Co-Arising Stars: Formula for the Paranatellonta

Paranatellonta and the formula for calculating rising and setting stars

The paranatellonta are another major feature of astrology that is occasionally utilized by classical authors, such as Firmicus Maternus, Manilus, and so on. We also find examples of these rising stars in later authors, influencing works such as the Astromagia of Alfonso and the Astrolabium Planum (or Astrological Optics as the English edition is known) of Johannus Angelus. They even find themselves appearing in the works of William Lilly and later renaissance authors, by their tables of “bright, dark and empty” degrees.

Now, the word paranatellonta (παρανατέλλοντα), literally “parallel rising” (or alternatively, συνανατέλλοντα, “synanatellonta” – rising simultaneously according to the Brill’s New Pauly) describes the rising of the fixed stars that occurs over the horizon. This is contrasted with the method of Ecliptic Projection given by Ptolemy, which sees use in various works including Anonymous 379 – in which, though he might imply use of the paranatellonta by his language, is in practice giving the ecliptic projections of the fixed stars in his work. This ecliptic projection method is also used by many astrologers today and in the classical period. It’s the most common and popular method of using the fixed stars, and itself one of much use and virtue.

Yet it is not the sole method of observing them, and, alongside the heliacal phases of the stars, the paranatellonta make up one of the three approaches to using the fixed stars in classical astrology.

Now, in ecliptic projections for the fixed stars, even if 8° Leo is rising in the Ascendant, it doesn’t mean that a fixed star whose ecliptic projection is at 8° Leo will also have its physical body also rising, because it is not directly on the Ecliptic (though some are and thus will). In other words, its body might be elsewhere, under or above the horizon. The paranatellonta however, refers to the more precise astronomical observation that we can use to determine the ascendant degree at the time a particular star’s physical body appears over the horizon. The name itself – co-arising, parallel rising, or rising alongside, however you spin it – is chosen because they rise at the same time as a particular degree of the zodiac, over the circle of the horizon. Thus they share a sympathetic relationship to the same said degree and exert their influence over it. The precise astronomical relationship here is going to be a subject in my upcoming Astrological Course, and we’ll leave those finer details for another time, though those familiar with the basics of astronomical coordinate systems will be able to understand these things just by what has already been written.

The concept of paranatellonta does indeed have a relationship to the “parans” of modern astrology, though there are also some distinctions and differences as well. Namely, as far as similarities go, the emphasis is upon the star’s physical body rising over the horizon (though we arguably also include culminating and setting stars as well) at the time of the Native’s birth.

As far as the differences, the first is that the paranatellonta emphasise the horizon. In other words, a star must be in the Ascendant or (arguably) culminating, setting and so on, to be considered relevant. It does not need to be regarding a planet to be effective or to have influence over the figure.

The second is that the paranatellonta are only effective at the time of birth itself – the exact measure of this seems to be based on the exact degree that the star co-rises with, if we consider the ‘images of the degrees’ given in Johannes Angelus & the Astromagia to give us an indication of this, although its worth thinking that Firmicus is clearly ascribing Sirius multiple degrees of influence as well. On the other hand, contemporary use of the parans considers them in relationship to the planets and horizon, and through a 24 hour period, whilst the paranatellonta emphasise the moment and degree of birth itself.

So we can see that the fundamental concept is very similar, and holds a similar basis; but there are distinctions as well.

Now, these paranatellonta also have a relationship with the decans, which makes sense considering that the decans were observed in relation to the horizon as well as their heliacal phases by the Egyptians, creating a nocturnal stellar clock – though we should distinguish the earlier Egyptian decans and the later use of decans in horoscopy.

We see them as especially important in judgements of physiognomy and the body in Astrology – which, is a far more important topic than is given credit at times. The body itself is a Nativity – a natal chart in and of itself – and the divinatory art of physiognomy as expressed in premodern Europe, much as chiromancy (or palmistry), was essentially looking at the celestial influences upon a person using the body as the medium and making divinatory statements based on their appearance. Likewise the paranatellonta are given a clear relationship to the body by Firmicus Maternus in his Mathesis – who, drawing on (I believe, as I am quoting from memory here) a lost work by Nechepso and Petosiris, gives the degrees that he calls full, and the degrees that he calls empty. These are related to the “bright, empty, dark, smoky” degrees found in later Astrological literature.

