A History of Mathematics From Mesopotamia to Modernity

156 A History ofMathematics

For example: in the above the second part, which is a third of the first, differs from the first by

two-thirds, therefore the ratio of the whole to the first part or the assumed quantity is as three to

two and this is sesquialterate.

The third proposition is this: It is possible that an addition should be made, though not

proportionally, to any quantity by ratios of lesser inequality, and yet the whole would become

infinite. For example, let a one-foot quantity be assumed to which one-half of a foot is added during

the first part of an hour, then one-third of a foot in another, then one-fourth, then one-fifth, and so

on into infinity following the series of numbers, I say that the whole would become infinite, which is

proved as follows: There exist infinite parts any one of which will be greater than one-half foot and

[therefore] the whole will be infinite. The antecedent is obvious, since one-quarter and one-third

are greater than one-half; similarly from one-fifth to one-eighth is greater than one-half, and from

one-ninth to one-sixteenth, and so on into infinity...

Appendix B

(From Descartes 1954, pp. 51–5)

Suppose the curveECto be described by the intersection of the rulerGLand the rectilinear plane

figureCNKL, whose sideKNis produced indefinitely in the direction ofC, and which, being moved

in the same plane in such a way that its sideKLalways coincides with some part of the lineBA

(produced in both directions), imparts to the ruler a rotary motion aboutG(the ruler being hinged

to the figureCNKLatL).

[See Descartes’s picture (Fig. 4.). The triangleCNKL, more properlyNKL, moves up and down

AB; the ruler (as the picture more or less shows) is fixed to the triangle atLand passes through a

loop or curtain-ringGwhich is fixed to the lineAG. The curve is traced by the intersectionCof the

ruler and the (produced) sideKNof the triangle.]

G

K

N

L

B

C

E

A

I

Fig. 4Descartes’s curve drawing machine.