Discussion:
New Waites Compendium / Natural Graduation System
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S***@aol.com
2005-06-04 05:18:49 UTC
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Does anyone happen to have a copy of The New Waites Compendium Of Natal
Astrology, which has "universal tables of houses" that include Colin
Evans' "natural graduation" method? Or does anyone happen to know the
formulas for that method? I'm trying to work them out (I'm taking stock
of different systems), and I've come up with four different possible
ways to calculate them-- one that gives the same results as StarLogin
(and which seems to match a brief description that I found on the web),
another that gives the same results as Astrolog (which I think may be
wrong), and two other ways. I have Waites Compendium, but can't find
it, and I want to check the different results against the tables that
Colin Evans included in that book.

Michael Rideout
Ray Murphy
2005-06-07 06:13:08 UTC
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----------
Post by S***@aol.com
Does anyone happen to have a copy of The New Waites Compendium Of Natal
Astrology, which has "universal tables of houses" that include Colin
Evans' "natural graduation" method?
[....]

RM: As a matter of interest, I've never heard of the book, or of
universal tables of houses, so it's probably pretty rare.
Post by S***@aol.com
Michael Rideout
Ray
S***@aol.com
2005-06-16 05:51:08 UTC
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Post by Ray Murphy
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Post by S***@aol.com
Does anyone happen to have a copy of The New Waites Compendium Of
Natal Astrology, which has "universal tables of houses" that
include Colin Evans' "natural graduation" method?
[....]
RM: As a matter of interest, I've never heard of the book, or of
universal tables of houses, so it's probably pretty rare.
Post by S***@aol.com
Michael Rideout
Ray
The term "universal tables of houses" refers not to a specific house
system, but to a set of tables which include many different house
systems-- in other words, Placidus, Regiomontanus, Campanus, Natural
Graduation, and so on.

I do have the book in a box somewhere, but can't find it right now.
When I finally locate it, I'll post the results of my investigation.

In the meantime, I received an email from Martin that quoted a brief
passage from another book, which described the Natural Graduation well
enough to confirm my tentative understanding of it, and which
essentially confirmed another brief description that I had found on the
internet. The results of this method-- which I call "Natural Graduation
A" (see below)-- agree with the house cusps given by the StarLogin
freeware program, but I was trying to see if they agree with the values
in the house tables which Colin Evans gave in "The New Waite's
Compendium."

The Janus commercial software program also includes the Natural
Graduation house system, so if anyone here happens to have Janus, or
access to it, I'd love to get some confirmation from you, so I can
determine whether the results from Janus agree with the results from
StarLogin.

In trying to "reverse engineer" the Natural Graduation system, I have
come up with at least five possible variations of it, which I am
calling A, B, C, D, and E. Variation A is the one that agrees with
StarLogin, and which conforms to every description of the system that
I've yet seen. Variation B is like A, but is based on whole houses
instead of half houses (see below). Variations A and B are based on a
geometric progression, or multiplication (see below). Variation C is
like variation A, but uses an arithmetic progression, or addition.
Variation D is like variation C (arithmetic), but is based on whole
houses instead of half houses. And variation E is similar to variation
D (arithmetic, and using whole houses), but the progression is not
constant. Variation E gives the house cusps referred to as
"Neo-Porphyry" by the Astrolog freeware program, which I had always
assumed was just another name for the Natural Graduation system, but it
appears that I may have been mistaken about that. If so, then I don't
know who devised the Neo-Porphyry system, or if it might have been a
faulty implementation of the Natural Graduation system.

Anyway, the basic concept behind Colin Evans' Natural Graduation system
is that the Porphyry system is generally correct, but that the division
of the quadrants should produce houses which gradually and smoothly
increase in size, starting from the center of the smaller quadrants,
and ending at the center of the larger quadrants. If we bisect the
quadrants, we can label their midpoints as M1, M2, M3, and M4, where
M1=ASC/IC, M2=IC/DSC, M3=DSC/MC, and M4=MC/ASC (i.e., M1 is the
midpoint of Quadrant I, M2 is the midpoint of Quadrant II, M3 is the
midpoint of Quadrant III, and M4 is the midpoint of Quadrant IV). It
should be noted that these four points, or midpoints, will be exactly
square or opposite each other. And if we determine the cusps of the
houses in the Eastern Hemisphere (or Quadrant I and Quadrant IV), then
the house cusps of the Western Hemisphere will be their opposites. So
we only need to worry about the MC, ASC, M1 (or ASC/IC), and M4 (or
MC/ASC), with M1 being exactly square M4.

The idea is that, if we know the ASC and the MC, then we can easily
calculate M1 (ASC/IC) and M4 (MC/ASC), and use the distances between
the ASC and M1, as well as the distance between the ASC and M4, to
determine the sizes of houses 11, 12, 1, and 2. Then we can use these
house sizes, along with M1 or M4, to determine the cusps of all twelve
houses.

Variation A -- This is consistent with the descriptions I have seen for
the Natural Graduation system, so it is presumably the "correct"
variation. The region between M1 and M4 is divided into six half houses
(such that the entire circle is divided into 24 half houses), with a
constant ratio or multiplying factor existing between each subsequent
half house, from the middle of the smaller quadrant to the middle of
the larger quadrant.

In other words, let's suppose that Quadrant I is smaller than Quadrant
IV. Then the progression will start at M1 (which is the midpoint of the
2nd House), and continue to M4 (which is the midpoint of the 11th
House). If we divide the 90 degrees between M1 and M4 into six half
houses, then we can label these half houses as A (first half of 2nd
House), B (second half of 1st House), C (first half of 1st House), D
(second half of 12th House), E (first half of 12th House), and F
(second half of 11th House). In that case, A will be the smallest half
house, and F will be the largest half house, with each half house being
a constant multiple of the previous half house, such that
B/A=C/B=D/C=E/D=F/E=X, where X is the constant multiple. Thus, B=A*X,
C=B*X, D=C*X, E=D*X, and F=E*X. We can simplify this to B=A*X, C=A*X^2,
D=A*X^3, E=A*X^4, and F=A*X^5.

