Early next year in the night sky, Mars will glow like a brilliant ember. As it nears earth for its January opposition, I thought I’d try to step into the ancient sandals of Alexandrian astronomer and mathematician Claudius Ptolemy who observed Mars’s return nineteen centuries ago. I wanted to get a flavor of how ancient peoples experienced the paradox of the seeming *mechanical* regularity of the stars (a word that encompassed the moon and planets) in the unfathomable heavens.

Mars’s regular visitations have long been part of human history and scientific curiosity. It was the subject of my article in the August 2021 issue of *Sky & Telescope* magazine about Mars's as seen through the eyes of the keen-eyed Babylonian observers. It's return was regarded as a maleficent, red-eyed omen, yet the Babylonians discovered how to predict its returns, partly taming the demon to reason. Centuries later Ptolemy sought to model its near-far, fast-slow motion across the sky with the geometry of Euclid and Apollonius. Ptolemy was a mathematician, but he was also an occasional observer. He used an astrolabe to sight planetary positions.

Ptolemy observed Mars at its oppositions in 130, 135 and 139 AD [1]. He gave an account of his observations in his *Almagest*, that great work of antiquity which was completed about a decade later [2]. We’ll focus on the 139 opposition as an example of his method. In May of that year, Mars was in the constellation of Sagittarius, not far from the moon, just three days past opposition (from the sun). Ptolemy’s astrolabe measurements of Mars together with ancient Babylonian records allowed him to determine observational parameters for his mathematical model of the planet.

Ptolemy’s sighted Mars on May 30, 139 at three hours before midnight, Alexandria time. He recorded it in the *Almagest *[3]:

*We took an observation which we obtained by sighting with the astrolabe about three days after the third opposition, that is, in the second year of Antoninus, Epiphi XI 15/16 in the Egyptian calendar [139 May 30/31), 3 equinoctial hours before midnight [4].*

We need not be an expert on ancient calendar systems to use this information [5]. Using *Stellarium* planetarium freeware for example, anyone can see that Mars was indeed a few degrees “to the rear of the moon’s center” as Ptolemy described it a little after, and that it had come to its opposition standstill point on the nights of May 26-27, 139 AD.

*Computers and the Almagest*

Harvard historian of ancient astronomy Owen Gingerich wrote about this particular observation in his excellent 1993 book, *The Eye of Heaven: Ptolemy, Copernicus, Kepler *[6].He had access to the computers of that era, and used a computer to check Ptolemy’s data and conclusions. With the remarkable changes in technology since then, anyone now can use a laptop, online databases, and software to relive these ancient events. I have used some of these materials, together with scholarly resources such as Otto Neugebauer’s monumental *History of Ancient Mathematical Astronomy *[7] to evaluate Ptolemy’s Mars data and to help visualize how his complex mathematical model of Mars worked. Whereas many will impatiently cast the *Almagest* aside as far too complex for understanding by any but specialists, we find that with these tools, new doors of accessibility are opened. In particular, I want to introduce you to the ingenious *Almagest Calculator* developed by Robert van Gent at the Mathematical Institute of Utrecht University in the Netherlands [8]. I have tapped it as a quick access tool to investigate Ptolemy’s mathematical methodology in a way that would be far too burdensome on the reader (and me) without it. It will also guide you to Ptolemy’s calendar dates in modern terminology [9]. And I’ve used simulations I’ve written with Maple mathematical software and configured with NASA’s JPL/Horizon parameters for ancient eras; and ephemeris software *Solex* developed by Aldo Vitagliano of the University of Naples, Italy [10]. The main benefit of having a variety of programs is cross-checking. When they all give you the same answer, you know you’re on the right track!

With these preliminaries out of the way, let’s go back 1,885 years in time to the rooftop of the Museum by the Great Library of Alexandria, Egypt – that great repository of learning in the ancient world – and sit with Claudius Ptolemy as he explains the challenges he faced in modelling Mars.

*Ptolemy’s tasks*

Ptolemy had two tasks for his project to succeed. First, he had to confirm the *mean motions* of all the bodies in the solar system, sun, moon, and planets, on the assumption of their uniform, circular, clock-like motions. To better understand what that means, imagine a planet turning (around the earth) like the hand on a clock. If you have a starting time – called the *epoch* – and you know the planet’s mean (average) motion (i.e., one revolution per hour, day, year . . .etc.), you can spin the hypothetical clock forward or backwards and get a decent idea where it would be now, then, or whenever was your chosen target date. For this task Ptolemy relied on Babylonian records that went back centuries. Indeed, Ptolemy’s starting epoch for planetary mean motions was the first month (Thoth, in the Egyptian calendar used by Ptolemy) of the first year of the reign of King Nabonassar of Babylon, in -746 BC!

