1 TThhee PPAATTTTEERRNNSS ooff GGAANNNN BByy GGrraannvviillllee CCoooolleeyy 11999999 Preface Introduction to the Series Book I The Cycle of Mars Book II The Great Cycle Book III The Book With No Name[.]
The PATTERNS of GANN B y G nville Coole y 1999 Preface Introduction to the Series Book I-The Cycle of Mars Book II-The Great Cycle Book III-The Book With No Name Book IV-On The Square Book V-The Cycle of Venus Book VI-The Triangular Numbers Book VII-The Cycle of Mercury Book VIII-The Single Digit Numbering System Book IX-Gann and Fibonacci Book X-The Cubes and the Hexagon Book XI-Gann and the Teleois Book XII-Gann's Magic Square Appendix AUTHOR The author of the series "The PATTERNS of Gann" is Granville Cooley, a journalist who has studied the market material of the legendary W D Gann since 1983 PREFACE Those of you who read my article in the "Gann and Elliott Wave Magazine" several years ago know that I not claim to be a mathematician My background is journalism Like most of you I had some algebra and geometry in high school and some algebra in college I went into journalism and the theorems and equations slipped from my mind Being reared in a Mom and Pop grocery store since I was 10, I was always pretty good with figures (I finished the bookkeeping course in high school three months before the rest of the class) but when I started studying Gann I realized how much I had forgotten One market writer did some interesting work on the markets, but the book was so full of algebra that it was hard to follow A big trader, well known in the markets, said he too found the book hard to follow Magazine articles dealing with systems seem to be written only for advanced mathematicians or for computer programmers I will assume that most of you are like me Some mathematical background, but not a great deal This series of books, "The PATTERNS of Gann," is dedicated to you When I started studying Gann several years ago it was with calculator in hand and I suggest that you the same thing Some believe Gann can be solved without looking at the numbers, but I'm sure that by this time you have learned differently The answer has to be in his numbers! So, with calculator in hand for several years I ran thousands of numbers through it Adding, subtracting, multiplying and dividing the Gann numbers As I did so, PATTERNS emerged PATTERNS! They made no sense at first I had no background for them During my reading of the Gann material I could see that his keys were based on math (specifically that of the ancient systems), astronomy, astrology, Masonic and Biblical material and related writings So, I started reading the ancient math, etc As I did the PATTERNS began to come into focus That will be our watchword as you read this series of books PATTERNS! I will make reference many times to PATTERNS and you too will learn to recognize them and how to uncover more You will also see something else as the PATTERNS emerge Even though Gann said he used algebra and geometry simple observational arithmetic can be used to solve most Gann problems No algebra is needed! This series of books, "The PATTERNS of Gann" were originally to be titled "Exploring the Numbers of Gann," but I thought a better title was "The PATTERNS of Gann" since the exploration of the numbers of Gann is really an exploration for PATTERNS Early on in the writing of this series of books I could see that I was really writing about PATTERNS After all, isn't that what market analysis, especially technical analysis is all about, a search for PATTERNS? When Elliott looked at a chart of prices and saw the waves, he was looking for PATTERNS Followers of Elliott today are still looking for those same PATTERNS When a fundamentalist looks at supply and demand figures he is looking for PATTERNS which make prices go up and down When financial astrologers look at planet configurations which caused price movement in the past they are looking for PATTERNS When market followers look at moving averages they are looking for PATTERNS When they look at divergences in the relative strength index or the stochastics they are looking for PATTERNS When volume followers look at volume or the rise and fall of the number of contracts outstanding they are looking for PATTERNS When the black box makers put together a series of highs, lows and closes, they are looking for PATTERNS When the fundamentalists study the various air currents for signs of drought they are looking for PATTERNS When the cyclists look at the cycles, whether they be astronomical or Fourier, they are looking for PATTERNS When the neuralogist (no not a nerve doctor but a person who studies computer neurals) seeks the right neurals they are looking for PATTERNS I don't know about the butcher or the baker, but when the candlestick makers study candlesticks they are looking for PATTERNS Again I say The study of Gann must be a search for PATTERNS! Introduction to the Series I once heard someone say that they were looking for something with which to pull the trigger in the commodities market I'm sure a lot of traders are looking for that something with which to pull the trigger But since