Since M1 - M4 = 90 degrees, and the arc between M1 and M4 is divided
into the half houses A, B, C, D, E, and F, it follows that
A + B + C + D + E + F = 90 degrees, or
A + A*X + A*X^2 + A*X^3 + A*X^4 + A*X^5 = 90 degrees, or
A * (1 + X + X^2 + X^3 + X^4 + X^5) = 90 degrees, or
A = 90 / (1 + X + X^2 + X^3 + X^4 + X^5).

Also, M1 - ASC is half the width of Quadrant I (which for now we are
assuming is the smaller quadrant), so if we define M=(M1-ASC)/2, we
also get
A + B + C = M, or
A + A*X + A*X^2 = M, or
A * (1 + X + X^2) = M.

Note that, if we know the ASC and the MC, then we also know M, but we
want to find A and X. I suppose we can solve this somehow, but I am
doing it by a recursive routine. That is, I pick a value for X, divide
90 by (1+X+X^2+X^3+X^4+X^5) to find A, and then multiply A times
(1+X+X^2) to see how close it is to M. Then I pick another value for X,
solve for A, see how close this comes to M, and keep picking new values
of X until I can find the value which satisfies the two equations given
above.

For example, suppose that ASC = 0 Aries, and MC = 0 Sagittarius. Then
M1 = 0 Taurus, and M4 = 0 Aquarius. Also, M = 30 degrees. It happens
that X must be greater than or equal to 1, so we can start by
considering X=1. Then we get A=90/(1+X+X^2+X^3+X^4+X^5)=90/6=15. This
then gives us A*(1+X+X^2)=15*3=45, which is greater than 30 (or M). If
we keep trying new values of X, and increase or decrease X until we can
match M, we end up with a value between 1.2599210 and 1.2599211 for X.
(My spreadsheet program doesn't go beyond 7 decimal places, so I am
stopping there.) I'll use 1.2599211 for X. Then, if we plug that value
into A=90/(1+X+X^2+X^3+X^4+X^5), we get a value of 7.7976305 for A.
This means that
A = 7.7976305 degrees,
B = 9.8243992 degrees,
C = 12.3779679 degrees,
D = 15.5952629 degrees,
E = 19.6488008 degrees, and
F = 24.7559387 degrees.

For the sizes of the houses, we get
H2 = A + A = 15.5952610 degrees,
H1 = B + C = 22.2023671 degrees,
H12 = D + E = 35.2440637 degrees, and
H11 = F + F = 49.5118774 degrees.

Note that the sizes of the other houses are H3=H1, H4=H12, H5=H11,
H6=H12, H7=H1, H8=H2, H9=H1, and H10=H12.

If we start at the MC, and add the sizes of each house, we get
MC = 240 = 0:00 Sag
Cusp 11 = MC+H10 = 240+35.2440637 = 275.2440637 = 5:15 Cap
Cusp 12 = C11+H11 = 275.2440637+49.5118774 = 324.7559411 = 24:45 Aqu
ASC = C12+H12 = 324.7559411+35.2440637 = 360.0000048 = 0:00 Ari
Cusp 2 = ASC+H1 = 0.0000048+22.2023671 = 22.2023719 = 22:12 Ari
Cusp 3 = C2+H2 = 22.2023719+15.5952610 = 37.7976329 = 7:48 Tau.

Variation B -- This is the same as variation A, except we use whole
houses instead of half houses. Thus, if we divide Quadrant IV and
Quadrant I into six houses, the sizes of the houses (beginning at the
IC and moving backward to the MC) can be given as B, A, B, C, D, and C,
where B is the size of houses 3 and 1, A is the size of house 2, C is
the size of houses 12 and 10, and D is the size of house 11, with A
being the smallest, D being the largest, and B=A*X, C=B*X, and D=C*X.
If we call M the size of the smaller quadrant (which we are supposing
is Quadrant I), then we have
B + A + B + C + D + C = 180 degrees, or
A + 2*A*X + 2*A*X^2 + A*X^3 = 180, or
A * (1 + 2*X + 2*X^2 + X^3) = 180, or
A = 180 / (1 + 2*X + 2*X^2 + X^3).
We also have
B + A + B = M, or
A + 2*A*X = M.

I'm not going to discuss this variation further for now, but it could
be solved recursively as in variation A.

Variation C -- This is like variation A, but it uses an arithmetic
progression instead of a geometric progression. Thus, if we label the
half houses and other points as we did in variation A, we get
B = A + X,
C = B + X,
D = C + X,
E = D + X, and
F = E + X.
Thus,
A + B + C + D + E + F = 90 degrees, or
A + A+X + A+X+X + A+X+X+X + A+X+X+X+X + A+X+X+X+X+X = 90, or
6*A + 15*X = 90, or
6*A = 90 - 15*X, or
A = (90 - 15*X) / 6, or
A = 15 - 5*X / 2.
Also,
A + B + C = M, or
A + A+X + A+X+X = M, or
3*A + 3*X = M, or
3*(15-5*X/2) + 3*X = M, or
45 - 15*X/2 + 3*X = M, or
90 - 15*X + 6*X = 2*M, or
90 - 9*X = 2*M, or
90 - 2*M = 9*X, or
9*X = 90 - 2*M, or
X = (90 - 2*M) / 9, or
X = 10 - 2*M / 9.

This variation is much simpler to solve, because if we know ASC and MC,
then we know M, and we can easily find X and A, then get B, C, D, E,
and F. I won't give an example right now.

