you have 2.3025850929933 the logarithm of 10. Hence by one addition are found the logarithms of 9 and 11: And thus the logarithms of all these prime numbers 2, 3, 5, 11 are prepared. Moreover, by only depreffing the numbers, above computed, lower in the decimal places, and adding, are obtained the logarithms of the decimals 0.98, 0.99, 1.01, 1.02; as alfo of thefe 0.998, 0.999, 1.001, 1.002: And hence by addition and subtraction will arife the logarithms of the primes 7, 13, 17, 37, &c. All which logarithms. being divided by the above logarithm of 10, give the common logarithms to be inferted in the table. And again a few pages farther on in the fame letter he refumes the construction of the logarithms, thus: Having found, as above, the hyperbolic logarithms of 10, 0.98, 0.99, 1.01, 1.02, which may be effected in an hour or two, divide the last four logarithms by the logarithm of 10, and adding the index 2, you will have the tabular logarithms of 98, 99, 100, 101, 102. Then by interpolating nine means between each of thefe, will be obtained the logarithms of all numbers between 980 and 1020; and agam interpolating 9 means between every two numbers from 980 to 1000, the table will be fo far conftructed. Then from thefe will be collected the logarithms of all the primes under 100, together with thofe of their multiples: all which will require only addition and subtraction; for = 61, 3×49. 9894 = 89,6x17 = 97• This quadrature of the hyperbola, and its application to the conftruction of logarithms, are ftill farther explained by our celebrated author in his treatife on Fluxions, publifhed by Mr Colfon in 1736, where he gives all the three feries for the areas AD, Ad, Bd, in general terms, the former the fame as that published by Mercator, and the latter by Gregory; and he explains the manner of deriving the latter feries from the former, namely by uniting together the two feries for the spaces on each fide of an ordinate, bounded by other ordinates at equal diftances, every 2d term of each feries is cancelled, and the refult is a feries converging much quicker than either of the former. And, in this treatise on fluxions, as well as in the letter before quoted, he recommends this as the most commodious method of constructing a canon of logarithms, computing by the feries the hyperbolic fpaces anfwering to the prime numbers 2, 3, 5, 7, 11, &c, and dividing them by 2.3025850929940457, which is the area corresponding to the number 10, or elfe multiplying them by its reciprocal 0.4342944819032518, for the common logarithms." Then the logarithms of all the numbers in the canon which are made by the multiplication of these, are to be found by the addition of their logarithms, as is ufual. And the void places are to be interpolated afterwards by the help of this theorem: Let n be a number to which a logarithm is to be adapted, the difference between that and the two nearest numbers equally distant on each fide, whofe logarithms are already found, and let d be half the difference of the logerithms; then the required logarithm of the number " will be obtained by adding d+ &c to the logarithm of the lefs number." This theorem he demonftrates by the hyperbolic areas, and then proceeds thus; " The two first terms d+ of this dx dx3 - + dx 2n feries I think to be accurate enough for the construction of a canon of logarithms, even though they were to be produced to 14 or 15 figures; provided the number whofe logarithm is to be found be not Jefs than 1000. And this can give little trouble in the calculation, because x is generally an unit, or the number 2. Yet it is not neceffary to interpolate all the places by the help of this rule. For the logarithms of numbers which are produced by the multiplication or divifion of the number laft found, may be obtained by the numbers whofe logarithms were had before, by the addition or fubtraction of their logarithms. Moreover by the differences of the logarithms, and by their 2d and 3d differences, if there be occafion, the void places may be more expeditiously supplied; the foregoing rule being to be applied only when the continuation of fome full places is wanted, in order to obtain thofe differences, &c." So that Sir I. Newton of himself discovered all the feries for the above quadrature which were found out, and afterwards published, partly by Mercator and partly by Gregory; and these we may here exhibit in one view all toge ther, and that in a general manner for any hyperbola, namely putting CA a, AF = b, and AB = Ab= x; then will ab BD = a+ x2 and bd = ab a-x ; whence the area AD = bx + 3a2 26x5 584 In the fame letter also, above quoted, to Mr. Oldenburg, our illuftrious author teaches a method of conftructing the trigonometrical canon of fines