The list given by Firmicus Maternus includes Sirius as the most reliably identifiable star, and gives it to the 7th to 11th degrees of Cancer. However, the ecliptic projection of Sirius was in Gemini during the period Firmicus was writing, and the same for any sources he would have been using. On the other hand, if we look at the rising of Sirius over the horizon, during 100 BC the value we have is roughly 12 degrees of Cancer using the modern zodiac. Of course, there would naturally be offset from the zodiac used by our classical authors by some measure, but, it is still approximately close enough.

Correcting these tables (assuming they need it – as I think they do) is a difficult task; the paranatellonta vary as to which star rises, and which stars set over the horizon at any given time based on latitude. Thus any such efforts to do so would need to be able to calculate them for the given latitude as well as precession. Probably an impossible task, all in all.

But we can see here the relationship the fixed stars and paranatellonta have with the appearance. This is resembling to the decans who themselves, in astrological literature, tend to represent the corporeal form or body of a thing that they signify, as well injury to the same by disease or accidents. We find examples of this relationship in the works of Sahl Bin Bishr (again quoting from memory, please comment if a correction is needed!) who cites Al Andrazaghar on the use of the Face-Lord in judging appearance and also Julian of Laodicea, recently translated by the Horoi project. Thus we see that the co-arising stars have a strong share and influence over the body of the Native, as well as their activity and behaviour.

This brief cursory look is just a beginning, but I leave you with the formula below – so that you can calculate the time that a particular fixed star might rise or set over the horizon, as well as the formula to determine which stars will rise or set.

Notes to the Formulary

A quick note: I do not have a background in mathematics, save for entry level programming in Javascript, BASH & a small amount of C#. As such please forgive my crude formatting and the fact that I am probably breaking several mathematical conventions. Furthermore, I wish to note my sources used here. Firstly, I have drawn upon the formula given by Robson in her work on the Fixed Stars, and the formula for the Ascendant and Midheaven given by Radixpro. I have removed the conversion of LST to RAMC (or vice versa) in both cases as a result, and redacted Robson’s use of the Logarithmic Trigonometry since it is unnecessary – though those who wish to make use of them will find modernized formula for these logarithmic tables following the example as a helping hand to those who are interested in using parans and calculating them by hand.

The Formula

Sin-1 (Tan (δ) x (Tan (φ)) = AD

90N + AD = H

or

90S – AD = H

α – H = Rising RAMC

α + H = Setting RAMC

sin RAMC / cos RAMC x Cos ε = ƛMC

Tan1 (cos RAMC / -(sin ε x tan φ + cos ε x sin RAMC)) = ƛAsc

1. Breakdown of the Formula

First we need to take the declination of the fixed star (δ), the right ascension of the fixed star (α), and the terrestrial latitude (φ).

2. Calculate the Ascensional Difference (AD) for the star, as follows:

Sin-1 (Tan (δ)) x (Tan (φ))

In the windows calculator for example, this is as follows:

Tan δ x Tan φ = Dif

Sin-1 (Dif) = AD

3. Use one of the following, depending on whether the star is southern or northern

90N + AD = H

or

90S – AD = H

4. Calculate the rising RAMC of the star as follows, using the Right Ascension (α)

α – H = Rising RAMC

5. Calculate the setting RAMC of the star as follows, using the Right Ascension (α)

α + H = Setting RAMC

6. Now we calculate the MC, before calculating the Ascendant; so that we can determine the rising time of the star. This is a simple formula, divided into a few steps, and should give you the chance to familiarize yourself with these calculations before the more complicated Ascendant. The formula is as follows: to calculate the zodiacal longitude (ƛ) of the MC, using the RAMC we calculated above, and the obliquity of the ecliptic, which we can approximate to 23.4371 for modern dates but ideally, we use a more precise amount, especially in dealing with trigonometry.

ƛMC = sin RAMC / cos RAMC x Cos ε

Simplified, it is as so:

Step 6.1:: sin RAMC = SinRam

Step 6.2: cos RAMC x Cos ε = CosRE

Step 6.3: SinRam / CosRE = Result

Step 6.4: tan-1 (Result) = ƛMC

Note that if the RAMC is under 180, it will fall between 0 Aries and 29°59’ of Virgo. If it is greater than 180, then it will be between 0° Libra and 29°59’ of Pisces. If it is not meeting these requirements, add or subtract 180 as necessary to produce the result sought.

7. Now we use the following formula to calculate the Ascendant from the RAMC. To do this we need our RAMC from earlier steps, and terrestrial latitude (φ) from step 1. Finally we need the obliquity of the ecliptic (ε). This can be found easily online or via its own relevant formula. Then we consider the following formula:

Tan1 (cos (RAMC) / -(sin ε x tan φ + cos ε x sin RAMC)) = ƛAsc

However, we can simplify this as follows into six steps.