Variation D -- This is like variation C, except using whole houses
rather than half houses. Thus,
B + A + B + C + D + C = 180 degrees, or
A + A+X + A+X + A+X+X + A+X+X + A+X+X+X = 180, or
6*A + 9*X = 180, or
6*A = 180 - 9*X, or
A = (180 - 9*X) / 6, or
A = 30 - 3*X / 2.
Also,
B + A + B = M, or
A + A+X + A+X = M, or
3*A + 2*X = M, or
3*(30-3*X/2) + 2*X = M, or
90 - 9*X/2 + 2*X = M, or
180 - 9*X + 4*X = 2*M, or
180 - 5*X = 2*M, or
180 - 2*M = 5*X, or
5*X = 180 - 2*M, or
X = (180 - 2*M) / 5, or
X = 36 - 2*M / 5.

Note that in this case, M is equal to the whole smaller quadrant,
rather than half of the smaller quadrant.

Variation E -- This is similar to variation D, and also uses whole
houses as in variation D, but we have
B = A + X,
C = B + 2*X, and
D = C + X.
In this case, the amount of the increase (or X) is not constant, but is
instead doubled between B and C. Thus,
B + A + B + C + D + C = 180 degrees, or
A + A+X + A+X + A+X+X+X + A+X+X+X + A+X+X+X+X = 180, or
6*A + 12*X = 180, or
6*A = 180 - 12*X, or
A = (180 - 12*X) / 6, or
A = 30 - 2*X.
Also,
B + A + B = M, or
A + A+X + A+X = M, or
3*A + 2*X = M, or
3*(30-2*X) + 2*X = M, or
90 - 6*X + 2*X = M, or
90 - 4*X = M, or
90 - M = 4*X, or
4*X = 90 - M, or
X = (90 - M) / 4, or
X = 22.5 - M / 4.

Remember, in this case (as in variation D), M is the whole smaller
quadrant, rather than half of the smaller quadrant. This variation
produces the house cusps which the Astrolog freeware program calls the
Neo-Porphyry system.

The interesting thing about the arithmetic-progression variations (or
variations C, D, and E) is that they produce the same type of pattern
as found in the Porphyry system, where the degrees and minutes of
houses 2, 6, 8, and 12 are the same, and the degrees and minutes of
houses 3, 5, 9, and 11 are the same. On the other hand, the
geometric-progression variations (or variations A and B) do not fit
that pattern. The reason I mention this is because, the way I remember
it (and my memory is admittedly hazy), the house tables which Colin
Evans gave in "The New Waite's Compendium" fit the Porphyry pattern,
which would indicate that he actually used an arithmetic progression
rather than a geometric progression, even though his description of the
method clearly suggests a geometric progression. That's why I want to
check with the tables of houses in "The New Waite's Compendium."

Until I manage to locate my copy of that book, I would appreciate it if
someone with the Janus program could erect a chart using the Natural
Graduation system, and post the house cusps as given by the Janus
program, so I can compare them with the results of the five variations
described above. Please use the following chart data:

January 1, 1950
6:00 p.m. GMT
0W00, 50N00

Michael Rideout
Ed Falis
2005-06-16 12:10:12 UTC
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Post by S***@aol.com
January 1, 1950
6:00 p.m. GMT
0W00, 50N00
X: 11 Ar 45
XI: 14 Ta 50
XII: 15 Ca 00
I: 03 Le 20
II: 29 Le 05
III: 16 Vi 00
S***@aol.com
2005-06-17 07:01:25 UTC
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Post by S***@aol.com
January 1, 1950
6:00 p.m. GMT
0W00, 50N00
X: 11 Ar 45
XI: 14 Ta 50
XII: 15 Ca 00
I: 03 Le 20
II: 29 Le 05
III: 16 Vi 00
Wow, that was fast! I was worried that maybe no one in this newsgroup
has the Janus program! Thank you, Ed!

Okay, this is what the StarLogin freeware program says:

X: 11.762 (or 11:45:43 Ari)
XI: 45.856 (or 15:51:22 Tau)
XII: 89.253 (or 29:15:11 Gem)
I: 123.347 (or 3:20:49 Leo)
II: 147.953 (or 27:57:11 Leo)
III: 167.156 (or 17:09:22 Vir)

As you can see, the Natural Graduation house cusps given by StarLogin
do not agree with those given by Janus.

This is what the StarLogin Help says about the Natural Graduation house
Colin Evans
House system based on an algebraic "geometric" determination of
the houses angular size.
[StarLogin Help, STARLOGIN Version 5.1.0, Joseph Stroberg and Francois
Deschamps, 1998-1999.]

This is how the Natural Graduation house system is described by one web
11. Natural Graduation System
Von dem Astrologen Colin Evans erfundenes Haussystem und in der
Hauptsache eine Verbesserung des Haussystem des Porphyrius. Das erste
Haus entspricht dem Aszendenten, das Zehnte dem MC, "the rest by
joining half-houses which increase in a constant ratio from midpoint
of a smaller arc to mid-point of larger arc".
[http://www.homoeopathie-online.com/bunkahle.com.usenet.1994.1995/Haus.sys.04.03.1994.txt]

I can't read German, but following is my translation-- really Google's
automatic translation, but I changed the words and phrasing as seemed
11. Natural Graduation System
House system invented by astrologer Colin Evans, and primarily an
improvement of the Porphyrius house system. The First House begins at
the Ascendant, the Tenth at the MC, "the rest by joining half-houses
which increase in a constant ratio from midpoint of a smaller arc to
mid-point of larger arc".
And here is what Ralph Holden says in "Elements of House Division," as
If the second half-house from the middle of a quadrant is say one and
a half times as "big" (in the zodiac) as the one nearest the middle
of the quadrant, then the one that is third from the middle of the
quadrant will be one and a half times as great as the one second from
the middle.
[Ralph Holden, Elements of House Division, LNFowler Ltd., 1977]

In other words, if ASC minus MC is greater than 90 degrees, and IC
minus ASC is less than 90 degrees (as in the example chart that we are
using for this test), and if we define six half-houses beginning at the
ASC/IC midpoint to the ASC/MC midpoint as having sizes of A, B, C, D,
E, and F, then we get B/A = C/B = D/C = E/D = F/E = X, such that B=A*X,
C=B*X, D=C*X, E=D*X, and F=E*X.

