3J9 



LOGARITHMS. 



LOGARITHMS. 



330 



computations, and on the details which still remained for the first 

 calculators after they had applied all the analysis which they had, we 

 have not here to speak ; but we shall now show how the table may be 

 formed by mere labour, and how the word logarithm arises. 



Let us suppose that our system is to be such that 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 l+t, it 

 would be found that 2 is very nearly the 30103rd power of (1 + 1), that 

 3 is very nearly the 47712th power of l + t, and so on. If then, 

 beginning with 1 , we increase it in the ratio of 1 to 1 + 1, giving 1 + 1 ; 

 if we increase this in the ratio of 1 to 1 + 1, 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 1 to 10 by 

 100000 steps, increasing each time in the same ratio, we shall come 

 nearest to z in 30103 steps, which is therefore the number of times the 

 increase is made in a cert dn ratio, or the number of the ratios, the 

 \oywv aptOfuis, 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 (I + f)* 17 '" very 

 nearly, must be very nearly (1 + r)' :5is . Xor is this property altered, 

 if we divide or multiply all the logarithms by the same number. If 

 then we divide every logarithm by luOOOO, the logarithm of 10 becomes 

 1, that of 2 becomes '30103, and that of 3 becomes '47712, as in the 

 common tables. 



The tirst step of importance which was made in the logarithmic 

 analysis was the following. If t be very small, the lower power.? o 

 l + t, the square, cube, &e., are 1 + 2', 1 + Zt, &c , very nearly ; or if m 

 and n be not so great but that mt and nt are still small, the mth anc 

 nth powers of 1 + 1 are 1 + mt and 1 + nt very nearly. But the logarithms 

 of these powers are m and n ; that is, if k and / be small, the logarithms 

 of 1 + k and 1 4- 1 are very nearly in the proportion of k to /. If then 

 we take two numbers, a and l>, and extract a very high root (say the 

 rth) of both, so that the results are very near to unity, say 1 + k and 

 1 + 1, we have (nearly) 



log. V : log. \/i : : k : I. 



But the two first terms are in the same ratio as log a : log. A, 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 lie 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 INTER- 

 FOLATioy,'which Briggs used with success, we have before us the liases 

 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 o + b be the logarithms of A 

 and AB, then a + 24, a + 36, &c., must be the logarithms of AB-, AB 3 , Ac. 

 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, he 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 f a quantity 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 

 hiv 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 1 become (l+r) in a year, and consequently, at the 

 te of interest, it becomes A'(l-t-)-)" in n years. Suppose ho 



same 

 wever 



rate of interest, it becomes A'(l-t-)-)" in n years. Suppose 

 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 



(r\' / r \ 



I + T ) at the end of one year, and ( 1 + } at the end of 



n years. 



If we may makes as great as we please, that is, if we may make 

 payments of interest follow one another as quickly as we please, we 

 mjy m.ike th 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 6 c. Divide 6 c into 



r~ 



A 



B P Q 11 



K V' Q' 



1 p q r s t n v e 



-+-H 

 *' P' < 



any number of equal parts, and in the successive equal times bp,pq, 

 qr, Ac., let a point move through BP, I>Q, QII, &c. 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 P 

 in the time bp ; at p 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 P, A Q &c., be 

 a continued set of proportionals, namely, AB:AP::AP:AQ::AQ: 

 A R, &c. Increase the number of subdivisions of 4 c 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 ant/ successive equal times, 

 shall be in continued proportion. To show this, suppose we compare 

 the motion from B to c with any other part of the motion described 

 in some sxibsequent time b'c' (equal to be), and which carries the 

 moving point from B' to c'. Divide the time b c' into as many equal 

 parts, b'ji', p'q', &c., as before, and let B'P', P'Q', &c., be the lengths 

 described in the second set of subdivisions. Then by the law of the 

 motion A B : A P : : A B' : A p', whence B p and B' p' are in the ratio of 

 AB to AB'; and similarly PQ and P'Q' are in the ratio of A P to A P', 

 that is, of A B to A B' ; and so on. Consequently, the sum of B P, P Q, 

 &c., or B c is to the sum of B' p', P'Q', &c., or B' c', in the same ratio of 

 AB to AB'; whence also A c is to AC' as AB to AB', or AB : AC : : 

 A B' : A c'. That is, if in any one time the distance from A increases 

 from x to y, and in any other equal time from x' to y', then x : y : : 

 x' : Y'. 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 B' are as 

 B p to B' p (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 B 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 [TABLES], AB being a unit, the point starts 

 from B at the rate of a unit of space (A B) 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 1) it shall have attained 10 times A B in one unit of time This 

 requires, as we shall see, an initial velocity of 2 302585 . . . times A B 

 in one unit of time. 



In addition to the principles here hid down, a known property of 

 the hyperbola very early showed that logarithms would become 

 applicable to geometry : 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 AE and AC be the asymptotes of an hyperbola, and let A B, AC, 

 A D, &c., be in continued geometrical progression. Draw B K, c L. D 11, 

 &c., parallel to the other asymptote A o, then the hyperbolic trapezia 

 B K L C, C L M D, D M N E, &c., are equal, or B K L C, B K M D, B K N E, &C., 

 are in arithmetical progression. So that any trapezium B K M D is a 

 logarithm to its terminal abscissa AD. This property was the dis- 

 covery of Gregory St. Vincent, who published it in his ' Opus Geome- 

 tricum,' Antwerp, 1647. It was therefore unknown both to Napier 

 and Briggs. 



\Ve shall now take the question of logarithms, availing ourselves of 

 the power of modern algebra. 



Definition, By the logarithm of a number let any such function of 

 ;hat number be understood as has the following property. When x 

 s to y as x is to y', the logarithm of x exceeds or falls short of the 

 logarithm of y by as much as the logarithm of x' exceeds or falls short 