by an easier method of multiple angles than that before delivered by Briggs for the fame purpofe, becaufe that in Sir Ifaac's way radius or 1 is the firft term, and double the fine or cofine of the first given angle is the 2d term of all the proportions by which the feveral fucceffive multiple fines or cofines are found. The fubftance of this method is thus: The best foundation for the construction of the table of fines, is the continual addition of a given angle to itself or to another given angle. As if the angle A be to I NYP A E G S a L be added; infcribe HI, IK, KL, LM, MN, NO, OP, &c each equal to the radius AB; and to the oppofite fides draw the perpendiculars BE, HQ, IR, KS, LT, MV, NX, OY, &c; fo fhall the angle A be the common difference of the angles HIQ, IKH, KLI, LMK, &c; their fines HQ, IR, KS, &c; and their cofines IQ, KR, LS, &c. Now let any one of them LMK, be given, and the reft will be thus found: Draw Ta and Kb perpendicular to SV and MV; then because of the equiangular triangles ABE, TLa, KMb, ALT, AMV, &c, it will be AB: AE: KT: Sa (LV + LS) LT: Ta (= MV + KS,) and AB : BE :: LT : La (LS-LV) :: KT (=‡KM) : ‡Mb (=MV~KS.) Hence are given the fines and cofines KS, MV, LS, LV. And the method of continuing the progreffions is evident. Namely AB: 2AE :: MV NXLT NX OY + MV &c SLV : MT+ MX :: MX ; NV + NY, &c. 2 or AB : 2BE : LV NX - LT :: MX : OY - MV &c MX NX : NV - NY &c And on the other hand, AB :: 2AE :: LS: KT+KR &c. Therefore put AB = 1, and make BEx LT-La, AE× KT = Sa, Sa La LV, 2AE x LV-TM MX, &c. where the 4th terms of these proportions are the fums or dif The sense of these general theorems is this, that if P be any one among a series of angles in arithmetical progreffion, the angle'd be ing their common difference, then as radius or I: 2 cof. d:: [ cof. P: cof. P+d+cof. P-d fin. P: fin. P+d+ fin. P-d Scof. P: fin. P+d-fin. P-d I: 2 fin. d:: fin. P: cof. P+d—cof. P-d ferences of the fines or cofines of the two angles next lefs and greater than any angle Pin the feries; and therefore fubtracting the lefs extreme from the fum, or adding it to the difference, the refult will be the greater extreme, or next fine or cofine beyond that of the term P. And in the fame manner are all the reft to be found. This method it is evident, is equally applicable whether the common difference d or angle A be equal to one term of the series or not: when it is one of the terms, then the whole series of fines and cofines becomes thus, as 1 : 2 cof. d :: Sn.d: fin. 2d :: fin. 2d: fin, d+fin. 3d :: fin. 3d : fin. 2d+fin. 4d : : fin. 4d: fin. 3d+fin. 5d &c, cold: 1+col. 2d:: cof. 2d: col.d+cof.3d:: cof. 3d : col.2d+cof.4d :: cof.4d: cof. 3d+cof.5d&c. which is the very method contained in the directions given by Mr. Abr. Sharp for conftructing the canon of fines. 2 ≈2 z3 24 &c Sir I. Newton remarks that it only remains to find the fine and cofine of a first angle A by fome other method, and for this purpose he directs us to make use of fome of his own infinite feries: thus, by them will be found 1.57079 &c for the quadrantal arc, the fquare of which is 2.4694 &c; divide this fquare by the fquare of the number expreffing the ratio of yo degrees to the angle A, calling the quotient z; then 3 or 4 terms of this feries 1-- + will give the cofine of that angle A. Thus we may first find an angle of 5 degrees, and thence the table computed to the feries of every 5 degrees; then these interpolated to degrees or half degrees by the fame method; and thefe interpolated again; and fo on as far as neceflary. But two-thirds of the table being computed in this manner, the remaining third will be found by addition or substraction only, asis well known. Various other improvements in logarithms and trigonometry are owing to the fame excellent perfonage; fuch as the feries for expreffing the relation between circular arcs and their fines, cofines, verfedfines, tangents, &c; namely, the arc being a, the fine s, the verfed, fine v, cofine c, tangent t, radius 1, then is Many other improvements in the conftruction of logarithms are alfo derived from the fame doctrine of fluxions, as we shall shew hereafter. In the mean time proceed we to the ingenious method of the learned Dr. Edmund Halley, Secretary to the Royal Society, and the fecond Aftronomer Royal, having fucceeded Mr. Flamfteed in that honourable office in the year 1719 at the Royal Obfervatory at Greenwich, where he died the 14th of January 1742, in the 86th year . of his age. His method was first printed in the Philofophical Tranfactions for the year 1695, and it is entituled "A moft compendious and facile method for conftructing the logarithms, exemplified and demonftrated from the nature of numbers, without any regard to the |