I created a spreadsheet to help me calculate the sizes of these six
half-houses, by plugging in values for the "constant ratio" X, until
the results satisfied the two equations A+B+C+D+E+F=90 and A+B+C=M,
where M is half of the smaller quadrant. Astrolog32 gives the following
10th house: 11.7625058 or 11Ari45'45"
[...]
1st house: 123.3470466 or 3Leo20'49"
If I plug those decimal values into my spreadsheet, and try different
values of X until I get results which satisfy the above two equations,
I come up with the following:

X = 1.1771088
A = 9.6016466
B = 11.3021827 = A * X
C = 13.3038987 = B * X
D = 15.6601362 = C * X
E = 18.4336841 = D * X
F = 21.6984518 = E * X

The two half-houses immediately on either side of the midpoint of the
smaller quadrant (i.e., in this case, on either side of the ASC/IC
midpoint) have a size of A, so the 2nd House has a size of 2*A. The 1st
House has a size of B+C, and the 12th House has a size of D+E. The two
half-houses immediately on either side of the ASC/MC midpoint have a
size of F, so the 11th House has a size of 2*F. The 8th House has the
same size as the 2nd House; the 3rd, 7th, and 9th Houses have the same
size as the 1st House; the 4th, 6th, and 10th Houses have the same size
as the 12th House; and the 5th House has the same size as the 11th
House. Thus, the sizes of the eastern houses are as follows:

Size of X = D + E = 34.0938203
Size of XI = F + F = 43.3969036
Size of XII = D + E = 34.0938203
Size of I = B + C = 24.6060813
Size of II = A + A = 19.2032931
Size of III = B + C = 24.6060813

Consequently, the cusps of the houses are as follows:

X = 11.7625058 or 11:45:45 Ari = MC
XI = 45.8563261 or 15:51:23 Tau
XII = 89.2532297 or 29:15:12 Gem
I = 123.3470500 or 3:20:49 Leo = ASC
II = 147.9531314 or 27:57:11 Leo
III = 167.1564245 or 17:09:23 Vir

It can be seen that these are the same values given by StarLogin, so
StarLogin is calculating the Natural Graduation house cusps correctly--
or rather, in a manner consistent with the descriptions of the Natural
Graduation system which were quoted above.

If I use my spreadsheet to calculate the "variation B" method which I
described in my previous post-- where we use whole houses instead of
half-houses, such that A is the size of the 2nd House, B is the size of
the 1st and 3rd Houses, C is the size of the 12th and 10th Houses, and
D is the size of the 11th House, with B=A*X, C=B*X, and D=C*X-- then I
get the following:

X = 1.3406301
A = 18.5847925
B = 24.9153323 = A * X
C = 33.4022444 = B * X
D = 44.7800542 = C * X

Size of X = C = 33.4022444
Size of XI = D = 44.7800542
Size of XII = C = 33.4022444
Size of I = B = 24.9153323
Size of II = A = 18.5847925
Size of III = B = 24.9153323

X = 11.7625058 or 11:45:45 Ari = MC
XI = 45.1647502 or 15:09:53 Tau
XII = 89.9448044 or 29:56:41 Gem
I = 123.3470488 or 3:20:49 Leo = ASC
II = 148.2623810 or 28:15:45 Leo
III = 166.8471735 or 16:50:50 Vir

For "variation C" (using half-houses, but adding some fixed value X
each time, rather than multiplying by some fixed value X), I get the
following:

X = 2.3982823
A = 9.0042942
B = 11.4025765 = A + X
C = 13.8008588 = B + X
D = 16.1991412 = C + X
E = 18.5974235 = D + X
F = 20.9957058 = E + X

Size of X = D + E = 34.7965646
Size of XI = F + F = 41.9914116
Size of XII = D + E = 34.7965646
Size of I = B + C = 25.2034354
Size of II = A + A = 18.0085884
Size of III = B + C = 25.2034354

X = 11.7625058 or 11:45:45 Ari = MC
XI = 46.5590704 or 16:33:33 Tau
XII = 88.5504820 or 28:33:02 Gem
I = 123.3470466 or 3:20:49 Leo = ASC
II = 148.5504820 or 28:33:02 Leo
III = 166.5590704 or 16:33:33 Vir

For "variation D" (using whole houses, but adding some fixed value X
each time, rather than multiplying by some fixed value X), I get the
following:

X = 8.6338163
A = 17.0492756
B = 25.6830918 = A + X
C = 34.3169082 = B + X
D = 42.9507245 = C + X

Size of X = C = 34.3169082
Size of XI = D = 42.9507245
Size of XII = C = 34.3169082
Size of I = B = 25.6830918
Size of II = A = 17.0492756
Size of III = B = 25.6830918

X = 11.7625058 or 11:45:45 Ari = MC
XI = 46.0794140 or 16:04:46 Tau
XII = 89.0301384 or 29:01:49 Gem
I = 123.3470466 or 3:20:49 Leo = ASC
II = 149.0301384 or 29:01:49 Leo
III = 166.0794140 or 16:04:46 Vir

Finally, for "variation E" (using whole houses, but adding some fixed
value X each time, rather than multiplying by some fixed value X, with
the exception that X is added twice when going from B to C), I get the
following:

X = 5.3961352
A = 19.2077296
B = 24.6038648 = A + X
C = 35.3961352 = B + X + X
D = 40.7922704 = C + X

Size of X = C = 35.3961352
Size of XI = D = 40.7922704
Size of XII = C = 35.3961352
Size of I = B = 24.6038648
Size of II = A = 19.2077296
Size of III = B = 24.6038648

