- MEDIEVAL GEOMETRY^331
Uniformly uniform Uniformly difform Difformly difform
FIGURE 6. Nicole Oresme's classification of motions.
had at the midpoint of the time of travel. This is the case now called uniformly
accelerated motion. According to Clagett (1968, p. 617), this rule was first stated
by William Heytesbury (ca. 1313-ca. 1372) of Merton College, Oxford around 1335
and was well known during the Middle Ages.^12 It is called the Merton Rule. In his
book De configurationibus qualitatum et motuum, Oresme applied these principles
to the analysis of such motion and gave a simple geometric proof of the Merton
Rule. He illustrated the three kinds of motion by drawing a figure similar to Fig. 6.
He went on to say that if a difformly difform quality was composed of uniform or
uniformly difform parts, as in the example in Fig. 6, its quantity could be mea-
sured by (adding) its parts. He then pushed this principle to the limit, saying that
if the quality was difform but not made up of uniformly difform parts, say being
represented by a curve, then "it is necessary to have recourse to the mutual mea-
surement of curved figures" (Clagett, 1968, p. 410). This statement must mean
that the distance traveled is the "area under the velocity curve" in all three cases.
Oresme unfortunately did not give any examples of the more general case, but he
could hardly have done so, since the measurement of figures bounded by curves was
still very primitive in his day.
Trigonometry. Analytic geometry would be unthinkable without plane trigonome-
try. Latin translations of Arabic texts of trigonometry, such as those of al-Tusi and
al-Jayyani, which will be discussed below, began to circulate in Europe in the late
Middle Ages. These works provided the foundation for such books as De triangulis
omnimodis by Regiomontanus, published in 1533, after his death, which contained
trigonometry almost in the form still taught. Book 2, for example, contains as
its first theorem the law of sines for plane triangles, which asserts that the sides of
triangles are proportional to the sines of the angles opposite them. The main differ-
ence between this trigonometry and ours is that a sine remains a length rather than
a ratio. It is referred to an arc rather than to an angle. It was once believed that
Regiomontanus discovered the law of sines for spherical triangles (Proposition 16 of
Book 4) as well; but we now know that this theorem was known at least 500 years
earlier to Muslim mathematicians whose work Regiomontanus must have read. A
more advanced book on the subject, which reworked the reasoning of Heron on the
area of a triangle given its sides, was TrigonometrieE sive de dimensione triangu-
lorum libri quinque (Five Books of Trigonometry, or, On the Size of Triangles),
first published in 1595, written by the Calvinist theologian Bartholomeus Pitiscus
(^12) Boyer (1949, p. 83) says that the rule was stated around this time by another fourteenth-century
Oxford scholar named Richard Suiseth, known as Calculator for his book Liber calculatorum.
Suiseth shares with Oresme the credit for having proved that the harmonic series diverges.