Let us suppose that our system is to be such that 0 being the logarithm of 1, a hundred thousand shall be the logarithm of 10. If the hundred-thousandth root of 10 be extracted and called 1 + t, it would be found that 2 is very nearly the 30103rd power of (1+ t), that 3 is very nearly the 47712th power of 1+ t, and so on. If then, beginning with 1, we increase it in the ratio of 1 to 1 + t, giving 1 + t; if we increase this in the ratio of 1 to 1 + t. giving (1 and so on, it appears that we shall reach 2 (or very near to it, one way or other), when 30103 such ratios have been taken ; or if we pass from I to 10 by 100000 steps, increasing each time in the same ratio, we shall come nearest to 2 in 30103 steps, which is therefore the number of times the increase is made in a cert tin ratio, or the number of the ratios, the Xo-yrer tifsepais, or the logarithm, of 2.
In such a table it must of course follow that the logarithm of a product is exactly or very nearly the sum of the logarithms of the factors, since for instance 2 being (1 and 3 being (1+1)4;71a very nearly, 6 must be very nearly (1 + Nor is this property altered, if we divide or multiply all the logarithms by the same number. If then we divide every logarithm by 100000, the logarithm of 10 becomes 1, that of 2 becomes and that of 3 becomes as in the common tables.
The first step of importance which was made in the logarithmic analysis was the following. If t be very small, the lower powers of 1 + t, the square, cube, &c., are 1 + 2t, 1 + 3!, rte very nearly ; or if an and n be not so great but that int and nt are still small, the with and nth powers of 1 +t are 1 + nit and 1 + nt very nearly. But the logarithms of these powers are an and n ; that is, if k and I be small, the logarithms of and I + I are very nearly in the proportion of k to I. If then we take two numbers, a and b, and extract a very high root (say the rth) of both, so that the results are very near to unity, say 1 and 1+ I, we have (nearly) But the two first terms are in the same ratio as log a : log. 6, since the multiplication of the former terms by r gives the latter. Con sequently, when the logarithm of one number is known, that of any other can be found to any degree of nearness. We shall presently see this in a clearer form ; it is sufficient here to show how the theorem was first obtained. If to the preceding methods we add that of livren rorasriorg,'which Briggs used with success, we have before us the bases of the original computations of logarithms.
It was evident from the first that the connection between a logarithm and its number must be of the following kind : when the logarithm increases in arithmetical progression, the number must increase in geometrical progression; so that if a and a+b be the logarithms of A and AB, then a +26, a + 3b, &c., must be the logarithms of AB=, &c. Several mathematicians had formed this conception ; but the prelimi nary difficulty which stopped their progress was their being unable to present the series of natural numbers or fractions of a high degree of nearness to them), in the shape of terms of a geometrical progression. The great merit of Napier is threefold : first, lie distinctly saw that all numbers, within any given limit, may be either terms, or as near as we please to terms, of a geometrical progression ; secondly, he had the courage to undertake the enormous labour which was requisite for the purpose ; thirdly, he made an anticipation of the principle of the differential calculus in developing the primary consequences of the definition.
The predecessors of Napier probably did not well understand the notion of a quaritity varying in geometrical ratio, while another varied simultaneously, but in an arithmetical ratio. The difficulty is that which a beginner finds in seizing the notion of compound interest carried to its extreme limit, so that every fraction of interest, however small, begins to make interest from the moment it becomes due. 'We have preferred to omit this consideration in the article INTEREST, where it would have been of no practical use, and to introduce it here, where it may aid in the explanation of the first principles of logarithms.
Let XI become .£(1 +r) in a year, and consequently, at the same rate of interest, it becomes £(1 + r)? in n years. Suppose however
that interest, instead of being payable yearly, is paid z times in a year, and that interest makes interest from the moment it is paid. Conse quently, at the end of the first, second, &c. fractions of a year, the pound first put out becomes a years.
I f we may make z as great as we please, that is, if we may make payments of interest follow one another as quickly as we please, we may make the increase of the pound approach as nearly as we please to a gradual increase, of which it must be the characteristic that in successive equal times the amounts are in geometrical progression. Let A B become A c in a time reprepresented by b c. Divide b e into any number of equal parts, and in the successive equal times bp, p q, q r. &c., let a point move through B P, r Q, Q it, itc. In the article ACCELERATION is explained the manner in which a succession of impulses, sufficiently small in amount, and often repeated, may be made to give, as nearly as we please, the results of a perfectly gradual motion. At B let a velocity be given sufficient to carry the point to r in the time b p ; at I' let an impulse be given which would cause P Q to be described in the time p q, and so on. And let A B, A I", AQ &c., be a continued set of proportionals, namely, An:AP: Al': : A Q : A 0: A n, &c. Increase the number of subdivisions of b be without limit, and we approach as a limit to gradual motion of such a kind that the distances of the point from A, at the end of any successive equal times, shall be iu continued proportion. To show this, suppose we compare the motion from is to e with any other part of the motion described in some subsequent time b'e' (equal to b e), and which carries the moving point from to c'. Divide the time b e' into as many equal parts, b'p', p' q', &c., as before, and let n' &c., be the lengths described in the second set of subdivisions. Then by the law of the motion An:Ae::_s : A e', whence n and is' V are in the ratio of AB to Are; and similarly P Q and e'Q' are in the ratio of Al' to A V, that is, of A B to A n'; and so on. Consequently, the sum of B P, P Q, or B C is to the sum of n' r', &c., or B' c', in the same ratio of A B to A a' ; whence also A C is to A C' as A B to A B', or An:AC:: A B' : A c'. That is, if in any one time the distance from A increases from x to r, and in any other equal time from x' to T', then x : X' : From which it readily follows that the distances attained at the ends of successive equal times are in continued proportion.
More than this, the velocities of the moving point at B and 13' are as B P to is (these being spaces described in equal times) : and the ratio of these, however many may be the number of subdivisions, is always that of A B to A' B‘. Hence a gradual motion of the character described is one in which the velocity of the moving point increases in the same proportion as the distance from A.
In the preceding diagram, the time elapsed from B to C is the logarithm of A c, that of A s being 0. An infinite number of systems may be constructed, depending on the different velocities with which the moving point may be supposed to start. from B. In Napier's system, at least in that system stripped of certain peculiarities not worth noting at present [TA LES], A B being a unit, the point starts from 13 at the rate of a unit or space (A u) in a unit of time : obviously the most simple supposition which can be made, and which has pro cured for this system the distinctive title of natural logarithms. In Briggs's system the point starts from B with such a velocity that (A B being I) it shall have attained 10 times A B in one unit of time. This requires, as we shall see, an initial velocity of 2302585 . .. times A in one unit of time.
In addition to the principles here laid down, a known property of the hyperbola very early showed that logarithms would become applical•le to : and thus it happened that the first decidedly algebraical step in the computation of logarithms was announced in Mercator's Logarithmotechnia,' as the quadrature of the hyperbola. Let A E and A G be the asymptotes of an hyperbola, and let A B, A C, A D, &e., be in continued geometrical progression. Draw B K, C L. D &c., parallel to the other asymptote A 0, then the hyperbolic trapezia