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

LOGARITHMS

11

On another line, AZ, not definite as the former, he conceived a point
beginning from A to move over equal parts, AB, BC, CD, &c., of
this line, with the velocity uniform, the same as the initial velocity of
Ihe other point moving in az. Now at the end of the first, second,
third, &c., equal portions of time, the moving point in az is found at
b, c, d, &c.; and that in AZ at B, C, D, &c., respectively. The lines
za, zb, cz, &c., will be a series of lines in a decreasing geometrical progression, and o, AB, A C, AD, &c., will be a corresponding series of lines
in an increasing arithmetical progression. The lines o, AB, A C, AD,
&c., may be made the logarithms of the series of lines za, zb, zc, &c.,
supposing the point in az moving with a velocity decreasing in
proportion to its distance from z, while the velocity of the point in
AZ moves with the same constant velocity as it had at the beginning of
its motion.
As two independent conditions are necessary to limit every system
of logarithms, as if 0 be taken for the logarithm of 1, and any
definite number be assumed for tie logarithm of some other number,
a system of logarithms can be computed under these conditions.
Napier, however, assumed the logarithm of the whole sine to be 0, and
hence as the series of the logarithms of the sines increase, the sines
themselves decrease. He also assumed that the points beginning to
move from a and A in the lines az and AZwith equal velocities,
the increments described in the first small portions of time are equal,
or that the natural sines and their logarithms near the whole sine
have equal differences, but different affections. With these limitations,
there is explained in his posthumous work his methods of computing
his canon of logarithms, which at first he styled artificials, or artificial
numbers, to distinguish them from the numbers which denoted the
natural sines.
There is a singular coincidence of ideas exhibited in Napier's
invention of logarithms, and in Sir Isaac Newton's invention of
fluxions, as will appear from the following account by Newton himline of the whole sine decreased proportionally into that sine, both
motions being equal-timed, and the beginning equally swift.
A Consequent. Therefore the logarithme of the whole sine 1000000 is
nothing or 0; and consequently the logarithmes of numbers
greater than the whole sine, are lesse than nothing.
Therefore we call the logarithmes of the sines, Abounding, because they are alwayes greater than nothing, and set this mark +
before them, or else none. But the logarithmes which are less
than nothing, we call Defective, or wanting, setting this mark -
before them.
C.i-vr. II. Of the Propositions of Logarithmes:
Prop. 1. The logarithmes of proportionall numbers and quantities are
equally differing.
Prop. 2. Of the logarithmes of three proportionals, the double of the second
or meane, made lesse by the first, is equall to the third.
Prop. 3. Of the logarithmes of three proportionals, the double of the second,
or middle one, is equall to the summe of the extremes.
Prop. 4. Of the logarithmes of foure proportionals, the summe of the second
and third, made lesse by the first, is equal to the fourth.
Prop. 5. Of the logarithmes of foure proportionals, the summe of the middle
ones, that is, of the second and third, is equall to the logarithme
of the extreames, that is to say, the first and fourth.
Prop. 6. Of the logarithmes of foure continuall proportionals, the triple of
either of the middle ones is equall to the summe of the further
extreame, and the double of the neerer.