C O N I C A L    H Y P O T H E S I S
Dr Robert A. Hatch - Copyright 1998

 
When he first glimpsed his Conical Hypothesis in bare outline, Boulliau was seduced by its form, intoxicated, he later confessed, by its subtle geometry. But Boulliau was not the first astronomer to be so smitten, not the first to report the effects of Ambrosia on his critical faculties. Ptolemy, Copernicus, and Kepler had also sampled the Nectar of the Gods, they too confessed to a certain drowsiness that made them muse under its influence. Staggered by beauty, the soul could not resist. And if Plato is a guide in such matters, the pleasure of laying bare the beauty of the world, of revealing her hidden secrets, was reserved for students of geometry. But wait. This seems strange. How does such talk relate to early-modern cosmology?  The answer is clear if not simple.

In what follows I attempt to suggest several strange but no less seductive ideas by means of Boulliau's Conical Hypothesis.  In hinting at his 'context of discovery' I hope to betray assumptions that may have blinded Boulliau's vision, and I aim to do this with elementary geometry.  But Boulliau's model, like the problem it sought to solve, was in fact not simple. In context, what became known as 'Kepler's Problem' was a mathematical nightmare that involved calculating double-false positions based on tedious trial and error methods. The method was not only clumsy and questionable but ugly and impractical.  As Newton was quick to recognize, there is no direct solution to Kepler's Problem.  This, in part, explains why Newton continued to use a later variation of the modified elliptical hypothesis, as a practical means to an end, long after the publication of his Principia.  But that is a different story.
For present purposes, suffice it to say that Boulliau was drawn to his Conical Hypothesis because it offered a crisp reply to Kepler's plea for geometrical assistance, and because it offered an simple solution to a practical problem. In practice, Boulliau's method represents a direct method for determining the position of a planet at a given time, that is, a solution to the 'Kepler Problem'.   It also offered, in context, an entirely new cosmology.   This said, the problem that remains is historical. Much of Boulliau's story remains untold, while other parts, alas, have been badly muddled or entirely misunderstood (indeed, by such talented thinkers as Jean-Baptiste Delambre, Jerome de la Lande, and even Adam Smith).  So it seems the story of Boulliau's Conical Hypothesis is worth telling again, albeit briefly.  In addition to the following overview (which will continue to develop 'under construction'), the reader is also directed to other materials at this WebSite, which include original texts by Boulliau and G-A Borelli (translated here from Latin) and to computer-generated illustrations (2D stills, 2D animations & 3D Flash Movies) of the Conical Hypothesis.

 
 

A modern and relatively succinct description of how to construct Boulliau's Conical Hypothesis (according to Boulliau, how to demonstrate elliptical orbits from the 'general circumstances of planetary motion') would be based on several carefully stated assumptions and, not insignificantly, certain prejudices Boulliau obtained early in his career.  Briefly, evidence suggests Boulliau was deeply committed to astronomy and astrology from his earliest teen-age years.  Second, good evidence shows he had accepted the so-called Copernican hypothesis during his early 20's; and finally, overwhelming evidence shows that by his mid-20s Boulliau was struck the mathematical implications of Kepler's work in astronomy and optics. Central to these interests were conic sections. This interest led Boulliau immediately to study the works of Apollonius, François Viete, and his friend (and defender) Claude Mydorge.  In sum, by way of background, Boulliau was a confirmed Copernican, certainly by 1630, and in the next few years he circulated a manuscript version of his second book, the Philolaus (1639). All of this in the immediate wake of Galileo's condemnation.  So much for Boulliau's personal history and private convictions.

More formally, Boulliau betrayed a number of his assumptions about astronomy in his first book on astronomy, the Philolaus (1639).  Here Boulliau made it clear he was a confirmed Copernican, that the Ptolemaic model was entirely unacceptable, the Tychonic model unpersuasive, if not contradictory in principle.  Boulliau's positive arguments were based on simplicity and symmetry, at bottom, geometrical, optical, and aesthetic.  His formal assumptions, described in greater detail in his Astronomia Philolaica (1645), were as follows:

