This Table I fometimes make ufe of for finding the Logarithm of any Number propos'd, and vice verfa. Suppofe I had Occafion to find the Logarithm of 2000. I look in the first Class of my Table (the whole Table confifts of 8 Claffes) for the next lefs to 2, which is 1.995262315, and against it is 3, which confequently is the first Figure of the Logarithm fought. Again, dividing the, Number propos'd 2, by 1.995262315 the Number found in the Table, the Quotient is 1.002374467; which being look'd for in the fecond Clafs of the Table, and finding neither its Equal, nor a Leffer, I add o to the Part of the Logarithm before found, and look for the faid Quotient 1.002374467 in the third Class, where the next lefs is 1.002305238, and againft it is I, to be added to the Part of the Logarithm already found; and dividing the Quotient 1.002374467, by 1,002305238, laft found in the Table, the Quotient is 1.000069070; which being fought in the fourth Clafs gives o, but being fought in the fifth Clafs gives 2, to be added to the Part of the Logarithm already found; and dividing the laft Quotient by the Number laft found in the Table, viz. 1.000046053, the Quotient is 1.000023015, which, being fought in the fixth Clafs, gives 9 to the Part of the Logarithm already found: And dividing the last Quotient by the new Divisor, viz. 1.000002072, the Quotient is 1.000000219, which being greater than 1.000000115 fhews that the Logarithm already found, viz. 3.3010299 is lefs than the Truth by more than half an Unit; wherefore adding 1, you have Brigg's Logarithm of 2000, viz. 3.3010300. ..... If any Logarithm be given, fuppofe 3,3010300, throw away the Characteristic, then over against thefe Figures 3...0..I..0 ..o, you have in their refpective Claffes 1,995262315.....0..... 1,002305238.....o 1,000069080.... o...o which multiplied continually into one another, the Product is 2.000000019966, which, by reason the Characteristic is 3, becomes 2.000,000019966, &c. that is, 2000. the Natural Number defired. I fhall not mention the Method by which this Table is fram'd, because you will eafily fee that from the Ufe of it. It is obvious to the intelligent Reader, that thefe Claffes of Numbers are no other than fo many Scales of mean Proportionals: In the first Clafs, between 1 and 10; fo that the laft Number thereof, viz. 1,258925412 is the tenth Root of ro, and the reft in order afcending are the Powers thereof. So in the fecond Class, the laft Number 1,023292992 is the hundredth Root of 10, and the reft in the fame Manner are Powers thereof. So 1.002305238 in the third Clafs, is the tenth Root of the laft of the fecond, and the the reft its Powers, &c. Or, which is all one, each Number, in the preceding Clafs, is the tenth Power of the corresponding Number in the next following Class: Whence 'tis plain, that to conftruct these Tables requires no more than one Extraction of the fifth or furfolid Root for each Clafs, the reft of the Work being done by the common Rules of Arithmetick. Their Conftruction, according to the common Rules, given by many Extractions of Roots, is tedious; the best Way yet known is this which follows. To make a Table of Logarithms. First, Put for the Logarithm of 1, a Cypher for the Index, and a competent Number of Cyphers for the Logarithm, according to the Number of Places you would have your Logarithms confift of; for 10 an Unit, with the fame Number of Cyphers; for 100, 2, with as many Cyphers; for 1000, 3, with as many Cyphers, &c. Secondly, Find the Difference between fome two Logarithms above 1000, or rather above 10000, that differ by Unity; thus multiply the two Numbers together, and that Product you must multiply again by 43429448190325183896* which laft Product divided by the Arithmetical Mean between both Numbers, the Quotient is the Difference fought. Suppose we would find the Difference between the Log. 10000, and 10001, the Product of these two Numbers is 1.000 10000. which multiplied by 4343 produced 43434343; this divided by 10000.5, quotes 4343. Now if to the Logarithm of 10000, which is 4.0000000, you add the Difference before found, to wit, 434, the Sum 4.0000434 is the true Logarithm of 10001 to 7 Places. Thirdly, Having thus found the Difference of any two Logarithms differing by Unity, and confequently fome of the Logarithms by dividing the Difference found by the Arithmetical Mean, between any two Numbers differing by Unity, you fhall have the Difference of the Logarithm of thofe two Numbers. Thus to find the Difference betwixt the Logarithm of 274, and 275; divide 4343, the Difference of the Logarithm of 10000, and 10001 by 2745 the Quotient 1582 1, is the Difference fought. Fourthly, Having by this Means found a few of the prime Logarithms, the reft are made by Addition and Subtraction, and having *Which is the Subtangent of the Curve expreffing Brigg's Logarithms. See Keil' Trig. Pag. 135, 140, &c. ing made the Canon upward, above 1000 to 10000, by Confequence it is made for all inferior Numbers. The prime Numbers to which Logarithms must be found, in firft Place, are thefe, 2.3.7. 11. 13. 17. 19.23.29.31.37. 