Elementary arithmetic, with brief notices of its history... by Robert Potts.

10

LOGAIRITMIIS.

system of logarithms.1 Hte was led to his invention solely by geometrical
considerations. Napier's object was not to calculate the logarithms
of the natural numbers, but to facilitate calculations in trigonometry.
This is the reason of his computing only the logarithms of the sines,
and the logarithms of the complements of the sines, (the word cosine not
yet having been devised,) from which the logarithms of the tangents,
&c., could be easily deduced. The conception he formed was that
of two points, generating straight lines by very small increments,
and so regulated that in one case the successive increments should be
equal; and in the other, that they should differ proportionally from
each other in an indefinitely small degree.2 He knew that the
indefinitely small ratios which he imagined to be generated between
the natural numbers were not exact, but only approximations, and the
excess or defect would become less than any quantity he might wish
to assign, and therefore would not materially affect the results of hi-,
calculations, so far as he intended them to apply.
As the arcs of a circle were considered as the measures of the
angles subtended at the center, and the sines of all arcs as parts of
the radius, which was the sine of the whole quadrant, he conceived
a line az equal to the radius, and a point beginning to move from a,
so that in equal times it moved over successive parts ab, be, cd, &c., of
the line az in a decreasing geometrical progression, leaving the
successive remainders, bz, cz, dz, &c., also in geometrical progression.
1 In Dr. Booth's Treatise on some new Geometrical Methods, vol. i., ch. 32,
on the geometrical origin of logarithms, he has exhibited the value of the
Napierian base in terms of the functions of the angle which a focal perpendicular
on a tangent makes with the axis of a parabola. He remarks:-" We are thus
(for the first time, it is believed) put in possession of the geometrical origin of that
quantity so familiarly known to mathematicians-the Napierian base." The value
of the angle is 49~ 36' 18".
2 The first two chapters of the Canon Mirificus afford a view of Napier'e-;
method. The following are his Definitions and Propositions, but without hi3
illustrations, taken verbally from the translation of Edward Wright:CHAP. I. Of the Definitions:
Def. 1. A line is said to increase equally, when the poynt describing the
same goeth forward equall spaces, in equall times, or moments.
Cor. Therefore, by this increasing, quantities equally differing, munnU
needes be produced, in times equally differing..Def. 2. A line is said to decrease proportionally into a shorter, when the
poynt describing the same in equall times, cutteth off parts continually of the same proportion to the lines from which they are
cut off.
Cor. Hence it followeth that by the decrease in equall moments (or
times) there must needes also be left proportional lines of the
same proportion.
Def. 3. Surd quantities, or unexplicable by number, are said to be defined, or
expressed by numbers very neere, when they are defined or e::pressed by great numbers which differ not so much as one unite
from the true value of the surd quantities.
Def. 4. Equal-timed motions are those which are made together, and in the
same time.
ZDef. 5. Seeing that there may be a slower and a swifter motion given than
any motion, it shall necessarily follow, that there may be a motion
given of equall swiftnesse to any motion (which we define to be
neither swifter nor slower).
Det. 6. The logarithme therefore of any sine is a number very neerely expressing the line, which increased equally in the meane time, whiles the