RICE UNIVERSITY STUDIES
which globally intersects the curve at one point only, and he pretends to
observe this definition even in his essay on (archimedean) spirals r = cθ
(polar coordinates). Archimedes is aware of the fact that any straight line
in the plane of the spiral intersects it in more than one point, and he ap-
parently observes the euclidean requirement only Aα⅛⅛lobally, for a half-
coil of the curve. But there is no tendency in the essay to make the require-
ment a properly local one.
Furthermore, Archimedes’ law of the lever is a “conservation law” for
the rotational momentum
(i) /(pJ)≡p∙/,
(/ =Iength of the arm,p = the suspended weight), meaning that
(2) pl ■ I1 = p2 ∙ ∕2.
However, Archimedes could not envisage “operationally” a “function”
like (1). Thus, he was unable to conceptualize the physical datum of rota-
tional momentum, and he had to express the equality (2) in the euclidean
(that is, Greek) manner as a proportion
Pi : Pi = ʃz ɪ ʃr
This explanation of the intellectual limitation of Archimedes is seemingly
different from, yet in fact very cognate with, our previous explanation that
Archimedes was unable to conceptualize a product like p ■ I as a ring opera-
tion within the semi-ring of positive real numbers [4, pp. 181 ff.].
We note that this ring operation was first introduced by Descartes at the
head of his La Géométrie (1637), and that the first formal definition of a
mechanical momentum was given by Newton in his treatise. Newton
introduced not the rotational momentum but the translational momentum.
He called it “quantity of motion” and characterized it as the product of
mass and velocity [⅛, ρ. 1], or rather as a bilinear functional on the cartesian
product of mass and velocity.
Middle Ages. In the Middle Ages there were some stirrings of the kind
of analysis in which functions are domiciled, and the most function-oriented
medieval mathematician was Nicole Oresme (1323-1382). He devised a ver-
sion of graphing for which he is renowned, and he also envisaged exponen-
tiation ar (for fractional exponents r = p∣q) [5, pp. 288-295]. These two
achievements, when taken together, certainly suggest functions of the kind
that occur in later developments. Oswald Spengler in his Decline of the West
(1922-1924) — whatever the shortcomings of the work as a whole —