Natal Chart Math
Mark Pottenger
This article presents the most direct procedure that I am familiar with for doing a basic natal chart. I use a calculator with conversion from decimal degrees to degrees, minutes and seconds and back, and several memories. I urge anyone who does not already own such a calculator to buy one—they don’t cost much and are a pleasure to use.
If you have a calculator without dms (or hms) conversion, the manual conversion procedure is still fairly straightforward. To convert a figure in degrees (or hours), minutes and seconds to decimal degrees (or hours), divide the number of minutes by 60 (which gives a decimal) and add on the whole number of degrees (or hours). For example, if you have a time of 11:21 and want to convert it to hours plus decimal fraction, divide 21 by 60, getting 0.35 and add 11, getting 11.35. Thus, 11 hours (or degrees) 21 minutes equals 11.35 hours (or degrees). The same procedure works with minutes and seconds—divide the seconds by 60 and add on the minutes. To convert 11:21:32, for example: 32/60=0.5333 + 21=21.5333 /60=0.3589 + 11=11.3589. The above example shows a continuous set of operations converting seconds to minutes, then converting that result to degrees (or hours). The continuity is important because it reduces the amount of punching (none of those decimals were entered by hand), and increases the accuracy (the calculator carries more digits than it displays). Expressed in general terms, the rule for converting to decimal is: divide the smaller unit by 60 and add on the larger unit.
To convert from decimal to degrees, minutes and seconds, simply reverse the above procedure. Write down the whole number, then subtract it from the number on the calculator (which leaves the decimal remainder). Multiply the decimal by 60. If the answer is a whole number, that is your smaller unit. If the number still has parts after the decimal point, write down and then subtract out the whole number portion. Multiply the decimal remainder by 60 to get your next smaller unit. Working our earlier example the other way, 11.3589 [record 11] -11=.3589 x60=21.534 [record 21] -21=.534 x60=32.04 [record 32]. The numbers in brackets, which you write down as you go along, give 11:21:32. The procedure here is: always write down and subtract out the whole number, and multiply the remainder by 60 to get the next smaller unit.
A note on rounding. It is pointless to carry a calculation to levels of accuracy greater than that of the data one starts with, so round to the level of the original data. If I start with seconds, I continue to carry things to seconds, but if I just start with minutes, I only get my answer to the nearest minute.
Times of Birth
For each step that can be done in more than one way, I will try to present alternatives. The book I will use for almost everything is The American Ephemeris 1931-1980 & Book of Tables, which I will abbreviate as AE. This is the most comprehensive astrological reference in existence. I will use my own data as an example as I go along.
A habit to develop right from the start is to use the 24 hour (European, military, 0, 1, 2, 3, ... 12, 13, 14, ... 23, 24) clock to specify all times. Instead of wondering if an A.M. or P.M. got left off, you always know where you are.
1. The first step, of course, is to get the birth data: name (or quality being researched), date, time and place. It is also a good idea to write down the source of the data as an indication of trustworthiness (e.g. birth certificate, baby book, memory, etc.).
My data: Mark Pottenger; February 9, 1955; 3:23 (A.M.); Tucson, Arizona; Birth Certificate (copy).
If the person was born in a large city, it is a good idea to determine which neighborhood, borough or hospital.
2. Look up the latitude and longitude of the place of birth. Many atlases have an index with coordinates. There are two atlases from the American Federation of Astrologers (AFA) that have coordinates, and longitude-time corrections for Greenwich and for the time zone meridian (no maps). There is a list of cities with coordinates and the longitude-time correction to Greenwich in the back of the AE. If none of these sources list a city, you can read coordinates off of a map.
Tucson, AZ (from AE): 32n13; 110w58
Indicating your source here is also a good idea.
3. Look up time changes—determine what time zone the city was in (in the year of birth—many cities have changed zones between 1883 and the present) and whether daylight, war or double summer time was being observed. The best currently available reference on this subject is the Time Changes trilogy (U.S.A., Canada & Mexico, World) by Doris Chase Doane. If the books are ambiguous about the time and place you are looking up, check with local authorities as Doane suggests. If they don’t know, as sometimes happens, about the only other possibility is a search through old newspapers. Sometimes it is impossible to be sure of the birth time.
My zone & type of time (by Doane): Mountain Standard Time—meridian 105° W—7 hours from Greenwich.
4. Subtract one hour from recorded time if daylight time was in effect (two hours if double summer). Result is Standard Time:
(no correction needed for me—already Standard time).
5. Add the zone number (in hours) to the time of birth if west of Greenwich (Western hemisphere). Subtract the zone number (in hours) from the time of birth if east of Greenwich (Eastern hemisphere). The result is the Universal Time (UT) of birth (and the UTI (Interval) with a midnight ephemeris).
For me: 3:23 MST + 7 hours = 10:23 UT.
