330 11. POST-EUCLIDEAN GEOMETRYnavigators and explorers for many centuries)^11 and a discussion of different ways
of using similar triangles to determine distances to inacessible objects.
3.1. Late Medieval and Renaissance geometry. In the late Middle Ages Eu-
ropeans from many countries eagerly sought Latin translations of Arabic treatises,
just as some centuries earlier Muslim scholars had sought translations of Hindu and
Greek treatises. In both cases, those treatises were made the foundation for ever
more elaborate and beautiful mathematical theories. An interesting story arises in
connection with these translations, showing the unreliability of transmission. The
Sanskrit word "bowstring" [jya) used for a half-chord of a circle was simply bor-
rowed by the Arab translators and written as jb, apparently pronounced jiba, since
Arabic was written without vowels. Over time, this word came to be interpreted as
jaib, meaning a pocket or fold in a garment. When the Arabic works on trigonom-
etry were translated into Latin in the twelfth century, this word was translated as
sinus, which also means a pocket or cavity. The word caught on very quickly, ap-
parently because of the influence of Leonardo of Pisa, and is now well established
in all European languages. That is the reason we now have three trigonometric
functions, the secant (Latin for cutting), the tangent (Latin for touching), and the
sine (Latin for a concept having nothing at all to do with geometry!).
We think of analytic geometry as the application of algebra to geometry. Its
origins in Europe, however, antedate the high period of European algebra by a
century or more. The first adjustment in the way mathematicians think about
physical dimensions, an essential step on the way to analytic geometry, occurred in
the fourteenth century.
Nicole d 'Oresme. The first prefiguration of analytic geometry occurs in the work of
Nicole d'Oresme (1323-1382). The Tractatus de latitudinibus formarum, published
in Paris in 1482 and ascribed to Oresme but probably written by one of his students,
contains descriptions of the graphical representation of intensities. The crucial
realization that he came to was that since the area of a rectangle is computed
by multiplying length and width and the distance traveled at constant speed is
computed by multiplying velocity and time, it follows that if one line is taken
proportional to time and a line perpendicular to it is proportional to a (constant)
velocity, the area of the resulting rectangle is proportional to the distance traveled.
Oresme considered three forms of qualities, which he labeled uniform, uniformly
difform, and difformly difform. We would call these classifications constant, linear,
and nonlinear. Examples are provided in Fig. 6, although Oresme realized that the
"difformly difform" constituted a large class of qualities and mentioned specifically
that a semicircle could be the representation of such a quality.
The advantage of representing a distance by an area rather than a line appeared
in the case when the velocity changed during a motion. In the simplest nontrivial
case the velocity was uniformly difform. In that case, the distance traversed is
what it would have been had the body moved the whole time with the velocity it
(^11) The FVench explorer Samuel de Champlain (1567-1635) apparently lost his astrolabe while
exploring the Ottawa River in 1613. Miraculously, that astrolabe was found 254 years later,
in 1867, and the errors in Champlain's diaries were used by an author named Alex Jamieson
Russell (1807-1887) to establish the fact and date of the loss (Russell, 1879). The discovery of
this astrolabe only a month after the founding of the Canadian Federation was of metaphorical
significance to Canadian poets. See, for example, The Buried Astrolabe, by Craig Stewart Walker,
McGill-Queens University Press, Montreal, 2001, a collection of essays on Canadian dramatists.