That is to say, the geometrical mean of any two positive numbers is the same as the geometrical mean of their arithmetical and harmonical means.

The arithmetical, geometrical and constructed graphically as in Fig. 90. (a + b) = OM + MK. Then the

harmonical means may be Draw the circle of diameter radius is the arithmetical

mean A. Erect a perpendicular at M. Then MG is the geometMG and draw CG'.

rical mean.

Make OG'

=

Draw G'H perpen

dicular to CG'. Then OH is the harmonical mean, since

OG' = NOC X OH

Now A>G > H; for from the figure, MG < CA. Therefore, the angle G'CO is less than 45° and also its equal HG'O is less than 45°. Therefore, HO < OG' which establishes the inequality.

Exercises

1. Continue the harmonical progression 12, 6, 4.

3. If the arithmetical mean between two numbers be 1, show that the harmonical mean is the square of the geometrical mean.

CHAPTER VIII

THE LOGARITHMIC AND THE EXPONENTIAL

FUNCTIONS

128. Historical Development. The almost miraculous power of modern calculation is due, in large part, to the invention of logarithms in the first quarter of the seventeenth century by a Scotchman, John Napier, Baron of Merchiston. This invention was founded on the simplest and most obvious of principles, that had been quite overlooked by mathematicians for many generations. Napier's invention may be explained as follows:1 Let there be an arithmetical and a geometrical progression which are to be associated together, as, for example, the following:

2,

3,

4, 5,

0, 1, 2, 3, 6, 7, 8, 9, 10 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 Now the product of any two numbers of the second line may be found by adding the two numbers of the first progression above them, finding this sum in the first line, and finally taking the number lying under it; this latter number is the product sought. Thus, suppose the product of 8 by 32 is desired. Over these numbers of the second line stand the numbers 3 and 5, whose sum is 8. Under 8 is found 256, the product desired. Now since but a limited variety of numbers is offered in this table, it would be useless in the actual practice of multiplication, for the reason that the particular numbers whose product is desired would probably not be found in the second line. The overcoming of this obvious obstacle constitutes the novelty of Napier's invention. Instead of attempting to accomplish his purpose by extending the progressions by continuation at their ends, Napier proposed to insert any number of intermediate terms in each progression. Thus, instead of the portion

1 Merely the fundamental principles of the invention, not historical details, are given in what follows.

0.

1/2, 1, 111, 2, 21, 3, 31,

4

1, √2, 2, √8, 4, √32, 8, 128, 16 by inserting arithmetical means between the consecutive terms of the arithmetical series and by inserting geometrical means between the terms of the geometrical series. Let these be computed to any desired degree of approximation, say to two decimal places. Then we have the series

Again inserting arithmetical and geometrical means between the terms of the respective series we have:

By continuing this process each consecutive three figure number may finally be made to appear in the second column, so that, to this degree of accuracy, the product of any two such numbers may be found by the process previously explained. The decimal points of the factors may be ignored in this work, as for example, the product of 2.38 × 14.1 is the same as that of 238 × 14.1 except in the position of the decimal point. The correct position of the decimal point can be determined by inspection after the

significant figures of the product have been obtained. Using the above table we find 2.38 X 14.1

=

33.6.

The above table, when properly extended, is a table of logarithms. As geometrical and arithmetical progressions different from those given above might have been used, the number of possible systems of logarithms is indefinitely great. The first column of figures contains the logarithms of the numbers that stand opposite them in the second column. Napier, by this process, said he divided the ratio of 1.00 to 2.00 into “100 equal ratios," by which he referred to the insertion of 100 geometrical means between 1.00 and 2.00. The "number of the ratio" he called the logarithm of the number, for example, 0.75 opposite 1.69, is the logarithm of 1.69. The word logarithm is from two Greek words meaning "The number of the ratios." In order to produce a table of logarithms it was merely necessary to compute numerous geometrical means; that is, no operations except multiplication and the extraction of square roots were required. But the numerical work was carried out by Napier to so many decimal places that the computation was exceedingly difficult.

The news of the remarkable invention of logarithms induced Henry Briggs, professor at Gresham College, London, to visit Napier in 1615. It was on this visit that Briggs suggested the advantages of a system of logarithms in which the logarithm of 1 should be 0 and the logarithm of 10 should be 1, for then it would only be necessary to insert a sufficient number of geometrical means between 1 and 10 to get the logarithm of any desired number. With the encouragement of Napier, Briggs undertook the computation, and in 1617, published the logarithms of the first 1000 numbers and, in 1624, the logarithms of numbers from 1 to 20,000, and from 90,000 to 100,000 to fourteen decimal places. The gap between 20,000 and 90,000 was filled by a Hollander, Adrian Vlacq, whose table, published in 1628, is the source from which nearly all the tables since published have been derived.

129. Graphical Computation of Logarithms. In Fig. 89 the terms of a geometrical progression of first term 1 and ratio 1N = r are represented as ordinates arranged at equal intervals along OX. Fig. 89 is drawn to scale for the value of r = 1.5. Fig. 91 is

a similar figure drawn for r = 2, in which a process is used for locating intermediate points of the curve, so that the locus may be sketched with greater accuracy. The lines y = x and y = rx 2x) are drawn as before, and the "stairway"

(in this case y constructed as through x

=

before (see §124).

=

Vertical lines drawn

and horizontal lines drawn

FIG. 91.-Graphical Construction of the Curve y = 2o.

through the horizontal tread of each step of the stairway divides the plane into a large number of rectangles. Starting at M and sketching the diagonals of successive cornering rectangles the smooth curve MNP is drawn. Intermediate points of the curve are located by doubling the number of vertical lines by bisecting the distances between each original pair, and then by increasing the number of horizontal lines in the following manner: Draw the line y = √rx (in the case of the Fig., y = √2x).

=