over 90 per cent of the traders lose in the commodity markets, those who pull the trigger either shoot themselves in the foot or in the head One is very painful and the other is very deadly This set of books in not about pulling the trigger It is not a system on how to make a million dollars in the market in the morning It is about certain mathematical and astronomical relationships between numbers and their possible application to the numbers of W D Gann I say possible because I not claim to be a market guru I know there are those who want to study Gann without studying the numbers, but since the answers to the Gann approach must be in the numerical systems and since all markets are made up of advances and declines of numbers, I can't see how it is possible to study Gann without studying his numbers For that matter, without studying numbers in general It is said of Gann that he was a Christian and a Mason and it is known of his use of astrology and geometry It is believed that his number system came from those sources This series of books, "The PATTERNS of Gann," is an exploration of the above sources (plus a few others) We will see how they apply to the Gann material I'm not a mathematician My field is journalism So you will not find any x+y=z in this material I'm sure that the average person will appreciate that approach as most persons are not mathematicians either BOOK I The Cycle of Mars Chapter 1- The Square of 144 Is Laid Out Like many of you, when I came to that section in chapter of Gann's commodity course that called for the laying out of the Square of 144 on the soybean chart of that 267-week period from Jan 15, 1948 to March, 1953, I put the charts on the floor, got down on my hands and knees and checked the places where he said to put the "square." Needless to say I spent many days and nights down on my hands and knees, at the end of which I said probably what you said "So what." I talked to a friend of mine who had started his Gann research months before I started and who had owned the material before turning it over to me to see what I could make of it His background was farming Mine was newspaper work He thought my curiosity as a newspaper man might lead me down some paths he had never gone I asked him if he had ever laid out the Square of 144 on this particular chart and what were his conclusions He had laid it out His conclusion was the same as mine "So what?" We decided to lay it out together and see if either one of us could discover something the other had missed We put the square on this top and that top and this bottom and that bottom and put it on the "inner square." The results seemed the same "So what." I did notice one interesting number "68" It was arrived at by subtracting 144 from another number and then going up the page by 68 I knew that the low on futures had been 67 but there were two or three times in the late 1930's that the low had been at 68 I also remembered that Gann had said in his discussion of the hexagon (page 113 in the "old" commodity course; section 10, The Hexagon Chart, page in the "new") that we have "angles of 66 degrees, 67 degrees, 67 1/2 degrees and 68 degrees." The 67 1/2 degree angle was easy enough to figure out since it was one of his divisions of the circle But the other two were mysteries as far as I was concerned So I put up the chart for awhile But after going through the "private papers" and seeing how Gann had marked the paths of Jupiter and Mars on his 1948 bean chart, I decided to mark all the planets on my chart to see if I could discover anything I did! Chapter 2- The Lines Are Laid Out About this time I happened to be flipping through a copy of an astrology magazine which showed the position of the planets and their relation to each other by a few lines drawn across a square, which represented that particular month, instead of on a circular zodiac (I'm not an astrologer and didn't know such a representation could be made Since that day and a long time after I had laid out my own chart I found that others had done some similar work If I laid out the lines geocentric style (the planets as seen from the earth) the lines would have to dip down at times because of the retrograde action of the planets Retrograde means that a particular planet seems to move backward in the degrees of the zodiac circle such as when the faster moving earth goes by a slower planet like Jupiter When the planet seems to move back its position becomes less in degrees in the circle, like moving from 225 degrees back to 2l9 degrees If I was representing that movement on a piece of graph paper like Gann's and had the paper numbered from the bottom to the top, to 360 degrees, then I would have to dip my line from 225 down to 2l9 to show the change So I decided to lay out the lines heliocentric style The heliocentric position of a planet is its position that would be seen if you were standing where the sun stands The planets would go in a circle around you and there would be no retrograde movement The lines that are drawn on the chart would be straight lines and would have no dips The positions of the planets for any