In Honor of Mike Flynn: The Great Ptolemaic Smackdown Part I

This may be a breach of internet protocol, but I wish to reprint and spread this column by my dearly beloved friend, fellow writer, and staunch Catholic, Mike Flynn. From here.

For Part II, see here.

The Great Ptolemaic Smackdown (Part I)

So, it occurs to TOF that a much-altered version including the artwork might not be amiss.  In particular, because it is often believed that the opposition to heliocentrism was religiously-inspired, a bit of attention to You-Know-Who(*) is warranted.  First, a bit of background.

(*) You-Know-Who.  Galileo.  You knew that, right?

1. Our Ancestors Were Stoopid

Before you laugh at your ancestors, TOF invites you to prove that the earth is, contrary to your senses, in wild and careening double motion: spinning like a top and whipping around the sun without (somehow) leaving the Moon and Air behind, and without everyone stumbling around like dunkards.  You are not allowed to appeal to authority or to the success of NASA, or suchlike things.  You’ve got eyeballs and armillaries, and that’s pretty much it.  Go. TOF will wait here.

2. Sundry Proofs of the Stationary Earth

Astonishingly, Late Moderns, who hold heliocentrism as a sort of holy doctrine, are generally unaware of the empirical evidences that would justify it; while Early Moderns, who thought geocentrism dough-face obvious, were well aware of the evidences that falsified heliocentrism. These evidences, plucked variously from Aristotle, Oresme, and Riccoli follow; but be it noted that both Oresme and Riccoli also supplied rebuttals for most of them and Aristotle cautioned against taking his cosmology as more certain than he himself did:

“We are far away from the things we are trying to inquire into, not only in place but more so in that we have sensation of exceedingly few of their accidents. ….  It is good to inquire about these things and so to deepen our understanding, although we have little to go on and we are situated at such a great distance from the attributes of these things.  Nevertheless, from contemplating such things nothing [we infer] should seem to be unreasonable, holding them now as fraught with difficulties.– Aristotle, On the heavens, 2.3.286a5-7 and 2.12.292a14-18

2.1 The Argument of the Winds.

If the Earth is rotating, “we and the trees and houses are moved toward the east very swiftly, and so it should seem that the air and wind blow continuously and strongly from the east, much as it does against a quarrel shot, only much more strongly.…. But the contrary appears by experience.”  Therefore, the earth does not turn.  (Oresme, “On the book of the heavens and the world by Aristotle.”)

2.2 The Argument of the Arrow.

“If a person is on a ship moved rapidly eastward and an arrow were shot directly upward, it ought not to fall on the ship but a good distance westward from the ship. Similarly, if the earth is moved so swiftly in turning from west to east, and it has been posited that one throws a stone directly above, then it ought to fall, not on the place it left, but a good distance to the west. But in fact the contrary is clear.”  (Oresme, “On the book of the heavens and the world by Aristotle.”)

2.3 The Argument from Coriolis

If the Earth is rotating objects at the tops of tall buildings will be moving eastward at a higher velocity than those at the base of the building.  Therefore, an object dropped from the top of a tower will fall east of the plumb line.  No such deflection is observed.  (This argument was unknown to Aristotle and the medievals, but was included in Riccioli’s Almagestum novum, summarized by Graney in “126 Arguments Concerning the Motion of the Earth.”

2.4. The Argument from Parallax.

“The earth, then, also, whether it move about the center or as stationary at it, must necessarily move with two motions. But if this were so, there would have to be passings and turnings of the fixed stars. Yet no such thing is observed. The same stars always rise and set in the same parts of the earth.” (Aristotle, On the heavens, Book II, part 14)

Parallax.  From one end of the year to the other, nearer stars will appear to shift against the background of more distant.  In January, A and B appear far apart in the western sky.  In July, they appear close together in the eastern sky.

2.5 The Weird Argument from Motion.

“The lowest place belongs to the heaviest and lowest of bodies. The Earth is the heaviest body. The center of the world [universe] is the lowest place. Thus Earth lies at the center of the world. …  If Earth were shifted towards the moon, heavy bodies would still tend toward the center of the world, not towards the Earth.  (So if the sun were at the center of the world, all heavy objects would naturally fall toward the sun, which is contrary to experience.)   (Summarized by Graney in “126 Arguments Concerning the Motion of the Earth.”)

