PTOLEMY DAY 22 PERIODIC JOINT RETURNS; THE EQUANT

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1 PTOLMY DAY 22 PRIODIC JOINT RTURNS; TH QUANT We want to get the least periodic joint returns for the 5 planets that is, we want to be able to say, for a given planet, the number of times the epicycle goes around the deferent in the same time that the star goes around the epicycle some whole number of times (and we want to express this in least numbers). We will put up with some slight inequality, though, since the two speeds (of star and of epicycle) might be such that they never complete whole numbers of cycles in exactly the same time (as we noted in Day 20), or else they might do so, but the smallest number of times the epicycle goes round in the same time that the star completes some whole number of cycles on the epicycle is 6 trillion. We don t have time to wait around for that! So Ptolemy instead gives us very nearly joint returns for the 5 planets, as follows: SUN LONGITUD ANOMALY # times # times epicycle s # times star mean sun center orbits deferent orbits epicycle orbits earth Saturn 59 (+ 1 45/60 days) 2 (+ 1 43/60 ) 57 Jupiter 71 ( 4 54/60 days) 6 (+ 4 50/60 ) 65 Mars 79 ( /60 days) 42 (+ 3 10/60 ) 37 Venus 8 ( 2 18/60 days) 8 ( 2 15/60 ) 5 Mercury 46 ( + 1 2/60 days) 46 ( + 1 ) 145 In other words, during the time it takes the mean sun to go around us 59 times (plus a little bit), Saturn s epicycle goes around its deferent 2 times (plus a tiny bit) and Saturn itself goes around its epicycle 57 times (precisely). And so on with the other entries. 155

2 What kinds of observations would Ptolemy have had to make in order to derive these numbers? In the case of an inner planet, like Venus, he could look at tables of data (of his own, or from Hipparchus and others) correlating where Venus was appearing in the zodiac and the date and what its elongation from the mean sun was at the time. If we look at such tables, we will notice that the value of Venus s greatest elongation from the sun varies depending upon where it is in the zodiac. And it is not the case that if it is at a greatest western elongation from the sun in place X in the zodiac, then a year later, when it is at that place in the zodiac, it will again be at a greatest elongation. Rather, we have to wait 8 years. Then it will be at a greatest western elongation again, and in pretty much the same part of the zodiac, and the size of that elongation will be what it had been 8 years before. But during that time, we can see that the mean sun went around us 8 times (of course), and so Venus s epicycle, which ties Venus to the mean sun and hence itself has the same speed as the mean sun also went around us 8 times, and we can count the number of times that Venus went from greatest western elongation back to greatest western elongation in those 8 years, i.e. its cycles of anomaly, namely 5 times. Let that suffice for an understanding (somewhat oversimplified) of how to derive the table of periodic returns above from the raw observations. NOT TH OTHR PATTRNS: S = L, and S = L + A. If we get rid of the plus a little or minus a little in the table, and present a simplified version of it, we have: SUN LONGITUD ANOMALY # times # times epicycle s # times star mean sun center orbits deferent orbits epicycle orbits earth Saturn Jupiter Mars Venus Mercury We have already seen that the number of solar cycles, S, equals the number of longitudinal cycles (times the epicycle goes round), L, for inner planets. But what about the outer planets? Do you notice something about their numbers? 59 = = =