particular date can be found in a heliocentric ephemeris Using the same style chart paper used by Gann I let one-eighth of an inch equal one cent in price I also let the one-eighth of an inch equal one degree in the zodiac Or one degree equaled one cent Going up the page I went from degrees in Aires, marking each 30 degrees for each sign Using two sheets of paper, since one was not big enough, I continued on up to the high on soybeans at that time, $4.36 per bushel In the zodiac that was 76 cents beyond 360 degrees or 76 cents in the second cycle of the zodiac or at l6 degrees in the sign of Gemini Right under the price of $4.36 on Jan 15, 1948 I placed a dot for the position of each of the outer planets on that date Since this is a weekly chart the faster moving planets Mercury and Venus would show up pretty much as straight up and down lines and since my interest was mainly in the planet Mars, I didn't put in the faster moving planets Chapter 3-An Immediate Discovery As stated in chapter 2, I had placed dots to represent the planets' positions on Jan l5, l948 Instead of listing them in their signs I listed them at their absolute degree in the 360 degree circle: Mercury 318 Venus 12 Earth 113 Mars 133 Jupiter 251 Saturn 138 Uranus 84 Neptune 191 Pluto 133 In his discussion of the Square of 144 chart Gann never mentioned the planets In fact he never mentioned the planets in any of the work in the course although there are many hints at their use It was in his "private papers" that evidence of his use of the planets was found However, his use of astrology was public information as far back as the 1920's as shown in his book, "The Tale of the Tape." But he never mentioned it in the course which is now available to the public When I extended the line of Mars out it connected with Jupiter at 276, which was also the price of soybeans at that time, $2.76 in the week of 11-26-1948, which shows on his famous "private paper" soybean chart I extended the line further and found an immediate discovery that I probably would not have known if I had not put on the lines Mars crossed Jupiter again in this time period in the week of Feb 6, 195l This crossing at 344 degrees was also at the price of soybeans, $3.44, although the price came a short time later Chapter 4-Looking at the High Although both of these crossings of Jupiter by Mars occurred at the exact price of beans, neither one of these crossings was at the real high of this time period Remember we started this 267-week study as presented in Gann's discussion of the Square of 144 on Jan 15, 1948 when the high was $4.36 Did you look at the planetary positions on Jan 15, 1948 that I listed in chapter and find something interesting? If you did not, try comparing the number of Mars with the other planets Now what did you find? Correct You found Mars and Pluto at conjunction (at the same degree) at: 133 That's an interesting number because of its relationship to a number in "The Tunnel Thru the Air," Gann's novel, and its relationship to the Great Cycle But that's another work for another time and there is no need to go down that path now It is also interesting because of its position on the Square of Nine chart in relationship to a triangle of the Teleois and their relationship to a paragraph in Gann's planetary discussion of resistance lines on soybeans in his "private papers." But that again is for another work and that path would take us down lots of roads with many forks and the work we have at hand is enough to fill this book Chapter 5-Subtracting 360 Degrees Just like in a single digit numbering system (another path we will explore later) where "you cannot go beyond without starting over" Gann noted that you cannot go more than 360 degrees in a circle without starting over (We will discover why later in our study of "Natural Squares.") He illustrates this in his discussion of the price and time chart of to 360 degrees on page 153 of the course Actually the high on beans was $4.36 3/4, but Gann often rounded off numbers for convenience sake So, subtracting 360 from 436 I got 76 As I said in the preface, I ran thousands of numbers through my calculator looking for PATTERNS Here, I went one better than Gann Instead of subtracting 360 from 436, I subtracted 76 from 436 and got 360 and kept subtracting 76 until I could not subtract any more in this manner: 436-76=360 360-76=284 284-76=208 208-76=l32 132-76=56 That 56 was very interesting Why? From the top at 436 on Jan l5, l948 to the low of 20l on Feb 14, 1949 was: 56 weeks! Gann never pointed out this time frame in his discussion of the Square of l44 I would not have known it if I had not laid out the chart It was confirmed by my layout of the weeks on the 38 columns as suggested in chapter of the course Both the numbers 56 and 76 as well as the number 133 are connected with the "Great Cycle." And since Gann seemed to give a clue when he said his 2x1 line coming down would cross at 303 (436133) and the fact that his 267-weeks was just one more than 2x133, I concentrated a long time on 133 As I noted earlier, it's position on the Square of Nine chart and its