2.6  The Argument from Heavenly Motions.

“We see with our senses the sun and moon and many stars rise and set from day to day, and some stars turn around the arctic pole. This could not be except by the movement of the heavens.”  (Oresme, “On the book of the heavens and the world by Aristotle.”)

IOW, all the empirical evidence seemed to be against a mobile earth and in favor of a stationary earth.

So what were the arguments in favor of geomobility?

3. Mystical Woo-woo

But… didn’t Aristarchus and the Pythagoreans propose heliocentrism in ancient times?  If only they had prevailed, we might have had Real Science™ millennia sooner!  We’d be on freaking Mars by now!  What was their evidence?

Well, you see, Fire is nobler than earth and the center is a nobler position.  So fire has to be in the center.  QED.

There are many names for this sort of thinking, but “scientific” is not one of them.  Aristotle says of the Pythagoreans:

In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own.

 – Aristotle, On the heavens II.13.293a

Today, we have answers to the objections listed above, many of them developed in the Middle Ages; but those answers depend on the most part on measurements and concepts that were not then available: force, mass, inertia, etc.  For example, Oresme answered the Argument of the Winds by postulating common motion: the sphere of the air is also moving to the east along with the sphere of the earth.  But he had no propter quid to explain why the air and the earth shared a common motion.  You can’t just say that if only A and B were true, then observation C would follow.  You actually have to show that A and B are true.  The Pope said as much to Galileo, and got mocked for his pains.

Oresme did undermine the one positive argument for a stationary earth when he appealed to relativity:

In the fourth book of The Perspective of Witelo, [he says] that one can perceive movement only in such a way as one perceives one body to be differently disposed in comparison with another. I say, then, that if the lower of the two parts of the cosmos … should today move with a diurnal movements while the upper (that is, the heavens) should not, we could not perceive this change in any way, but everything would seem the same today and tomorrow. It would seem to us continually that the part where we are situated was at rest and that the other part was always moved, just as it seems to a person who is in a moving ship that the trees outside are moved. Similarly, if a person were in the heavens and it were posited that they were moved with a diurnal movement, and [furthermore] that this man who is transported with the heaven could see the earth clearly and distinctly and its mountains, valleys, rivers, towns, and chateaux, it would seem to him that the earth was moved with a diurnal movement, just as it seems to us on the earth that the heavens move. Similarly, if the earth and not the heavens were moved with a diurnal movement, it would seem to us that the earth was at rest and the heavens moved. This can be imagined easily by anyone with good intelligence. For this [reasoning] is evident the response to the [apparent motion of the heavens], since one could say that the sun and the stars appear thus to set and rise and the heavens to turn as the result of the movement of earth and its elements where we are situated. 

In short, motion is relative to the inertial reference frame in which the observer is situated.  Oh, them unscientific medieval dark agers!

There were similar rebuttals to most of the arguments: plausible-sounding but unevidenced.  The most serious objections were those dealing with Coriolis-like effects and with parallax.

The Copernicans answered the Argument from Parallax by claiming that the stars are much farther away and the parallax is therefore too small to detect.  But you cannot save one unproven hypothesis by adding a second unproven hypothesis to it, and there were sound scientific reasons for supposing the stars to be closer.

4. Another Fine Math You’ve Gotten Us Into — the 1500s

In the question of mobile vs. stationary earth, you have a 50/50 chance of guessing right.  But if science were no more than lucky guesses, we’d credit Jonathon Swift with discovering the moons of Mars.  There were only two practical reasons for studying the heavens:

  • to prepare calendars and
  • to cast horoscopes.

Later, oceanic navigation became important.  In China, astronomers were called “calendar-makers” and in Europe, one was called a “mathematicus.”  You need more than a “designated center,” you need a complete mathematical model.  Aristarchus, so far as we know, did not have a mathematical system for his heliocentrism.  But Claudius Ptolemaeus perfected such a model for the geocentric theory with Syntaxis Mathematiké (a.k.a. The Almagest) – plus the Tetrabiblos on astrology.  It was an awesome accomplishment, and his system for recording the positions and movements of the stars (right ascension and declination) is still in use today.  His mathematics did an excellent job of predicting when and where celestial phenomena would take place.  So, calendars could be made, and the fates of kings predicted.