X = 11.7625058 or 11:45:45 Ari = MC
XI = 47.1586410 or 17:09:31 Tau
XII = 87.9509114 or 27:57:03 Gem
I = 123.3470466 or 3:20:49 Leo = ASC
II = 147.9509114 or 27:57:03 Leo
III = 167.1586410 or 17:09:31 Vir

These are the same values which Astrolog32 gives for the "Neo-Porphyry"
10th house: 11Ari45'45" or 11.7625058
11th house: 17Tau09'31" or 47.1586410
12th house: 27Gem57'03" or 87.9509114
1st house: 3Leo20'49" or 123.3470466
2nd house: 27Leo57'03" or 147.9509114
3rd house: 17Vir09'31" or 167.1586410
The cusps given by these five variations are summarized below:

Hs. | Var. A | Var. B | Var. C | Var. D | Var. E
X | 11Ar46 | 11Ar46 | 11Ar46 | 11Ar46 | 11Ar46
XI | 15Ta51 | 15Ta10 | 16Ta34 | 16Ta05 | 17Ta10
XII | 29Ge15 | 29Ge57 | 28Ge33 | 29Ge02 | 27Ge57
I | 3Le21 | 3Le21 | 3Le21 | 3Le21 | 3Le21
II | 27Le57 | 28Le16 | 28Le33 | 29Le02 | 27Le57
III | 17Vi09 | 16Vi51 | 16Vi34 | 16Vi05 | 17Vi10

And the cusps given by the three programs are summarized below:

Hs. | StarL. | Astro. | Janus
X | 11Ar46 | 11Ar46 | 11Ar45
XI | 15Ta51 | 17Ta10 | 14Ta50
XII | 29Ge15 | 27Ge57 | 15Cn00
I | 3Le21 | 3Le21 | 3Le20
II | 27Le57 | 27Le57 | 29Le05
III | 17Vi09 | 17Vi10 | 16Vi00

StarLogin clearly uses "variation A," and Astrolog/Astrolog32 clearly
uses "variation E." It seems very interesting to me that these two
variations give almost exactly the same results as each other for the
house cusps in the smaller quadrants, despite the differences in how
they are calculated. ("Variation A" gives 27:57:11 Leo for the 2nd
House cusp, and 17:09:23 Virgo for the 3rd House cusp. "Variation E"
gives 27:57:03 Leo and 17:09:31 Virgo, respectively. The differences
between them are only 8 arcseconds!)

Janus seems to be using something closest to "variation D" for the
smaller quadrants, but what is it using for the larger quadrants? The
11th House cusp is closest to "variation B," but the 12th House cusp is
about 15 to 17 degrees different than any of the variations I've tried.
It would appear that Janus has a bug in it with regard to the Natural
Graduation system.

Michael Rideout
Ed Falis
2005-06-17 16:21:33 UTC
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Raw Message
Post by S***@aol.com
Janus seems to be using something closest to "variation D" for the
smaller quadrants, but what is it using for the larger quadrants? The
11th House cusp is closest to "variation B," but the 12th House cusp is
about 15 to 17 degrees different than any of the variations I've tried.
It would appear that Janus has a bug in it with regard to the Natural
Graduation system.
Difficult to tell what they're doing - they do not describe the method
they use. They do refer to Mike Munkasey's Astrological House Formulary
(among others), but it seems to be offline right now.

- Ed
Todd Carnes
2005-08-06 15:31:39 UTC
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Raw Message
Post by Ed Falis
Post by S***@aol.com
Janus seems to be using something closest to "variation D" for the
smaller quadrants, but what is it using for the larger quadrants? The
11th House cusp is closest to "variation B," but the 12th House cusp is
about 15 to 17 degrees different than any of the variations I've tried.
It would appear that Janus has a bug in it with regard to the Natural
Graduation system.
Difficult to tell what they're doing - they do not describe the method
they use. They do refer to Mike Munkasey's Astrological House
Formulary (among others), but it seems to be offline right now.
- Ed
Fortunately, I have Munkasey's House Formulary backed up on a cdrom.

BUT, this is all Michael has to say on the subject...

"The Natural Graduation House System
A complicated mathematical variation of the Porphyry House System, as
described on pp. 46- 47 in "New Waite's Compendium" by Colin Evans."

Todd
S***@aol.com
2005-08-06 21:38:31 UTC
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Raw Message
Post by Todd Carnes
Post by Ed Falis
Post by S***@aol.com
Janus seems to be using something closest to "variation D" for the
smaller quadrants, but what is it using for the larger quadrants? The
11th House cusp is closest to "variation B," but the 12th House cusp is
about 15 to 17 degrees different than any of the variations I've tried.
It would appear that Janus has a bug in it with regard to the Natural
Graduation system.
Difficult to tell what they're doing - they do not describe the method
they use. They do refer to Mike Munkasey's Astrological House
Formulary (among others), but it seems to be offline right now.
- Ed
Fortunately, I have Munkasey's House Formulary backed up on a cdrom.
BUT, this is all Michael has to say on the subject...
"The Natural Graduation House System
A complicated mathematical variation of the Porphyry House System, as
described on pp. 46- 47 in "New Waite's Compendium" by Colin Evans."
Todd
Thank you, Todd, but I have Michael Munkasey's "House Formulary," so I
already knew that it doesn't give any formulas for the Natural
Graduation house system.

However, I just figured out how to do the calculations directly (rather
than using a recursive procedure), and as usual it is very simple, but
I'd been trying to make it too difficult.

Let's suppose that the MC-to-AS arc (quadrant 4) is wider than the
AS-to-IC arc (quadrant 1). That means we want to find the following
points, three of which are already given for the chart. Since we're
going to be talking about points and arcs, I'll use capital letters to
indicate the arcs, and add a "prime" symbol (') to the letters to
indicate the points:

G' = middle of house 11 = MC/AS midpoint
F' = cusp of house 12
E' = "middle" of house 12
D' = cusp of house 1 = AS
C' = "middle" of house 1
B' = cusp of house 2
A' = middle of house 2 = AS/IC midpoint

(Points E' and C' are not the *mathematical* midpoints of houses 12 and
1, because the "half-houses" which we join to get house 12 and house 1
are not equal in size.)