relation to the Teleois angle and for the reasons given above, it was difficult not to concentrate on it But that concentration, interesting as it was, led nowhere at that time so I rolled up the chart and it was untouched for several years Because of sickness in the family (I'm now the sole caregiver for two elderly parents), I had pretty much put aside my Gann studies for several years, only going through it occasionally One night in early 1989 I called my farmer friend and told him not to be long on beans I had not seen a chart of beans for many years and the only thing I knew about current prices was picked up from the commodity news on TV They were somewhere in the middle $8 range He asked me why he should not be long I told him Mars and Jupiter would be in conjunction the next day I told him he needed to buy some puts Beans would probably be down the next six to eight weeks They never went up from that date Five days later they dropped limit down and went on down 90 cents during the next six to eight weeks and started recovering I called my friend a few weeks later and asked him if he had bought the puts He said no He thought I was crazy He had looked up the date in his ephemeris and the conjunction of Mars and Jupiter didn't come up until March I told him he was looking in the wrong book The date I had given him was the helio date and he had been looking in his geo book A short time later I was visiting a small commodity office where my friend and a few others were present They were talking about what a good call I had made One asked me to tell him when I had something coming up I called him that night and told him not to be long (the market on the charts looked strong at that point and he and the others at the commodity office thought the direction was up) I told him the geo Mars and Jupiter conjunction was coming up the next day That call was off by five days The market moved up another five cents during those five days and then headed down Not only did it go down for six or eight weeks like a I predicted for another 90 cents, but kept on going down from there for several months I didn't know where it bottomed as I was still not looking at charts I not relate this to brag about a couple of good calls, but to show that Gann's Mars and Jupiter conjunction still seems to depress the bean market I checked this out on a weekly chart several years ago for a 10-year period and it was working then And it still seems to be working now It did not always catch the exact top but from the point where the conjunction was made the market moved down or sideways for at least six to eight weeks if not for longer As I told my friend, "you might not want to be short, but you'd better not be long." By the way, the man did not take my advise on the geo because, as I said earlier, the market on the charts looked strong and friendly Needless to say, he lost You can confirm those two calls by looking them up on your charts The recent (1991) conjunctions with the same results renewed my interest in the 267-week chart under question And although we will return to the Mars-Jupiter conjunction later in this book, let's look again at the Mars-Pluto conjunction of January, 1948 With some tinkering and running numbers back and forth through the calculator I came up with a most interesting set of numbers Chapter 6-133 or 132? The cycle of Mars is 687 days in its 360 degree circle around the sun So it goes approximately 524 degrees a day So two days before its conjunction with Pluto, Mars would be at 132 degrees Since this is a weekly chart maybe the operative number is 132 and not 133 If so, it sets up a whole list of interesting number combinations Remember how we subtracted 76 from 436 several times and got 56 as the final answer? If we add 76 and 56 we get: 132 If we place the Square of 144 on 132 what we get? 276 (132+144) 10 to the number uses the powers of I believe you know what powers are to the second power means to multiply 2x2 or to the third power means to multiply 2x2x2 or 8, etc You begin the search for perfect numbers by multiplying the first power of times the second power of (-1) I not have a power symbol here so I will use this sign (^) 2^1 times ((2^2 (-1)) 2^1 is 2^2 is when we subtract the from we have So now we have times and the answer is Now let's the next one with our first multiplier being to the power of and the next to the power of from which is subtracted: 2^2 times ((2^3 (-1)) 2^2 is and (2^3 (-1) is and times is 28, our next perfect number Let's the next one (each time we will use the next power of 2.) 