One wee problem.  It conflicted with Aristotelian physics.

You see, the Ptolemaic model was not strictly geocentric.  Each planet was embedded in an orb made of dark matter (aether) which carried it along a circular path called a deferent.

  • To account for the changes in speed that we now associate with Kepler’s equal area law, the deferent was centered not on the Earth but on a point halfway between the Earth and an imaginary locus called an equant.
  • To account for retrograde motion and changes in size and brightness, planets moved on a second circle called an epicycle riding along the deferent.

Unwittingly, Ptolemy seems to be prodding circles to act like ellipses, with the Earth and the equant acting as the foci.

Reading the orbituary.  The deferent is to account for changes in speed.
The epicycle is to account for retrograde motion and changes in size and brightness.

Since each planet was solved as a separate problem, each orbit had a different center! But the mathematicians weren’t trying to formulate an overarching physical theory.  They were only trying to figure out when the next eclipse would be, or the date of Easter, or (in China) lucky and unlucky days for sundry activities.

These mathematical devices really, really bugged the physicists.  In Aristotelian physics all the orbs are homocentric on the Earth.  The orbs were like nested ball bearings made of aether, which carried the planets within them.  There was no room for such foo-foo as epicycles; and no philosophical justification for the @#$%* equants!  (Æther — from aei thein: “always running” — also explains why the Michelson-Morley experiment failed.  Earth doesn’t fly through the aether; it is carried along within it.)

The astronomers’ only excuse was that their calculations worked.  The physicists groused, “Sure, in practice, but do they work in theory?”

There was only one solution.

5. Astronomy is not Physics

More than any other unexamined assumption, this one startles us moderns.  Except for the Sun and Moon, inexplicably of the same apparent size, the planets and other stars are little more than dots in the sky.  “We sense little of the heavens,” said Aristotle, except luminous stars and planets, and perceive none of their properly sensible attributes (color, smell, sound) and even the common sensibles (magnitude and motion) are difficult to sense without the propers. (De Anima, 2.6)

So astronomy was not about making physical discoveries about physical bodies in the sky.  It was a specialized branch of mathematics for making predictions about sky events.  (And now you know why Osiander added that unauthorized foreword to Copernicus’ book.)  Of course, making correct predictions does not mean a theory is physically true.  Aristotle was seconded in this by Thomas Aquinas, when he wrote:

“The suppositions that these astronomers have invented need not necessarily be true; for perhaps the phenomena of the stars are explicable on some other plan not yet discovered by men.”
— De coelo
, II, lect. 17

Anticipating Duhem and Quine, Thomas also noted the underdetermination of science regarding astronomical models:

“The theory of eccentrics and epicycles is considered as established because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.”

– Summa theologica, I, q.32, a.1, ad. 2

Copernicus would later supply “some other theory” to explain the same appearances.  Too bad he wasn’t around a couple hundred years earlier.

6. The Return of Mystical Woo-woo — the 1530s

Ptolemy’s math worked fine for a thousand years.  It was “settled science,” as we say today.  But gradually, as the star tables were copied and recopied, copyist errors crept in and multiplied like loaves and fishes.  This was not due to bugs in the Ptolemaic model, but to errors in the data itself.

The Italian Renaissance was a humanist reaction against Aristotelian obsessions with logic, reason, and natural philosophy.  Greco-Roman art and literature were “rediscovered.”  Platonic mysticism was revived, along with astrology, magic, Hermeticism, and Pythagoreanism.  Natural science faltered*; but since astronomy was only mathematics, it prospered.

(*) faltered.  Did someone say ‘DaVinci’?  He was an artisan engineer.
You need more than sketches for SF cover art to be a scientist.

In particular, the humanists were unhappy with the Earth’s position in the scheme of things.  She was in the bottom of the world, farther from Heaven than anything but Hell itself.  Hell was situated in the center of the Earth, which was assumed therefore to be incredibly hot.  The humanists wanted to raise the position of the Earth (and hence, of humans) by elevating her into the heavens.  Since the Ptolemaic system was beginning to falter from those accumulative copyist errors, they decided to make fresh new accurate observations of the heavens.

Ha-ha.  TOF is joking, of course.  They were humanists!  No, they decided instead to turn the entire universe inside out.