41.43.47.53.59.61.67.71.73.79.89.97, &c. or the fame Numbers with Cyphers. But fince it was very tedious and laborious, to find the Logarithms of the prime Numbers, and not eafy to compute Logarithms by Interpolation, by firft, fecond, and third, &c. Differences, therefore the great Men, Sir Ifaac Newton, Mercator, Gregory, Wallis, and laftly, Dr. Halley, have published infinite converging Series, by which the Logarithms of Numbers to any Number of Places may be had more expeditiously and truer : Concerning which Series, Dr. Halley has written a learned Tract, in the Philofophical Tranfactions, wherein he has demonftrated thofe Series after a new Way, and fhews how to compute the Logarithms by them. But I think it may be more proper here to add a new Series, by Means of which may be found, eafily and expeditiously, the Logarithms of large Numbers. Let z be an odd Number, whofe Logarithm is fought; then shall the Number 2-1 and z+1 be even, and accordingly their Logarithms and the Difference of the Logarithms will be had, which let be called y: Therefore, alfo the Logarithm of a Number which is a Geometrical-Mean between 2-1 and z+1 will be given, viz. equal to the half Sum of the Logarithms. Now the Series 42 + 2121 + 36021 181 &c. fhall be equal to the Logarithm of the Ratio, which the GeometricalMean between the Numbers z-I and z+1, has to the Arithmetical- Mean, viz. to the Number z. If the Number exceeds 1000, the firft Term of the Series is fufficient for producing the Logarithm to 13 or 14 Places of Fi gures, and the fecond Term will give the Logarithm to 20 Places of Figures. But if z be greater than 10000, the first Term will exhibit the Logarithm to 18 Places of Figures; and fo this Series is of great Ufe in filling up the Logarithms of the Chiliads omitted by Briggs. For Example; It is required to find the Logarithm of 20001. The Logarithm of 20000 is the fame as the Logarithm of 2, with the Index 4 prefix'd to it; and the Difference of the Logarithms of 20000 and 20002, is the fame as the Difference of the Logarithms of the Numbers 10000 and 10001, viz. 0.000043427 27687. And if this Difference be divided by 4%, or 80004, the Quo Quotient y shall be 42 0.00000 0000542813 And if the Logarithm or the Geometri- 4. 30105 1709302416 cal-Mean be added to the Quotient, the Sum will be the Logarithm of 20001. 4. 30105 1709845230 Wherefore it is manifeft, that to have the Logarithm to 14 Places of Figures, there is no Neceffity of continuing out the Quotient beyond 6 Places of Figures. But if you have a Mind to have the Logarithm to 10 Places of Figures only, as they are in Vlacq's Tables, the two firft Figures of the Quotient are enough. And if the Logarithms of the Numbers above 20000 are to be found by this Way, the Labour of doing them will moftly confift in fetting down the Numbers. Note, This Series is eafily deduced from that found out by Dr. Halley; and thofe who have a Mind to be inform'd more in this Matter, let them confult his abovenam'd Treatise. Mr. WARD's Eafy Method of making the Canon of Sines, Tangents, &c. FIR IRST, let me premise two Things, that the Periphery of a Circle, whofe Radius is Unity or 1, is 6.283185, &c. and that the natural Sine of one Minute doth fo infenfibly differ from the Length of the Arch of one Minute, that it may be taken for the fame. Confequently, As the Periphery in Minutes is to the That is, 21600': 6,283185:: I': 0,000290888 the Natural Sine of one Minute; which agrees with the largest Table of Sines I ever faw. Having thus got the Sine of one Minute, its Co-fine may be thus found: Suppofe Suppofe RA RS the Radius of any Circle, SN the Sine of the Arch SA. Then RN-CS is the Co RS Sine of that Arch. But That is, From the Square of the R ΝΑ 0,0000000846120,999999915388, the fquare Root thereof is 99999995 the Co-Sine required. The Sine and Co-Sine of one Minute being thus obtain'd, all the rest of the Sines in the Quadrant may be gradually calculated by Mr. Michael Dary's Sinical Proportions; which I fhall here infert, to the fame Effect as they are in his Mifcellanies; and then explain and demonftrate the Truth of thofe Proportions. If a Rank of Arches be equi-different ; [ As the Sine of any Arch in that Rank: is to the Sum of the Sines of any two Arches equally remote from it on each Side:: Then fo is the Sine of any other Arch in the faid Rank: to the Sum of the Sines of two Arches next it on each Side; having the fame common Distance. Immediately after thefe Proportions, he lays down the following Æquations: Three Arches equi-different, being propofed; if (faith he) you put Z= the Sine of the great Extreme, y the Sine of the leffer Extreme; M the Sine of the Mean; m = the Co-Sine thereof; D the Sine of the common Difference; d=the Co-Sine thereof; and R the Radius. = 2 Md 1. Then Z+y=2. Then Z-y= 3. Then Zy MM-DD. 4. Then 2mD Ꮓ Md+mD. - = y Md-vid From the foregoing it is evident (faith he) that if two Thirds, viz. either the former or latter 60 Degrees, or the former 30 Degr. and the latter 30 Degr. of the Quadrant be completed with Sines; the remaining Part of the Quadrant may be completed by Addition, or Subtraction only. PPP Thus |