Boulliau:  Planetary Theory: Boulliau's Conical Hypothesis (1645), Primary Documents:  Borelli's response to Boulliau; Boulliau's response to Borelli to Prince Leopold of Tuscany
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Theoricae  Mediceorum  Planetarum   (1666)
BORELLI'S  CRITIQUE  OF  BOULLIAU'S
Astronomia  Philolaïca (Paris 1645)

Johannes Kepler was the first, by his boldness, and in opposition to the ancient philosophers, to give the order which banished perfectly circular orbits from the sky.  He proved most clearly from Tycho Brahe's observations that confirmation was provided in the case of the orbit of Mars, and he then noticed that the ellipticity of the orbit ought of necessity to occur in the case of Mercury; but in the case of the Sun he was unable to demonstrate the elliptical shape of its orbit, although he ascribed such to all the planets on the basis of extremely ingenious, but false, physical arguments.  This view was so agreeable to all the learned that it was readily accepted, especially as the most learned and famous astronomer Boulliau had largely perfected it, though he had deduced it from different principles, and had formed the said elliptical figures from different elements.  For this reason, it will perhaps be useful to give a brief account of what Boulliau proposed so we may later show in what manner this doctrine of ellipticity is confirmed (Figure 1).
Imagine a scalene cone with its apex at A and having a circular base of diameter BC.  Let its axis be AI; and let the triangle through the axis perpendicular to the circular base be ABC, so that angle AIC is acute, its complementary angle obtuse.  Draw the straight line EK to subtend an angle at the apex so that EK is divided into two equal parts at point X by straight line VT, which is equal to EK, is parallel to the base BC, and cuts the axis at point Z.  It follows that the triangle MXZ will be isosceles, having MX equal to ZX.  Therefore triangle AEK will not be sub-contrary to triangle ABC.  Through straight line EK raise a plane surface perpendicularly to the plane of triangle ABC, which surface will develop the ellipse ERK in the section of the cone.  The transverse axis of this ellipse will be EK, its conjugate axis will be ON, its center will be X, and one of the foci or poles will be the point M on the axis of the cone.  The segment XH being equal to XM, the other focus of the ellipse will be H. 
This being presupposed, Boulliau then assumes that the Sun is at point H, and that the planet moves with uniform motion about the axis AMI of the cone in circles which are always equidistant from the circle of the base BC of the cone, which circles may be called equant circles.  The point M, or rather, indeed, the entire axis, will be called the center of uniform motion.  Seeing that it is the nature of uniform circular motion to sweep out equal angles at the center in equal times, and given that these angles at the center belong to similar circumferences, which are, nevertheless, proportional to their radii, it thus follows that when the planet passes through point E belonging to the circle whose semi-diameter is SE, then its motion will be slowest, because this circle is the smallest of those described by the celestial body in its proper period, so too, when the celestial body arrives at point Y and describes the circumference of circle FG on the cone (which circle passes through the focus M) its motion will be rapid because this circle is larger. 
Later, when it traverses the peripheries of other circles, the greatest of which passes through K, its motion will be fastest because it describes the periphery of the largest circle PK.  At position K the planet is nearest to the pole H.  Consequently, from aphelion E to perihelion K, the celestial body will traverse the peripheries of innumerable circles [parallel to the base] which increase successively [in diameter], and for this reason, the uniform motion which sweeps out equal angles about axis AI in equal times is [also simultaneously] represented on the elliptical periphery ERK by increases which will have the same relationship to each other as the radii of the said circles above the minimum [SE = HK]. 
Although, by deducing the physical equation as well as the optical equation from this hypothesis, Boulliau, as pointed out by Seth Ward, omitted certain things concerning the mean motion; nevertheless, it cannot be denied that this first invention of his is admirable, ingenious, and praiseworthy.  It is no doubt true that Cosmologists raise two objections; in the first place, these cones for each planet are fictitious, consequently, it is not clear how it happens that the planet moves on a certain conical surface which has no existence in the world; and second, it seems contrary to fact that these motions are performed about a certain point [M] and a certain line of uniform motion passing through M, for this point is indivisible and assumed by the imagination as being in the aether, and has absolutely no substance or [physical] faculty.  Consequently, there is absolutely no reason why the planet should revolve about the said point and imaginary line in accordance with a perfectly constant rule and, on the contrary, move in a quite irregular manner with respect to the very large globe of the Sun itself, placed at point H, as if the main object of the star were not to turn round the Sun itself but to turn around the said imaginary, fantastic point, which is without perfection or faculty. Indeed, this is such a strong objection that it seems very difficult to answer. 
With regard to the first point, I believe I can not only deal with it satisfactorily but, perhaps, also give enlightenment on some things concerning the secrets of nature.  In the first place, we shall imagine the planet to move under two motions, the one circular, the other, on the contrary, linear, and we shall show from these two motions, taken as elements, that an elliptical motion can result.  Let us assume the Sun is at H and the planet, to begin with, is at aphelion E but has two motions, the first is orbital about the Sun, the second is linear in the direction from A towards P.  Let us also assume the said motions are commensurable with each other in such a manner that when the planet describes a semi-circle starting from E, it must go from E to P in the same time with a linear motion; but that during the following semi-circle the planet returns from P to E.  We must assume also that the plane ED of the circular motion is always inclined with respect to line EP of linear motion, whence it follows that the planet, by its motion in a straight line, traverses the circumferences of innumerable circles which are always equidistant from each other, and if, during this time, the circular motion were uniform, especially if equal angles were swept out in equal times with respect to the center, then the planet would describe an elliptical orbit as we have said above.  We see therefore, even though no real cone be supposed to exist in the world, how it is nevertheless possible for elliptical motion to take place in exactly the same way as if we assumed a solid cone of that kind. 
It can be shown that the hypothesis of the aforesaid two motions is possible, in the first place, by the example of all those planets that have orbits similar to circles, but never follow the exact periphery of circles.  Furthermore, the curves described by them incline more uniformly to the plane of the planet's orbit than is postulated by the inclination on which its latitude depends. 