Nicholas Copernicus, a canon lawyer at Frauenberg cathedral, was a medical practitioner, a financial advisor, and was once shortlisted for the bishop’s seat.  He was also a gifted “mathematicus.”   Hey, he was a Renaissance Man™.  But he was not a scientist in our modern sense.  He made few empirical observations, instead doing new math on existing data: viz., Peuerbach’s Epitome in Almagestum and Gerard of Cremona’s 12th century Latin translation of the Almagest.  He defended heliocentrism by quoting Hermes Trismegistus, “a nonexistent propagator of more woo than you pack into an articulated truck.”  And he wanted to save the Platonic axiom of purely circular orbits by ridding the World of those @#$% equants.

Cardinal Nicolaus von Schönberg and Bishop Giese urged him to publish, but he had already been satirized on the stage and dreaded the mockery of those who “on account of their natural stupidity hold the position among philosophers as drones among bees.”(*)   De revolutionibus caused great excitement among mathematicians when it appeared.  However, the enthusiasm quickly dried up; and for a reason that startles us Moderns.

(*) drones among bees.  From the Dedication of De revolutionibus. 
The Renaissance replaced the syllogism with the witty insult.

The new system was no improvement.

7. The Copernican Flop

It’s not enough for a new model to equal the standard model in predicting phenomena; it must do better.  Otherwise, why bother changing?  And the Copernican model did not do that.  Nor were its calculations simpler.  To preserve pure Platonic circles, Copernicus used twice as many epicycles as Peuerbach’s then-current edition of Ptolemy!  That’s right: epicycles.  The Earth revolved around the Sun on two circles; the Moon ran on an unprecedented double epicycle, and Mercury librated idiosyncratically across the center of an epicycle!  Try explaining that with a theory of universal gravitation!

Technically, Copernicanism wasn’t even heliocentric: The Sun was off-center, and planetary motions were referenced to the center of the Earth’s orbit instead.  And because each planet was solved as a separate problem, each planet orbits a different center!

Ptolemy vs. Copernicus.  The Copernican model (right) is not notably simpler than the Ptolemaic model (left).  It uses more epicycles; the Sun– like Ptolemy’s Earth – is off-center; and each planet’s orbit has a different center.  Note also the double epicycle for the Copernican Moon and the curiosity that, for Mercury, Venus, and Earth, their orbital centers run around epicycles!. Image after (De Santillana 1955)

At least he got rid of those @#$% equants.

There were two reasons for the epic fail of the Copernican model:

  • Copernicus insisted on pure Platonic circles; and
  • Accumulated copyist errors in the Alfonsine Tables carried into his Prussian Tables.

What a let-down.  If only the data were better!

8. Enter Tycho Brahe — 1570s

Irritated by both models, Tycho Brahe set out to do what the Copernican humanists should have done:  gather new, precise data.  Tycho designed and (more importantly) calibrated new instruments and compiled meticulous observations with errors as small as the width of a quarter seen from a football field away.

And produced a geo-heliocentric system regarded today as a kludge.

How could the greatest astronomer of his age be so stoopid?

Tycho’s updated Heraclidean model.

Like any good scientist, Tycho followed the data.  (Graney, 2012)

  1. Procyon has the same diameter and brightness as Saturn.
  2. If Procyon is much farther than (say) 100 times Saturn’s distance, simple geometry proves its actual size would dwarf the sun.
  3. All the stars would dwarf the Sun, which would then be the only pea in a universe of melons, which is absurd.
  4. But if Procyon were any closer, there would be visible parallax from the Earth’s revolution.
  5. There is no visible parallax.
  6. Lack of parallax plus the apparent size of the stars therefore requires a stationary Earth. QED.
Geometric optics.  The farther an object is, the bigger it must be to present
a disk of a given size.

Did anyone notice the pea under the thimble?  Go ahead, think on’t.  TOF will wait.

Because Tycho otherwise admired Copernicus’ treatment but empirical data and solid science forced him to conclude that the Earth was stationary, his solution was an updated Heraclidean model: all planets circling the Sun, but the Sun and Moon circling the Earth!

The Imperial Mathematician, Nicolai Reymers Bär, styled “Ursus,” proposed a similar model – but with a rotating Earth.  Tycho accused him of plagiarizing his data and a feud developed that caught a young math teacher named Johann Kepler smack in the middle.

