In the Sagittarius 1980 issue of The Mutable Dilemma I discussed some of the basics of current pattern math (how to get the derived date and time—the time to use to get positions). This article is a continuation on that subject, covering angles for Secondary progressions. In the examples, I will use my own chart and birth data. The data is: February 9, 1955 at 3:23 A.M. MST in Tucson, AZ (32 N 15 9, 110 W 52 36) with relocation to Los Angeles, CA (34 N 3 24, 118 W 19).
In all systems of progressions or returns, once you have gone through the math to determine the derived date and time you can simply follow normal natal interpolation procedures (see the Pisces and Gemini 1978 issues of The Mutable Dilemma for these) for planets and cusps. For returns, the resulting cusps are the ones you want. For progressions, the resulting cusps are only one of several options—and not the most widely used one.
All systems of progressed angles and house cusps work by first getting a progressed Midheaven or sidereal time and then calculating the rest of the cusps. In directions, on the other hand, all cusps are moved by the same amount (the amount you are moving everything in the chart). The systems of progressed angles I will discuss in this article are Quotidian, Solar arc, degree-for-a-year, Naibod, and Right Ascension of Mean Sun. Most of these systems move the Midheaven by some arc, so the progressed MC is the same as the MC in the corresponding system of directions.
The only angles that actually use the day-for-a-year ratio used to get planetary positions are the Quotidian angles. (Quotidian is from a Latin root for day.) Quotidian angles are the angles mentioned above that result if you do standard natal math for the derived date and time. They are the angles you get by calculating the Local Sidereal Time to match the time you are interpolating to in the ephemeris. They move 361 degrees in a progressed year—the normal daily motion of angles. Because they move so fast they are impractical for looking at long periods but good for narrowing in. These angles are very easy to get with a computer: run a natal chart for the derived date and time and the angles are Quotidian. With my chart as an example (as usual), progressions for September 1 will involve interpolating to a UT of 23:48 because that is only 3 days before my ACD. For September 1, 1981, I will interpolate to 23:48 in the ephemeris for March 7, 1955. The 0 hour GST from the ephemeris of 10:55:45 plus 23:48:0 plus a solar sidereal correction of 0:3:55 gives a GST of 34:47:40 or 10:47:40. Subtracting my Tucson (110 W 52.6) longitude correction of 7:23:30 gives an LST of 3:24:10. My Quotidian progressed house cusps for September 1, 1981 will be cusps for a sidereal time of 3:24:10. If I want local angles, I just use the Los Angeles (118 W 19) longitude correction instead to get an LST of 2:54:24.
Solar Arc Midheaven angles are a very popular system. To get these angles, get the Solar Arc by subtracting the longitude of the natal sun from that of the progressed sun. Add this arc to the natal MC. From this MC, calculate the other cusps.
To continue using September 1 as an example, my progressed sun is at 16 Pisces 38. Subtracting my natal sun of 19 Aquarius 54 from this gives a solar arc of 26 degrees 44 minutes. Adding this to my natal MC of 3 Libra 56 gives a solar arc progressed MC of 0 Scorpio 40. To get the rest of the angles (and intermediate cusps if they are wanted), I need to be able to interpolate in a table of houses using this MC. One option interpolates directly without bothering to determine what sidereal time produces this MC. To do this, just determine the sidereal time interpolation fraction directly from the Midheavens in the table of houses. This fraction is the difference between the earlier MC in the table and the progressed MC divided by the difference between the two MCs in the table of houses that bracket the progressed MC. The closest MCs before and after 0 Scorpio 40 are at 0 Scorpio 6 and 1 Scorpio 8. The difference between the progressed MC and the earlier MC is 0°40’ - 0°6’ = 0°34’. The difference between the two table entries is 1°8’ - 0°6’ = 1°2’ = 0°62’. The fraction to interpolate between sidereal times is the partial difference divided by the total difference: 34/62 = .5484. If the sidereal time is wanted, this fraction can be multiplied by the difference in time between table entries and the result added to the earlier table time. In this case, 4 minutes times .5484 = 2.1936m = 2m12s plus the earlier table time of 13:52:0 gives a sidereal time of 13:54:12. Thus, the rest of my progressed angles to go with a solar arc MC of 0 Scorpio 40 are obtained by interpolating in the table of houses between the times 13:52:0 and 13:56:0 using a fraction of .5484.