The zone number is really just a special (by virtue of being an even hour) longitude-time correction. If the UT obtained is more than 24, it is already the next day in Greenwich. Subtract 24 hours from the UT and add one day to the date of birth. If the UT is less than zero, it is the previous day in Greenwich. Add 24 hours to the UT and subtract one day from the date of birth. Always use the Greenwich date thus obtained when using an ephemeris. It is important to remember when doing manipulations like this that all we are really doing is saying the time of birth a different way—we are not changing it.
6. Look up the Sidereal Time in the ephemeris for the date of birth—this is the Greenwich Sidereal Time at 0 hours UT (in the AE or any other midnight ephemeris—12 hours UT with a noon ephemeris):
2/9/1955 GST at 0 hr UT = 9h13m15s.
7. Get the Solar-Sidereal Correction: Convert the UT (UT interval from the preceding noon with a noon ephemeris) from hours and minutes (and seconds, if any) to hours plus a decimal. This just means pressing a button on calculators that do the converting—otherwise, follow the procedure described above. Multiply this number of hours by 9.856 seconds per hour to get an answer in seconds. This is the amount that has to be added to an interval of solar time to convert it to an equivalent interval of sidereal time.
10:23 UT(I) = 10.3833 hr x 9.856 sec/hr =102.34 sec = 1 min 42 sec
This correction can also be obtained directly from table II in the AE. The table is simply the working out of the above procedure for each minute of a day.
8. Add the results of steps 5,6 and 7. That is: Universal Time (Interval) of birth plus Greenwich Sidereal Time at 0 hr plus Solar-Sidereal Correction. A solar time interval plus the sidereal time at the start of the interval plus the correction necessary to change the solar interval into a sidereal interval gives the sidereal time at the end on the interval. The result is the Greenwich Sidereal Time of birth.
10:23 UT
+ 9:13:15 GST 0 hr
+ 1:42 S-SC
=19:37:57 GST of birth
9. Determine the Longitude-Time Correction. This is the longitude of birth converted from degrees and minutes to hours, minutes and seconds. Longitude (in degrees) divided by 15 gives time (in hours)—convert from and to minutes and seconds in the usual manner. If, instead of dividing by 15, you multiply degrees of longitude by 4 you get an answer in minutes of time.
110w58 = 110.9667 /15=7.3978 = 7h 23m 52s
This value, as already mentioned, is in the AFA atlases and in the back of the AE, so it only needs to be calculated for cities not in these references. It can also be obtained from Table III in the AE which gives values for every degree up to 180 and separate values for every minute up to 60.
10. If west of Greenwich (Western hemisphere) subtract this Longitude-Time Correction from the Greenwich Sidereal Time obtained in step 8. If east of Greenwich (Eastern hemisphere) add the L-TC to the GST. This gets the Local Sidereal Time of birth, which is what will be used to look up house cusps.
19:37:57 GST of birth
- 7:23:52 L-TC (west)
=12:14: 5 LST of birth
11. This step and the next apply only with more modern ephemerides (such as the AE) which give planetary positions for midnight ephemeris time (ET). In older ephemerides positions are given in terms of Universal Time (UT), which was already obtained in step 5.
Look up for the year of birth the difference between UT and ET, which is called delta T (delta is a Greek letter used in math to stand for change), in a table (Table IV in AE).
delta T (for 1955) = +31 seconds
12. Add delta T to the Universal Time to obtain the Ephemeris Time, which will be used to obtain planetary positions.
10:23 UT + 31 sec delta T = 10:23:31 ET.
Planetary Positions
An Ephemeris is a table or book giving the positions of celestial bodies at regular intervals. Astrological ephemerides give the positions of the Sun, Moon and the planets other than Earth (and the ascending Node of the Moon). The positions are given once a day, and are from a geocentric viewpoint. The Moon is sometimes given every twelve hours, since it moves much faster than any other natural body in Earth’s sky. It is possible to make ephemerides with positions given more (or less) frequently, but one giving positions for every minute of time would not be very practical.
Older ephemerides give positions once a day at either midnight or noon Universal (Greenwich) Time. Modern ephemerides use midnight (0 hours) Ephemeris Time. Ephemeris Time is more uniform than Universal Time. The AE also gives the Moon position at noon ET, so it is every 12 hours. Since very few people are born at exactly midnight or noon ET, we usually have to interpolate between positions given in the ephemeris to get planetary positions at birth. (Interpolation is just getting numbers between those listed in a table.) You can interpolate to any fraction of the whole distance between two listed numbers, so you need to know what fraction you want.
13. The first step in interpolating planetary positions is to express the Ephemeris Time of birth as a fraction of the time interval between listings. When positions are given once a day, the fraction is ET/24 hours. If listed more or less often, the fraction is (ET since earlier listing) / (ET between listings).