Like many new authors, Kepler had sent copies of his book (Mysterium Cosmographicum, “a strange Renaissance piece of Platonic Pythagorean mathematical mysticism”)  to famous people, including Tycho and Ursus, hoping for blurbs.  Ursus used Kepler’s thank-you note to make it seem that Kepler favored the Ursine over the Tychonic model.  Since Kepler was asking Tycho for a job at the time, this caused some embarrassment.

Kepler also received an unsolicited fan letter from an unknown math professor at Padua who had got hold of a copy of the book, and whose name amused Kepler.  His forename and family name are the same! Kepler wrote to a friend.  “Galileus Galileus.”  In Italian, Galileo Galilei.

Tycho’s great kludge was thus the best model that accounted for the data.  Further, it was mathematically equivalent to the Copernican model.  Anything the one model could do the other model could do as well.  Even today, however, there are those who believe that Tycho was living in fear of the all-powerful Church — even though he lived in Lutheran Denmark and had the favor of the king, who showered money on his Uraniborg observatory proportionate to the kingdom’s budget as NASA was to the US budget.  After he and the king quarreled, Tycho moved to Prague, where the Catholic Emperor made him Imperial Mathematician, succeeding Ursus.  He died of a burst bladder in…

Hey!  Wait a gol-danged minute!  (TOF hears you say)  Procyon doesn’t have a disk!

9. Airy Abstractions: a brief excursion into the future

That’s right.  The apparent disks of the stars are caused by the diffraction pattern resulting from a uniformly-illuminated circular aperture.  The bright region in the center is known as the Airy disk, not because they are caused by the air, but because they were identified and studied by George Biddell Airy…

…In 1835.  So we can’t blame ol’ Tycho for not realizing that the diameters of stars were optical illusions.  In an 1828 article for Encyclopedia Metropolitana, astronomer John Herschel described the appearance of a bright star seen through a telescope under high magnification:

Herschel’s star.  Even with 19th cent. telescopes, stars appeared as disks.

…the star is then seen (in favourable circumstances of tranquil atmosphere, uniform temperature, &c.) as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders. They succeed each other nearly at equal intervals round the central disc….

Once the illusory nature of Airy disks was realized, the stars could be as distant as you might like.  Their diameters were not real and implied nothing about their sizes.  But no one will know this for another quarter millennium.

10. The Last Hurrah of Eyeball Astronomy — the 1600s

Copernicus had given the Earth three distinct motions, which struck many as excessive.  Heck, even one motion struck many as excessive.  Some, like Ursus, accepted the rotation of the Earth, at least as a mathematical gimmick.  Two motions, throwing in the revolution around the sun, seemed pushing things.  But three was way over the top.

Remember, Nick was using bad data.  He had made very few actual observations and was simply reworking published tables.  Tycho was able to show with his pristine and precise new data that the “trepidations of the equinoxes” was simply observational error.

Now for the orbit of Mars, which Nicky had really messed up.  The sucker was way out of position; so either Copernicus was wrong (falsified) or Mars was wrong.  Tycho hired Kepler to fix the orbit of Mars; though, remembering Ursus, he would not allow Kepler to make copies of the double-plus Secret Data of Uraniborg.  He had to use them on-site.  But in 1601, Kepler succeeded Tycho as Imperial Mathematician, and negotiated better access to the data with Longomontanus, Tycho’s heir and long-time assistant.

Kepler worked Mars in the Ptolemaic, Copernican, and Tychonic models, and none of them gave a good account.  He then assumed (as Copernicus had not) that all orbital planes passed through the Sun, which reduced the error to eight or nine arc minutes.  Still not good enough.  He even tried re-introducing the @#$%^; equants, though his heart wasn’t in it.  (cf. Crombie 1959, .pp. 176-182.)

A Neoplatonic mystic, Kepler was convinced that physics must reduce to simple mathematical forms, but he was more liberal than either Copernicus or Galileo.  He began to try ovals.