Spinning the “Mars clock” forward to May 30, 139 meant adding 885 years to the epoch position of 3⁰ 32’ longitude, measured from the vernal equinox [11]. This data and tables of computations are easily found using “Mars’ Mean Motion tables” in Book IX.4 of the *Almagest*, or you can use the *Almagest Calculator*. Once you have your result (I get 252⁰ 40’ mean longitude, both from the *Almagest* and the *Almagest Calculator*), you must then add in precession (the slow drift of the background stars discovered by Hipparchus) which was assumed to be a 1⁰ per century movement of the Martian apogee. The apogee has its own longitude at epoch (found in the Mars Apogee tables of XI.11 of the *Almagest*),* *to which the accumulated precession must be added – all values of which are also automatically generated for you by the *Almagest Calculator*. It is clear from the *Almagest* what an immeasurable head-start these ancient observations gave Ptolemy! Ptolemy supplemented them with a smattering of his own observations.

His second, and trickier task was to ascertain *deviations* from mean motion. What variations – *anomalies* they’re called – make their motions non-uniform? That is, what further additions or subtractions must then be made to mean motion account for the anomalies? This in a way has been astronomy’s central problem for over two millennia. Why, at 9 PM, May 30, 139, wasn’t the planet *exactly* where its mean motion predicted it would be, at 252⁰ 40’ mean longitude? Why, after a certain number of turns, a clock that was supposed to point precisely to 3:14 PM actually reads 3:21 PM? Or why is it behind by 43 seconds on some days and ahead 27 seconds on others? All planets, and most noticeably Mars, exhibited these sorts of irregularities. Adjusting for them required the elaborate machinery of Ptolemy’s models, and more tables, to tweak and shift positions subtly to correct the observed anomalies.

*Going in circles*

Ptolemy, like everyone else then and for centuries after, believed that the sun, moon, and planets all orbited the earth uniformly in *circles*. The “axiom of astronomy,” as it was traditionally called, was that all celestial bodies moved circularly and regularly around our stationary earth. But planetary movements do not always *appear* circular. Mars, in particular, created hair-pulling problems for those who actually observed the heavens and tried to model planetary motions mathematically [12].

For example, if Mars moves in a circle around the earth, why does it appear bright as a beacon in some years and dim in others? If it were moving in a circle around the earth, shouldn’t it be evenly bright throughout its orbit? Another problem known to Ptolemy, and for anyone who watched the skies carefully (as the Babylonians did and discovered) was that Mars moved about forty percent faster when it was moving through the constellation of Capricorn than it did on its slower sojourn through the constellation of Cancer. Such acute deviations from uniform circular motion were hard to ignore! How could a model that was based entirely on circular motion deal with that? Even the idea of an apogee or perigee makes no sense in perfectly circular orbits because such a planet would always be equidistant from the earth.

Finally, and no less perplexing, was the curious matter of retrograde motion. Mars (as with the other outer planets) moves patiently eastward across the sky but as it nears opposition from the sun, it slows, stops, reverses course, then halts and resumes its easterly travels, tracing an elegant loop in the sky like the practiced stroke of a calligrapher’s pen. Why, if the planet is moving around the earth in a perfect, eternal circle, would it ever do that?

The answer to all of these questions, of course, is that Mars, like the other planets, orbits not the earth but the sun, and moves not in a circle but an *ellipse*: As with the other planets, Mars’s speed increases at perihelion (closest to the sun) and decreases at aphelion (farthest from the sun). This seminal fact announced by Kepler in 1609 revolutionized astronomy. We also know that retrograde motion occurs when the earth overtakes Mars in its orbit and Mars *appears* for a while to go backwards as we pass it. (Spoiler: Mars doesn’t really *ever* go backwards.) From an earthly perspective, Mars over time exhibits curls and loops, as illustrated in this woodcut image from Kepler's 1609 book, *Astonomia Nova*,* * of the planet's path from 1580 to 1596:

*Babylon revisited*

Ptolemy’s model had to resolve the anomalies in planetary movements, but also and crucially, satisfy the *Babylonian period relations *[13]. These were the mathematical model’s anchors to the real happenings in the sky. Discerned during centuries of observation, the so-called Goal-Year periods told when a planet would return to the same place in the sky [14]. Mars’s period of revolution is 686.98 days; the interval needed for Mars to move from opposition (opposite the sun) to opposition is 779.93 days, called Mars’s *synodic period* [15]. This means *Mars completes 37 synodic occurrences in 42 orbital revolutions of that planet over the course of 79 years *[16]*.* This statement is an instance of the most important planetary equation of antiquity: “Thus it holds for an outer planet [according to antiquities scholar Otto Neugebauer:] the number of occurrences + number of sidereal rotations = number of years. This is the well-known relation which underlies the Greek epicyclic models for the outer planets” [17]. These Babylonian empirical criteria were as critical to Ptolemy as Tycho Brahe's Mars data were to Johannes Kepler.

