jected in the Factors m-1, m-2, &c. (as before) our Equation will become 1 + e = 1 ~ mx — mx2 m x3 &c. whence x + + + &c. = ne = the hyperbolic Logarithm of I -x Which Series, it is manifeft, will always converge, will be always less than Unity. But it is further obfervable that this Series has exactly the fame Form (except in its Signs) with that above for the Logarithm of 1 + x; and that, if both of them be added together, the Series 2 x + 2x3 2x5 2x7 + + &c. thence arifing, will be more 7 3 5 fimple than either of them; fince one Half of the Terms will be intirely destroyed thereby. Therefore, because the Sum of the Logarithms of any two Numbers is equal to the Logarithm of the Product of thofe Numbers, (fee Article 1.) it is mani2x3 2x5 &c. will truly exprefs feft that 2x + + 3 5 or > Series converges, still, faster than x + - x2 x3 not only because the even Powers are here deftroyed, but becaufe x, in finding the Logarithm of any given Number (N), will have a leís Value. But now, to determine what this Value must be, make =N, and then x will be found = N •I ; but if the Quantity propofed be a Fraction N+I Values, fubftituted in the foregoing Series 2 x + 2x3 2x5 3 + 5 &c. will give the hyperbolic Logarithm of the refpective Number. Example. Let it be proposed to find the hyperbolic Logarithm of the Number 2. Which Values being refpectively divided by the Numbers, 1, 3, 5, 7, 9 &c. and the feveral Quotients added together, (fee the general Series) we fhall have ,346573590 &c. whofe Double, being ,693147180 &c. is the hyperbolical Logarithm of the Number 2. any After the very fame Manner the hyperbolic Logarithm of other Number may be determined; but, as the Series converges, flower and flower, the higher we go, it is ufual, in computing of Tables, to derive the Logarithms we would find, by Help of others already known; for which there are various Methods; but the following is the most commodious and fimple, that has occur'd to me, efpecially, when a great Degree of Accuracy is required. It is thus. Let a, b and c denote any three Numbers in arithmetical Progreffion, whofe common Difference is Unity; then, a being b-1 and c = b+1, we fhall have ac = b2-1, and con of Logarithms, we likewife have 2 Log.b-Log. ac+1: But the Logarithm 2ac + I &c. (by what has been already fhewn): Which being denoted by S, we fhall (Log. b = 1 Log. a + have Log. a 2 Log. bLog.c2 Log. b Log. c + 1 S. As an Example hereof, let it be proposed to find the hyperbolic Logarithm of 3. Then, the hyperbolic Logarithm of 2 being already found,693147180 &c. that of 4, which is the Double thereof, will also be known. Therefore, taking a = 2, = 3, and C=4, we fhall, in b this Therefore ÷S (*+*+* &c.)=,058891517 &c.and 5 confequently hyp. Log. 3. (hyp. Log, 2+byp. Log. 4• +S) = 1,098612288 &c. 2 2. Let the hyperbolic Logarithm of 10 be required. The Logarithms of 8 and 9 being given, from thofe of 2 and 3 (already found), a may, here, be = 8, b= 9 and c = 10; and then 2+1 being x + &c.) = 3 5 ,006211180 &c. +,000000079 &c. &c. = ,006211259 &c. And therefore Log 10 (2 Log 9 — Log. 8-S) 2,302585092 &c. Hitherto we have had Regard to Logarithms of the hyperbolic Kind: But those of any other Kind may be derived from thefe, by, barely, multiplying by the proper Multiplicator, or Modulus. Thus, in the Brigean (or common) Form, where an Unit is affumed for the Logarithm of 10, the Logarithm of any Number will be found, by multiplying tiplying the hyperbolic Logarithm of the fame Number by the Fraction,434294481 &c. which is the proper Modulus of this Form. For, fince the Logarithms of all Forms preferve the fame Proportion with refpect to each other, it will be, as 2,302585092 &c. the hyperbolic Log. of 10 (above found) is to (H) the hyperbolic Logarithm of any other Number, fo is 1, the common Logarithm of 10, to ( H 2.302585092 &c. Hx,434294481 &c. the common Logarithm of the fame Number. But (to avoid a tedious Multiplication, which will always be required when a great Degree of Accuracy is infifted on) the best Way to find the Logarithms of this Form is from the Series 2x+ 2x3 2x5 3 + &c. x 0,434294481&c. which expreffes the common Logarithm of (by what has been I-X already fhewn), and which, by making R = ,868588963 &c. will ftand more commodiously Rx3 Rxs Rx7 thus, Rx+ 3 + + &c. 5 7 For an Example hereof, let the common Logarithm of 7, be required: In which Cafe (the Logarithms of 8 and 9 being known, from thofe of z and 3), we fhall have Log. 72 Log. 8-Log. 9 Rx3 Rxs S (by the Theor.), S being = Rx + + 3 5 2 |