This bugged Longomontanus, who accused Kepler of shoveling shit and gave the world a colorful phrase to replace “your argument is lacking in merit.”   In a letter of 6 May 1604, he told Kepler he was “submerged in shit in the Augean stable of old…”  Kepler replied in early 1605:

If you are angry that I cannot eliminate the oval path, how much more ought you to be angry with the spirals [epicycles], which I abolished.  …  This is like being punished for leaving behind one barrow full of shit although I have cleaned the rest of the Augean stables.

Scientists have been accusing one another of shoveling it ever since.

The mathematical difficulty of generating ovals had led Kepler to complain earlier that year (4 July 1603) to David Fabricius: “I lack something: knowledge of the geometrical generation of the oval path …  If the figure were a perfect ellipse…!”

Yeah, if only.

Kepler decided to chuck two basics of physics:

  • the motion of heavenly bodies is uniform 
  • the motion of heavenly bodies is circular  
 An elliptical comment.

He wondered if the reason why Mars seemed to speed up or slow down was that – wait for it – it was speeding up and slowing down, and not moving uniformly around a circular epicycle riding along a circular (but off-center) deferent.  This almost worked.

In 1604 he gave up Platonic circles.

He was able to show geometrically that movement along an ellipse was mathematically equivalent to movement along an epicycle on a deferent.  Shazaam! – the Martian orbit suddenly made sense!  And btw, the other focus of the ellipse does kinda sorta look like that @#$^&;% equant…

Without Tycho’s precise new data, Kepler would never have found his ellipse.  No one before Tycho could possibly have done so.  The old tables were just too badly corrupted. The interesting corollary to this is that as soon as it was possible to do so, European astronomers discovered elliptical orbits. 

Kepler wrote this up in Astronomia nova (1609).  Since he believed mathematics caused physics, he decided that there must be a universal cause of planetary motions: the Sun projected a field, which by rotating would chivvy the planets around their ellipses with an impetus inversely proportional to their distance.  Okay, you can’t get everything right; but this prepared the way for Newton.  Kepler thought the field was the Holy Spirit, which proceeded from the Father (the Sun) toward the Son (the fixed stars).  This did not prepare the way for Newton.

Game, set, and match, dudes and dudettes!  Kepler had the correct mathematical answer before the telescope was invented.  Hear the applause?

That’s the sound of one hand clapping while a tree falls in the forest.  In later years, Kepler will admit ruefully that he tried to read his own book once and couldn’t make heads or tails out of it.  As a writer, his scintillating prose can be summarized as “WTF?”

Besides, Kepler had only Platonic number mysticism to back himself up.  He had neither empirical evidence nor physical theory that his model was physically true.  It predicted the heavens; but then, so did the Ursine/Tychonic system. And the latter explained the lack of parallax and Coriolis directly, without making additional assumptions.

Kepler sent a copy of Astronomia nova to Galileo – but Galileo never read it.


So what happened next? (TOF hears you ask).

A troll and glory hog named Galileo swooped in, claimed credit for everything in sight, and delayed acceptance of heliocentrism for the rest of his lifetime.  But that is a story for another time; viz.,


Aristotle. On the Heavens.
Christie, Thony. But it doesn’t move! June 22, 2011.
Copernicus, Nicholas; Charles Wallis (trans). On the Revolutions of the Heavenly Spheres.
Crombie, A. C. Medieval and Early Modern Science, vol. II. Garden City, NU: Doubleday Anchor, 1959.
Flynn, Michael.  The Great Ptolemaic Smackdown and Down ‘n Dirty Mud Wrassle.  Analog (Jan/Feb 2013)
Franklin, James.  “The Renaissance Myth”  Quadrant 26 (11) (Nov. 1982), pp. 51-60
Graney, Christopher M. 126 Arguments Concerning the Motion of the Earth. Mar. 14, 2011.
—. Tycho was a scientist, not a blunderer. Mar. 6, 2012.
Oresme, Nicholas. On the Book of the Heavens and the World by Aristotle. Feb. 1999.
Osiander, Andreas.  Foreword to Copernicus’ Revolutionibus.  unsigned.
Ptolemy, Claudius. Syntaxis Mathematiké. In The Great Books Series. Chicago: Encyclopedia Britannica/Univ. of Chicago, 1952.
Thomas Aquinas.  De coeloII, lect. 17
—.  Summa theologicaI, q.32, a.1, ad. 2
Wikipedia.  Airy disks and patterns.