Mars advances about a sign and a half along the twelve signs of the zodiac each synodic period. That’s about 48.72° of arc along the ecliptic, a measure called the planet’s *mean synodic arc*. Recovered tablets from Uruk, in sexagesimal notation, show the same value: 48;43,18,30 (48.72⁰) [18]. This is the *mean* arc, and the actual arc will vary plus or minus from that because Mars’s orbit is not circular but (unknown to the ancients) elliptical.

In counterpoint to Ptolemy’s philosophical constraints of uniform circular motion around the center, therefore, were these empirically derived relations particular to each planet. After accounting for non-substantial errors in them (small deviations from purely integer relationships), Ptolemy needed to make sure, to a first approximation, that every planetary model conformed to the empirical constraints. These would be refined by the working model.

*Ptolemy’s model*

I generated the image below with *Maple* mathematical software using Ptolemy’s *Almagest* data for May 30, 139, confirmed by the *Almagest Calculator*. I exaggerated the distances between the three central points and the small circle is not at scale. The picture below shows Ptolemy’s geometrical depiction of the Martian movements – it’s difficult to say it’s an “orbit” since there are really two orbits involved:

Earth is at E, Mars at M, and the Sun at S. The vernal equinox off to the right is along the usual *x* axis of geometry, and is the starting point for angular measure going counterclockwise, the direction of planetary motion. A is for apogee, the farthest point of the orbit from earth E. The P is for perigee, the closest point to earth. The large blue circle became known as the “deferent” (from Latin, *ferrre*, “to carry”) but I’ll call it the “main orbit.” It carries the smaller circle, the “epicycle,” once around the main orbit every 686.98 days, or 1.88 Earth years, which is Mars’s orbital period. The blue square marks the center of the large main orbit, but the actual center of revolution of the whole shebang is the point Eq, called the “equant.” The equant line from Eq to Ep sweeps the epicycle around on the main orbit with *uniform circular motion*, thus (potentially, theoretically) preserving the axiom of circular motion. The beige circle is the equant circle. The equant was one of Ptolemy’s most controversial inventions *since it was not at the center* (critics said the center of revolution should have been at E, of course). The equant functions as a regulator of uniform motion, to keep the mélange philosophically honest (or at least half-honest). But Ptolemy the scientist saw that offsetting the center of revolution to the equant *greatly* improved the accuracy of his model.

Mars itself is carried counterclockwise steadily around on the rim of its epicycle; that motion is called its *mean motion in anomaly*. The planet goes once around the epicycle every 779.93 days, the mean Martian synodic cycle. Mars is on the inside portion of its epicycle during oppositions, in line with the Earth and Sun – although recall that Ptolemy’s observation was three days after opposition.

Movement of the Ep counterclockwise around the main orbit is Mars’s *mean motion in longitude*. This motion from the epoch is what gave us 252⁰ 40’ mean longitude, measured from the vernal equinox. This satisfied step one of Ptolemy’s tasks. Tweaking the planet’s position on the epicycle then allowed Ptolemy to correct deviations from mean motion [19]. This finally yielded a quite accurate *true longitude* of 241⁰ 35’ [20].This is the position of Mars along the ecliptic seen along the line of sight from earth, where it appeared to Ptolemy looking skyward in the Spring of 135 AD.

*Checking the model with the Almagest Calculator*

If we advance the time interval on the model from May 30, 139 a bit more than two years until Mars is in exactly the position on its epicycle as it was then, we will theoretically have made the model turn through one synodic cycle [21]. Do it a few more times and we should see incremental increases in the angular distance of Ep from the vernal equinox on the diagram, reflecting the synodic arcs traversed by Ep from opposition to opposition. The angular differences we predict should be the modelled actual (vs mean) synodic arc lengths [22].

