firft Place to the Left-hand; for they are leffer than the Logarithm of the Number 10, whofe Beginning is Unity and the Logarithms of the Numbers between 10 and 100 begin with Unity; for they are greater than 1,0000000, and lefs than 2,0000000. Alfo the Logarithms between 100 and 1000 begin with 2; for they are greater than the Logarithm of 100 which begins with 2, and lefs than the Logarithm of 1000 that begins with 3. In the fame manner it is demonftrated, that the firft Figure to the Left-hand of the Logarithms between 1000 and 10000 must be 3; and the firft Figure to the Left-hand of the Logarithms between 10000 and 100000 will be 4; and fo on. The first Figure of every Logarithm to the Lefthand is called the Characteristic, or Index, because it fhews the highest or most remote Place of the Number from the Place of Units. For Example, if the Index of a Logarithm be 1, then the higheft or moft remote Place from Unity of the correfpondent Number, to the Left-hand, will be the Place of Tens. If the Index be 2, the most remote Figure of the correfpondent Number fhall be in the fecond Place from Unity, that is, it fhall be in the Place of Hundreds ; and if the Index of a Logarithm be 3, the laft Figure of the Number anfwering to it, fhall be in the Place of Thoufands. The Logarithms of all Numbers that are in decuple or fubdecuple Progreffion, only differ in their Characteristics, or Indices, they being written in all other Places with the fame Figures. For Example, the Logarithms of the Numbers 17, 170, 170, 17000, are the fame, unless in the r Indices; for fince I is to 17, as 10 to 170, and as 100 to 1700, and as 1000 to 17000; therefore the Distances between 1 and 17, between 10 and 170, between 100 and 1700, and be tween 1000 and 17000, shall be all equal. And fo, fince the Distance between 1 and 17, or the Logarithm of the Number 17 is 1.2304489, the Logarithm of the Number 170 will be 2.2304489, and the Logarithm of the Number 1700 fhall be 3.2304489, because the Logarithm of the Number ICC 2.0000000. In like manner, fince the Logarithm of the Number 1000=3.0000000, the Logarithm of the Number 17000 fhall be 4.2304489. So alfo the Numbers, 6748. 674,8. 67,48. 6,748. 0,6748. 0,06748, are continual Proportions in the Ratio of 10 to 1; and so 6748 3,8291751 67 4,8 2,8291751 67,48 1,8291751 6,7 4 8 0,8291751 0,6 7 4 8-1,8291751 0,0 6 7 4 8-2,8291751 their Distances from each other fhall be equal to the Distance or Logarithm of the Number 10, or equal to 1,00000Co. And fo, fince the Logarithm of the Number 6748 is 3,8291751, the Logarithms of the other Numbers shall be as in the Margin; where you may observe, that the Indices of the last two Logarithms are only negative, and the other Figures pofitive; and fo, when thofe other Figures are to be added, the Indices must be subtracted, and contrariwife. CHA P. II. Of the Arithmetic of Logarithms in whole Numbers, or whole Numbers adjoined to Decimal Fractions. Fig. 2. BEcaufe, in Multiplication, Unity is to the Multiplier, as the Multiplicand is to the Product; the Diftance between Unity and the Multiplier fhall be equal to the Distance between the Multiplicand and the Product. If therefore the Number G H be to be multiplied by the Number EF, the Distance between GH and the Product must be equal to the Diftance, A E, or to the Logarithm of the Multiplier; and fo, if GL be taken equal to AE, the Number L M fhall be the Product; that is, if the Logarithm of the Multiplicand AG be added to the Logarithm of the Multiplier A E, the Sum fhall be the Logarithm of the Product. In Divifion, the Divifor is to Unity, as the Dividend is to the Quotient; and fo the Distance between the Divifor and Unity fhall be equal to the Distance between the Dividend and the Quotient. So if L M be to be divided by EE, the Distance E A shall be aqual to the Distance between L M and the Quotient; and and fo, if LG be taken equal to EA, the Quotient will be at G; that is, if from AL, the Logarithm of the Dividend, be taken GL, or AE, the Logarithm of the Divifor, there will remain A G, the Logarithm of the Quotient. And from hence it appears, that whatsoever Operations in common Arithmetic are performed by multiplying or dividing of great Numbers, may be done much eafier, and more expeditioufly, by the Addition or Subtraction of Logarithms. Log. 3.8801846 For Example, Let the Number 7589 be to be multiplied by 6757. Now, if the Logarithms of thofe Numbers be added together, as in the Margin, their Sum will be the Logarithm of the Product, whofe Index 7 fhews, that there are feven Places of Figures, befides Unity, in the Product; and in feeking this Logarithm in Tables, or the neareft equal to it, I find that the Number answering thereto, which is leffer than the Product, is 51278000; and the Number greater than the Product is 51279000; and if the adjoined Differences, and proportional Parts, be taken, the Numbers that must be added to the Place of Hundreds and Tens in the Product are 87; and that which must be added in the Place