These seven points give us six arcs, as follows:

F = arc F'G' = 2nd half of house 11
E = arc E'F' = 1st "half" of house 12
D = arc D'E' = 2nd "half" of house 12
C = arc C'D' = 1st "half" of house 1
B = arc B'C' = 2nd "half" of house 1
A = arc A'B' = 1st half of house 2

And when we finish, we'll add the arcs of the "half-houses" to get the
houses, as follows:

F + F = full arc of house 11
D + E = full arc of house 12
B + C = full arc of house 1
A + A = full arc of house 2

By simple math, we know that arc A'G', or the arc between the AS/IC
midpoint and the MC/AS midpoint, is always equal to 90 degrees (I'm
using the ">" symbol to keep Google from eating the leading spaces, so
Post by Todd Carnes
MC/AS midpoint = (MC + AS) / 2 = G'
AS/IC midpoint = (AS + IC) / 2
= (AS + MC + 180) / 2
= (AS + MC) / 2 + 180 / 2
= MC/AS midpoint + 90
= G' + 90 = A'
And by the definition of the Natural Graduation house system, we are
looking for some factor, X, such that the following are true:

B = A*X
C = B*X = A*X*X
D = C*X = A*X*X*X
E = D*X = A*X*X*X*X
F = E*X = A*X*X*X*X*X

And finally, by our definitions of the points and arcs, we also know
that the following are true:

A + B + C = A'B' + B'C' + C'D' = A'D' = A' - D' = AS/IC - AS
D + E + F = D'E' + E'F' + F'G' = D'G' = D' - G' = AS - MC/AS

By substitution, we get the following:

AS/IC - AS = A + B + C = A + A*X + A*X*X
AS - MC/AS = D + E + F = A*X*X*X + A*X*X*X*X + A*X*X*X*X*X

Factoring these, we get the following:

AS/IC - AS = A * (1 + X + X*X)
AS - MC/AS = A*X*X*X * (1 + X + X*X)

Now it's a simple matter to find the value of factor X, because if the
MC and AS are given, and thus the MC/AS and AS/IC midpoints are also
given, then we also know what AS/IC - AS and AS - MC/AS are. So we can
divide them as follows:

(AS - MC/AS) / (AS/IC - AS)
= [A*X*X*X * (1 + X + X*X)] / [A * (1 + X + X*X)]
= X*X*X

Furthermore, since the quadrants are twice as big as the half
quadrants, and 2 / 2 = 1, we could have just divided the two quadrants,
as follows:

(AS - MC) / (IC - AS) = X*X*X

So if we divide the larger quadrant by the smaller quadrant, and then
take the cube root, we have found factor X.

Once we have X, we must calculate the points B', C', E', and F', which
Post by Todd Carnes
IC - AS = 2*A + 2*B + 2*C
= 2*A + 2*A*X + 2*A*X*X
= A * (2 + 2*X + 2*X*X)
A = (IC - AS) / (2 + 2*X + 2*X*X)

A' = AS/IC (by definition)

A = A'B' = A' - B'
B' = A' - A = AS/IC - A

B = B'C' = B' - C'
C' = B' - B = B' - A*X

C = C'D' = C' - D'
D' = C' - C = C' - A*X*X = AS (by definition)

D = D'E' = D' - E'
E' = D' - D = D' - A*X*X*X = AS - A*X*X*X

E = E'F' = E' - F'
F' = E' - E = E' - A*X*X*X*X

F = F'G' = F' - G'
G' = F' - F = F' - A*X*X*X*X*X = MC/AS (by definition)

I know this all seems very complicated, because I'm trying to show the
rationale behind the system, as opposed to just giving simple steps for
calculating it. Also, in practice it really doesn't matter whether we
calculate from the midpoint of the smaller quadrant to the midpoint of
the larger quadrant, or vice versa, because starting with the larger
quadrant will simply result in a factor X which is less than 1. So we
can simplify the whole process by just using the MC and AS, and
ignoring which quadrant is larger, as follows (the letters have no
connection to the letters used above):

Step 1: Calculate the MC.

Step 2: Calculate the AS.

Step 3: Calculate A = AS - MC
(the size of quadrant 4).

Step 4: Calculate B = 180 - A
(the size of quadrant 1).

Step 5: Calculate C = B / A
(the ratio between the two quadrants).

Step 6: Calculate D = cube root of C
(the "X" factor).

Step 7: Calculate E = 2 + 2*D + 2*D*D.

Step 8: Calculate F = A / E
(either half of house 11).

Step 9: Calculate G = F * D
(the 2nd "half" of house 10, and the 1st "half" of house 12).

Step 10: Calculate H = G * D
(the 1st "half" of house 10, and the 2nd "half" of house 12).

Step 11: Calculate I = H * D
(the 1st "half" of house 1, and the 2nd "half" of house 3).

Step 12: Calculate J = I * D
(the 2nd "half" of house 1, and the 1st "half" of house 3).

Step 13: Calculate K = J * D
(either half of house 2).

Step 14: L = MC
(cusp of house 10).

Step 15: M = L + H
("middle" of house 10).

Step 16: N = M + G
(cusp of house 11).

Step 17: O = N + F
(middle of house 11, which should be = MC/AS midpoint).

Step 18: P = O + F
(cusp of house 12).

Step 19: Q = P + G
("middle" of house 12).

Step 20: R = AS
(cusp of house 1).

Step 21: S = R + I
("middle" of house 1).

Step 22: T = S + J
(cusp of house 2).

Step 23: U = T + K
(middle of house 2, which should be = AS/IC midpoint).

Step 24: V = U + K
(cusp of house 3).

Step 25: W = V + J
("middle" of house 3).

The remaining house cusps and "middle" points can be found by simply
adding 180 degrees to the cusps and "middles" of houses 10, 11, 12, 1,
2, and 3.