Seeing that the plane of the solar whirlpool is definitely inclined to the plane of the ecliptic, then, if it were true, as Kepler thought, that the planets are in some way caught up and carried round the Sun by the solar rays as they revolve, the planets should necessarily be carried round along circle LQ and others parallel to it.  In fact, if we imagine the Sun to remain at the point H and to describe circles parallel to LQ by its own whirlpool, in such a way that the axis of this whirlpool is raised perpendicularly to the plane LQ of the whirlpool, the solar rays would move in the plane of the said circle LQ and others parallel to it; moreover, the planets, being carried around by these rays, would advance on this plane and, since in the meantime the planet would execute its proper motion from E to P and from P to E, it would be obliged to travel first in circle PK, then in circle LQ, and so on, without remaining for any definite time in any one of them (because motion on EP is supposed continuous).  Consequently, the planets must be carried round along the peripheries of innumerable circles of different sizes, which provide a measure of the irregular velocity with which the planet moves round the Sun; and we see that such a motion, far from being impossible, is on the whole rational and probable, provided it does not encounter other objections. 
Furthermore, it must be noted that while the solar body always revolves in the same place, nevertheless, the equidistant circles described by its rays as the Sun itself revolves on its axis are always in the same plane with respect to the situation and extent of the world.  Therefore, in order that the elliptical paths described by the different planets may be saved, we need only:  1) place the aphelia at various distances from the Sun in various places in the starry sky and Zodiac in such a way that the line of aphelion drawn from the Sun to the planet is more or less inclined with respect to the plane of the Sun's whirlpool; and 2) ascribe to each of them a motion along the straight line inclined to the plane of the solar whirlpool, and 3) having the exact value necessary to form an ellipse suitable to the planet in question, with its periods and all other circumstances which have been observed in its motion. 
One final difficulty remains, namely, if it is possible for the planet to move around the focus of the ellipse, or the point of equality, while the Sun is placed at the other focus or pole.  Most certainly, it seems difficult and incomprehensible that the planet, whether carried around by its own innate power or by some external power, should be moved around this point of equality which possesses no power or entity.  This view is indeed refuted by the reality of nonuniform speed of planetary motion, which speed, according to this hypothesis, should increase in exactly the same proportion as the distances drawn from the axis of the cone through the focus increase; or else to the same extent as the semi-diameters of the equant circles increase.  This clearly cannot occur either in the case where the motion of the planet on the circumferences of the equant circles is effected by a proper force existing in the planet itself; or in the case where it results from an external faculty which propels the planet.  For in the former case, the speed of the planet should always be uniform and constant, whereas in the latter case, the velocity of the planet should decrease in proportion to the increase in the semi-diameters and circumferences of the equant circles, as will be shown later.  For these reasons we have been obliged to abandon the above hypothesis, and if possible, to try to find another more probable one, or to show that the same elliptical path of planetary motion can be retained but only on the basis of firmer hypotheses more conformable to physical arguments. 

§§§
 

 
Extract, G-A Borelli (1608-1671), Theoricae Mediceorum Planetarum (1666) ff. 31, reprinted in A. Koyré, The Astronomical Revolution, pp. 474-478, translation after R.E.W. Maddison (translation modified and corrected from the original Latin by Robert A. Hatch.  Diagram from Boulliau's APFCE, Paris 1657). 

 
 rah.viii.98

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