3 For the outer planets, for some reason, the number of solar cycles is equal to the sum of the number of longitudinal cycles plus the number of cycles of anomaly in other words, S = L + A. A remarkable coincidence! For the inner planets, too, it is a coincidence that S = L. There is no reason why these things should be so in Ptolemy they just are. They are cosmic coincidences which need not be. But Ptolemy is aware of them. He is so aware that S = L, for instance, that the entry on the table for Venus s mean movement in longitude (i.e. the motion of the center of its epicycle) in 1 day is exactly the same value as that for the mean sun. In fact, that whole table is just the table of the mean sun s movement reproduced. S = L follows for Ptolemy only because the inner planets are each tied to the sun, and hence their epicycles must have the same average speed as the sun, i.e. the speed of the mean sun. But why should that be, if they do not orbit the sun, but make epicycles in front of it, closer to us? S = L + A is even more arbitrary-sounding right now. But there it is! GNRATING TH TABLS OF PLANTARY MAN MOVMNTS IN LONGITUD AND ANOMALY. As we did with the sun, so we do now with the planets. If you tell me how far the epicycle moves in longitude in any amount of time, then, since I know it is uniform, I can tell you how far it moves in any other time, or, conversely, given how far it has moved, I can tell you how long it took. So too with the mean motion in anomaly, i.e. the uniform motion of the star on the epicycle. CHAPTR 5 DTRMINATION OF TH GNRAL PLANTARY HYPOTHSIS. We have already seen that the planetary phenomena are highly suggestive of epicycles on eccentric deferents. And we have also seen how to determine the relative speeds of the star on the epicycle and of the epicycle on the deferent by means of the least periodic joint returns. But are we talking about same-direction epicycles, or opposite-direction epicycles? Do the star and epicycle rotate in the same directions about the centers of the epicycle and deferent respectively? Or do they go in opposite directions? If we are to explain the planetary phenomena by epicycles on eccentric deferents, we must do so by means of SAM-DIRCTION PICYCLS. The following explanation will bring out the reason for this. First, recall that the greatest passage of a star means its fastest apparent speed, or the moment dividing the time in which it was speeding up from the time in which it will be slowing down, while the least passage of it means its slowest apparent speed, or the moment dividing the time in which it was slowing down from the time in which it will be 157

4 speeding up. The mean passage of a star means when it appears to be moving with its mean speed (e.g. with the uniform speed which its epicycle actually has around the center of the deferent), or, alternatively, the moment dividing the time in which it had a speed less than the mean speed from the time in which it will have a speed greater than the mean speed (or else the moment dividing the time in which it had a speed greater than the mean speed from the time in which it will have a speed less than the mean speed). Let G = Greatest passage M = Mean passage L = Least passage In the case of all 5 planets, for the heliacal anomaly, Time [G to M] > Time [M to L] and therefore we are dealing with a SAM-DIRCTION PICYCL, where greatest passage is at apogee. In the case of all 5 planets, for the zodiacal anomaly, Time [L to M] > Time [M to G] and therefore the CCNTRIC (deferent) will do to explain that anomaly. Ptolemy is very terse about the observations which justify these claims. The strange thing is that he is separating the appearances for the two anomalies of a single planet, whereas a planet has only one set of appearances, not two. It seems he is assuming that the planet is moving on an epicycle, and then isolating the appearances due to the planet s motion on the epicycle, and again isolating the appearances due to the epicycle s motion on the deferent. Let s see briefly how he does this for the heliacal and zodiacal anomalies. 158

5 ISOLATING TH HLIACAL ANOMALY. At some time the planet is G seen making a greatest passage when it is M Z against some spot, X, in the zodiac. Some whole number of cycles of longitude later, L M the center of the epicycle, C, is back at the same spot and we can know when that is F the case, since we know the period of C around the center of the deferent, F, thanks to our periodic joint returns. Very well, the next time we know C is back at the same spot it was in when we observed the planet in greatest passage against X in the zodiac, we observe the planet again, and this time we see it moving with its mean speed (which is the same as the speed of C around F, which we know by the periodic joint returns). When it is at mean passage, it is viewed against Q in the zodiac. We do this again still later, and we observe the planet moving with its least speed, and it is once again viewed against X in the zodiac. Since we are viewing the apparent speed of the planet (over a couple of nights, so that the epicycle does not move much) for one location of the epicycle, the effect of the zodiacal anomaly is removed, and all the differences in apparent speed are due to the direction in which the planet is moving on the epicycle. And although the planet does not go from its greatest passage to its mean passage without the epicycle moving meanwhile, we know the rate at which the star really moves on the epicycle (thanks to the periodic joint returns again). Hence after observing the planet at greatest passage on one date, and then at mean passage at another, and then at least passage at still another (at which dates C is back at the same spot on the deferent), it is just a matter of number-crunching to see that the time from greatest to mean is greater than the time from mean to least. And that is a property of the SAM-DIRCTION PICYCL. Hence we must employ that simple hypothesis in order to explain the heliacal anomaly. Q X 159

