Let's now take the triangle of each of the signs in the zodiac.
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
9 45
10 55
11 66 12 78 Now add them.
The total is 364 or days in the year as figured in weeks or 7x52.
Coincidence? Maybe. And then again, maybe not.
Since we can make squares by adding any two successive triangles, let's do that:
1+3=4 6+10=16 15+21=36 28+36=64 45+55=100 66+78=144 PATTERN?
Yes, adding the even squares from the square of 2 through the square of 12 also equals the year when figured in weeks.
C C ha h a pt p te e r r 1 1 5 5 -T - T he h e T T r r ia i an ng gl le e s s a an nd d t th he e C C ub u be es s
We have seen the relationship between the triangles and the squares, but there is also an interesting relationship between the triangles and the cubes.
In Book V-"The Cycle of Venus," I noted that the heliocentric cycle of Venus is 225 days and when multiplied by the square of 3 or 9 we get:
3x3x15x15 since 225 is the square of 15.
This can also be read as:
3x15x3x15 or 45x45
which equals 2025.
Now let's put down some cubes:
1x1x1=1 2x2x2=8 3x3x3=27 4x4x4=64 5x5x5=125
6x6x6=216 7x7x7=343 8x8x8=512 9x9x9=729
Now, let's add our answers and we find that they equal 2025!
PATTERN?
Think hard. Think of the cubes. Think of the square. Think of the triangles!
Got it now?
The square 45x45 equals the sum of the cubes. And the cubes are 1 through 9.
Got it now?
The triangle of 9 is 45.
How about now?
That's right. The square of any triangular number equals the sum of the cubes up through the number (root) for which we find triangle.
Example again.
We know from our work that the triangle of 7 is 28. So if we square 28 which is 28x28=784, then 784 will be the total of all the cubes from 1 through 7.
Pick out a number, find its triangle, square the triangle and work out the cubes.
Those familiar with Masonry might know of something called the cubic stone.
A representation of this cubic stone can be found in A. E.
Waites's book, "A New Encyclopedia of Masonry," Book I, page 405.
You can find numbers around this stone, the cubes of 9, 7, 5, 3 and I believe the cube of 1 is to be understood.
The cubes are:
1x1x1=1 3x3x3=27 5x5x5=125 7x7x7=343 9x9x9=729
When we add them we find that they total 1225.
Look at your triangular number list. You will find 1225 opposite
the number 49 so 1225 is the triangle of 49.
Note that the cubes we listed are the odd cubes from 1 through 9. The same list as we had before except this time we left out all the even cubes.
Is it possible that all odd cubes add to a triangular number?
Let's go to the bottom of the ladder again.
1x1x1=1
And the cube of 1 is in the triangle of 1.
1x1x1=1 3x3x3=27
Add them and we find that 28 is indeed a triangular number, the triangle of 7.
1x1x1=1 3x3x3=27 5x5x5=125
The total is 153 and from our triangular list we know that number is the triangle of 17.
That should be enough of PATTERN search to tell us we are right.
There is a couple of ways of knowing what triangular number contains the odd cubes.
Let's look at the cubes above, 1, 3, and 5.
Can you make a PATTERN?
Try this. Add 1 to 5 and get 6 and then multiply 6 by half of 6.
Now can you make it?
That's right. Subtract 1 from 18 and we have our answer.
Now try the ones where we are trying to find the total of the odd cubes 1 through 9 which we know to be the triangle of 49.
First, add 1 to 9 and get 10. multiply 10 by half of 10 or 5 and get 50. Now subtract 1 and we will have 49. From our work above we know that the triangle of 49 is 1225 and 1225 contains all the odd cubes from 1 through 9.
That's the way for doing it from scratch, but if you know the triangle of one set of odd cubes, it is easy to know how to get the next triangle.
Through trial and error I found that the triangle which contains the odd cubes from 1 through 11 is the triangle of 71.
PATTERN?
1 through 9 was 49. 71-49=22.
Got it now?
22 is two times 11.
