Planetary Cycles and their Various Kinds I: Ascending in the Apogee of the Eccentric Circle

The following series of posts is intended to be a basic guide to the various cycles of the seven planets within Medieval Astrology, including both Persian-Arabic and Latin sources. In particular, throughout this we will be paying special attention to the motion of the planets, and the role this plays in their condition, which goes beyond just retrograde and direct! For this post, we’ll be observing what we call “ascending in the apogee of the eccentric” or the deferent circle.

So to begin with, we most often find the techniques relating to the apogee alongside a scattering of other techniques in medieval texts. We can find them in works such as Abū Maʿšar’s Great Introduction (Yamamoto and Burnett, 2019), Al-Qabisi’s Introduction to the science of Astrology (Dykes, 2010), Al-Biruni’s Book of instruction in the elements of the art of Astrology (Wright, 1934), and also Ibn Ezra’s work On Nativities (Sela, 2013). It is especially prominent in works of Medieval Perso-Islamic Astrology, however by the 1500’s in Europe, it seems to have considerably fallen out of favor and to have been ignored as a dignity or power of the planet.

What does ascending in the apogee mean? Essentially, it refers to the planet and its distance to the earth. The further away from the terrestrial earth a planet is, the more dignified it was considered. Conversely, a planet closer to us, becomes closer to the nature of the terrestrial, more corruptible and perishable. This technique then, aims to assess whether a planet is close to us, or distant from us, in order to judge its strength and quality. It is a laborious process, but alongside the Solar Phases and strength by Latitude (and planetary dragons, otherwise called nodes) they comprise some facets of Astrology that are often neglected today. Hence I have selected them to be the first in this series of posts.

Now, to understand this technique, we do need to know some Astronomical terms. Geocentric Astronomy, especially in this period, often draws on the Almagest of Ptolemy (usually accompanied by a lengthy commentary from its translator) and what we are engaging with here relies on the model presented by Ptolemy (Toomer, 1984). In particular we need to understand epicycles, and the deferential Circle. I am not going to present the entire theory of epicycles here, as it would distract from the main points. However, hopefully some explanation in the form of the following diagram will be sufficient.

As you can see, the circle of the deferent encircles the eccentric point in the centre. Conversely, the epicycle moves itself along the circle of the deferent. When a planet is “outside” the deferent circle (far from the earth) we call the planet direct. When a planet is “inside” the deferent circle via his epicycle, we call him retrograde; when he’s on the circle itself, we say he is in his stations. The deferent ring moves around the Ecliptic, which means each part of the deferent has its own “Zodiacal Longitude,” i.e: 0 degrees of Aries, 5 degrees of Cancer and so on, with the “Start” always beginning at the Vernal Equinox, or 0 Aries. The next part that is important for us to note is that the centre of this epicycle is called the mean longitude of a planet. This mean longitude, or the middle of the epicycle, moves across the deferent in its standard secondary motion, from east to west. Its motion is uniform, constant, unceasing and unchanging in Ptolemaic thought. It does not retrograde, there is no tangible body to be found here. It is an invisible axis point around which the planet circles, whilst the epicycle itself circles around the eccentric point. This uniformity of movement, is also the reason we use the mean longitude in considerations like the solar revolutions.

Now, in the above diagram, you’ll note that we also defined the apogee and perigee of the eccentric circle. But we also need to note that there is an apogee and perigee of the epicycle. They are depicted in our image, marked as PE and AE. These are based on the location relative to us on the earth. The pink line cutting through the middle of the epicycle, is known as the apsidial axis in modern Astronomical terms. The same term is applied to the one cutting through the deferent.

Thus, we have two apogees and two perigees. The first is the epicycle’s apogee; the second is the deferent’s apogee. We’re going to talk about the deferent’s apogee for now, saving the epicycle’s apogee for the future, as the calculations are considerably laborious and involve us having to find the verus locus of the planet using tables of anomalies if we want to make use of the technique itself via Ptolemy. Whilst I do intend on writing on this topic, it is more properly treated on its own once we have become acquainted with the calculation of mean longitude, which is a pre-requisite for the calculation of the “epicycles” anomalies and how we might consider this, in Modern Astronomy where there is no such thing as an epicycle.

I also want to add a brief note here. Judging from Al-Biruni’s work, it was common for contemporary astrologers to mainly emphasize the deferential motion. On the one hand, he criticizes this and seems to consider the epicycle’s apogee more important. On the other, it does show that the deferential apogee was thought to play an important role in the planetary motions regardless. When we look in these older texts and see the various terms “equation of centre” and “increasing in number,” these are referring to the tables of anomaly used to calculate a planets true position in the epicycle. Conversely, the tables of mean motion are relatively easy to understand with a little engagement, and so I have chosen to start with the deferential motion.