7.1: cos RAMC = CosRam

7.2: sin ε x tan φ = sintanEL

7.3: cos ε x sin RAMC = cossinERAM

7.4: sintanEL + cossinERAM = Negative Number1

7.5: – Negative Number1 = Negative Number2

7.6: CosRam / Negative Number2 = Result

7.7: Tan-1 Result = ƛAsc

From this we have our ascendant axial degree – and remember that 00.00 is 0 Aries; Whilst 359.99 is 29.59’ of Pisces. It must be in a sign following, or to the left; of the MC. So if the result is not appropriate, then you must apportion 180 to the ascendant longitude as necessary.

Examplum

Let us take the star of Regulus, on 7 Sep 2022; and as for our local horizon, let us say that the Native is from Winchester, Hampshire.

1. First we need to take the declination of the fixed star (δ), the right ascension of the fixed star (α), and the terrestrial latitude (φ). [I have also included the obliquity of the Ecliptic (ε) since we will need it later.]

φ: 51°05’ or 51.08 North

δ: 11°51’ or 11.85

α: 10h9m32s, or 152.25

ε: 23.4382260812

2. Calculate the Ascensional Difference (AD) for the star, as follows:

Sin-1 (Tan (δ)) x (Tan (φ))

In the windows calculator for example, this is as follows:

Tan δ x Tan φ = result

Sin-1 (result) = AD

Let us take the Ascensional difference, by observing the formula given, and it gives us:

tan(11.85) x tan(51.08) = result 0.2598493562969941021326313835425

We then use the arcsin on result to generate our AD:

sin-1 0.2598493562969941021326313835425 = AD 15.061123671264834812596990682949 (or 15°03’)

3. Use one of the following, depending on whether the star is southern or northern

90N + AD = H or 90S – AD = H

Now, since the star of Regulus is of northern declination; and we likewise are northern, we add his ascensional difference to 90, so we can make H.

90.00 + 15.061123671264834812596990682949 = 105.06112367126483481259699068295

4. Calculate the rising RAMC of the star as follows, using the Right Ascension (α)

α – H = Rising RAMC

Then following this, we calculate the RAMC of the star by subtracting the same from his RA, and it gives us a sum of 47.18887632873516518740300931705 in RAMC. Thus the midheaven shall have the Right-Ascension of 47 degrees and 11 minutes, when the star rises over the Ascendant. Note that we will dispense with looking for the setting time (step 5) here to keep the example simple.

6. Now we calculate the MC, before calculating the Ascendant; so that we can determine the rising time of the star. This is a simple formula, divided into a few steps, and should give you the chance to familiarize yourself with these calculations before the more complicated Ascendant. The formula is as follows, to calculate the zodiacal longitude (ƛ) of the MC, using the RAMC we calculated above, and the obliquity of the ecliptic, which we can approximate to 23.4371 for modern dates but ideally, we use a more precise amount, especially in dealing with trigonometry.

ƛMC = sin RAMC / cos RAMC x Cos ε

Simplified, it is as so.

Step 6.1:: sin RAMC = SinRam

Step 6.2: cos RAMC x Cos ε = CosRE

Step 6.3: SinRam / CosRE = Result

Step 6.4: tan-1 (Result) = ƛMC

So, using the RAMC to determine the longitude of the midheaven in the zodiac, we must proceed as follows:

The Sine of the RAMC is 0.73359794075541645499224468998881

The obliquity of the ecliptic is 23.4382260812 and when cosined and multiplied with the cosine of the RAMC, it gives us 0.62351091696510862919932339243141.

So, we divide this sum from the sine of the RAMC, giving us 1.1765598978220749303357600844386. Make an arc-tan with it and we are given 49.637609676335657842990870332858.

Converted to DMS (this means at the time of the stars rising) it will be at 49°38’ degrees of absolute longitude. IE: 19°38’ Taurus will be on the MC.

Note, that if we consider the Swiss ephemeris table of houses for the right ascension of the Midheaven, the answer is approximately the same. (Possibly give or take an insignificant deviation of a few minutes – I haven’t interpolated the table of houses).

Intermission

Now, before we move onto calculating the Ascendant (which is more useful if we would construct a table, for example) we ought to examine the figure, and determine the evidence to show that this works.

Here we have the Chart as I have designed it. We can see the Ascendant is 29’45’’ Leo. The Midheaven is at 19’38’’ of Taurus, as we observed earlier.

