152 A History ofMathematics
in these early stages of what later becomes the calculus, there is a general sense of exploration,
of trying out statements to see how they sound, and (by contrast with algebra) of a loosening
rather than a sharpening of definition. While there are certainly traces in Stevin’s work, the most
interesting introduction is in Kepler’sAstronomiaNova, which presents itself (only semi-realistically)
as the account of the various false trails he followed until his final discovery of his famous planetary
laws. (The account here is largely based on a detailed analysis by Bruce Stephenson (1987).
Any analysis of such a complex text as theAstronomia Novais contestable, but the broad lines
seem persuasive enough.) Kepler was faced by a new problem almost from the beginning. Both
Ptolemy and Copernicus explained the motion of planets—Mars in particular—as being composed
of uniform motions in a circle around a point which was not the real centre (sun for Copernicus,
earth for Ptolemy) but a point in empty space, the ‘equant’.^15 The advantage of such a scheme is
that it is relatively easy to calculate using uniform motion in a circle. A modern physicist would
point out that this is ‘unphysical’, in that one is postulating a force linking the planet to the equant,
where there is no matter. Kepler’s version of this, which stemmed from his own ideas on planets,
was that the souls which animated them could perfectly well perceive the sun (for example) and its
distance and adjust their movement accordingly; but it was unreasonable to suppose them capable
of perceiving the equant. He therefore had to suppose that motion was dependent on distance from
the sun, and slowed down when the planet was further.
At this point we have two problems about velocity. The first is the old question of whether Kepler
or his contemporaries could define, or even think of ‘velocity at an instant’, as we register at an
instant that a car is travelling at 45 miles per hour. It has been suggested that Th ̄abit ibn Qurra and
al-B ̄ir ̄un ̄i used such an idea (see for example Hartner and Schramm 1963), but it does not seem
to have been in general use among Islamic astronomers. The first steps towards understanding
what this might mean were taken by Galileo—see below. The second was the old problem that even
velocity over a time interval was the ratio of quantities of different kinds—distance and time—
and so unacceptable in a Greek framework. Kepler’s way of avoiding these problems was to use the
‘delay’—the time taken by the planet to travel along a small interval of its orbit—in circumstances
where the intervals were approximately equal. He had to frame a hypothesis about how the delay
depended on the distance, and he tried several; but in each case, he faced the problem of adding a
large number of very small delays to arrive at the measurable time-intervals given by observations.
Here, in chapter 40, he is introducing the difficulty, supposing that the planet moves in a circular
eccentric orbit around the sun:Since, then, the delays of the planet in equal parts of the eccentric are in the inverse proportion to the distances of
those parts [from the eccentre], but the individual points are changing their distance from the eccentre all around the
semicircle; I thought it would be no easy work to find out how the sum of the individual distances could be arrived at.
For unless we had the sum of all, which would be infinite, we could not say what was the delay of each. And so we
should not know the equation. For as the whole sum of the distances is to the whole period, so any part of the sum of
the distances is to its own time.
[Kepler began by dividing the circle into 360 degrees, but found the calculation tedious; but then he had an idea.] For
I remembered that once Archimedes also, when he was seeking the proportion of the circumference to the diameter,
divided the circle into an infinite number of triangles, but this scheme was hidden by his proof by contradiction. Hence
where before I had divided the circumference in 360 parts, now I cut the plane of the eccentric in as many lines from
the point from which the eccentricity was computed. [Fig. 3]. (Kepler 1990, pp. 263–4)- Even this is a serious oversimplification of the system of epicycles and equants which both needed: points which were occupied
by no real bodies, but whose rotation was a necessary part of the description of the ‘phenomena’, the observed motions.