LAW I: THE ELLIPSE LAW
(1609)
1. Modern form
The orbit of a planet around the Sun is an ellipse with the Sun at
one focus.
2. Kepler stated his
result in Astronomia nova (1609), in Chapter
58 (KGW
III p.366 line 4; Donahue p.575): 'an ellipse is the
path of the planet [Mars]'.
3. Background for
Kepler's approach
Figure (a) shows the planet at a position P on its path CFD whose
(major) diameter is CD, circumscribed by the circle diameter CD centre B. Q is a typical point of that circle,
determined by the angle ĐQBC = b at the
centre, with QPH the ordinate linking the associated points Q and P. Kepler established that the path was an
ellipse by finding a (geometrical) formula for the distance AP of the planet P
from the Sun at A, whose position Kepler described as 'the eccentric point'
(the word 'eccentric' is used in astronomy in the sense of 'off-centre'). This point A was not then known to possess
any special properties as far as the ellipse was concerned. We call AB the eccentric distance, and when
we introduce symbolic notation for convenience we shall express the length AB
as a proportion of the radius of the circle; this length was defined by
(frequent) astronomical observations of the distances from the extreme points,
or apsides, C and D, so (because of symmetry along the major diameter) we have:
Figure (a)
Note that the eccentric distance has to be greatly exaggerated in
every figure, because the separation between the ellipse and the circle is
undetectable on the scale of a printed page – which is why discovery of the
ellipse could not have happened before Tycho Brahe (1546-1601) provided
observations that were accurate enough – and then only in the case of Mars.
Kepler defined the ellipse he (eventually) found by the
ratio-property of the ordinates which Archimedes had stated in his work On Conoids and Spheroids, Prop.4:
(This definition obviously does not involve the focus.)
4. How Kepler discovered
the path of Mars
From his student days Kepler enthusiastically adopted the
Sun-centred theory of Copernicus (1473-1543), and applied it rigorously by
insisting that all planetary distances should be measured from the Sun
itself. After much preliminary
investigation, which began around 1601, Kepler started from the framework he
had derived from Ptolemy (second century CE), adapted to a heliocentric view;
so that scheme depicted the path of Mars initially as the circle CQD, of radius
a, whose centre B was eccentric to
the Sun at A, as shown in Figure (b).
Figure (b)
Kepler had inherited a superb set of naked-eye observations from
Tycho Brahe and he used them as a criterion against which to assess his
geometrical theory at every stage.
Though he found that putting the planet at Q made the distances from A
generally too long, he continued to work from this traditional framework, using
straightedge and compasses – the only geometrical tools permitted by
Euclid. After a second attempt gave
distances that were too short, Kepler eventually managed to apply his
characteristic method to construct distances that fell more-or-less in the
middle, by drawing a perpendicular from Q to meet AN extended at K, as shown in
Figure (c). This produced a typical
distance of the right length AK along AN, swung round in a circular arc to get
it in the right direction (AP). (The
resulting point P was satisfactory because the planet was then positioned as
precisely as could be achieved in relation to the accuracy of the
observations.)
Figure (c)
Thus Kepler's characteristic construction produced AP = AK, and
also we have AK = QR because of the rectangle ARQK. Hence Kepler found that the position of the
planet was given by:
Kepler proved this, actually in these geometrical terms. (Symbolic notation had hardly been developed
in that era, and Kepler did not use it.)
However, for the convenience of readers, the result can be expressed in modern
terms as:
.
Even nowadays, this particular formula for the radius vector is
seldom recognized as representing an ellipse – because it is expressed in terms
of the angle b at the centre – though we shall prove in the
next section that this representation is entirely valid and exact. However, Kepler did not have the faintest
idea what curve his invented construction had produced, until he suddenly
thought of trying to identify the unknown curve with the ellipse defined by
Archimedes. When this turned out to be
successful, Kepler provided the mathematical proof.
5. Kepler's proof of
the result (transcribed for modern readers)
In Chapter 56 Kepler specified the position of A on CD by
establishing that the important distance from the Sun (shown in Figure (c)) to
the point F of the ellipse lying at the end of the perpendicular axis (through
B) was Then we apply Pythagoras' theorem to the
right-angled triangle AFB shown in Figure (b), to obtain:
or alternatively, in modern notation, when we call the minor
semi-axis
Kepler confirmed this identity in Chapter 59, Protheorema VII: it
was as much as he needed to know about the position of A. Indeed A was not identified as the focus of
the ellipse until a few years later – we do not know when, but Kepler stated
the fact in 1621 in Epitome Book V, Part I,
Section 3 (KGW VII p.372). It is also
interesting to notice that the special length shown in Kepler's
own diagram of Chapter 59, appears in Figure (b) as a radius of the circle
centred at the Sun, being the (arithmetic) mean of the two extreme (apsidal)
distances. Hence the mean distance was
geometrically illustrated here ready for application in Law III.
We shall now determine the unknown curve starting from Kepler's
characteristic construction which gave a position P for the planet:
For modern readers we shall apply the method of elementary
coordinate geometry which is anachronistic (by half a century or so) – but the
statements made in this proof can be identified in Kepler's own proof of Law I
(he did not call it that) in Protheorema XI of Chapter 59 (KGW III p.371). This proof depends entirely on the geometry
of Euclid, using Pythagoras' theorem and ratios from similar triangles
(nowadays we use trigonometry for the same effect). We start with what Kepler had discovered
about the unknown curve.
From
DPHA, applying
Pythagoras' theorem, we have:
Hence,
using the result (1) from Protheorema VII as set out above.
Now from DQHA,
Thus we obtain:
Hence the path of P is the ellipse defined by the ratio-property
of the ordinates stated by Archimedes – and this path was the one produced by
Kepler's characteristic construction.
Thus we have provided a sound geometrical proof that an exact
ellipse can be constructed to satisfy the observations in the case of Mars,
subject to the limits of accuracy imposed by those observations. [Unfortunately it is not possible to
establish the modern result that all planetary orbits are ellipses without knowledge
of the underlying dynamics, which did not become available until Newton's
synthesis. However, Kepler did go on to
demonstrate, later, that the orbits of all the other naked-eye planets were
compatible with an elliptic orbit.]
For more details of Kepler's method, see:
A E L Davis
ael.davis@imperial.ac.uk