I.  Planets have a simple motion in a simple line.
II.  Planetary revolutions are equal, perpetual, uniform.
III.  They should be regular revolutions or composed of regular revolutions.
IV.  They can only be circular.
V.  Or they are composed of circles.
VI.  Motions should have a principle of equality.
VII.  Because they admit a certain inequality, the center of the zodiac must be the reference point of inequality.
VIII.  This point is in the Sun.
IX.  Half of the inequality is attributed to eccentricity, the other to another cause which makes the planet slower at aphelion, less slow at perihelion, without disturbing the equality of motion or transposing it to some other location, whether the circle or surface.
X.  When the planet, moving from aphelion, comes to quadrature on the same surface with equal motion, it should differ from the apparent motion of the first inequality completely or nearly so; but because the other half [of the inequality] is due to the distance [between] the circles, the center of planetary motion must be between the points of true and apparent motion.
XI.  Since the equal motion in the first quadrant is greater than the apparent motion, that part of apparent motion must be greater; hence, from the first quadrant to perihelion the arc described moving toward perihelion must be larger than the first.
XII.  All revolution is composed of circular parts; the same is true of each part.
XIII.  Equal motion is uniform; thus, the motion in coming from aphelion corresponds to the larger parallel circles, which increase from aphelion to perihelion. This equal motion does not correspond to a single circle but to several unequal circles to which the apparent motion also corresponds; the apparent motion includes all the circles on the same surface. The motion must also be eccentric and inclined.
XIV.  These circles follow one another in a continuous series and are all parallel among themselves; they do not overlap or enclose one another; the apparent motion forms a solid surface containing larger and smaller circles.



 

Finally, having sketched Boulliau's biographical background and his more formal assumptions about planetary motion, it is time to provide a succinct modern account of his Conical Hypothesis.  In brief, the construction goes like this.  It is a model of simplicity in shortest form.
 

Imagine an oblique cone where ABC is a plane section through the axis AI and perpendicular to the plane of the base whose trace is BC.  Let EK be the trace of the cutting plane [an ellipse] which is perpendicular to AB.  It follows that EK will be the major axis of the ellipse ENKO where M is the intersection of EK and AI, and X is the midpoint of EK.  It follows that M will be a focus of the ellipse if and only if angle IMK is equal to angle AIC, or equivalently (provided WRK is parallel to BC) if and only if the triangle MKR is isosceles [Apollonius].  It follows that the eccentricity must be bisected.  Other aspects are implied, for example, the motion of the planet on ellipse ENKO can be considered to be produced by an element of the cone (e.g. AB) rotating about axis AI in the conic surface, its angular motion measured in any circle parallel to the base of the cone being uniform.  The intersection of this line with the inclined plane EK fixes the instantaneous position of the planet.  At any instant the planet is on a circle parallel to the base, its center is on axis AI, and it  moves with uniform angular motion; the planet also moves, at any instant, along ellipse ENKO, its slowest speed is at E [aphelion], it accelerates from E toward K [perihelion] and decelerates from K toward E, the equality and inequality are conjoined.  [Planets move with uniform angular motion around a central point (Plato's Dictum) but they also move on an ellipse where they accelerate toward perihelion and decelerate toward aphelion.]
 


 
If we venture somewhat further back in time, it is possible to uncover a brief description of Boulliau's Conical Hypothesis by Adam Smith.  In his Essays on Philosophical Subjects the author of the Wealth of Nations published a remarkably sophisticated essay on the 'History of Astronomy'.   Here his hope was to uncover and provide examples of persistent problems in science and philosophy, most notably how various thinkers had proceeded in the face of intellectual challenge.

Regarding Boulliau's Conical Hypothesis Adam Smith provided the following summary:

The Planets, according to that astronomer [Boulliau], always revolve in circles; for that being the most perfect figure, it is impossible they should revolve in any other.  No one of them, however, continues to move in any one circle, but is perpetually passing from one to another, through an infinite number of circles, in the course of each revolution; for an ellipse, said he, is an oblique section of a cone, and in a cone, betwixt the vertices of the ellipse there is an infinite number of circles, out of the infinitely small portions of which the elliptical line is compounded.  The Planet, therefore, which moves in this line, is, in every point of it, moving in an infinitely small portion of a certain circle.  The motion of each Planet, too, according to him, was necessarily, for the same reason, perfectly equable.  An equable motion being the most perfect of all motions.  It was not, however, in the elliptical line, that it was equable, but in any one of the circles that were parallel to the base of that cone, by whose section this elliptical line had been formed:  for, if a ray was extended from the Planet to any one of those circles, and carried along by its periodical motion, it would cut off equal portions of that circle in equal times; another most fantastical equalizing circle, supported by no other foundation besides the frivolous connection betwixt a cone and an ellipse, and recommended by nothing but the natural passion for circular orbits nd equable motions.  It may be regarded as the last effort of this passion, and may serve to show the force of that principle which could thus oblige this accurate observer and great improver of the Theory of the Heavens, to adopt so strange an hypothesis.  Such was the difficulty and hesitation with which the followers of Copernicus adopted the corrections of Kepler.

Adam Smith, History of Astronomy, IV.55-57
 


 
 
Finally, the last selection presented here (at least for the time being) is extracted from Theoricae Mediceorum planetarum, ex causis physicis deductae (Florence 1666) by G-A Borelli, a contemporary of Boulliau.  

 
 
[p. 31]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 [p. 32] 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. 
[p. 33]  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. 

 
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Dr Robert A. Hatch - Copyright 1998 - All Rights Reserved

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