Understanding the‘ScientificRevolution’ 157
If I wish to find out to what class this curve belongs, I choose a straight line, asAB, to which
I refer all its points, and inABI choose a pointAat which to begin the investigation. I say ‘choose
this and that’, because we are free to choose what we will, for, while it is necessary to use care in
the choice in order to make the equation as short and simple as possible, yet no matter what line
I should choose instead ofABthe curve would always prove to be of the same class, a fact easily
demonstrated.
Then I take on the curve an arbitrary point, asC, at which we will suppose the instrument
applied to describe the curve. Then I draw throughCthe lineCBparallel toGA[and meetingBA
inB]. SinceCBandBAare unknown quantities, I shall call one of themyand the otherx. To the
relation between these quantities I must consider also the known quantities which determine the
description of the curve, asGA, which I shall calla;KL, which I shall callb; andNLparallel toGA,
which I shall callc. Then I say that asNLis toLK,orascis tob,soCB,ory,istoBK, which is
thereforebcy. ThenBLis equal tobcy−b, andALis equal tox+bcy−b. Moreover, asCBis toLB,
that is, asyis tobcy−b,soAGorais toLAorx+bcy−b. Multiplying the second by the third, we
getabcy−abequal toxy+bcyy−by, which is obtained by multiplying the first by the last. [This is
the usual ‘multiplying out’ of an equation between ratios.] Therefore the required equation is
yy=cy−cx
by+ay−acFrom this equation we see that the curveECbelongs to the first class, it being in fact a hyperbola.
[Without setting it as an exercise, you are encouraged to follow through this calculation to see
how Descartes has derived his equation.]
Appendix C
(From Kepler,Nova stereometria doliorum(New measurement of wine-barrels), 1615 (in 1999).
Since, the wine-barrels are made up of the circle, the cone and the cylinder which are regular
figures, they are suitable for geometrical proofs; and I shall gather these together in the first part of
this investigation. Since they were investigated by Archimedes, to read a part of his work is enough
to delight a lover of geometry. For absolute proofs which are exact in every number can be sought
in these same books of Archimedes, if anyone is not frightened by the thorny reading of them.
However, we can pass the time in certain regions which Archimedes did not reach; and even the
wiser readers can find things to please them there.
Theorem I.First we need to know the relation of the circumference to the diameter. And Archimedes
taught:
The ratio of the circumference to the diameter is very near to the ratio which the number 22 has to 7.
[Note that this is not really what Archimedes said, and the use of terms like ‘very near’ would
have been quite unacceptable in Greek mathematics. However, if this is bad enough, Kepler’s ‘proof ’
is even worse. I shall omit it, since it is not calculus so much as crude approximation using inscribed
and circumscribed hexagons. (Remember Archimedes used 96-sided figures!)].