10:23:31 ET=10.3919 / 24=.4330 (fraction of day)
10.3919 / 12 = .8660 (fraction of 12 hours)
The fraction of a day (24 hours) will be used for all planets except the Moon, which will use the fraction of 12 hours (because it is listed every 12 hours). When using a calculator with a memory, store the fraction of the day in memory (otherwise, write it down).
14. The next step is to get the daily motion of each planet (12 hour motion for the Moon with the AE). Daily motion is the change in position from the start of one day to the start of the next day. To get it, subtract the position at the start of the day of birth from the position at the start of the day after birth. (Remember that “day of birth” in dealing with planets really means “Greenwich date of birth”.) If the planet is retrograde (appearing to move backward), you can perform the subtraction just as with prograde (forward moving) planets and get a negative answer, or subtract the smaller number from the larger while noting the fact that the motion is retrograde.
Sun: Feb 10 20:28:22 Aquarius
Feb 9 -19:27:39 Aquarius
Motion 2/9 = 1:00:43
It is a good idea to express the daily motion in a single kind of unit—usually minutes (or degrees for the Moon). The answer above would be 60.7167 minutes (or 1.0119 degrees or 3643 seconds).
15. Next we multiply the daily motion by the fraction of the day to get the motion up to the time of birth.
60.7167 min x .4330 = 26.2902min = 26:17 min:sec
16. Now add the planet’s motion up to the time of birth (15) to the planet’s position at the start of the day (from the ephemeris or your scratch sheet).
Sun Feb 9 0 hr 19:27:39 Aquarius
Motion to birth + 26:17
Natal Sun = 19:53:56 Aquarius
The answer is the position of the planet at the moment of birth.
17. The procedure for every planet is a repeat of steps 14, 15 and 16: get daily motion, multiply by fraction of day (from memory of calculator) and add result to start of day position. If a planet is retrograde, and you work only with positive numbers, then subtract the partial motion from the start of day position. If the Moon is given every 12 hours, as in the AE, get the 12 hour motion and multiply by the fraction of 12 hours, then add the result to the position before birth. If something is listed less often than once a day, get the motion for the whole period and multiply by the fraction of the whole period.
The basic principle is the same for all the above variations: motion in total interval (14) x fraction (from calculator memory) of interval to ET of birth (13) = fractional motion or increment to ET of birth (15). (15) + position before birth (from ephemeris) = birth position (16). It doesn’t really matter whether the motion is positive or negative or how long the interval is.
Further examples:
Moon 12 hr Feb 9 20: 2:14 Virgo
0 hr Feb 9 -13:29:28 Virgo
12 hr Motion =6:32:46=6.5461deg
fraction of 12 hrs x .8660
increment 5:40:8= 5.6689 deg
earlier +13:29:28 Virgo
Natal Moon 19: 9:36 Virgo
Mean Node
0h Feb 9 3:21 Capricorn
0h Feb 10 -3:18 Capricorn
motion 3 min R
fraction x.4330
increment 1.3 min R (round to 1)
earlier 3:21 Capricorn
increment - 1
Natal 3:20 Capricorn
All the interpolations done here with a calculator can also be looked up in a table of diurnal motions, but the answers are not quite as precise. With a table you work out the daily (diurnal) motion as before, then look up the hours and minutes of the ET of birth. Where the column for a given motion meets the row for a given time, you have the fractional motion (increment). Table V in the AE is a diurnal motion table for the Sun. Table VI is a Semi-Diurnal (12 hour) motion table for the Moon (to go with the Moon being given every 12 hours). Table VII is a diurnal motion table for the rest of the planets.
Asteroid positions are obtained through the same procedures from a separate ephemeris.
House Cusps
A Table of Houses is a table or book giving the positions of house cusps at regular intervals. House cusps change with both time and latitude, so getting a cusp calls for double interpolation (planets just required single interpolation). Some Tables list cusps for every 1 o change of Midheaven, some for every 4 minutes of Sidereal Time. Most tables list cusps every 1 o of latitude. Unless someone’s Sidereal Time and latitude of birth both fall exactly on values in the table, interpolation is necessary.
18. The first step of interpolation, as before, is to get a fraction—two fractions now. The natal Sidereal time will fall somewhere between two Sidereal times listed in the table, so the very first thing to do is find those two closest entries. The Sidereal time fraction (STF) is the time between the earlier table entry and the natal time divided by the time between the earlier and later table entries.
My natal ST of 12:14:5 is between 12:12:0 and 12:16:0 in the AE.
Natal - earlier: 12:14:5 - 12:12:0 = 0:2:5 = 2.0833 min.
Later - earlier: 12:16:0 - 12:12:0 = 0:4:0 = 4 min.