2^3 times ((2^4 (-1)) 2^3 is and (2^4 (-1) is 15 and times 15 is 120 But 120 is not a perfect number So what happened? What happened is we did not observe a rule that I didn't tell you about earlier If our answer from the numbers in parentheses does not equal a prime number, it is not used in our calculations A "prime number" in case you have forgotten is a number that is divisible only by itself and the number one Examples of prime numbers are 7, 11, 13, 19, etc The number 15 is not a prime number since it can be divided by and so we have to pass by this calculation It will not give us a perfect number calculation We have to go on to the next one 2^4 times ((2^5) (-1)) Let's calculate that: 16 times (32-1) or 16 times 31 or 496 This was the next perfect number after 28 as we saw earlier And 496 is the triangle of 31 Can you see now why these perfect numbers are triangular numbers? Look at the calculations above In Book VI-"The Triangular Numbers" I showed that triangular numbers both natural and unnatural could simply be made by multiplying the root of the triangle by the next number and dividing by For example to make the triangle of which is 10, all we need to 363 is multiply 4x5 and divide by But I also explained that it could be made in another way Take half of any of the two numbers, the root or the next number and multiply it by the remaining number As an example again of finding the triangle of which is 10, we could take half of the which is and multiply times or we could take half of the which is 2.5 and multiply by The answer will always be the same We can see that when we take one power of and multiply it by the next power of (-1) we are doing the same thing! We could make the triangle of 31 by multiplying it by 32 and dividing by Or we could make it by taking half of 31 or 15.5 and multiplying it times 32 or we could take half of 32 which is 16 and multiply it by 31 And that is exactly what we did when we used the 4th power of or 16 and multiplied it by the 5th power of or 32 (-1) So why don't you now try to find the next perfect number after 496 and see what it is the triangle of Give it a try before you look at the answer below: The first sequence you would try would be: 2^5 time ((2^6 (-1)) and that answer would be: 32 times 63 and that would not be right since 63 is not a prime number as it can be divided by and The next you would try would be: 2^6 times ((2^7 (-1) which would be: 64 times 127 Since 127 is prime that would be the next number which is a power of minus that we could use The answer is 8128 I told you earlier that the next perfect number after 496 was in the 8,000 range And we have found it And what is 8128 the triangle of? It is the triangle of 127 We could multiply 127 times 128 and divide by and get 8128 Or we could take half of 128 which is 64 and multiply it times 127 which is exactly what we did in our calculation with the powers of above Some experimenting will get you the next perfect number, but I would suggest you get a computer program that prints out primes to help or you will be dividing a lot of numbers to see if they are primes! Believe me, I did that No, we have not found the "perfect" number We have seen what some call perfect numbers and have seen what mathematicians call perfect numbers Maybe we will have to have a definition of a perfect number in order to find one 364 Until then that one perfect number is still elusive Appendix DECANATES OF THE PLANETS I had read in a book somewhere a long time ago about the decanates Each planet is suppose to have influence over 10 degrees of a sign It started with Mars for the first 10 degrees, etc and ended with Mars Then I saw in "The Astrology of the Old Testament" that these decans or decanates are also called the "faces" of the planets They didn't seem to be in any particular order But then I discovered that they are in the same order as the planets are arranged to make the planetary hours as shown in "The Astrology of the Old Testament." Whey the decanates start with the planet Mars is something I have not found out as yet I will be checking these decanates with Gann and Bayer stuff to see if they used them (Note: a number of years ago I read in the Encyclopedia Britannica about Magic Squares Recently in 1996, I read about them again and found that Cornelius Agrippa (1486-1535) constructed squares of the order 3, 4, 5, 6, 7, and and associated them with the seven astrological planets When I read this years ago I did not pay any attention to the order of the planets But this time I noticed that the order was the same as for the decanates So the planets and their magic squares would be Saturn (3), Jupiter (4) Mars (5), Sun (6), Venus (7), Mercury (8) and the Moon (9) Also note that the Sun is associated with and the Moon with just as it is in Karl Anderson's book, "The Astrology of the Old Testament." Might be coincidence and then again maybe not These are the "faces" of the planets or the decanates: Aires (1-10) March 23-April Mars Aires (11-20) April 2-April 11 Sun Aires (21-30) April 12-April 22 Venus Taurus (1-10) April 23-May Mercury Taurus (11-20) May 3-May 12 Moon Taurus (21-30) May 13-May 23 Saturn Gemini (1-10) May 24-June Jupiter Gemini (11-20) June 3-June 12 Mars 365 Gemini (21-30) June 13-June 23 Sun Cancer (1-10) July 24-July Venus Cancer (11-20) July 4-July 14 Mercury Cancer (21-30) July 15-July 24 Moon Leo (1-10) July 25-Aug Saturn Leo (11-21) Aug 5-Aug 14 Jupiter Leo (21-30) Aug 15-Aug 25 Mars Virgo (1-10) Aug 26-Sept Sun Virgo (11-20) Sept 5-Sept 14 Venus Virgo (21-30) Sept 15-Sept 24 Mercury Libra (1-10) Sept 25-Oct Moon Libra (11-20) Oct 6-Oct 15 Saturn Libra (21-30) Oct 16-Oct 25 Jupiter