Knowing that the triangle of 71 contains all the odd cubes from 1 through 11, how would we know what number to find the triangle of which would equal the odd cubes 1 through 13.
That's right. 2x13=26. Add 26 to 71 and get 97. Now if we found the triangle of 97, it would contain all the odd cubes from 1 through 13. Want to find the triangle which would contain the odd cubes from 1 through 15. Add 30 to 97 and get 127.
Quick now. How would we find the triangular number which would contain all the odd cubes from 1 through 49?
Right again. Multiply 50 times 25 and subtract 1. Then find the triangle of your answer and you'll have the right number.
50x25=1250 1250-1=1249
Then 1249x1250 and divide by 2.
We have found the triangular numbers that include the odd cubes and we have seen how the square of triangular numbers include both the odd and even cubes.
The search for the even cubes is a little different. It took me several minutes to figure it out.
Let's look once again at all the cubes contained in the square of the triangular number 45.
We found that the square of 45 was 2025. We also found that all the cubes 1 through 9 would be contained in the square.
We know now that the total of the odd cubes from 1 through 9 equal 1225.
That leaves the even cubes, 2, 4, 6 and 8. By subtracting 1225 from 2025 we can find that the even cubes total 800.
PATTERN?
If we added the cube of 10 to the even cubes they would total 1800 since the cube of 10 is 1000.
PATTERN now?
Let's put down the numbers for both situations:
The cubes of 2, 4, 6, 8 equal 800 The cubes of 2, 4, 6, 8, 10 equal 1800
The Masons have something called the double square. Does that help?
Lets add the cube roots in both situations:
2+4+6+8=20 2+4+6+8+10=30 Got it now?
Let's square 20. 20x20=400 Let's square 30. 30x30=900 How about now?
Double each:
2x400=800 2x900=1800 PATTERN made!
So, to find the total of all the even cubes simply add the cube roots, square your answer and then double it.
There is an easy way to find the total of the cube roots without adding them. Simply take half of the largest cube root and multiply by the next number.
Above we found that 2+4+6+8=20. That can be found by taking half of 8 and multiplying by the next number (4x5=20). Then find the square of 20 and double it. For the even cubes up through 10 the answer would be 5x6=30. Square it and double it.
If we wanted to find the total of the even cubes up through 24,
we would divide 24 by 2 and get 12 and then multiply that by 13. Then square that answer and double it.
We will see that cubic stone number, 1225, again when we look at the single digit numbering system and the TELEOIS.
C C ha h a pt p te e r r 1 1 6 6 -T - T ri r ia an ng gl le e s s a a nd n d t th he e C C a a r r di d in na al l N N um u mb be e rs r s
The triangular numbers also have a close relationship to the
cardinal numbers in the cube.
Look at your Square of Nine chart again. Count the numbers between the heavy lines that make up the cardinal points.
Since this is the square of 33 then there are 33 numbers across and counting down there are 32 since the one in the middle was counted when going across, for a total of 65 numbers .
Now imagine that this is not a square but a cube and we are going to bring in a crane and lift out all the numbers in the cardinal lines.
Since there are 65 numbers involved and since the numbers are 33 deep then we are removing 33x65 and from our previous work we know that the answer is a triangular number, the triangle of 65 since one way of finding triangles is to take half of the next number (66) and multiply times the original number (65).
That works for any odd cube.
If we wanted to remove the cardinal numbers, the center numbers or the core, in the cube of 9 we would be removing 9x17 since the cube is 9 deep and there are 9 numbers across and 8 numbers down.
We know from our work that 9x17=153 or the triangle of 17.
The numbers that remain are also interesting, but I'll let you play with that. After all why should I be the only one rowing the boat!
I know you are asking yourself what does all this business about extracting the cardinal numbers have to do with Gann. As I said earlier, I believe that we have to explore every avenue in order to uncover his methods. They are very well hidden in spite of what some other writers might claim.
We will look at some well hidden relationships among just three numbers in his soybean chart at a later date.
We will see more of the triangles in the work on the single digit number system and the TELEOIS of the triangles.
