P. HISTORY OF ' AATHEMATICAL - School of Mathematics

TRIGONOMETRY
151
(P. 360) Rad. sin. ai. sm. a. sin. ie.
(P. 364) Sin. yu sin. ys sm. w sin. it
(P. 378) Rad. tan. com. a. tan. com. i sin. com. ia.
(P. 381) Rad. sin. com. ie. sec. ia. sec. ea.
It will be observed that the four terms of each proportion are written
one after the other, without any symbols appearing between the terms.
For instance, the first proportion yields sin ie=sin ai.sin a. An angle
of a spherical triangle is designated by a single letter, namely the
letter at the vertex; a side by the letters at its extremities. Accordingly,
the triangle referred to in the first proportion is marked a i e.
Glaisher remarks that Finck is not at all uniform in his manner of
appreviating the names of the trigonometric lines, for one finds in his
book also "sin. an.," "sin. ang.," "tang.," "sin. comp.," "sin. compl.,"
"sin. com. an.," "tan. comp.," etc. Finck used also "sin. sec." for
sinus secundus, which is his name for the versed sine. The reader will
observe that Finck used abbreviations which (as we shall see) ultimately
became more widely adopted than those of Oughtred and Norwood
in England, who are noted for the great emphasis which they
laid upon trigonometric symbolism.
Similar contractions in treating spherical triangles are found not
long after in Philipp van Lansberge's Triangulorum geometricorum libri
quator (1591), previously referred to. In stating proportions he uses
the abbreviations' "sin.," "sec.," "tang.," "sin. cop.," "tang. comp.,"
"tang. compl.," "sec. comp.," but gives the names in full whenever
there is sufficient room in the line for them. For instance, he writes
in a case of aright spherical triangle (p. 87), "ut radius ad sinum basis
ita sec. comp, lat. ad sec. comp. ang."
A few years later, one finds a few abbreviations unrelated to those
of Finck, in a manuscript of Jostel, of Wittenberg, whom we have
mentioned earlier. He was a friend of Longomontanus and Kepler.
In 1599 he writes "S." for Sinus,2 as does also Praetorius of Altdorf.
Later still one encounters quite different symbols in the Canon
triangulorum of Adriaen van Roomen (Adrianus Romanus), of the
Netherlands. He designates the sine by "S.," the tangent (which he
called, as did Vieta, the Prosinus) by "P.", the secant or Transsinuosa
by "T." The co-functions are indicated by writing ct. after
the function. A rectangle or the product of two factors is indicated by
1 Ph. van Lansberge, op. cit., p. 85-88.
2 Tycho Brahe Dani Opera Omnia (ed. J. L. E. Dreyer; Hauniae, 1913),
Vol. I, p. 297 f.; A. von Braunmtihl, Vorlesungen uber Geschichte der Trigonometrie,
1. Tl!il (Leipzig, 1900), p. 230 n.