Spherical Trigonometry

The relations of perpendicularity among lines and planes show that also the first is polar to the second trihedral angle. In solid geometry it is shown, as is visible if one will construct a model, that each dihedral angle in either Is the supplement of a face-angle in the other. If the first is a right-trihedral angle (spherical tri angle), the second is quadrantal, and vice versa. There is no need of new formulae for a quadrantal triangle, since the ten given already for a right triangle can be altered by merely ex changing small letters for capitals and reciprocally, and changing the sign prefixed to every cosine.

Oblique Spherical Triangles.

When no angle is right, then from any point in an edge OC two planes can be passed, each perpendicular to one other edge, both therefore containing a line CX, of length p, perpendicular to the plane AOB (fig. 5). From Hyperbolic Functions.—Trigonometric functions are alge braically related to the sine or cosine. Sine and cosine of a real variable quantity, represented as rectangular coordinates in a plane, define a point whose locus is a circle of unit radius about the origin as centre. They may be defined either by this geo metric property; or better, by two absolutely convergent series : Misled by apparent implication in the names, some suppose that the Jacobian elliptic functions sin am u, cos am u, etc., re semble the trigonometric and hyperbolic functions as closely as these sets resemble one another. That this is far from correct may be seen at once by consulting the article on ELLIPTIC The beginnings of trigonometry in explicit form may perhaps be traced to the lost work of a Greek, Hipparchus of -Nicaea (c. 140 B.c.), which is said to have comprised twelve books with tables On Chords of Circles. Apparently Egyptian builders of the pyramids knew of fixed ratios in similar triangles, and they had a name seqt for the cosine of an angle, but there is no known record of tables or of further theory. Three books of Menelaus of Alexandria show that interest in astronomy had induced more progress in spherical than in plane trigonometry. To advance toward its modern form, the science needed more named ratios or functions tabulated, being like a language scantily supplied with nouns. In India, six centuries later, chords were halved, and became what we now call sines, appearing in the writings of Aryabhata (c. A.D. 500) and Brahmagupta (c. A.D. 620). The knowledge of India passed into the keeping of the Arabs, and Al Battani (c. A.D. goo) gave names to functions now called cotangent and secant. Still later (c. A.D. 1250, there came a systematizer, the Persian Nasir ed-din al-Tusi, who collected and supplemented older knowledge into a coherent whole. The sine theorem was employed wherever possible; but, lacking the cosine theorem, he resorted when necessary to auxiliary right triangles to solve problems in oblique triangles.

Among modern writers, the first to exhibit trigonometry as a science was the German Johann Muller, better known as Regio montanus (c. 146o). He seems to have invented the tangent, traces of which have been suspected, however, in the ancient Egyptian Ahmes ms. (c. 165o B.c.) while the secant was re introduced c. 1500 by Copernicus. Vieta, in Paris in the latter part of the sixteenth century, brought algebra and particularly algebraic transformations into the service of trigonometry. Euler,

in the mid-eighteenth century, recast, simplified, and gave ele gance to the body of rules and technique, by this time so neces sary for navigation and the physical sciences. The name, trig onometry, probably originated with Pitiscus (Heidelberg, 1593).

The first discoverers of leading theorems and formulae must of course be named with an implied proviso. For plane triangles the sine theorem narrowly missed by Al Battani and the Spanish Moor Jabir Ibn Aflah (c. I140), was enunciated by Nasir ed-din (c. 125o). The cosine theorem in complete form is first found in Vieta's work of 1593, Variorum de rebus mathe maticis responsorum liber octavus. The same author restated the tangent theorem in the accepted form of today, in 1593. It stands however unmistakable in the work of a Dane, Thomas Fincke, ten years earlier, save that the angle is expressed as 2(I8o— C), a form immediately applicable to the usual data. Formulae for finding angles directly from the sides, without auxiliary right triangles, appear first with Rhaeticus of Witten berg in a work written in 1568, published in 1596. He gives the rule for tan2A. Rules for the sine and cosine of the half-angle are certainly as old as Oughtred's Trigonometrie of 1657. Pro portions for sines and cosines, of 1(A-1-8) and of 1(A —B) are found, the sines in Newton's writings, 1707, and both functions appear in F. W. Oppel's Analysis Triangulorum, 1746.

The six triadic relations in spherical right triangles evolved during long centuries, first of course in words stating proportions, later in the short-hand of equations. Menelaus of Alexandria (c. A.D. 100) and Ptolemaeus (c. A.D. 140) give most of them, and all were known to Nasir ed-din al-Thst (c. 125o). For oblique spherical triangles, the sine theorem was found by the earl) Arabians. It was known to Abu'l-Wea (c. 98o), and possibly to his contemporaries Abi.1 Na.sr or al-Khojendi (al-Chodschendi), in the tenth century of our era. The cosine theorem was implied in rules known to the early Indians, but was exhibited more fully by Regiomontanus (c. 1460), and ultimately by Tycho Brahe (be fore 159o). Gauss's formulae of 1809 were found earlier by Delambre (1807) and Mollweide (1808). Napier's analogies, curiously enough, arrived much earlier, 1619, in Napier's work, and were practically exhibited by Briggs in 162o.

BIBLIoGRAprr.—Fuller accounts are best found in R. Wolf, Ge schichte der Astronomie (1864) ; M. Cantor, Vorlesungen fiber Geschichte der Mathematik, 4 vols. (Leipzig 188o-19o8) ; W. W. R. Ball, A Short Account of the History of Mathematics (1888) ; F. Ca jori, A History of Mathematics (1893) ; D. E. Smith, History of Mathe matics (s v., 1923, 1925) ; Tropfke, Geschichte der Mathematik (2nd ed. Leipzig 1922) ; A. Braunmilhl, Geschichte der Trigonometrie 2 vols. (Leipzig 2900, 1903). For detailed directions and checks in the accuracy of computation see E. W. Hobson, A Treatise on Plane Trigonometry (Cambridge, 1897) ; W. Chauvenet, Plane and Spherical Trigonometry (9th ed. 1879) ; for computing sine and cosine values see L. Euler, Introductio in Analysin Infinitorum, ed. F. Rudio, etc. in Opera omnia, vol. 8 (I 91I) ; for a full statement of series con nected with Trigonometry, see G. Chrystal, Algebra, pt. 2 (Edinburgh, 2889). (H. S. W.)