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LOGARITHMS. Mr. Bonnycastle has communicated the following new method of making these useful numbers:

Logarithms. The series now chiefly used in the computation of Logarithms were originally derived from the hyperbola, by means of which, and the logistic curve, the nature and properties of these numbers are clearly and elegantly explained.

The doctrine, however, being purely arithmetical, this mode of demonstrating it, by the intervention of certain curves, was considered, by Dr. Halley, as not conformable to the nature of the subject. |

He has, accordingly, investigated the same series from the abstract principles of numbers; but his method, which is a kind of disguised fluxions, is, in many places, so extremely abstruse and obscure, that few have been able to comprehend his reasoning.

An easy and perspicuous demonstration, of this kind, was therefore still wanting; which may be obtained from the pure principles of Algebra, independently of the doctrine of Curves, as follows:

The Logarithm of any number, is the index of that power of some other number, which is equal to the given number.

Thus, if , the logarithm of a is x, which may be either positive or negative, and r any number whatever, according to the different systems of Logarithms.

When a = 1, it is plain that x must be = 0, whatever be the value of r; and consequently the Logarithm of 1 is always 0 in every system.

If x = 1, it is also plain that a must be = r; and therefore r is always the number in every system, whose Logarithm in that system is 1.

To find the Logarithm of any number, in any system, it is only necessary, from the equation , to find the value of x in terms of r and a.

This may be strictly effected, by means of a new property of the binomial theorem of Newton; which is given under its proper article in this Appendix. The general Logarithmic equation being , let , &c. See Binomial Theorem, Appendix.

And if p - p²/2 + p³/3 - p⁴/4 + p⁵/5 &c be put = s, we shall have , which let be put = q; then, by reverting the series z or 1/x will be found =(q-(1/2)q²+(1/3)q³-(1/4)q⁴+(1/5)q⁵ &c)/s=(q-(1/2)q²+(1/3)q³-(1/4)q⁴ +(1/5)q⁵ &c)/(p-(1/2)p²+(1/3)p³-(1/4)p⁴+(1/5)p⁵ &c) and consequently .

The Logarithm of a, or 1 + p, is therefore =(p-(1/2)p²+(1/3)p³-(1/4)p⁴+(1/5)p⁵ &c)/(q-(1/2)q²+(1/3)q³-(1/4)q⁴+(1/5)q⁵ &c); or, since , and , the Logarithm of a is =((a-1)-(1/2)(a-1)²+(1/3)(a-1)³-(1/4)(a-1)⁴+(1/5)(a-1)⁵)/ ((r-1)-(1/2)(r-1)²+(1/3)(r-1)³-(1/4)(r-1)⁴+(1/5)(r-1)⁵) &c; Which is a general expression for the Logarithm of any number, in any system of Logarithms, the radix r being taken of any value, greater or less than 1.

But as r in every system, is a constant quantity, being always the number whose Logarithm in the system to which it belongs is 1, the above expression may be simplified, either by assuming r = to some particular number, and from thence finding the value of the series constituting the denominator; or by assuming this whole series = to some particular number, and from thence finding the value which must be given to the radix r.

By the latter of these methods, the denominator may be made to vanish, by assuming the value of the series of which it consists = 1, in which case, the Logarithm of 1 + p becomes = p-p²/2+p³/3-p⁴/4+p⁵/5 &c, or the Logarithm of , and r, by reversion of series is found = 2.7182818 &c.

The system arising from this mode of determining the value of the radix r, is that which furnishes what have been usually called hyperbolic Logarithms; and appears to be the simplest form the general expression admits of.

If, on the contrary, the radix r be assumed = to some particular number, as for instance 10, the value of the series q-(1/2)q²+(1/3)q³-(1/4)q⁴+(1/5)q⁵ &c, or its equal (r-1)-(1/2)(r-1)²+(1/3)(r-1)³-(1/4)(r-1)⁴+(1/5)(r-1)⁵ &c will become = 2.30258509 &c, and the or the , which gives the system that furnishes Briggs's or the common Logarithms.

And, in like manner, by assuming any particular value for r, and thence determining the value of the series q-(1/2)q²+(1/3)q³-(1/4)q⁴+(1/5)q⁵ &c, or its equal (r-1)-(1/2)(r-1)²+(1/3)(r-1)³-(1/4)(r-1)⁴+(1/5)(r-1)⁵ &c; or by assuming the same series of some particular value, and thence determining the value of r, any system of Logarithms may be derived.

The series q-(1/2)q²+(1/3)q³-(1/4)q⁴+(1/5)q⁵ &c, or its equal (r-1)-(1/2)(r-1)²+(1/3)(r-1)³-(1/4)(r-1)⁴+(1/5)(r-1)⁵ &c, which forms the denominator of the above compound expression, exhibiting the Logarithms of numbers according to any system, is what was first called, by Cotes, the Modulus of the system, being always a constant quantity, depending only on the assumed value of r.

And, as the form of this series is exactly the same as that which constitutes the numerator, and which has been shewn to be the hyperbolic Logarithm of a, it follows that the Modulus of any system of Logarithms is equal to the hyperbolic Logarithm of the radix of that | system, or of the number whose proper Logarithm in the system to which it belongs is 1.

The form of the series here obtained for the hyperbolic Logarithm of a, is the same as that which was first discovered by Mercator; and if the series of Wallis be required, it may be investigated in a similar manner as follows:

The general Logarithmic equation being , as before, let and ; then , and .