For work with computers, which calculate cusps directly instead of using a table of houses, the sidereal time needs to be calculated directly from the progressed MC. The basic form of this calculation is: Y=SIN(MC) * COS(OBLIQUITY) : X=COS(MC) : ST=ARCTAN(Y/X) : IF ST<0 THEN ST=ST+180 : IF Y<0 THEN ST=ST+180. This returns a value for sidereal time in the correct quadrant which can then be used by the standard house cusp formulas.
NOTE: if you use relocation house cusps or angles, be sure to add the Solar Arc to the natal relocation MC to get the progressed relocation MC. Doing the relocation from the natal location progressed MC will produce a different, and incorrect, result. Continuing with the same example, my natal relocated MC is 25 Virgo 49. Adding the solar arc of 26°44’ gives a progressed relocated MC of 22 Libra 33.
Degree for a year (or 2 1/2 or 5 or any number of degrees for a year) adds the specified arc to the natal MC to get a progressed MC and calculates the rest of the cusps as described above for Solar Arc. Just add one degree for every year of age and a fraction of a degree for the fraction of the year since the last birthday. If you want extreme precision in the fraction of the year, use a table of days between dates and divide the resulting number by 365 (or 366 in leap years). Otherwise, assuming 30 days in every month as is done in getting the ACD will usually be close enough. Still working with the example of September 1, I would count September 9 as 7 months or 210 days and subtract 8 days to get back to the first (or take August 9 as 6 months and count forward 23 days—getting an answer 1 day different). Dividing 202 days by 365 days gives a fraction of a year of .5534, making my age 26.5534. The arc to add to the natal MC using the degree for a year system would thus be 26.5534 degrees or 26°33’. This is 11 minutes of arc less than the solar arc for the same date. To use any of the other constant arcs, just multiply the travel per year by the age. 2 1/5 degrees per year would be an arc of 2.5° times 26.5534 = 66.3835° = 66°23’. Five degrees per year would be 5 times 26.5534 = 132.767° = 132°46’. And so on for other arcs.
Naibod’s measure is a special constant arc like the degree for a year family. The arc is 59 arcminutes per year—the average motion of the sun. (Actually 360 degrees divided by 365.2422 days gives an arc of 59’ 8”.) Using the same age as with the degree for a year, the Naibod arc is 59’ times 26.5534 = 1566.65’ = 26.11° = 26°7’. (Using 59’ 8” instead of an even 59 minutes produces an arc of 26°10’.) This is again less than the solar arc since the sun is moving faster than average in Aquarius and Pisces.
The last technique I will discuss here does not work by adding a longitudinal arc to the natal MC. The RAMS techniques directly calculates a progressed sidereal time, making it more closely related to Quotidians. RAMS is short for Right Ascension of Mean Sun. The RAMS is the 3 minute 57 second increase in sidereal time from one day to the next. In fact, sidereal time is actually defined in terms of the position on the equator (Right Ascension) of a fictitious sun that always moves with exactly the average speed (Mean Sun) of the true sun. If you do Quotidians for the same day every year you will only see the one degree excess motion instead of the full 361 degree motion that actually occurs. The RAMS progressions use this one degree excess as the entire motion for a year. You can get RAMS motion by calculating the sidereal time at your time of birth on each successive day in the ephemeris and subtracting the natal sidereal time. A slightly easier technique is to just read the 0 (or 12) hour GST from the ephemeris for each day, subtracting the GST from the ephemeris for your day of birth. Be careful not to mix the two methods! Whichever way you get the RAMS motion, just add that to the natal LST to get a progressed sidereal time to calculate the cusps from.
Continuing my example, the change in RAMS for progressions for February 9, 1981 would be the March 7, 1955 GST minus the February 9, 1955 GST. (Whole days equated to whole years means we get birthday values.) The GSTs are: 10:55:45 - 9:13:15 = 1:42:30. To get up to September 1, I will multiply the change in GST from March 7 to March 8, 1955 by the fraction of a year from February 9 to September 1, 1981 that I determined above for the degree for a year system. (There is a shortcut for working within the year, but I use RAMS so seldom (never) that I have forgotten it.) The change in GST on March 7, 1955 is 10:59:42 - 10:55:45 = 0:3:57 (does this number look familiar—it should). The change in RAMS for the fraction of the year up to September 1 is 3m57s times .5534 = 2.186m = 2m11s. The total change in RAMS is thus 1:42:30 plus 0:2:11 = 1:44:41. Adding this to the natal LST of 12:14:26 gives a progressed LST of 13:59:7.
The above list does not exhaust the subject (just the writer and reader). There are many variations on the above techniques differing in just how the day and the year are defined. For most purposes, I suggest simply sticking to solar arc and not worrying about the rest.