Finding the Degree of the Mean (Eccentric) Apogee

With this out of the way, how do we find the degree of the deferent’s apogee? In Islamic Astronomy, we find one method presented by Al-Biruni, building on Ptolemy’s theory and adjusting it for precession. The theory he puts forth in his The Book of Instruction in the Elements of the Art of Astrology is that the planetary apogee (upon the deferent) moves according to precession. The rate of motion for the deferent is the rate of precession. According to Al-Biruni, that is 1˚ for every 66 Arabic years, or 64 years in the Gregorian calendar; Astrologers today typically use 1˚ per 72 years for precession of the fixed stars, we also have the modern Astronomical rates of procession which are as follows.

The Apsidial lines of the planet’s today follows the following key, according to Mohammed Mozzafari.

Saturn: 1˚ 50.8y,
Jupiter: 1˚ 61.2y,
Mars: 1˚ 54.1y,
Sun: 1˚ 58.2y,
Mercury: 1˚ 64.9y,
Venus: 1˚53.2y

Below is an example of Biruni’s calculations. The year in which he wrote this portion of his text was 1029AD, or 420AH. Thus we can surmise (sticking to his technique) the following longitudes for the apogee of the deferent. You can also find alternative values in Introductions to Traditional Astrology (Dykes, 2010).

Planets and their Deferent Apogee, a table according to Al-Biruni

Apogee Longitude in 420 AH;
According to Biruni


Apogee Longitude
Values adjusted for precession according to Al-Biruni

Year: 2020 Key: 1˚ per 64 Gregorian Years
Value added: 15˚29’
Perigees
(180˚ from the Apogee)
Saturn6˚48′ Sagittarius22˚17’ Sagittarius22˚17’ Gemini
Jupiter16˚43′ Virgo02˚12’ Libra02˚12′ Aries
Mars:08˚33’ Leo24˚02’ Leo24˚02’ Aquarius
Sun24˚32′ Gemini09˚01’ Cancer09˚01’ Capricorn
Venus24˚32′ Gemini09˚01’ Cancer09˚01’ Capricorn
Mercury23˚43′ Libra08˚12’ Scorpio08˚12’ Taurus

*Note, there may be some inaccuracy as pertaining to the precise minute ’ of the table. However, the degree itself should be fine.

A planet is considered to be in its eccentric apogee when its mean longitude (the middle of it’s epicycle) is within the eccentric apogee. The same, of course, applies for the perigee. Thus, we cannot for this technique apply the true longitude of the planet as we usually see it in our Astrological software, but instead must calculate this ourselves via the mean longitude of the planet.

Calculating the Mean Longitude of the planets

I will now present the following method, utilized by Ptolemy, in observing the mean longitude of the planets. Following it will be a modern table of the orbital cycles of the planets using modern astronomy, usable should you wish to adapt these values to more modern ones. I’d also note that mean longitude has other uses than simply the relationship of a planet with the apogee of the deferent, including a role in the mean conjunctions of Saturn and Jupiter, often phrased as the “Great Conjunctions,” as Ben Dykes succinctly puts forth [here]. There is also the option of calculating these mean longitudes using Ptolemy’s values online, thankfully due to this excellent [tool].

Abbreviations used in calculation:

  • Difference in Time = DT
  • Difference Times Motion = DTM
  • Stored Value = SV; this is the number you add to the next calculation
  • Preserved Value = PV; this is the final value of that particular position.
  • DTMF = Difference Times Motion Final, i.e.: DTM + SV, (Do not calculate this for sixths, simply use the DTM)

Formula

Calculate difference in time, IE: How many days and hours between the two dates?

Consult the table below, starting from the far right hand side. Take the value on the far right there and times it by the DT. This is called the DTM, or difference times motion.

Then divide the value of DTM by 60, take the integer of this answer, and add it to the next calculation. We call this the SV, stored value. (an integer is a whole number, IE: You need to ignore decimal places and make sure not to round it up or down)

Then take the same DTM, and modulus 60, this gives us the remainder, or how much is left in this time.

Then move onto the next calculation (IE: From sixths to fifths) and repeat the process, making sure to add the SV from the previous before calculating the remainder.

The formula is as follows, and an example is provided also.

