TRACT 21.

LOGARITHMS.

341

was therefore called the sinus totus, or whole sine, he conceivedthe line of the radius to be described, or run over, by a pointmovingalong it in suchamanner,that in equal portions of timeit generated, or cut off, parts in a decreasing geometrical pro-gression, leaving the several remainders, or sines, in geome-trical progression also; while another point, in an indefiniteline, described equal parts of it in the same equal portions oftime; so that the respective sums of these, or tile whole linegenerated, were always the arithmeticals or logarithms ofthese sines. Thus, az is the given radius on whichall the sines are to betaken, and a&c the indefinite S| P es -Off-line containing the logarithms; these lines being “each generated by the motion of points, beginning - 1 -1

at a, a. Now, at the end of the 1 st, 2 d, 3d, &c, . 2 -2

moments, or equal small portions of time, the mov- ,,ing points being found at the places marked 1,2, -3

3, &c; then za, zl, z2, z3, &c, will be the series of ’5natural sines, and AO, or 0, Al, a2, a3, &c, will be -®.their logarithms; supposing the point which gene- 1 & c .rates az to move every where with a velocity de- " s

creasing in proportion to its distance from z, namely > e -7

its velocity in the points 0, 1, 2, 3, &c, to be re- &c

spectively as the distances z 0, zl , z2, z3, &c, whilethe velocity of the point generating the logarithmic line a&cremains constantly the same as at first in the point a or 0.

Hitherto the author had not fully limited his system or scaleof logarithms, having only supposed one condition or limita-tion, namely, that the logarithm of the radius az should beO:whereas two independent conditions, no matter what, arenecessary to limit the scale or system of logarithms. It didnot occur to him that it was proper to form the other limit,by affixing some particular value to an assigned number, orpart of the radius : but, as another condition was necessary,he assumed this for it, name])’, that the two generating pointsshould begin to move at a and A with equal velocities; or thatthe increments «1 and Al, described in the first moments,should be equal; as he thought this circumstance would be