ST fraction: 2.0833/4=.5208
The latitude usually also falls between two entries in the table, and has to be interpolated. The latitude fraction (LF) is the distance between the latitude below birth and that of birth divided by the distance between table entries. In any table giving cusps for every degree (60 min) of latitude, this is just minutes of natal latitude divided by 60.
Natal - below: 32:13-32:0=13 min
Above - below: 33:0-32:0=1:0=60min
Latitude fraction: 13/60=.2167
If your calculator has the memories for it, store both of these fractions—otherwise, write them down.
The Midheaven is independent of latitude, so it just has to be interpolated between Sidereal times. Also, if a person is born exactly on a latitude given in the table, one interpolation is all that is needed.
19. After you have your fractions, the next step is to get the cusp interval between sidereal times. Cusps always increase as Sidereal time increases so you don’t have to worry about backwards motion. Always subtract the cusp at the earlier sidereal time from that at the later sidereal time in the table.
MC from AE
later 4:22 Libra
earlier -3:16 Libra
interval 1: 6 = 66 minutes
20. After you have your cusp interval between sidereal times, multiply it by the sidereal time fraction to get the cusp increment up to the natal sidereal time.
66 min x .5208 = 34.4 min.
21. Add this cusp increment to the cusp in the table at the earlier sidereal time to get the cusp at the natal sidereal time.
MC earlier 3:16 Libra
increment + 34
Natal 3:50 Libra
For the Midheaven, which is independent of latitude, this is the complete process.
22. For other cusps, which change with latitude, there is more to do. Repeat steps 19, 20 and 21 for the latitude below birth and for the latitude above birth. This will get the cusp at the natal sidereal time for each latitude.
ASC 32N 33N Sagittarius
later 19:31 18:58
earlier -18:38 -18: 6
interval = 53min = 52min
ST fraction x.5208 x .5208
increment= 27.6min = 27.1min
earlier +18:38 +18: 6
natal ST =19: 5.6 =18:33.1
23. With the two sidereal time corrected cusps, you can now interpolate to the natal position. Going between latitudes, cusps sometimes increase and sometimes decrease. The situation is much like dealing with planets which can go either retrograde or direct. It is usually easier to work with positive numbers, so subtract the smaller cusp from the larger to get the cusp interval between latitudes.
ASC 19:5.6 - 18:33.1 = 32.5min interval.
24. Multiply the cusp interval between latitudes by the latitude fraction to get the latitude increment.
32.5 x .2167 = 7 min
25. Here it matters whether the cusp was increasing or decreasing as latitude increased. If the cusp was increasing, add the increment to the ST corrected cusp at the lower latitude. If the cusp was decreasing, subtract the increment from the ST corrected cusp at the lower latitude. Again, the principle is the same as with retrograde planets. (One way to remember is that the latitude fraction was the fraction of the distance going from the lower latitude toward the higher. So the increment gotten with that fraction should go from the cusp at the lower latitude toward the cusp at the higher latitude.)
lower lat cusp + or - increment = natal cusp
ASC 19:5.6 - 7min = 18:59 Sagittarius
26. The procedure for all house cusps other than the MC is a repetition of steps 19, 20 and 21 for the latitudes below and above birth, then steps 23, 24 and 25.
The AE has a procedure for using tables to get the increments in house cusp interpolation.
Summary
All interpolation involves just 4 basic steps: Determine what fraction of the interval between entries will reach the natal position. Get the total interval or distance change of the object or cusp. Multiply the interval by the fraction to get an increment of change. Move by this increment from the first entry toward the second.
1 Name, date, time, place
2 Latitude & Longitude (atlas)
3 Zone & type of time (Doane)
4 Standard time (-1 hour from recorded if daylight)
5 UT (Standard + zone) & Greenwich date
6 GST at 0 hr (ephemeris)
7 Solar-Sidereal Correction (calculator or table)
8 GST of birth (5+6+7)
9 Longitude-time correction (calculator or table)
10 LST (8 + or - 9)
11 delta T (table)
12 ET (5+11) Planets
13 Fraction of interval between entries (calculator)
14 (Daily) Motion (next day - birth day [Greenwich])
15 Part of day Motion (14 x 13)
16 Natal position (15 + [Greenwich] birth day position)
17 Repeat 14, 15, 16 for each planet (& 13 if different total intervals) Houses
18 a Sidereal time fraction & b Latitude fraction (calculator)
19 Cusp interval (later - earlier)
20 Increment (19 x 18a)
21 ST corrected cusp (earlier + 20)
22 For other than MC, repeat 19, 20, 21 for latitudes below & above birth—get a) cusp below & b) cusp above
23 Interval (a] 22a-22b or b] 22b-22a)
24 Increment (23 x 18b)
25 Natal cusp (a] 22a-24 or b] 22a+24)
26 Repeat 19-25 for all cusps other than MC.