(2.) Hence we have the Logarithms of 3.1622777 and 10 given. Log. of 10 Log. of 3.1622777 = 0.5 Hence the Log, of 10 X 3.1622777, 1.5 viz. of 5.6234132 is 0.75 0.75 (3.) We have now the Logarithms of 10 and 5.6234132 given. 10. Numbers { .6234132 its Log. 0.75 1.75 2 10x5.6234132 7.4989421 its Log. =0.875 (4.) Here we have the Log. of 10 and 7.4989421 10X7.4989421 8.6596431 its Log. = 0.9375 (5.) Hence we have the Logarithms of 10 and 8.6596431 given. Their SI. Numbers {8.6596431 Logarithms 10.9375 2 1.9375 ✔10X8.65964319.3057204 its Log. 0.96875 (6.) We have now the Log. of 8.6596431 and 9.3057204 given. 9.30572042 Their S0.96875 8.6596431 S Logarithms 0.9375 9.3057204X8.6596431 its Log. 0.953125 And proceeding after this Manner, after 25 Extractions the Logarithm of 8.9999998 will be found to be 0.9542425, which we take for the Logarithm of 9, because 8.9999998 differs but 1 or 3000000 from 9. By the fame Method the Logarithms of the prime Numbers were found. I ΤΟ II. The Logarithms of the prime Numbers being found, the Logarithms of the Numbers compofed of them are found by adding the Logarithms of the component Parts; the Reafon of which is evident from the Definition, in Art. 1. Take an Example, Given the Logarithm of 2 = 0.30103, and the Logarithm of 3 = 0.4771212, to find the Logarithm of 2 X 3 or 6. The Log. of 2 = 0.30103 Log. of 30.4771212 Log. of 60.7781512 12. The Logarithm of any Number being given, the Logarithm of any Number equal to the Square, or Cube, &c. of that Number, may be found, by multiplying the Logarithm of the Root by 2, 3, &c. refpectively, according as the Power, whofe Logarithm is to be found, is a Square, or Cube, &c. This is manifeft by Art. 4. 13. Laftly, the Logarithm of any Power being given, the Logarithm of its Root is found by dividing that Logarithm by the Index of the given Power: By Art. 5. 14. After the Method juft now defcribed the Tables of Logarithms were at firft made. And if we confider confider the great Labour and Patience neceffary: to the Accomplishment of fo great a Work, how much are we beholden to Lord Neper and Mr. Brigs? Certainly their Names will, and ought to be perpetuated from one Age to another, even to the End of the World! There have lately been invented much easier Methods of conftructing Logarithms than the above; but as they depend on Fluxions, Conic Sections, &c. they cannot be understood in this Place. And it is fufficient to have fhewn the Method by which the Tables were made, as we have now no Occafion to construct new ones. 15. Having explained the Nature of making a Table of Logarithms of Numbers greater than Unity, the next Thing to be done is to fhew how the Logarithms of fractional Numbers may be found. In Order to which let it be observed, that, as we have hitherto supposed a geometrical Series to increase from an Unit on the right Hand, we may now fuppose it to decreafe from an Unit towards the left Hand, as here annexed. Log.3.2.-I. o. I A. B. C. Numbers. TOO. TO⋅ I. IO. 100. 1000, &t. Here the Series from the Point A, or Unity, increafes towards B, and decreases towards C. It is evident by a, bare Infpection, that, if the Numbers are both Fractions, the Sum of their Logarithms is equal to the, Logarithm of their Product. But if one Number is on the Right-hand of the Point A, and the other on the Left of it; that is, if one Number is greater than an Unit, and the other a Fraction, then the Difference of their Logarithms will be equal to the Logarithm of their Product; and the Logarithm of the Product will be on the fame Side of the Point A, as is the greater of the two Logarithms. 16. 16. In Order to make a proper Diftinction, we may denote the Logarithm of any Number greater than an Unit by the Sign+, for they may be reckoned affirmative; and then the fractional Quantities muft be confidered as negative. 17. It is allò evident from Inspection of the above Series, that if a Number, having an affirmative Logarithm, is divided by another Number whofe Logarithm is a greater affirmative Number, the Difference of their Logarithms taken negatively will be the Logarithm of their Quotient. Thus, e. g. the Logarithm of 100= 2, and the Logarithm of 1000 = 3; their Difference taken negatively is the Logarithm of the Quotient of 100 1000. This might have been deduced from the Nature of Fractions. For the Numerator of any Fraction may be esteemed as a Dividend, and the Denominator as a Divifor: Therefore, by the Nature of Logarithms, (Art. 1.) the Logarithm of the Numerator minus the Logarithm of the Denominator is equal to the Logarithm of the Fraction: But fince the Denominator of a proper Fraction is always greater than the Numerator, its Logarithm will be greater than that of the Numerator, and confequently it will be impoffible to take the Logarithm of the Denominator from that of the Numerator; therefore, we are obliged to take the Logarithm of the Numerator from that of the Denominator, and exprefs the Difference negatively, for the Logarithm of the Fraction. 18. The Logarithm of 1, 10, 100, 1000, &c. being 0, 1, 2, 3, &c. refpectively, it is manifeft, that the Logarithms of the intermediate Terms may be confidered as a mixed Number, made up of an Integer and decimal Fraction; and that the integral Part, called by fome Writers the Index, (Lat.) by Others the Exponent, (from expono, Lat.) and by Others the Characteristick, (from Character, Lat.) is one lefs than the Number which expreffes how many Figures the natural Number confifts of; e. g. if the natural natural Number has 4 Places, the Characteristick of the Logarithm will be 3 if of 3 Places, 2; if of 2 Places, 1, &c. 19. Divifion being performed by Subtraction of the Logarithms, and the Logarithm of 10 being i, it follows, that the Logarithm of of any Number, must be one less than the Logarithm of that Number. Thus, e. g. the Logarithm of 7851 being + 3.894925, The above affirmative Logarithms were found by fucceffively fubtracting the Logarithm of 10, viz. + from the preceding Logarithm; then, to find the first negative Logarithm, +1 (the Log, of 10) was to be taken from the Logarithm +0.894925 ; Sut as the Log. to be fubtracted is greater than that from which it was to be taken, we are obliged to fubtract the leffer 0.894925 from the greater 1, and express their Difference negatively, for the required Logarithm, (as directed in Art. 17,) which gives -0.105075. Then to find the other negative Lo4 garithms, the Logarithm + 1 being to be fubtracted from a negative Logarithm, we have only to add fucceffively, and exprefs the Sum negatively; for by Art. 46, Efay on Arithmetick, to fubtract an Affirmative is the fame as to add a Negative. |