Philosophy of Astronomers

To make models of the Ptolemaic and Copernican world systems, the LYM animations team first had to figure out what values they used for their various equants and circles.  On reading parts of Ptolemy's Almagest and Copernicus's De Revolutionibus, some particularly telling sections were spotted, which were copied here, along with the values they used to construct their systems.

Ptolemy

Ptolemy, in Book I of his Almagest gives impeccable reasoning for the earth's being in the center of the heavens: If it weren't, then the clearly observed, daily circular motion of all the fixed stars and the planets around a center would appear irregular to us! How simple: consider the first impressions of your senses -- circular motion -- as an unassailable reality, and then say (truthfully) that you could not move the observer while still maintaining appearances.  Clearly, this man is a genius; he need only assume something to prove it! The basic assumption here is that seeing is believing.  But, it were best to allow Ptolemy to hang himself, rather than doing the work ourselves.  So, from the Master's own pen:

From Ptolemy's The Almagest, I.7:

“And so it also seems to me superfluous to look for the causes of the motion to the center when it is once for all clear from the very appearances that the earth is in the middle of the world and all weights move towards it.  And the easiest and only way to understand this is to see that, once the earth has been proved spherical considered as a whole and in the middle of the universe as we have said, then the tendencies and movements of heavy bodies (I mean here their proper movements) are everywhere and always at right angles to the tangent plane drawn through the falling body's point of contact with the earth's surface.  For because of this it is clear that, if they were not stopped by the earth's surface, they too would go all the way to the center itself, since the straight line drawn to the center of a sphere is always perpendicular to the plane tangent to the sphere's surface at the intersection of that line.”

Kepler disagrees:
     “Heavy bodies (most of all if we establish the earth in the center of the world) are not drawn towards the center of the world qua center of the world, but qua center of a kindred spherical body, namely, the earth.  Consequently, wherever the earth be established, or whithersoever it be carried by its animate faculty, heavy bodies are drawn towards it.
     “If the earth were not round, heavy bodies would not everywhere be drawn in straight lines towards the middle point of the earth, but would be drawn towards different points from different sides.” (New Astronomy, p. 55)
Again, from Ptolemy:

“For thus it seems possible for that which is relatively least to be supported and pressed against from all sides equally and at the same angle by that which is absolutely greatest and homogenous.” (Almagest, I.7)

Kepler:
“It is likewise impossible for heavy bodies to tend towards the center of the world simply because they are seeking to avoid its spherical extremities.  For, compared with their distance from the extremities of the world, the proportional part by which they are removed from the world's center is imperceptible and of no account.  Also, what would be the cause of such antipathy? With how much force and wisdom would havy bodies have to be endowed in order to be able to flee so precisely an enemy surrounding them on all sides? Or what ingenuity would the extemities of the world have to possess in order to pursue their enemy with such exactitude?” (New Astronomy, p.54)
Again, Ptolemy:

“And if it [the earth] had some one common movement, the same as that of the other weights, it would clearly leave them all behind because of its much greater magnitude.  And the animals and other weights would be left hanging in the air, and the earth would very quickly fall out of the heavens.  Merely to conceive such things makes them appear ridiculous.” (Almagest, I.7)

And just as you are about to say: "Well, but surely he cannot object to the earth's spinning:"

     “Now, some people, although they have nothing to oppose to these arguments, agree on something, as they think, more plausible.  And it seems to them there is nothing against their supposing, for insteance, the heavens immobile and the earth as turning on the same axis from west to east very nearly one revolution a day; or that they both should move to some extent, but only on the same axis as we said, and conformably to the overtaking of the one by the other.
     “But it has esscaped their notice that, indeed, as far as the appearances of the stars are concerned, nothing would perhaps keep things from being in accordance with this simpler conjecture, but that in the light of what happens around us in the air such a notion would seem altogether absurd.  For in order for us to grant them what is unnatural in itself, that the lightest and subtlest bodies either do not move at all or no differently from those of contrary nature, while those less light and less subtle bodies in the air are clearly more rapid than all the more terrestrial ones; and to grant that the heaviest and most compact bodies have their proper swift and regular motion, while again these terrestrial bodies are certainly at times not easily moved by anything else -- for us to grant these things, they would have to admit that the earth's turning is the swiftest of absolutely all the movements about it because of its making so great a revolution in a short time, so that all those things that were not at rest on the earth would seem to have a movement contrary to it, and never would a cloud be seen to move toward the east nor anything else that flew or was thrown into the air.  For the earth would always outstrip them in its eastward motion, so that all bodies would seem to be left behind and to move towards the west.”

