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HISTORICAL VIEW OF THE
geometry with any thing having the appearance of arelation to numerical computation; and it was with anobject of this kind in view that trigonometry first pre-sented itself to them. Partly, again, the sort of inves-tigation was in itself of a new class, and of a kindwhich, perhaps, would not exactly harmonise with thestyle and manner adopted in their demonstrations.
Yet the whole science is nothing but a continued andhighly elegant application of one simple idea, that ofgiving names to the several ratios subsisting between thesides and angles of a right-angled triangle. The ratio ofone of the rectangular sides to the hypothenuse is calledthe sine of the angle opposite to it, while that of the otherside to the hypothenuse is the cosine of the same angle.Then, again, that of the sine to the cosine, is named thetangent; its inverse, the cotangent. These simple namesimposed upon ratios often recurring, added to the ideaof applying the triangle in different positions, so that itshypothenuse is always the radius, and its sides coincidentwith the diameters of a circle, whose arc measures theangle, are the elements out of which the whole super-structure of the science is directly reared; and from suchsimple principles a machine is constructed, which, inthe hands of the skilful analyst, has exceeded in powerperhaps every other invention of geometry, and hasmainly achieved the triumphs of the mathematician inevery department to which his science has been applied.
The etymology of the terms above defined has beena subject of question, especially as bearing upon theorigin of the science. Some writers have contended forthe derivation of the term “ sine ” from the Arabic .And though others have dwelt upon the Greek andLatin etymologies of the arc, or bow, its chord, and thesagitta, or arrow (a name sometimes applied to the por-tion of the diameter intercepted), as giving a sort offanciful description of the diagram, yet, certainly, nomeaning of the word " sinus ” will very aptly apply tothe case.