6 ISOLATING TH ZODIACAL ANOMALY. When the center of the epicycle, C, is appearing at some place in the zodiac, Z (although we can t really see it), let the planet be at P on the epicycle. Observe its speed (over a few nights chart its longitudinal progress or regress). When will the planet next be at that same spot on the epicycle, P? We know the answer from our tables of the planet s regular movement in anomaly (i.e. on its epicycle), thanks to our periodic joint returns. When that time has elapsed, we observe the planet s speed again. We keep doing this, and soon (well, actually after a long time) we have a table of the planet s apparent speeds throughout the zodiac when it is on a certain point P on the epicycle. That means all the differences in its apparent speeds will be due not to its motion on the epicycle (which has been removed from these considerations) but to some other irregularity in its Q motion around us. Doing as we did with the heliacal A anomaly a moment ago, we find that the time from P least passage to mean passage is always greater than C the time from mean to greatest. This means we can account for this apparent irregularity of speed by an eccentric circle. So we place our same-direction epicycle upon an eccentric deferent. NOT: Any slight inaccuracy in our table of F mean motions is too slight to be relevant to the crude C inequality Ptolemy needs, i.e. concerning the time [L to M] and time [M to G]. P A Z DIFFICULTIS. TH QUANT. But things won t be so simple as an eccentric deferent! Ptolemy notes this in Book 9 Chapter 5. There are two principal complications which will force us to make our basic model for planetary motion a bit more sophisticated: [1] The lines of apsides for the planets precess eastward with the speed of the precession of the equinoxes (here s a bit of a spoiler: this phenomenon which Ptolemy notes is just caused by the rotation of earth s axis, too. Real precession of perihelion for a planet completes a back-and-forth cycle once every 100,000 years or so, and is a small oscillation. The planetary apehelia are basically at rest; so much so that Newton refers to the resting of the aphelia as an astronomical phenomenon). [2] The epicycle for each planet is carried around a circle with one center, but the center of uniform motion is another point (which I will call the equant or equalizing point). As Ptolemy puts it: The epicycles centers are borne on circles equal to the eccentrics effecting the anomaly [i.e. the equant circles] but described about other centers [i.e. the centers of the eccentric deferents], and these other centers [i.e. the centers of the eccentric deferents] in the case of all except Mercury [which is more complex] bisect the straight lines between the centers of the eccentrics effecting the anomaly [i.e. the centers of the equant circles] and the center of the ecliptic [us]. That is characteristically obscure of Ptolemy. Let s see if we can get more clarity about this complication he is describing. A diagram might help things: 160