To give an example, the Natural Graduation house tables in "The New
Waite's Compendium of Natal Astrology" give the following:

60N00, ST 14:00:00
MC = 2:11 Sco
XI = 19 Sco
XII = 28 Sco
AS = 15:06 Sag
II = 22 Cap
III = 26 Pis

To replicate these cusps in an astrology program, I'm using the
following chart data:

January 1, 1950, 7:18:30 AM GMT, 0W00, 60N00

Astrolog32 gives the following:

MC = 2:11:02 Sco = 212.1839952 (absolute longitude, decimal)
AS = 15:05:46 Sag = 255.0962107 (absolute longitude, decimal)

Using the steps listed above, I get the following:

Step 1: MC = 212.1839952
Step 2: AS = 255.0962107
Step 3: A = 255.0962107 - 212.1839952 = 42.9122155
Step 4: B = 180 - A = 180 - 42.9122155 = 137.0877845
Step 5: C = B / A = 137.0877845 / 42.9122155 = 3.194609808
Step 6: D = cube root of C = cube root of 3.194609808 = 1.472784733
Step 7: E = 2 + 2*D + 2*D*D
Post by Todd Carnes
= 2 + 2 * 1.472784733 + 2 * 1.472784733 * 1.472784733
= 2 + 2.945569466 + 4.338189739
= 9.283759204
Step 8: F = A / E = 42.9122155 / 9.283759204 = 4.622288726
Step 9: G = F * D = 4.622288726 * 1.472784733 = 6.807636266
Step 10: H = G * D = 6.807636266 * 1.472784733 = 10.02618276
Step 11: I = H * D = 10.02618276 * 1.472784733 = 14.7664089
Step 12: J = I * D = 14.7664089 * 1.472784733 = 21.74774158
Step 13: K = J * D = 21.74774158 * 1.472784733 = 32.02974177
Step 14: L = MC = 212.1839952
Post by Todd Carnes
= 2:11:02.38 Sco (cusp of house 10)
Step 15: M = L + H = 212.1839952 + 10.02618276 = 222.210178
Post by Todd Carnes
= 12:12:36.6 Sco ("middle" of house 10)
Step 16: N = M + G = 222.210178 + 6.807636266 = 229.0178142
Post by Todd Carnes
= 19:01:04.13 Sco (cusp of house 11)
Step 17: O = N + F = 229.0178142 + 4.622288726 = 233.640103
Post by Todd Carnes
= 23:38:24.3 Sco (middle of house 11, or MC/AS)
Step 18: P = O + F = 233.640103 + 4.622288726 = 238.2623917
Post by Todd Carnes
= 28:15:44.6 Sco (cusp of house 12)
Step 19: Q = P + G = 238.2623917 + 6.807636266 = 245.0700279
Post by Todd Carnes
= 5:04:12.1 Sag ("middle" of house 12)
Step 20: R = AS = 255.0962107
Post by Todd Carnes
= 15:05:46.36 Sag (cusp of house 1)
Step 21: S = R + I = 255.0962107 + 14.7664089 = 269.8626196
Post by Todd Carnes
= 29:51:45.4 Sag ("middle" of house 1)
Step 22: T = S + J = 269.8626196 + 21.74774158 = 291.6103612
Post by Todd Carnes
= 21:36:37.3 Cap (cusp of house 2)
Step 23: U = T + K = 291.6103612 + 32.02974177 = 323.640103
Post by Todd Carnes
= 23:38:24.3 Aqu (middle of house 2, or AS/IC)
Step 24: V = U + K = 323.640103 + 32.02974177 = 355.6698447
Post by Todd Carnes
= 25:40:11.4 Pis (cusp of house 3)
Step 25: W = V + J = 355.6698447 + 21.74774158 = 377.4175863
Post by Todd Carnes
= 17:25:03.31 Ari ("middle" of house 3)
2:11:02.38 Sco (cusp of house 10) vs. 2:11 Sco
19:01:04.13 Sco (cusp of house 11) vs. 19 Sco
28:15:44.6 Sco (cusp of house 12) vs. 28 Sco
15:05:46.36 Sag (cusp of house 1) vs. 15:06 Sag
21:36:37.3 Cap (cusp of house 2) vs. 22 Cap
25:40:11.4 Pis (cusp of house 3) vs. 26 Pis
The advantage of doing the calculations ourselves is that we get more
accurate positions (as opposed to interpolating from positions that are
given to the nearest degrees in the house tables), and we also now know
the "middle" points of the houses, which the house tables do not list.

Of course, I don't expect a stampede of astrologers rushing to use the
Natural Graduation house system. But if anyone is interested in
experimenting with it, but your software doesn't include it, now you
know how to do the calculations!

Michael Rideout
e***@ernest79.wanadoo.co.uk
2005-07-07 12:30:01 UTC
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Hi Michael,

I am wondering if you have located that book by Colin Evans yet. If
not please try www.thevirtualbookshelf.co.uk they may well have
it. Good luck.

Cheers Ernest
S***@aol.com
2005-08-06 06:20:08 UTC
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Post by e***@ernest79.wanadoo.co.uk
Hi Michael,
I am wondering if you have located that book by Colin Evans yet.
If not please try www.thevirtualbookshelf.co.uk they may
well have it. Good luck.
Cheers Ernest
Tonight I went into my spare room, and pulled out most of my boxes of
books and other stuff. So I finally dug up my copy of "The New Waite's
Compendium of Natal Astrology."

The description of the Natural Graduation house system given by Colin
Evans in that book (pages 46-47) definitely fits what I had referred to
as "variation A," which also fits what the "StarLogin" program uses.

To recap, the Natural Graduation house system first divides the
horoscope into four quadrants using the Medium Coeli, Ascendant, Imum
Coeli, and Descendant.

