TRIGONOMETRY, (TOtovotwrgia from Toeovor a tri angle, and my:Teo I measure) in its original sense signified that part of mathematical science which treated of the admcasurement of triangles. angles were supposed to be either described upon a plane or upon the surface of a sphere, and hence the science was divided into plane trigonometry and spherical trigonometry.
Like every other department of science, the ob jects of trigonometry became more extended as knowledge advanced and discovery accumulated, and this department of mathematical science, which was at first confined to the solution of one general problem, viz. " given certain sides and angles of a triangle to determine the others," has now spread its uses over the whole of the immense domains of mathematical and physical science. In the wide range of modern analysis, there is scarce ly a subject of investigation to which trigonometry has not imparted clearness and perspicuity by the use of its language and its principles; and the physi cal investigations of philosophers of our times are still more largely indebted for their conciseness, elegance, and generality, to the symbols and estab lished formula of this science.
In its present improved and enlarged statc, trig onometry might not improperly be called the an gular calculus; for however extensive and various its more remote uses and applications may be, its immediate object is to institute a system of sym bols, and to establish principles by which angular magnitude may be submitted to computation, and numerically connected with other species of mag nitude; so that angles and the quantities on which they depend, or which depend on them, may be united in the same analytical formulae, and may have their mutual relations investigated by the same methods of computation that are applied to all other quantities.
It is not easy nor indeed possible to trace back trigonometry to its earliest origin. Most proba bly this science took its rise in the solution of some problems relative to heights or distances, so inac cessible as not to admit of direct measurement.- In the rudest state of geometrical knowledge the sim ilitude of equiangular triangles must have been known. This indeed is a property so obvious and striking, that the practical conviction of its truth must have long preceded its geometrical demon stration. This then being understood, it was an
easy step to perceive that a small and measurable triangle might be constructed similar to a large and immeasurable one, so that the sides of the large one might consist of as many leagues as those of the small one of inches. Thus if the number of leagues in any one side of the large one be mea sured, and a line be drawn consisting of as many inches, a triangle constructed upon that line hav ing the same angles as the large triangle, will have its other sides consisting of as many inches as there are leagues in the other sides of the large one. Such is in fact the foundation of the applica tign of trigonometry to the measurement of trian gles.
In the first instance, trigonometry must have been considered not as a separate part of science, but simply as a class of geometrical problems. As however their number increased and their useful application became extended, they gradually as sumed the form of a distinct science. The earliest work on this subject of which we have any record, is one by Hipparchus, who flourished about 150 years before the Christian era. Theon informs us that Hipparchus wrote a work on the Chords of Circular Arcs, a title from which we collect that the subject must have been trigonometry, or the doctrine of sines.
The earliest work extant on the subject is the spherics of Theodosius. In the first century Mene laus is said to have written nine books on trigo nometry, of which, however, only three have come down. The earliest trigonometrical tables which we have received arc those of Ptolemy, given in his Almagcst. In these he adopts the sexagesimal division of the arc, whose chord is equal to the radius, and reckons all arcs by sixtieths of that arc, and all chords by sixtieths of the ra dius. In the eighth century sines or half-chords were introduced by the Arabians, which was the only striking improvement which the science re ceived until the 15th century, when Purbach aban doned the sexagesimal division of the radius and constructed a table of sines to a radius of 600,000, computed for every ten minutes of the quadrant. Subsequently Regiomontanus adopted a radius of 1,Q00,000, and calculated sines for every minute.