Page:EB1911 - Volume 27.djvu/286

 nave of the cathedral or church, and being of less height gives more importance to the ground storey or nave arcade. In consequence of its less height it was usually divided into two arches, which were again subdivided into two smaller arches and these subdivisions increased the scale. On account of the richness of its mouldings and carved ornament in the sculpture introduced in the spandrels, it became the most highly decorated feature of the interior, the triforium at Lincoln being one of the most beautiful compositions of Gothic architecture. Even when reduced to a simple passage it was always a highly enriched feature. In the 15th-century churches in England, when the roof over the aisles was comparatively flat, more height being required for the clerestory windows, the triforium was dispensed with altogether. In the great cathedrals and abbeys the triforium was often occupied by persons who came to witness various ceremonies, and in early days was probably utilized by the monks and clergy for work connected with the church.

 TRIGLYPH (Gr. , three, and  , an incision or carving), an architectural term for the vertically channelled tablets of the Doric frieze, so called because of the angular channels in them, two perfect and one divided—the two chamfered angles or hemiglyphs being reckoned as one. The square sunk spaces between the triglyphs on a frieze are called metopes.

TRIGONOMETRY (from Gr. , a triangle,  , measure), the branch of mathematics which is concerned with the measurement of plane and spherical triangles, that is, with the determination of three of the parts of such triangles when the numerical values of the other three parts are given. Since any plane triangle can be divided into right-angled triangles, the solution of all plane triangles can be reduced to that of right-angled triangles; moreover, according to the theory of similar triangles, the ratios between pairs of sides of a right angled triangle depend only upon the magnitude of the acute angles of the triangle, and may therefore be regarded as functions of either of these angles. The primary object of trigonometry, therefore, requires a classification and numerical tabulation of these functions of an angular magnitude; the science is, however, now understood to include the complete investigation not only of such of the properties of these functions as are necessary for the theoretical and practical solution of triangles but also of all their analytical properties. It appears that the solution of spherical triangles is effected by means of the same functions as are required in the case of plane triangles. The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science, which is from analogy frequently called spheroidal trigonometry. Every new class of surfaces which may be considered would have in this extended sense a trigonometry of its own, which would consist in an investigation of the nature and properties of the functions necessary for the measurement of the sides and angles of triangles bounded by geodesics drawn on such surfaces.

Trigonometry, in its essential form of showing how to deduce the values of the angles and sides of a triangle when other angles and sides are given, is an invention of the Greeks. It found its origin in the computations demanded for the reduction of astronomical observations and in other problems connected with astronomical science; and since spherical triangles specially occur, it happened that spherical trigonometry was developed before the simpler plane trigonometry. Certain theorems were invented and utilized by Hipparchus, but material progress was not recorded until Ptolemy collated, amended and developed the work of his predecessors. In book xi. of the Almagest the principles of spherical trigonometry are stated in the form of a few simple and useful lemmas; plane trigonometry does not receive systematic treatment although several theorems and problems are stated incidentally. The solution of triangles necessitated the construction of tables of chords—the equivalent of our modern tables of sines; Ptolemy treats this subject in book i., stating several theorems relating to multiple angles, and by ingenious methods successfully deducing approximate results. He did not invent the idea of tables of chords, for, on the authority of Theon, the principle had been stated by Hipparchus (see ).

The Indians, who were much more apt calculators than the Greeks, availed themselves of the Greek geometry which came from Alexandria, and made it the basis of trigonometrical calculations. The principal improvement which they introduced consists in the formation of tables of half-chords or sines instead of chords. Like the Greeks, they divided the circumference of the circle into 360 degrees or 21,600 minutes, and they found the length in minutes of the arc which can be straightened out into the radius to be 3438. The value of the ratio of the circumference of the circle to the diameter used to make this determination is 62832 : 20000, or ＝3·1416, which value was given by the astronomer Aryabhata (476–550) in a work called Aryabhaiya, written in verse, which was republished in Sanskrit by Dr Kern at Leiden in 1874. The relations between the sines and cosines of the same and of complementary arcs were known, and the formula sin ＝√{1719(3438−cos)} was applied to the determination of the sine of a half angle when the sine and cosine of the whole angle were known. In the Sūrya-Siddhānta, an astronomical treatise which has been translated by Ebenezer Bourgess in vol. vi. of the Journal of the American Oriental Society (New Haven, 1860), the sines of angles at an interval of 3° 45′ up to 90° are given; these were probably obtained from the sines of 60° and 45° by continual application of the dimidiary formula given above and by the use of the complementary angle. The values sin 15°＝890′, sin 7° 30′＝449′, sin 3° 45′＝225′ were thus obtained. Now the angle 3° 45′ is itself 225′; thus the arc and the sine of of the circumference were found to be the same, and consequently special importance was attached to this arc, which was called the right sine. From the tables of sines of angles at intervals of 3° 45′ the law expressed by the equation sin (n + 1.225′) − sin (n.225′)＝sin (n.225′) − sin (n−1 . 225′)−sin$(n. 225′)⁄225$

was discovered empirically, and used for the purpose of recalculation. Bhaskara (fl. 1150) used the method, to which we have now returned, of expressing sines and cosines as fractions of the radius; he obtained the more correct values sin 3° 45′＝100/1529, cos 3° 45′＝466/467, and showed how to form a table, according to degrees, from the values sin 1°＝10/573, cos 1°＝6568/6569, which are much more accurate than Ptolemy's values. The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved right-angled plane and spherical triangles by geometry.

The Arabs were acquainted with Ptolemy’s Almagest, and they probably learned from the Indians the use of the sine. The celebrated astronomer of Batnae, (q.v.), who died in 929–930, and whose Tables were translated in the 12th century by Plato of Tivoli into Latin, under the title De scientia stellar um, employed the sine regularly, and was fully conscious of the advantage of the sine over the chord; indeed, he remarks that the continual doubling is saved by the use of the former. He was the first to calculate sin from the equation sin /cos ＝k, and he also made a table of the length of shadows of a vertical object of height 12 for altitudes 1°, 2°, of the sun; this is a sort of cotangent table. He was acquainted not only with the triangle formulae in the Almagest, but also with the formula cos a＝cos b cos c + sin b sin c cos A for a spherical triangle ABC. Abū’l-Wafā of Bagdad (b. 940) was the first to introduce the tangent as an independent function: his “umbra” is the half of the tangent of the double arc, and the secant he defines as the “diameter umbrae.” He employed the umbra to find the angle from a table and not merely as an abbreviation for sin/cos; this improvement was, however, afterwards forgotten, and the tangent was reinvented in the 15th century. Ibn Yūnos of Cairo, who died in 1008, showed even more skill than Albategnius in the solution of problems in spherical trigonometry and gave improved approximate formulae for the calculation of sines. Among the Vest Arabs, (q.v.), who lived