This is spot-on and the time intervals between these particular dates turns out to be exactly 779.937 days. What the Babylonians, without any particular planetary theory, saw and recorded over the centuries as synodic arcs of Mars from event to event, and what we now observe of Mars in our sky from opposition to opposition, Ptolemy could thus replicate in his model quite well. He could play with it, altering the initial conditions to whatever time and date he chose, including to a time long after he lived. They could later – even centuries later – be checked and the model parameters be adjusted, if necessary [23].

*Afterward: Problems (seemingly) solved*

Ptolemy’s model appeared consistent with empirical constraints but did damage to the Aristotelian underpinnings of circular motion around the center, a Ptolemaic heresy that would vex natural philosophers for generations. Fourteen centuries later it would inspire Copernicus to give the whole thing a fresh look.

As food for thought, I leave you with these notions of how the three empirical problems of Mars’s motion were neatly solved by this model at least as far as appearances were concerned:

§ The differential rotation of Mars on its epicycle and revolution of the epicycle on the main orbit will appear to us at E as retrograde motion at its oppositions.

§ During oppositions of Mars near apogee A, the planet will appear dim, being relatively far away from our viewpoint at E. Oppositions near perigee P (closer to E) will appear brighter.

§ The center of circular motion being at Eq, the epicycle (i.e., Mars) appears to move more quickly at P (perigee motion is faster) than at A (apogee motion is slower).

It’s no wonder the *Almagest* survived as the last word in astronomy till the late Renaissance.

NOTES

[1] *Almagest* X.7, p.484. All references to the *Almagest* are from G. J. Toomer’s acclaimed translation: Ptolemy, Claudius. (~ A.D. 150) 1998. *Ptolemy’s Almagest*. Translated and annotated by G.J. Toomer. Princeton: Princeton University Press.

[2] All dates herein are AD (or CE, depending on your preference) unless otherwise noted.

[3] *Almagest* X.8, p. 499.

[4] Ibid. The *Almagest Calculator* is set up to receive the inputs as Ptolemy has given us. The top three rows of the Calendar Module of the *Almagest Calculator* allow us to enter that information, as I have done here:

For Ptolemy’s observation of Mars referred to in the text, enter the 15th day of the eleventh (XI) Egyptian month (from the drop-down menu, it is *Epeiph*). Ptolemy refers to *the second year of Antoninus*. You may adjust the “Era Nabonassar” until “Era Antoninus” indicates 2. (The epoch of the tables is fixed at the first year of the reign of the Babylonian king Nabonassar who reigned 747-734 BCE.) The 886 means that that many years after King Nabonassar corresponds to the second year of the reign of Antoninus, who ruled the Roman Empire, which included Egypt, when Ptolemy was looking at Mars in Alexandria. The time, 3 equinoctial hours before midnight, means 9 pm, which we enter in the space “Alexandria time.” At the bottom of the Calendar Module, you may then “Update ephemeris” and the Julian calendar date and Julian Day Number will appear. You may then scroll to the modules for solar, lunar, or planetary bodies and pick off the information you seek.

[5] The *Almagest Calculator* is an abundant resource on the calendar system used by Ptolemy. “The tables in the *Almagest* are based on the ancient version of the Egyptian calendar with a constant year length of 365 days and no intercalary days. The epoch of the tables is fixed as noted above, as this predated the earliest (Babylonian) observations that were available to Ptolemy (cf. *Almagest* III.7). Compared with the Julian calendar, the dates in this calendar shift one day in every four years and it is therefore often referred to as the ‘mobile’ or ‘wandering’ Egyptian calendar.”

[6] Gingerich, Owen. 1993. *The Eye of Heaven: Ptolemy, Copernicus, Kepler*.* *New York: American Institute of Physics.

[7] Neugebauer, O. 1975. *A History of Ancient Mathematical Astronomy.* 3 vols. Studies in the History of Mathematics and Physical Sciences 1. Berlin: Springer (usually abbreviated as HAMA).

[8] The *Almagest Calculator* can be found at __https://webspace.science.uu.nl/~gent0113/astro/almagestephemeris.htm__.

[9] See note 4 above.

[10] His website is http://www.solexorb.it/index.html. I’ve also used NASA’s JPL/Horizon database (for more recent evaluations) as helpful to check and confirm what the other software is telling me. Ancient calculations in modern planetarium software are also typically quite accurate. I have used *The Sky X Professional* and the freeware *Stellarium*.

[11] From ancient records Ptolemy deduced that Mars moves 191⁰ 54’ 54” in a year. Given a starting epoch, one can multiply years and fraction of years elapsed from the epoch, times Mars’s annual angular progress, and eliminate the extra rotations (just as clock time starts fresh when the hands reach full circle). This will yield the remainder, which is Mars’s approximate mean longitude, to which presession accumulated during the interval must be factored in.