Date A – Date B = DT

DT x TableValue = DTM

DTM + SV = DTMF (Ignore this step the first time, IE: when calculating sixths)

DTMF / 60 = SVn (make sure only to use the Integer value, ignoring decimals)

DTMF % 60 = PV

The Example

For our example let’s say we are calculating the mean longitude of Jupiter. Let us say that the difference in time for our hypothetical motion to keep things simple, is measuring between 5 precise days. Thus, we observe the table of his daily motions as follows:

DegreesMinutesSecondsThirdsFourthsFifthsSixths
045914264631

The Example calculation with notae:

Sixths

5 x 31 = 155, thus we say that Jupiter moves 155 sixths in this time period.

Each time the number reaches 60, we add 1 to the next calculation. (IE: Fifths) and keep the remainder with the sixths. To see what we add to the 5ths, and what we keep in the sixths when we make our new table, consider the following calculations for the SV and the PV. If you wish, you may choose to ignore the PV until you begin to calculate the motion in seconds as we don’t typically consider them in the chart. But if you plan on observing new dates based on your new calculations as the starting date in the future, it might be wise to keep them.

Sixths, Calculating the Stored Value/SV

To see what you add to the next value, take the number obtained and divide it by sixty. Ignore decimals and use the actual integer or whole number given, (do not round it upwards, ever.)

155 / 60 = 2.58333333, so we will add 2 when we calculate the fifths.

Sixths, Preserved Value/PV

155 % 60 = 35, so our final value for sixths, if we were going to make a new table, is 35 Sixths.

Fifths

5 x 46 = 230 + 2 = 232

232 / 60 = 3.86666 (so SV = 3)

232 % 60 = 52 (So we keep 52 in our fifths position)

Fourths

5 x 26 = 130 + 3 = 133

133 / 60 = 2.216666666666667 (so SV = 2)

133 % 60 = 13

Thirds

5 x 14 = 70 + 2 = 72

72 / 60 = 1.2

72 % 60 = 12

Seconds

59 x 5 = 295 + 1 = 296

296 / 60 = 4.93333333333 (so SV = 4)

296 % 60 = 56

Minutes

4 x 5 = 20 + 4 = 24

24 / 60 = 0.4

We do not have any SV, so we do not need to calculate a PV. The total motion in minutes, is therefore 24

Degrees

0 x 5 = 0

Since there was no SV when we observed the Minutes, we do not add anything here and his motion in degrees remains zero. Thus, Jupiter, in the course of five days has not moved a full degree of mean longitude

With this, our final table for five days of motions, now looks something like this. You’ll note I haven’t included the thirds and fourths in his final position. However it is certainly valuable when they do, and are desirable when calculating over very long periods of time.

JupiterDegreesMinutesSecondsThirdsFourthsFifthsSixths
Starting Point 3˚2’3” Aries
—› Position now
32659
Amount of mean motion he has moved in 5 days:0245612135235

Adjusting to more precise values – Minutes & Julian Days

Now, most of the time when we consider two different dates, they will typically be more than five days apart, and also making use of hours, minutes, etc. When we want to consider the mean longitude over for a long period of time, to begin with, it is typically it is best if we use the smallest value we have (i.e. the hourly mean motions of the planets). Thus, you’ll note that what I described as difference in time does not necessarily equal to one day, but can also apply to hours, minutes, seconds and so on.

Ptolemy gives us hourly values, which is more than enough for most purposes. So, if we were to consider the above calculation for 5 days, rather than DT = 5, DT now = 120 (5 sets of 24 hours). But what if we need to calculate minutes? Well, a minute is a 1/60 fraction of an hour. Thus we just need to divide the hourly motion by 60 to get the result for one minute of mean motion.

Therefore, if we are considering a nativity, and the birth was at 5 hours and 20 minutes, we would calculate the first 5 Hours as was said above. For the remaining 20 minutes, we would calculate them separately, and then we would divide the result.

A quick way of calculating this formula would be to take the hourly mean motion and divide by 60. Then multiply the resulting answers based on how many minutes you had left, as per the following brief and easy formula.

Formula for Adjustments by minute

Mean motion per hour / 60 = Motion per minute.

Motion per minute x number of minutes desired = final result for the adjustment.

IE: We want to add on 15 minutes to our previous calculation.

Now, Jupiter’s hourly motion in seconds = 12”.

Therefore:

12 / 60 = 0.2

0.2 x 15 = 3

Therefore, we would need to add 3 more seconds to the calculation for Jupiter’s mean longitude. If we obtain a decimal number, but no integer (i.e. 0.35584484 as a random example) we can take the decimal number, multiply it by 60, and add the integer from that number to the next table, though this shouldn’t be a common occurrence for most planets.