B B oo o ok k V V II I I
Th T he e C C yc y cl le e o o f f M M er e rc cu ur ry y
C C ha h a pt p te e r r 1 1 -T - T he h e S Se ea ar rc c h h F F or o r " "1 17 7" "
My interest in the cycle of Mercury, as far as the works of W.
D. Gann is concerned, probably started when I read Gann's novel "The Tunnel Thru the Air." Especially that part which tells the birthdate of Robert Gordon, the hero of the novel.
But, to explain why that interested me, I will have to back up a little.
We all know (most Gann followers anyway) that Gann was a Mason.
So when I got interested in the Gann material I also got interested in any material on Masonry.
I was able to get my hands on a Masonry book called "Morals and
Dogma" by Albert Pike. He was an Arkansas judge who lived sometime in the middle or late 1800's.
It was an extremely large book, running close to 900 large
pages. Being new to Masonry I had never seen another Masonry book and I assumed this was the "main book" of Masonry. I have since learned that there are many other Masonry books, but Pike's book seems to be one of those held in highest regard by Masons.
The book is full of philosophy, but there are some parts on
numbers (some of which we will look at another time) and some parts deal with mythology, etc.
Pike tells the story of Osiris and Isis, the mythological gods
of Egypt. You might have heard of these two if you have read books on ancient history, etc. They were mentioned in a l995 program on the A&E Cable TV channel about the pyramids.
Let's have a look at that story as related by Pike beginning on page 375 (Note: The punctuation in this quote follows that in the book and is not necessarily the way a writer would punctuate today):
"Osiris, said to have been an ancient King of Egypt, was the Sun; and Isis, his wife, the Moon: and his history recounts, in poetical and figurative style, the annual journey of the Great Luminary of Heaven through the diffeent Signs of the Zodiac.
"In the absence of Osiris, Typhon, his brother, filled with envy and malice, sought to usurp his throne; but his plans were frustrated by Isis. Then he resolved to kill Osiris. This he did, by persuading
him to enter a coffin or sarcophagus, which he then flung into the Nile.
"After a long search, Isis found the body, and concealed it in the depths of a forest; but Typhon, finding it there, cut it into fourteen pieces, and scatterd them hither and thither. After tedious search, Isis found thirteen pieces, the fishes having eaten the other (the privates), which she replaced of wood, and buried the body at Philae; where a temple of surpassing mangificence was erected in honor of Osiris.
"Isis, aided by her son Orus, Horus or Har-oeri, warred against Typhon, slew him, reigned gloriously, and at her death was reunited to her husband, in the same tomb...
"In the legend of Osiris and Isis, as given by Plutarch, are
many details and circumstances other than those that we have briefly mentioned; and all of which we need not repeat here. Osiris married his sister Isis; and labored publicly with her to ameliorate the lot of men. He taught them agriculture, while Isis invented laws. He built temples to the Gods, and established their worship. Both were the patrons of artists and their useful inventions; and introduced the use of iron for defensive weapons and implements of agriculture, and of gold to adorn the temples of the gods. He went forth with an army to conquer men to civiliztion, teaching the people whom he overcame to plant the vine and sow grain for food.
"Typhon, his brother, slew him when the sun was in the sign of the Scorpion, that is to say, at the Autumnal Equinox. They had been rival claimants, says Synesius, for the throne of Egypt, as Light and Darkness contend ever for the empire of the world. Plutarch adds, that at the time when Osiris was slain, the moon was at its full; and therefore it was in the sign opposit the Scorpion, that is, the Bull, the sign of the Vernal Equinox."
Then on page 589:
"Osiris is a being analogous to the Syrian Adoni; and the fable of his history, which we need not here repeat, is a narrative form of the popular religion of Egypt, of which the Sun is the Hero, and the agricultural calendar the moral. The moist valley of the Nile, owing its fertility to the annual inundation appreared, in contrast with the surrounding desert, like life in the midst of death.