[12] Plato believed that planetary motions could be mathematically accounted for without disturbing the fundamental ideal of circular motion; he famously challenged his students to come up with a solution. His seemingly impossible challenge, according to Harvard historian of science Gerald Holton, “kept natural philosophers agitated for 2,000 years.” Holten, Gerard. 1988 (1973) *Thematic Origins of Scientific Thought: Kepler to Einstein* (Cambridge, Mass.: Harvard Univ. Press).

[13] HAMA, 170: “Any theory of the motion of the outer planets must satisfy two conditions: (a) it must explain the characteristic phenomena, e.g. retrogradations, stations, etc.; (b) it must maintain a fundamental numerical identity between the number of occurrences of the same phenomenon (e.g. opposition to the sun) and the number of sidereal rotations of the planet and of the sun. This relation, well known in Babylonian astronomy, states that Number of occurrences + number of sid. rot. of the planet = number of years.”

[14] See my article on Mars in the August 2021 issue of *Sky & Telescope* magazine, and my October 2020 blog article *Alas Babylon! Mars Draws Near*, __https://www.douglasmacdougal.com/post/alas-babylon-mars-draws-near__. My June, 2021 blog article called *A Farewell to Mars *will help acquaint you with the vast technical terminology on the subject, including sexagesimal notation. It contains "A Glossary of Babylonian Planetary Mathematics" and "Babylonian Math on your Calculator." It's at __https://www.douglasmacdougal.com/post/a-farewell-to-mars____.__ See also my journal article, “Exploring the Mysteries of Babylonian Astronomy with Maple,” in *Maple Transactions*, September 2022, a journal of computational mathematics, cited as Maple Trans. 2, 1, Article 14357 (2022), __https://doi.org/10.5206/mt.v2i1.14357__. This refereed paper was presented at the Maple 2021 Conference, November 2021 in Waterloo, Ontario, Canada.

[15] This number varies somewhat because Mars's orbit is not perfectly circular. The Babylonian records reflect mean motion. The interval that appears on a NASA/JPL website is 779.94 days. So the old records are remarkably accurate. Ptolemy called the 37 returns, “returns in anomaly.”

[16] More precisely for Mars, 37 cycles (synodic events, such as oppositions) occur in 79 earth years plus about 3 days and 5 hours (according to Ptolemy), which is 42 Martian revolutions, plus a little more than 3⁰. *Almagest*, IX.3, p. 424. Ptolemy called the 37 returns, “returns in anomaly,” and the 42 Martian revolutions (Mars’s years) “returns in longitude.” “As a general definition, we mean by 'motion in longitude' the motion of the center of the epicycle around the eccenter, and by 'anomaly' the motion of the body around the epicycle.” Ibid.

[17] HAMA, 389.

[18] HAMA, 455.

[19] The correction includes the “Mean Epicyclic Anomaly” of 172;46,19, according to the *Almagest Calculator*, which is measured counterclockwise from the apogee of the epicycle. The apogee line of the epicycle is shown by the faint dotted line extending the line connecting E to Ep.

[20] SOLEX gives a value of 242⁰ 13’ which is within about 40 arcminutes of Ptolemy’s value.

[21] Recognizing that we are starting as Ptolemy did a few days after opposition as our starting point. And remembering that synodic events include by definition any events where the sun, earth and planet are in sync in some way, e.g., risings, settings, first or last appearances, etc. Ptolemy choose oppositions for the outer planets because they are most easily observed, occuring high in the nighttime sky.

[22] We advanced the model each time by turning it (so to speak) until the "Mean Epicyclic Anomaly" (Mars’s position on the epicycle, measured from the apogee of the epicycle) is identical from the previous opposition, and noting the “Mean Longitude of Epicycle Center” each time. What changes here is Mars’s position in the sky by a little more than 48⁰ each opposition (the synodic arc). Note, we are comparing longitudes not measured around the center of the main orbit (the deferent circle), which Ptolemy called the “eccenter,” but from the *equant point* Eq used by Ptolemy. This was Ptolemy’s most controversial innovation.

[23] This result can be cross-checked by simple multiplication. Multiply the synodic interval we found (779.937 days) by Mars's daily motion 0;31,26,36,53,51,33⁰ (0.5240597114 degrees per day) found by Claudius Ptolemy (*Almagest* IX.3), and the result for the length of the synodic arc is the same.

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