Julian Days

The most precise way to calculate difference in time, when it is over a long period of time, is using Julian Days. This doesn’t refer to the Julian Calendar, but rather a system created in order to count days, with day 0 beginning from the date January 1, 4713 BC in the Julian Calendar. This topic has been spoken about at length by others elsewhere (see here) and so there isn’t much need for me to explain it, however if you are learning astrology I do advise at least attaining a cursory understanding of them, if not the formula as it is the preferred dating system in astronomical systems.

With that, here are the tables of mean motion, taken from G.J Toomer’s edition of Ptolemy’s Almagest.

Table of motion via mean longitudes for the seven planets taken from the Almagest (Toomer, 1984)

Note on the tables: Mercury and Venus in the Ptolemaic system are considered to share the same mean motion with the Sun, which is the centre of their epicycle. Hence Mercury never has more than 28 degrees of elongation from him, and Venus 48 degrees. Their difference with the Sun lies not in their mean longitude, but in the true and apparent longitude (that is, in motion along the epicycle). I would also note that when a planet moves over 360˚, Ptolemy keeps the remainder, much as we have done. Thus the Moon’s yearly motion in mean longitude is not the large number she actually travels, but her difference in location from the starting point from where we begin our measurement.

SaturnDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly motion12132356303015
Monthly (30 day) motion101645442530
Daily Motion02033312851
Hourly Motion0051234842
JupiterDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly motion30202252525835
Monthly (30 day) motion2293713231530
Daily Motion045914264631
Hourly Motion0012286656
MarsDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly motion191165427383545
Monthly (30 day) motion15431826554630
Daily Motion0312636535133
Hourly Motion011836321439
Sun/Venus/MercuryDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly motion35945244521835
Monthly (30 day) motion2934836361530
Daily Motion059817131231
Hourly Motion0227504331
MoonDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly motion129224613503230
Monthly (30 day) motion3517291645150
Daily Motion13103458333030
Hourly Motion0325627262323

Mean Longitudes of the Planet’s from January 1st, 2020, 12pm, Greenwich, England (Ptolemy method)

Julian day: 2458850.0000000

(Starting from 0 Aries, the vernal equinox)

Sun: 271˚ 25’ 48”

Moon: 342˚03’20”

Saturn: 287˚14’42”

Jupiter: 275˚58’26”

Mars: 214˚59’33”

Venus: 271˚ 25’ 48”

Mercury: 271˚ 25’ 48”

You may use these to calculate the mean longitudes of the planets at your own desired date. Note that these are considered using Ptolemy’s values, and so there are certainly arguments one can put forth that they are outdated. On account of this I have calculated a corrected mean motion of the planets using better values from NASA. Note that they may still lack precision.

Modern tables of orbital periods

Here is a modern table of the planetary orbits, taken from NASA’s planetary fact sheets for those who’d prefer more precise values. Note that I have included the inferiors here, but we need to remember: their mean longitude were considered equal to the Sun in Geocentric astronomy/astrology and so those particular values aren’t all that useful for our purposes in this particular context.

PlanetDays to complete a revolution in the Tropical Zodiac
Saturn10,746.94
Jupiter4,330.595
Mars686.973
Sun365.24217
Venus224.695
Mercury87.968
Moon27.3217

The formula of correction and notes to the table

Here follows the formula I have used in order to calculate this corrected longitude; with thanks to my friend, B. Key for his help in determining the best way to go about these initial corrections.

I will also note, that where the table has said year, it refers to a solar or tropical year, and thus is 365.24217 days, rather than simply 365 days.

Terms used:

  • VT = Value of time (I began with the solar year, 365.24217, to calculate daily motion)
  • PO = Planetary orbit (in days) value, as above
  • FR = Fraction result
  • POS = Position
  • POSI = Position Integer (IE: The integer number preceding a decimal point in POS)
  • POSD = Position Decimal points. (IE: the numbers following the integer)
  • NTPOSI = Integer to place in next table (as as POSI)
  • NTPOSD = Decimals to round to get the next tables POSD.

Formula for year

Year (or time)/ PO = FR

FR x 360 = POS

POS % 360 (if the POS is over 360. IE: the Moon)

Place POSI within table (the whole number)

POSDx60 = NTPOSI and NTPOSD

Repeat process until yearly table is filled out.