Ptolemy's results

After jumping ahead in the book, we find Ptolemy's values for the sizes of the eccentric and epicycle, and their relative motions.

“Likewise we find 37 cycles of anomaly of Mars in 79 of our solar years plus 3 + 1/6 + 1/20 days very nearly, and in 42 revolutions of the star plus 3 1/6° of a revolution, counting from a tropic point back to the same.” (Almagest, IX.3)

The time between oppositions of Mars is thus:

[(79 years * 365.25 days / year) + (3 + 1/6 + 1/20) days] / (37 cycles)
= 780 days.

The time of the second inequality-- Mars’ return to the same zodiacal position when effect of the first inequality is excluded, comes out to be

(780 days * 37 cycles) / (42 + 3 1/6°)
~ 687 days.

Section X.7 gives the speed ratios of the eccentric and the epicycle to be:

     135°39' for the eccentric to 171°25' for the epicycle.


We find the ratio of the eccentric to the epicycle to be, in X.8:
     60p to 39p30',

which, in our decimal notation, is:

     1.5190 to 1.

Copernicus

Andrew Osiander wrote a remarkably cowardly short introduction to Copernicus's De Revolutionibus, excerpts of which follow:

“For it is the job of the astronomer to use painstaking and skilled observation in gathering together the history of the celestial movements, and then -- since he cannot by any line of reasoning reach the true causes of these movements -- to think up or construct whatever causes or hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future too.
     “But since for one and the same movement varying hypotheses are proposed from time to time, as eccentricity or epicycle for the movemenet of the sun, the astronomer much prefers to take the one which is easiest to grasp.  Maybe the philosopher demands probability instead; but neither of them will grasp anything certain or hand it on, unless it has been divinely revealed to him.  Therefore let us permit these new hypotheses to make a public appearance among old ones which are themselves no more probable, especially since they are wonderful and easy and bring with them a vast storehouse of learned observations.  And as far as hypotheses go, let no one expect anything in the way of certainty from astronomy, since astronomy can offer us nothing certain, lest, if anyone take as true that which has been constructed for another use, he go away from this discipline a bigger fool than when he came to it.  Farewell.”

Now, to be fair, Copernicus himself did not write this introduction.  What does he say? He does refute the Ptolemaic objections to the motion of the earth, but insists on circular motions, or motions compounded of circular motions:

“We must however confess that these movements are circular or are composed of many circular movements, in that they maintain these irregularities in accordance with a constant law and with fixed periodic returns: and that could not take place, if they were not circular.  For it is only the circle which can bring back what is past and over with; and in this way, for example, the sun by a movement composed of circular movements brings back to us the inequality of days and nights and the four seasons of the year.  Many movements are recognized in that movement, since it is impossible that a simple heavenly body should be moved irregularly by a single sphere.  For that would have to take place either on account of the inconstancy of the motor virtue -- whether by reason of an extrinsic cause or its intrinsic nature -- or on account of the inequality between it and the moved body.  But since the mind shudders at either of these suppositions, and since it is quite unfitting to suppose that such a state of affairs exists among things which are established in the best system, it is agreed that their regular movements appear to us as irregular, whether on account of their circles having different poles or even because the earth is not at the center of the circles in which they revolve.”  (I.4 -- emphasis added)

While removing one wrong astronomical axiom, geocentrism, Copernicus solidly sticks with axiomiatic, mathematical-geometrical thinking.  Copernicus is a mathematician, not a physicist, and is trapped in looking at geometrical (e.g. uniform circular) motion.  He allows the earth to move, but "shudders" at anything non-geometric (e.g. physical) moving it.

Copernicus's results

Copernicus gives the ratios of the orbs of earth and Mars as:
     1p to 1p31'11",
which, in our decimal notation, is:
     1 : 1.5197 for Earth : Mars,
precisely the same as Ptolemy's eccentric - epicycle ratio.  (Ptolemy's epicycle is, in reality, the sun's motion.)  Now, we can use these values in making models of the systems.