Then it bisects the four quadrants using the MC/AS midpoint, AS/IC
midpoint, DS/IC midpoint, and MC/DS midpoint-- which will always be
exactly square or opposite each other (i.e., MC/AS is 90 degrees from
AS/IC, which is 90 degrees from DS/IC, which is 90 degrees from MC/DS,
which is 90 degrees from MC/AS).

It doesn't matter whether we work with the eastern, western, upper, or
lower half of the chart, but it's traditional to calculate the cusps of
the eastern houses, so that's what we'll do.

We want to divide the 90 degrees between the midpoint of the smaller
quadrant and the midpoint of the larger quadrant (i.e., from MC/AS to
AS/IC, or else from AS/IC to MC/AS).

We want to find seven points which will create six "half-houses," such
that the sizes of the half-houses form a geometric progression. For
example, if we label the midpoint of the smaller quadrant as point A,
label the Ascendant as point D, and label the midpoint of the larger
quadrant as point G, then we want to find the points B, C, E, and F
such that the proportion of arc A-B to arc B-C is equal to the
proportion of arc B-C to arc C-D, which is also equal to the proportion
of arc C-D to arc D-E, and the proportion of arc D-E to arc E-F, and
the proportion of arc E-F to arc F-G.

There may be a way to calculate the intermediate points B, C, E, and F
directly, but when I tried to work out a formula back in June, it made
my head swim, so I settled for a recursive calculation method.

Once we calculate the points to get the half-houses, we join them in
pairs to get full-size houses. (The reason Colin Evans decided to use
half-houses is because there's two opinions about where the houses
begin-- at the cusps, or in between the cusps-- so we can join the
half-houses together according to whichever method we prefer to use.)

For example, I suggested using the chart for January 1, 1950, 6:00 PM
GMT, 0W00, 50N00 to compare the different variations I'd come up with,
and see which variation gives results that agree with any program that
includes the Natural Graduation system. For "variation A," I calculated
the following house cusps, which agree with the "StarLogin" program:

X = 11:45:45 Ari = MC
XI = 15:51:23 Tau
XII = 29:15:12 Gem
I = 3:20:49 Leo = ASC
II = 27:57:11 Leo
III = 17:09:23 Vir

The sidereal time for the chart is 0:43:16. The table of houses for the
Natural Graduation system in "The New Waite's Compendium of Natal
Astrology" lists the following intermediate cusp values:

50N00, ST 0:36:00 - XI 14 Tau, XII 28 Gem, II 26 Leo, III 15 Vir
50N00, ST 0:48:00 - XI 17 Tau, XII 1 Can, II 29 Leo, III 18 Vir

If we interpolate for 0:43:16, we get the following:

XI 16 Tau, XII 0 Can, II 28 Leo, III 17 Vir

Obviously, it's difficult to be sure how accurate our results are, when
the table lists only whole degrees, but it's clear that our results are
in pretty good agreement with the calculations I'd done for "variation
A." So I'm convinced that the "StarLogin" program is doing the
calculations correctly, and the "Janus" program is doing the
calculations incorrectly. The "Astrolog" program is also doing the
calculations incorrectly-- at least, if its "Neo-Porphyry" house system
is supposed to be the Natural Graduation system.

Michael Rideout
Todd Carnes
2005-08-06 15:23:12 UTC
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Post by S***@aol.com
Does anyone happen to have a copy of The New Waites Compendium Of Natal
Astrology, which has "universal tables of houses" that include Colin
Evans' "natural graduation" method? Or does anyone happen to know the
formulas for that method? I'm trying to work them out (I'm taking stock
of different systems), and I've come up with four different possible
ways to calculate them-- one that gives the same results as StarLogin
(and which seems to match a brief description that I found on the web),
another that gives the same results as Astrolog (which I think may be
wrong), and two other ways. I have Waites Compendium, but can't find
it, and I want to check the different results against the tables that
Colin Evans included in that book.
Michael Rideout
I "thought" I had the formulas for this house system, but after looking
I discovered that I do not.

However, I have the formulas for a lot of other house systems in a word
file. If you'd like I can email what I have to you.

Todd
a***@gmail.com
2014-06-20 13:25:36 UTC
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Geo Series

Powers from 0 to n - 1

a(1 - r^n) / (1 - r) = sum

r is ratio of power of one term to the next
n is number of terms
a is first term (no power - a constant)

For example: If ratio is 5 and s0 = 2 and 4 terms, then 312

Sigma(2, 10, 50, 250) = 312 = Sigma(2 * 5^0, 2 * 5^1, 2 * 5^2, 2 * 5^3)

a = 2 r = 5 n = 4

2 times 4 successive powers of 5, i.e. 0 to 3.

Geometric is totally dependent on r and a

Say one quad = 3/7 and the other = 4/7 of 180

Nocturnal 77.1428
Diurnal 102.8571

If Diurnal > Nocturnal then

Houses increase from MC to Asc
Houses decrease from Asc to IC

And the converse is true as well if Diurnal < Nocturnal

Now change these arcs to arcs that cross the boundaries from the center of the quads.

E = 90 W = 90

Divide into 6th's

The fraction is .75 ((3/7) / (4/7)), the # of terms is 6

(1 - r^n) / (1 - r) = 0.822021484375 / .25 = 3.2880859375 is the ratio

90 / 3.2880859375 = 27.3715 27.3715 * .75 = 20.5286 and so on ...

Series = {27.3715, 20.5286. 15.3964, 11.5473, 8.660, 6.495}

Combine extrema 33.8655, 29.1886, 26.9437

26.9437 = 10th, 3rd, 4th, 9th
33.8655 = 11th, 2nd, 5th, 8th
29.1886 = 12th, 1st, 6th, 7th

Diurnal 1/2 Arc = Nocturnal 1/2 Arc = 90 MC to IC = 180 Asc to Dsc = 180

Increase from MC to Asc
Decrease from Asc to IC
Increase from IC to Dsc
Decrease from IC to MC

Sequence is a modulating wave.

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