Scorpio (1-10) Oct 26-Nov Mars Scorpio (11-20) Nov 5-Nov 14 Sun Scorpio (21-30) Nov 15-Nov 24 Venus Sagittarius (1-10) Nov 25-Dec Mercury Sagittarius (11-20) Dec 5-Dec 13 Moon Sagittarius (21-30) Dec 14-Dec 23 Saturn Capricorn (1-10) Dec 24-Jan Jupiter Capricorn (11-21) Jan 3-Jan 12 Mars Capricorn (21-30) Jan 13-Jan 22 Sun Aquarius (1-10) Jan 23-Feb Venus Aquarius (11-20) Feb 2-Feb 10 Mercury Aquarius (21-30) Feb 11-Feb 20 Moon Pisces (1-10) Feb 21-March Saturn Pisces (11-20) March 3-March 12 Jupiter Pisces (21-30) March 13-March 22 Mars Appendix Another Way to Make Cubes (Discovered July, 1995) This is another way to make cubes on a flat surface You can draw some boxes I'm just going to put some numbers in rows Of course we can put down: and that would be the cube of If we put down: 366 1, 2, and add those we will have the cube of two (or 8) eight Note that we are using numbers in each row Also note that we started with in the first row and in the second Now let's put down a series in which we have three numbers in three rows: 1, 2, 2, 3, 3, 4, When we add all the numbers we have the cube of or 27 Note how we began each row If we wanted to a cube of or 64, we would use four numbers in each row and we would use four rows Give it a try! There is another way to make cubes using triangles Take the number of any cube we want to make Subtract from it and get the triangle of your answer Now multiply that triangle by twice the number of the cube we want to make Now add the square of the original number and you will have the cube Example: We want to make the cube of which is 4x4x4 or 64 First we subtract from and get Now find the triangle of which is Multiply that by 2x4 which is The answer is 48 Now add the square of which is 16 and 48+16 is 64 I can never say enough about triangles You can so much with them Using the formula above why don't you try to make the cube of or 125 I will start you off The triangle of is 10 Appendix Triangles and the Arithmetic Mean (Discovered Jan 27, 1996) This is another property of triangles A number times its triangle equals the arithmetic mean (half-way point) between the square of the number and the cube of the number Let's use for our number Its triangle is 10 Multiply times 10, getting 40 40 is the arithmetic mean (halfway point) between the square of (16) and the cube of (64) We can check that by adding 16 to 64 and dividing by to get the halfway point 16+64=80 Divide by and get 40 which is times its triangle (4x10) This also works for other powers of numbers 367 Again let's use the number But this time instead of multiplying times its triangle (10), we will multiply the square of (16) times the triangle of (10) and we get 10x16=160 We will find that 160 is the arithmetic mean between to the 3rd power (4x4x4=64) and to the 4th power (4x4x4x4=256) Now lets add 64 and 256 and get 320 Now divide by and get 160 If we wanted to find the arithmetic mean between the 4th and 5th powers of we would multiply the triangle of (10) times to the 3rd power And this would work for any powers you wished to work out Appendix More Triangle Stuff Using even numbers, if we add up to a number (getting a triangle) and leave out half of the number, you will have a double square of half of the number Example: We will use the even number 12 If we add from through 12 we will get the triangle of 12, which is 78 If we leave out half of our even number we will leave out the number and that will give us a total of 72 And 72 is the double square of 36 is the square of and 72 is two times 36 or a double square of Try that with another even number to check it out Appendix DISSECTING A NUMBER Somewhere in my work I told you that I often ran numbers back and forth through my calculator looking for PATTERNS So here I will show you how I dissected a number (really two numbers) that gave me trouble for a long time At the local library there is a set of books put out by Time-Life called "Mysteries of the Unknown." You might recall seeing these advertised on TV in the late 1980's or early 1990's One of the volumes in that series was "Ancient Wisdom and Secret Sects." In had quite a bit of stuff in there on Masonry On page 88 was a figure of a man made up of the sun, moon and Masonic symbols That figure stood on two stones, cubes if you like I noticed that there were numbers on the stones One number was 3734 The other was 1754 368 I played with those number off and on for a long time I checked them to see if they were squares, triangles, etc But I got nowhere One day I added the two numbers 3734+1754=5488 Again I check to see if the total was a square or cube or triangle Still nothing Then I decided to start dividing 5488 by two (You might want to get out your calculator and start doing the same.) I divided by two until I could not divide by two anymore 5488 divided by 2=2744 2744 divided by 2=1372 1372 divided by 2=686 686 divided by 2=343 When I had reached an odd number I could no longer divide by 2, I noticed that my final answer was a cube, the cube of since 7x7x7=343 Interesting Since I had divided by four times then to the fourth power or 16 times 343 would