Gauquelin Sector Calculations
Mark Pottenger
Gauquelin Sectors are defined as divisions of the sky based on diurnal (daily) motion. The sectors start at rise and go clockwise through upper culmination, set, and lower culmination. Different studies over the years have divided the sky into 36, 18 or 12 sectors, all obtained using the same principles. Comparing 36 sectors to Placidus houses, sector 1 is the third of the 12th house closest to the Ascendant, sector 9 is the third of the 10th house closest to the Midheaven, and on around to sector 36 in the third of the 1st house closest to the Ascendant.
For any readers who have never heard of the Gauquelin research, an extremely brief summary: Michel and Francoise Gauquelin have collected birth data for tens of thousands of people and have found planetary placements in sectors 36, 1, 2, 3, 9, 10, 11 and 12 (the astrological 12th and 9th Placidus houses plus one third of the 1st and 10th) strongly statistically significant in many studies of profession, heredity and character. The areas around the Descendant and IC show up, but less strongly. Read books by either of them for a lot more detail.
A calculation matching the definition would be done by calculating (or looking up) the exact times of the rise and set bracketing the moment of birth, then getting a proportion: (birth  rise) / (set  rise) for points above the horizon or (birth  set) / (rise  set) for points below the horizon. This proportion (decimal fraction) is multiplied by 180 to get a system measurable in degrees (and 180 is added to the answer for points below the horizon). This use of a proportion gets a continuous 360degree measure clockwise from the Ascendant out of what would otherwise be unequal diurnal and nocturnal intervals. This 360degree sector value can then be divided by 10, 20 or 30 to get the 36, 18 or 12 sectors usually used in studies and graphs. For hand calculations, you look up rise, culminate and set times in an almanac. Unfortunately, calculating an exact rise or set time on a computer is a process requiring several calculations of a planet's position for successively refined times. Since this must be done for each planet, this kind of sector calculation can be time consuming.
We have had a printout of Gauquelin Sectors in the CCRS Horoscope Program for years, and have encouraged astrologers to pay attention to the work of the Gauquelins. In the process of getting a series of tallies for CCRS working in the spring of 1987, I worked with these formulae again for the first time in years. I also got Tom Shanks at Astro Computing Services to look at his formulae in trying to get results to match exactly and clear up where discrepancies between our results were coming from.
The formula we have been using since the late 1970s calculates diurnal placements using a proportion based on meridian distance divided by semiarc. (Meridian distance is how far a point is from the Midheaven or IC, measured along the equator in Right Ascension. A planet's semiarc is one half the distance it will travel in moving across the sky from horizon to horizon. Diurnal semiarc is above the horizon, and nocturnal semiarc is below the horizon.) This formula produces almost, but not quite, the same answer as calculating the exact rise and set times. The difference comes because the calculations are based on celestial latitude and longitude at the moment of birth, and the body being studied will move slightly between the birth time and the rise or set time. For all practical purposes, the only body significantly affected is the Moon. A rise or set time calculated from birth positions will be slightly off. To see this, look at a paran printout for a chart, calculate a new chart for a Moon rise or set time given, then look at a paran printout for the new chart. The Moon rise or set time shown will now be slightly different. Repeating this procedure of calculating a new chart for the time obtained in the previous try until the time stops changing is the kind of iterative solution used to get exact rise and set times with a computer.
Tom Shanks at Astro Computing Services checked their sector formula for me. The formula they were using was expressed as a ratio of rise and set times as described above. While looking, Tom changed the formula to do a full iterative calculation of the exact rise and set times. The ACS formula is now as close a match as a computer can give to the definition of Gauquelin sectors.
After Tom changed the calculations at ACS, the discrepancies which had been worrying me went away and a test of the CCRS program with the 1794 alcoholics collected by Michel Gauquelin gave average discrepancies from Tom's results of well under a degree of sector position for everything except the Moon and the Moon had an average discrepancy of 0.757 degree.
For the CCRS program, I did a fix for the Moon and left the formula unchanged for the rest of the planets. I did not attempt to do a full calculation of exact rise and set times because that would have required adding planetary routines to three different program modules where sectors are calculated, and I feel that my Moon fix is good enough. The fix consists of calculating Moon rise and set times based on longitudes and latitudes estimated for the first approximation rise and set times based on birth positions. The estimated longitudes are birth positions plus or minus velocity times time difference. The latitudes are trickier, involving a sine function and the inclination. I was surprised by how important the latitude estimate was in the final result. As you can see from the table below, given in 360degree sector measure, the results are good enough for most work. (Most of the discrepancies are in the neighborhood of the possible differences between our planetary longitudes and latitudes.)
Average for 1794 charts
new ACS 
versus old CCRS 
versus new CCRS 
Sun 
0.002 

Moon 
0.757 
0.050 
Mercury 
0.002 

Venus 
0.003 

Mars 
0.013 

Jupiter 
0.006 

Saturn 
0.005 

Uranus 
0.004 

Neptune 
0.004 

Pluto 
0.004 

Another reason I am willing to let things go with this level of accuracy, aside from the practical program limitations, is that I found out from Francoise that the Gauquelins themselves always used Placidus positions (not taking latitude into account) in the years they did things by hand. Sector positions based on rise and set times from almanacs were suggested by Francoise as a theoretical improvement, but were not used except for some short trials because they took too much longer than Placidus positions to calculate. Rise and set based sectors really only started to be used when the Gauquelins started having their computations done at ACS. This is what finally convinced me that it is OK for people to use Placidus houses as an approximation of Gauquelin sectors if they don't have a program available that does the true diurnal sectors.
Other people have suggested using prime vertical longitudes to measure diurnal position. Prime vertical longitudes start with 0 at rise, 90 at culmination, and so on around. They can be calculated directly from other coordinates without use of proportions or corrections for the motion of the Moon, which might make them more appealing on a theoretical basis. Tom Shanks is doing a study comparing results of this system to Gauquelin sectors and Placidus houses, the monograph for which will soon be published.
If anyone wants to see the actual sector formula, look it up in the CCEXTRA1 module in the 1987 release of the CCRS Horoscope Program, or write for a program listing.