of Unity, will neceffarily be 3, fince feven Times 963; and fo the true Product thall be 51278873. If the Index of the Logarithm had been 8 or 9, then the Numbers to be added in the Place of Tens or Hundreds could not be had from thofe Tables of Logarithms which confit of but 7 Places of Figures, befides the Characteristic; and fo, in this Cafe, the Valacquian or Briggian Tables fhould be used; in the former of which, the Logarithms are all to ten Places of Figures, and in the latter to fourteen. If the Number 78596 be to be divided by 276, by fubtracting the Logarithm of the Divifor from the Logarithm of the Dividend, the Logarithm of the Quotient will be Log. 4.8954004 Log. 2.4440448 Log. 2.4513556 had. And to this Logarithm, the Number 282, 719 anfwers; which therefore fhall be the Quotient. Becaufe Unity, any affumed Number, the Square thereof, the Cube the Biquadrate, &c. are all con Z 2 tinual tinual Proportionals, their Distances from each other fhall be equal to one another. And fo it is manifeft; that the Dittance of the Square from Unity is double of the Distance of its Root from the fame: Alfo the Distance of the Cube is triple of the Distance of its Root; and the Diftance of the Biquadrate is quadruple of the Distance of its Root from Unity, &c. And fo, if the Logarithm of any Number be doubled, we fhall have the Logarithm of its Square; if it be tripled, we fhall have the Logarithm of its Cube; and if it be quadrupled, the Logarithm of its Biquadrate. And contrariwife, if the Logarithm of any Number be bifected, we shall have the Logarithm of the Square Root thereof Moreover, a third Part of the faid Logarithm will be the Logarithm of the Cube Root of the Number; and a fourth Part, the Logarithm of the Biquadrate Root of that Number. Hence, the Extractions of all Roots are eafily performed, by dividing a Logarithm into as many Parts as there are Units in the Index of the Power. So if you want the Square Root of 5, the Half of 0,6989700 must be taken, and then that Half 0,3494850 will be the Logarithm of the Square Root of 5, or the Logarithm of 5, to which the Number 2,236068 nearly anfwers. CHA P. III. Of the Arithmetic of Logarithms, when the Numbers are Fractions. Fig. 3. WE HEN Fractions are to be worked by Loga. rithms, it is neceflary, for avoiding the Trouble of adding one Part of a Logarithm, and fubtracting the other, that Logarithms do not begin from an integral Unit, but from fome Unit that is in the Tenth or Hundredth Place of Decimal Fractions: For Example, let PO be 1, and from this let the Logarithm begin. Now this Fraction is ten Times more diftant from Unity to the Left-hand, than the Number 10 is diftant therefrom to the Right; for there are 10 proportional Terms in the Ratio of 10 to 1, from Unity to PO. And fo, if A B be Unity, the the Logarithm thereof, according to this Suppofition, will not be o, but OA=10.0000000. Now the Distance of Ten from Unity is 1.0000000, whence the Distance of the Number ro from PO will be 1-1.0000000. Alfo the Distance of the Number 100 from PO, or its Logarithm, beginning from PO, shall be 12.0000000; and the Logarithm of 1000, or the Distance from PQ, will be 13.0000000. And thus, the Indices of all Logarithms are augmented by the Number 10; and thofe Fractions whofe Indices are I, or—2, or—3, &c. are now made 9, 8, or 7, &c. But if Logarithms begin from the Place of a Fraction, whofe Numerator is Unity, and Denominator Unity with 100 Cyphers added to it (which they must do when Fractions occur that are less than P O), then that Fraction will be 100 Times more diftant from Unity, than 10 is diftant from it; and fo the Logarithm of Unity will have 100 for the Index thereof. And the Logarithm of any Tens will have 101 for the Index, that of any Hundreds 102, and so on; all the Indices being augmented by the Number 100. I 1000 The Logarithms of all Fractions that are greater than PO (whereat they begin) will be politive. And fince the Numbers 10, 1, 1, 100, Too, &c. are in a continual Geometrical Progreffion, they will be equally diftant from each other; and accordingly their Logarithms will be equidifferent: And fo, when the Logarithm of 10 is 11.0000000, and the Logarithm of Unity is 10.0000000; then the Logarithm of the Fraction will be 9.0000000, and the Logarithm of the Fraction will be 8.0000000; and, in like manner, the Index of the Logarithm of will. be 7. Alfo, for the fame Reafon, if the Index of the Logarithm of Unity be 100, and of 10 be 101, then will the Index of the Logarithm of the Fraction be 99, and the Index of the Logarithm of will be 98, and the Index of the Logarithm of the Fraction,0 fhall be 97, &c. And thefe Indices fhew in what Place from Unity the firft Figure of the Fraction, not being a Cypher, must be put. For Example, if the Index be 4, the Distance thereof from the Index of Unity (which is 10), viz. 6, fhews that the firft fignificative Figure of the Decimal is in the fixth Place from Unity; and therefore five Cyphers are to be prefixed Z3 TOO thereto |