TRACT 20.

LOGARITHMS.

307

sines, namely, when radius is the first term in the proportion,and the sines of two arcs are the second and third terms ; forhe showed, that the fourth term, or sine, would be found byonly taking half the sum or difference of the sines of two otherarcs, which should be the sum and difference of the less of thetwo former given arcs, and the complement of the greater.This is no more, in effect, than the following well-knowntheorem in trigonometry: as half radius is to the sine of onearc, so is the sine of another arc, to the cosine of the differ-ence minus the cosine of the sum of the said arcs. The au-thor published this ingenious device, in 1588, in his “ Fun-damentum Astronomies.” And three or four years afterwardsit w r as greatly improved by Clavius, who adapted it to allproportions in the solution of spherical triangles, for sines,tangents, secants, versed sines, &c; and that whether radiusbe in the proportion or not. All which he explains very fully 5 "in lem. 53, lib. ], of his treatise on the Astrolabe. See moreon this subject in Longomont. Astron. Danica. pa. 7, et seq.This method, though ingenious enough, depends not on anyabstract property of numbers, but only on the relations ofcertain lines, drawn in and about the circle; for which rea-son it was rather limited, and sometimes attended with troublein the application.

After perhaps various other contrivances, incessant endea-vours at length produced the happy invention of logarithms,which are of direct and universal application to all numbersabstractedly considered, being derived from a property inhe-rent in numbers themselves. This property maybe considered,either as the relation between a geometrical series of termsand a corresponding arithmetical one, or as the relation be-tween ratios and the measures of ratios, which comes to muchthe same thing, having been conceived in one of these waysby some of the writers on this subject, and in the other'bythe rest of them, as well as in both ways at different times bythe same writer. A succinct idea of this property, and of theprobable reflections made on it by the first writers on loga-rithms, may be to the following effect:

x 2