332 11. POST-EUCLIDEAN GEOMETRY
FIGURE 7. Pitiscus' derivation of the proportions in which an al-
titude divides a side of a triangle.
(1561-1613). This was, incidentally, the book that established the name trigonom-
etry for this subject. Pitiscus showed how to determine the parts into which a side
of a triangle is divided by the altitude, given the lengths of the three sides. To
guarantee that the angles adjacent to the side were acute, he stated the theorem
only for the altitude from the vertex of the largest angle.
Pitiscus' way of deriving this proportion was as follows. If the shortest side of
the triangle ABC is AC and the longest is BC, let the altitude to BC be AG, as in
Fig. 7. Draw the circle through C with center at A, so that Β lies outside the circle,
and let the intersections of the circle with AB and BC be Ε and F respectively.
Then extend Β A to meet the circle at D, and connect CD. Then ZBFE is the
supplement of ZCFE, which subtends the arc Ε DC, which in turn is an arc of
180° plus the arc CD. Hence ZBFE is the complement of the angle subtended by
the arc CD. That in turn is the angle subtended by the supplementary arc CE;
thus ZBFE = ZCDB, and so the triangles BCD and BEF are similar. It follows
that
AB^2 = AC^2 + BC^2 - 2AC.BCcos{ZACB),
which is what we now know as the law of cosines.
Pitiscus also gave an algebraic solution of the trisection problem discovered by
an earlier mathematician, Jobst Biirgi (1552-1632). The solution had been based
on the fact that the chord of triple an angle is three times the chord of the angle
minus the cube of the chord of the angle. This relation makes no sense in terms
of geometric dimension; it is a purely numerical relation. It is interesting that it is
stated in terms of chords, since Pitiscus surely knew about sines.
4. Geometry in the Muslim world
In the Western world most of the advancement of geometry in the millennium
from the fall of the Western Roman Empire to the fall of the Eastern Empire
occurred among the Muslim and Jewish mathematicians of Baghdad, Samarkand,
Cordoba, and other places. This work had some features of Euclid's style and
some of Heron's. Matvievskaya (1999) has studied the extensive commentaries
on the tenth book of Euclid's Elements written by Muslim scholars from the ninth
through twelfth centuries and concluded that while formally preserving a Euclidean