"The inundation was in evident dependent on the Sun, and Egypt, enviorned with arid deserts, like a heart within a burning censer, was the the female power, dependent on the influences personified in its God. Typhon his brother, the type of darkness, drought, and sterility, threw his body into the Nile; and thus Osiris, the "good,"
the "Saviour," perished, in the 28th year of his life or reign, and on the 17th day of the month Athor, or the 13th of November.
"He is also made to die during the heats of the early Summer, when, from March to July, the earth was parched with intolerable heat, vegetation was scorched, and the languid Nile exhausted. From
that death he rises when the Solstitial Sun brings the inundation and Egypt is filled with mirth and acclamation anticipatory of the second harvest. From his Wintry death he rises with the early flower of Spring, and then the joyful festival of Osiris found was celebrated."
C C ha h a pt p te e r r 2 2 -P - P re r ec ce es s s s io i o n n o o f f t th he e E Eq qu ui in no ox xe es s
Let's look back at Chapter 1 again. We read there that Osiris was slain at the Autumnal Equinox.
Then we read that he was slain on the "17th" day of Athor or the 13th of November.
I imagine that Athor was an Egyptian month which was November in the calendars of some other countries. And because of differences in calendars the 17th day of Athor in the Egyptian calendar was the l3th of November in other calendars.
But let's stick with the "17th" day of Athor since that is the number which caught my eye.
We also saw that the Autumnal Equinox was in Scorpio. For those that may not have much knowledge of astronomy or astrology some explanation is in order. I an neither an astronomer or astrologer, but have read some books on each. I would suggest that anyone pursuing a Gann study should read a little in these two fields.
Many of you may already know about the precession of the
equinoxes and those that don't know can find an explanation in most encyclopedias under the topic "Precession of the Equinoxes."
But for those who don't care to do that search I will offer a brief explanation here.
When the sun crosses the equator each spring going north (I know the sun doesn't really move across the equator, but the path of the earth around the sun makes it seem to do so) we call that the vernal equinox. We are told that the sun's position in the sky against the background of the zodiac is entering the first point of Aires or OO AIRES.
When it goes back south and crosses the equator in the fall, about Sept. 22 we are told that the sun is in the first point of
Libra or OO LIBRA (the symbol of which is a balance since it balances night and day).
But both of these positions are incorrect!
Astronomers will tell you that the sun at the Vernal Equinox is
somewhere in Pices and headed toward Aquarius. (Remember the song in the late 1960's about the age of Aquarius?)
The use of the first point of Aires and Libra for the equinoxes
is a pretty much modern day convenience we now use instead of using
the places where the sun really is on the first day of spring and the first day of autumn.
The idea that the sun is in the first points of Aires and Libra at the equinoxes is a matter of convenience. We use that convenience because the sun at the equinoxes is slowly moving "backwards" in the zodiac. Thus the term "precession of the equinoxes."
The sun is slowly moving "backwards" through the zodiac because it does not come back to the same place in the sky each year because of the wobble of the earth. (Here again I'm saying that the sun does not come back to the same place in the sky, when actually it is the earth, but because the earth does not come back to the same place our view of the sun makes it appear not to come back to the same place.) There is about 50 seconds of arc difference each year.
("Seconds" here is in terms of degrees and not time. There are 360 degrees in a circle. There are 60 minutes in each degree and there are 60 seconds in each minute of those degrees) It takes over 2,000 years for the sun to "precess" or go "backward" through a sign. So a couple of thousand years ago the sun really was at the first point of Aires at the equinox and a couple of thousand years earlier than that the sun was in Taurus at the Vernal Equinox. And when it was in Taurus at the vernal equinox it was in Scorpio (the opposite sign) at the Autumnal Equinox.
There are some computer programs available which show the position of the sun, moon, planets, stars, etc. for any day of the year. I have a registered program called "Skyglobe."
With it you can run the time forward or backwards. Run it
backwards 100 years at a time and you can see the sun move forward in the zodiac at the equinoxes and run it forward and you can see the
backwards 100 years at a time and you can see the sun move forward in the zodiac at the equinoxes and run it forward and you can see the