7 TH GNRAL PLANTARY HYPOTHSIS. = us, earth, the center of the ecliptic (a great circle on the celestial sphere) D = center of the deferent for a planetary same-direction epicycle Q = center of uniform motion for C, center of the epicycle [i.e. QC sweeps out equal angles in equal times around Q, not around D, although C always rides on the circle around center D] QD = Q (I have exaggerated the eccentricity in the figure.) Q D C There is something eerily reminiscent of an ellipse in this diagram! In an elliptical orbit for a planet there will be two foci and a geometric center right between them. But more on that when we come to Kepler. C, the center of the epicycle, is called the MAN PLANT. The RATIOS of speeds and of lengths in the figure will differ from planet to planet, and Ptolemy will be determining these. MRCURY has special problems which Ptolemy addresses, but we will stick to the less complicated planets. We will look at one inner planet, Venus, as far as the question What are the ratios? is concerned; and we will look at one outer planet, Saturn, as far as the question Where do stations occur? is concerned. In the case of an INNR PLANT, where QC moves around Q with the speed of the mean sun, Ptolemy will assume that the line from (arth) out to the mean sun is always parallel to QC. (To get the proportions right, we would have to shrink the eccentricity quite a bit, and maybe grow the epicycle; in the case of Venus, the line joining arth to the mean sun will always pass through the epicycle, almost through its center; and the apparent sun very nearly appears in line with the center of the epicycle!) In the case of an OUTR PLANT, Ptolemy assumes that the line joining C to the planet (i.e. the epicyclic radius drawn to the star) is always parallel to the line joining to the mean sun (i.e. from us to the mean sun). In either case, if the lines were VR parallel, they would always be parallel. But that they were parallel at any time in the past (or will be at some time in the future) is not known, or not well known, by naked-eye astronomical observation. But it certainly keeps things simple. 161

8 RMARK ON TH QUANT. To get a sense of how the equant works, you could make a circular track for a marble in a piece of plywood, and rotate an arm uniformly around some point other than the geometric center of the circle (but inside the circle). You will see the marble speed up and slow down in its circular groove. This mechanism is a bit of a shift away from the ancient astronomical ideal, from the astronomer s axiom that the heavenly bodies move uniformly on perfect circles. Until we introduced this idea of an equant point, we had stars moving on perfect circles with uniform speed around the centers of those circles. That had a nice simplicity to it. But now we are divorcing the circular path that the planet moves on from the center around which it sweeps out equal angles in equal times! This will later scandalize Copernicus, who refuses to accept equants. Kepler (still later) loves equants, at least at first. But they will eventually get replaced by the empty focus, in one way, and by the full focus, the sun, in another way (with uniform area-velocity, as we shall see). The need to posit an equant point will become clearer later. For now, Ptolemy is just preparing us for the idea, and I will generally include one in the diagrams. IGNORING LATITUDINAL DIFFRNCS. Ptolemy will be ignoring differences in latitude, since they make so little difference in longitudes. TH MICKY MOUS PROPOSITION. In Book 9 Chapter 6 of the Almagest, Ptolemy presents an argument which is needed later to find Venus s apogee in Book 10. It is not of great interest in itself, and might be called an overelaboration. It is really a matter of symmetry. The proposition is as follows. S L B G N A X D M F Given: Venus s eccentric deferent, line of apsides AC G = center of uniform motion (equant point) F = eye arcab = arcad (or AGB = AGD) FL & FM are tangents to the epicycle at positions B and D K C H Prove: GBF = GDF (angles of zodiacal anomalistic difference) BFL = DFM (greatest elongations from the mean planet) 162

9 Draw: N perpendicular to DGK X perpendicular to BGH Now XG = NG [bc AGB = AGD, bc arcab = arcad] and GX = GN [both right] and G is common so N = X [r GX = r GN] so KD = BH [equidistant from center] so ND = XB [halves of KD & BH] minus NG = XG [equal bc r GX = r GN] so GD = GB [remainders] but DGF = BGF [supplements to AGD & AGB] and GF is common so BF = DF [since r BGF = r DGF] so GBF = GDF But BL = DM [radius of epicycle] and BLF = DMF [both right] and BF = DF [proved above] so r BLF = r DMF [1.47] so BFL = DFM Q..D. So, given equal angles on opposite sides of the line of apsides, we have proved that the greatest elongations (i.e. G on one side, GW on the other) are equal. QUSTIONS about the diagram: Is BG parallel to LF? No. BG is a line moving uniformly around G, and so it is always parallel to the line out to the mean sun S (for an inner planet). S B D Is L on the deferent? No, not necessarily. But since F is not the center of the deferent, it is not impossible for L to be on the deferent. L G F 163