Now, if we look at the ‘parans’ in the Solarfire program reports, we can see that connecting to and rising over the Ascendant is Regulus, with only about a minute’s difference to our manual calculation.

In this respect then, we can see that it works effectively for the calculations. But still, if we had the wish to construct a table, for example, then it is good that we know the rising degree as well.

Back to our example!

7. Now we use the following formula to calculate the Ascendant from the RAMC. To do this we need our RAMC from prior steps, and terrestrial latitude (φ) from step 1. Finally we need the obliquity of the ecliptic (ε). This can be found easily online or via its own relevant formula. Then we consider the following formula

Tan1 (cos (RAMC) / -(sin ε x tan φ + cos ε x sin RAMC)) = ƛAsc

However, we can simplify this as follows, into six steps.

7.1: cos RAMC = CosRam

7.2: sin ε x tan φ = sintanEL

7.3: cos ε x sin RAMC = cossinERAM

7.4: sintanEL + cossinERAM = Negative Number1

7.5: – Negative Number1 = Negative Number2

7.6: CosRam / Negative Number2 = Result

7.7: Tan-1 Result = ƛAsc

So in this example, we have our RAMC, and we make the cosine of it to be as follows which we will save for later. (Cosine of the RAMC: 0.67958374121178950478721381305492)

Now we take the sine of the Obliquity of the Ecliptic and multiply it via the tangent of the Terrestrial Longitude, and it gives us this result. 0.49259755429671712792119030118711

We then take cosine of the obliquity of the ecliptic, and multiply this via the sine of the RAMC. This gives us the following. 0.67306837551544119543796432811591

We add these two values together to produce the following:

1.165665929812158323359154629303

But we then negate it, so that it is written as such instead:

-1.165665929812158323359154629303

Now we take that Cosine of the RAMC, and divide it by the negative number:

0.67958374121178950478721381305492 / -1.165665929812158323359154629303

This shall give us the following result:

-0.58300043248351719733908295153597

And this we make the arctan, or tan-1 and the value that is returned will be our absolute longitude. However, in this example we must also add 180 Degrees, since the result is -30 and would place us elsewhere.

tan₀⁻¹ ( -0.58300043248351719733908295153597 ) = -30.242203862335842420585487984185

-30.242203862335842420585487984185 + 180 = 149.75779613766415757941451201582

Converted to degrees this is 149 degrees and 45 minutes, 28 seconds of absolute longitude. So we thus know the Ascendant will have the 29th Degree of Leo, 45 minutes and 28 seconds, just as it is also affirmed in the figure above. And if we wished to include it in a table for the same star, we could do so.

Co-Latitude and which stars can rise or set

Note that not every star will rise and set however, depending on the latitude we look. Therefore we must look to its colatitude:

1. Determine the Co-Latitude. This is found by subtracting the latitude of a place from 90. Thus, Alexandria has the latitude of 32°10’ (or 32.17 decimal).

90° – 32°10’ = 57°50’N

2. Any star with a greater declination than the co-latitude in the same hemisphere cannot set. Any star with a greater declination than the co-latitude in the other hemisphere cannot rise. As an example, if our co-latitude for Alexandria is 57°50N, then a fixed star at 58°N cannot set; and a fixed star at 58°S cannot rise over the horizon.

A final note on Robson’s formularies

Though I don’t consider the aspects with the paranatellonta, I wanted to make a note for those who are interested in using the above formula for their own use of parans. The formula given by Robson was making use of logarithmic tables, and converting these to a calculator can be troublesome. Whilst I won’t give the full formula here, as a helping hand for the interested parties who might want to use the same, the method of deriving the logarithms is as follows. For example, if we wanted to observe the logarithmic tangent of a particular stars declination:

log(tan(δ)) + 10 = LTδ

In the Windows calculator this would be as follows (make sure you are using scientific mode):

Tan δ = δt

log δt = δtl

δtl + 10 = LTδ (logarithmic tangent of the declination)

For example:

log(tan(11.85)) + 10 = 9.3218506117750843878824221985422

And finally, a note on usage of these formulas

If any readers, students, and fellow astrologers are interested in using these formulas in their publications and programs, you have my permission and encouragement to do so! All I’d ask you to give me a shout out as well, and ideally a link to either this article/our blog, or the website of our upcoming platform for hosting classes, soon to be featuring work from myself, Sfinga, and Key: mercurii-school.com. The website isn’t live yet, but expect updates very shortly on the first course: a 106 lesson tour of traditional astrology and magic by yours truly. I named my sources, and would appreciate being named in turn!

Many thanks for your time, and I hope you find this useful!

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