And if p+p²/2+p³/3+p⁴/4+p⁵/5 &c be put = s, we shall have , which let be put = q; then, by conversion of series, z or 1/x will be found =(q+(1/2)q²+(1/3)q³+(1/4)q⁴+(1/5)q⁵ &c)/s=(q+(1/2)q²+(1/3)q³+(1/4)q⁴+(1/5)q⁵)/ (p+(1/2)p²+(1/3)p³+(1/4)p⁴+(1/5)p⁵) and consequently .

The Logarithm of a or 1/(1-p) is, therefore, = (p + (1/2)p² + (1/3)p³ + (1/4)p⁴ + (1/5)p⁵ &c)/ (q + (1/2)q² + (1/3)q³ + (1/4)q⁴ + (1/5)q⁵ &c); or since and , the Logarithm of a is = ((a-1)/a + (1/2)((a-1)/a)² + (1/3)((a-1)/a)³ + (1/4)((a-1)/a)⁴ + (1/5)((a-1)/a)⁵ &c)/ ((r-1)/r + (1/2)((r-1)/r)² + (1/3)((r-1)/r)³ + (1/4)((r-1)/r)⁴ + (1/5)((r-1)/r)⁵ &c) Which is another general expression for the Logarithm of any number a, in any system of Logarithms, that may be simplified in the same manner as the former, the denominator being still equal to the hyperbolic Logarithm of the radix r; or, which is the same thing, to the Modulus of the system.

For if the series q + (1/2)q² + (1/3)q³ + (1/4)q⁴ + (1/5)q⁵ &c, or its equal (r-1)/r + (1/2)((r-1)/r)² + (1/3)((r-1)/r)³ + (1/4)((r-1)/r)⁴ + (1/5)((r-1)/r)⁵ &c, be assumed = 1, the hyperbolic Logarithm of 1/(1-p) will be = p + (1/2)p² + (1/3)p³ + (1/4)p⁴ + (1/5)p⁵ &c, or the hyperbolic Logarithm of ; and r, by reversion of series will be found = 2.7182818, as before. And if, on the contrary, the radix r be assumed = 10, the value of the series q + (1/2)q² + (1/3)q³ + (1/4)q⁴ + (1/5)q⁵ &c, or its equal (r-1)/r + (1/2)((r-1)/r)² + (1/3)((r-1)/r)³ + (1/4)((r-1)/r)⁴ + (1/5)((r-1)/r)⁵ &c, will become = 2.30258509, as before; and the common Logarithm of , or the common Logarithm of .

Or the latter formula, for the Logarithm of 1/(1 - p), or its equal a, may be more concisely derived from the first, as follows:

The Logarithm of 1 + p has been shewn to be = (p - (1/2)p² + (1/3)p³ - (1/4)p⁴ + (1/5)p⁵ &c)/ (q - (1/2)q² + (1/3)q³ - (1/4)q⁴ + (1/5)q⁵ &c), and if - p be substituted in the place of + p, the logarithm of 1 - p will become = (- p - (1/2)p² - (1/3)p³ - (1/4)p⁴ - (1/5)p⁵ &c)/ (q - (1/2)q² + (1/3)q³ - (1/4)q⁴ + (1/5)q⁵ &c), whence the Logarithm of where the denominator is the same as in the first formula, q being here = r - 1.

If the denominator, in either of these general formulæ, be put = m, the Logarithm of 1 + p will be denoted by 1/m X (p - (1/2)p² + (1/3)p³ - (1/4)p⁴ + (1/5)p⁵ &c, or the Logarithm of a by (1/m) X : (a-1) - (1/2)(a-1)² + (1/3)(a-1)³ - (1/4)(a-1)⁴ + (1/5)(a-1)⁵ &c. And the Logarithm of 1/(1 - p) will be denoted by 1/m X (p + (1/2)p² + (1/3)p³ + (1/4)p⁴ + (1/5)p⁵ &c, or the Logarithm of a by 1/m X : (a-1)/a + (1/2)((a-1)/a)² + (1/3)((a-1)/a)³ + (1/4)((a-1)/a)⁴ + (1/5)((a-1)/a)⁵ &c.

And since the sum of the Logarithms of any two numbers is equal to the Logarithm of their product, the Logarithm of (1 + p)/(1 - p) will become | = (2/m) X (p + (1/3)p³ + (1/5)p⁵ + (1/7)p⁷ &c), or the Logarithm of Which is a third general formula, that converges faster than either of the former.

The Logarithm of any number may, therefore, be exhibited universally, or according to any system of Logarithms, in the three following forms: .

. And if a+b be put = s, and a <01> b = d, these general formulæ may be easily converted into the following: .

From which last expressions, if d or its equal a <01> b be put = 1, we shall have, by proper substitution, and the nature of Logarithms: .

And from the addition and subtraction of these series, several others may be derived; but in the actual computation of Logarithms they will be found to possess little or no advantage above those here given. The same general formula may be derived from the original Logarithmic equation in a different way, thus: Let , then ; or if r be put =1/(1-q), we shall have .

And by denoting q-(1/2)q²+(1/3)q³-(1/4)q⁴ &c in the first case, or its equal q+(1/2)q²+(1/3)q³+(1/4)q⁴, in the latter case, by m, these expressions will become ; and ; which are the two anti-Logarithmic series of Halley: from whence, by reversion of series, may be found the Logarithm of any number a, as before.