Formula for time when under one year in length

1 unit = days

365.24217 / 365 for tabledays;

tabledays x30 for months;

tabledays / 24 for hours

Therefore, years values in table: time = 365.24217

Month values in table: time = 30.019904383561643835616438356164

Days values in table: time = 1.0006634794520547945205479452055

For hours: time = 0.04169431164383561643835616438356

Corrected Tables of Mean Motion in Longitude for the Seven Planets

SaturnDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly Tropical motion1214527142517
Monthly (30 day) motion102010273035
Daily Motion0204020551
Hourly Motion0051405217
JupiterDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly Tropical motion30214434351637
Monthly (30 day) motion22943561630
Daily Motion04592752326
Hourly Motion001228394120
MarsDegreesMinutesSecondsThirdsFourthsFifthsSixths
Yearly Tropical motion1912425237517
Monthly (30 day) motion154353394024
Daily Motion031274719204
Hourly Motion011839281820
Sun, Mercury, VenusDegreesMinutesSecondsThirdsFourthsFifthsSixths
Tropical year360000000
Monthly (30 day) motion2935203252369
Daily Motion059104154512
Hourly Motion022756424423
MoonDegreesMinutesSecondsThirdsFourthsFifthsSixths
Tropical year132331738183732
Monthly (30 day) motion353385049127
Daily Motion1311617413824
Hourly Motion03321193737

Corrected Mean Longitudes of the Planets for January 1, 2000, 12:00PM

Saturn: 49°33’50”
Jupiter: 34°24′ 15″
Mars: 355°27’11”
Sun, Mercury, Venus: 280°27’36”
Moon: 218°18’58”

Ascending in the Apogee of the Deferent

With this information in our hands, it’s time to actually see whether according to what we have calculated so far, if a planet is ascending in the apogee of its eccentric circle (that is, the deferent). This is a fairly easy process, as what we do in this consideration is observe the mean longitude of the planet, and whether it is ascending (moving towards the apogee) or descending (moving towards the perigee). The values for these have already been given (using historical, not corrected values). Thus, quadrants 1 and 2 represent a planet descending from its apogee. It is a place of weakness, with quadrant 2 weakest of all, and quadrant 1 more like one who is descending towards weakness. Quadrants 3 and 4 represent the climbing of the planet, that is to say, it is a place of strength, and quadrant 4 is stronger than 3, because 3 is more like a recovering from weakness. Thus we might decide to label them as follows, in the same manner we see the Solar Phases treated in Guido Bonatti’s Liber Astronomiae (Dykes, 2010):

Quadrant 1) Strength moving towards weakness;

Quadrant 2) Most Weak

Quadrant 3) Weakness moving towards Strength;

Quadrant 4) Most Strong.

The Effects of a planet ascending in its Apogee and its implications

Ibn Ezra in his Book of Reasons (Sela, 2007) says of a planet ascending in its eccentric circle (the deferent) that for the planet, it is the same to a horseman as having a horse with excellent legs. Further, a planet in its apogee is close to the zodiac, thus it resembles a soul; when it is low (besides its perigee) and close to the earth, it is more like a body instead. We see then that we have two conditions, one of strength and one of weakness. These two conditions are divided between moving towards strength, or being naturally strong and continuing in increase; likewise, we also see decrease in strength, followed by weakness.

This motion within the deferent here is essentially very much like the solar phases of the planets (their relationship in distance with the Sun), in that this cyclical and uniform motion reflects a period of motion and travel. From strength to weakness to strength once again. It is unceasing and unchanging, undisturbed. In much the same way, the motion of a planet by its latitude (southern or northern within the ecliptic belt) also follows the same pattern. A planet ascending to become northern is strong, especially when beside its ascending dragon (called the mean north node of the planet). A Planet descending is sapped of strength, especially when besides its descending dragon (called the mean south node of the planet). Indeed, the mean nodes are calculated from the apogee’s longitude! So we can see there is a correlation between these techniques.

Now, each of these two topics (both solar phases, and latitude/nodes) will receive more discussion in time, and I do also intend on writing on the calculation of a planet’s true position, so that we might consider its place in the epicycle and its relation to the epicycle’s apogee. However for now, hopefully this will suffice. Astrology is one of my most beloved passions, and I deeply enjoy discussing and teaching its mechanisms.


If you would like to purchase any astrological services from me, I offer long, in-depth, customizable and clear readings on this page. It would be my honour to be of service to you!


References:

Ptolemy, G.J. Toomer, The Almagest (1984).
Abū Maʿšar, Al-Qabisi, Benjamin Dykes, Introductions to Traditional Astrology (2010).
Abū Maʿšar, Keiji Yamamoto, Charles Burnett, The Great Introduction to Astrology (2019).
Al-Biruni, Ramsey Wright, The Book of Instruction in the Elements of the Art of Astrology (1934).
Abraham Ibn Ezra, Schlomo Schea, On Nativities and Continuous Horoscopy (2013).
Abraham Ibn Ezra, Schlomo Schea, The Book of Reasons (2007).
NASA, Planetary Fact Sheets, https://nssdc.gsfc.nasa.gov/planetary/planetfact.html.

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