equal my starting number 5488 Since 16 is the square of we could say the square of four times the cube of seven would equal 5488 Still interesting, but I was looking for something a little more outstanding So I put down my numbers and started rearranging (that's how I a lot of my dissecting of numbers, since they can be expressed in a number of different ways.) 2x2x2x2x7x7x7 4x4x7x7x7 or 4x7x4x7x7 or 28x28x7 Ah hah Now that looks a little more interesting Why? Study it a minute What relationship does the number 28 have to 7? By now you should know that 28 is the triangle of So what we have here is a number times the square of its triangle, 7(28x28) But why this particular number times its triangle? Since there is a moon symbol among the drawings I presume it has to with the moon Admittedly, the number 28 is not exactly a moon cycle It takes slightly over 27 days for the moon to come back to the same point in the sky and it take 29.5 days to make a new moon But from what I have read the ancients divided the year 364 days (7 times 52 weeks) into 13 months of 28 days On the cover of Gann's little pamphlet "The Magic Word" he has a 369 triangle of or 28 The cubes are also involved here Remember, if you square any triangle the answer will include all the cubes up through the root of the triangle Since 28 is the triangle of then 28X28 includes all the cubes up through the cube of and 7(28x28) will include times all of the cubes up through Note that when I took the original number 5488 and divided by two, etc until I got to 343 I had found that the square of or 16 times the cube of or 343 equals 5488 Also remember that we can take half of (4) and multiply that times to get the triangle of 4x7 is 28 4x4x7x7x7=7X28x28 That works for any odd number Example, let's use the number instead of 3x5=15, the triangle of The square of is and the cube of is 125 9x125=1125 or we can multiply the square of the triangle of by 5: 5x15x15=1125 So there you have a sample of how I dissect numbers Appendix 10 Some Julian Days In "The Tunnel Thru the Air" Robert Gordon is born on June 9, 1906 which is Julian Day 2351 The novel ends on Aug 31, l933 which is Julian Day 12296 I found no significance in this, but maybe you can The difference is 9945 days 370 Appendix 11 Some More on the Cycle of Venus This is figuring the cycle of Venus Feb 15, 1906, Sun and Venus in conjunction at 25 Aquarius Julian Day is 2238 Since it takes 584 days for another conjunction we add that and get Julian Day 2822 On Sept 15, 1907 the conjunction is at 20 degrees Virgo, which is days before it should be since Julian Day 2822 is Sept 23 Don't know why this is Sept 15, 1907-Julian Day is 2814 Add 584 and get JD 3398 JD 3398 is April 21, l909 But the conjunction comes at degrees Taurus on April 29, which is eight days later So one conjunction came at 576 days and the other at 592, so I guess when the astro books lists the synodic period at 584, it is an average Appendix 12 Jewis h Calendar Based on 3.3 Seconds Time-Life Books, "Cosmic Connections" p 24 says lunar month is 29 days, 12 hours, 44 and 3.3 seconds Look up Jewish calendar based on 3.3 seconds Appendix 13 Geometric Means on Hexagon Chart Discovered February, 1993 Some geometric means on Hexagon chart on 90 degree angle, etc 3x3 6x7 9x11, etc first multiple increases by and second by 4x4 7x8 10x12 2x2 5x6 8x10 works off other squares too 371 Appendix 14 The Number Three and Triangles and Squares You know that the numbers 7, 40 and 77 have a PATTERN They are made by adding a triangle of a number to the square of a number Seven is made my adding the triangle of (3) to the square of (4) Forty is made by adding the triangle of (15) to the square of (25) Seventy-seven is made by adding the triangle of (28) to the square of (49) I have discovered that when you multiply those numbers, 7, 40 and 77 or any other that is the sum of a number's triangle and square, you end up with another triangle Let's look at the number 7, which is made up of 3+4 If we multiply by we get 21 which is the triangle of If we multiply 40, which is made up of 15 and 25, we get 120 which is the triangle of 15 If we multiply 77 by we get 231 which is the triangle of 21 How can we determine the triangle we will get? We simply multiply times the roots of the numbers The roots of above was the triangle of and the square of so multiply by and get and 21 is the triangle of The roots of 40 was the triangle of and the square of so simply multiply times and we get 15 and the triangle of 15 is 120 The roots of 77 is the triangle of and the square of Multiply times and that is 21 and the triangle of 21 is 231 Now can you it with 12 The triangle of 12 is 78 and the square of 12 is 144 Add them and get 222 When we multiply by we get 666 and we know that is the triangle of 36 And times 12 is 36 Appendix 15 The Number and the Difference in the Cubes I showed you that times any triangular number plus will always be the difference in two successive cubes and the cubs would be the cube of the root of the triangle and the root of the next triangle For instance we could multiply times 28 and add and get 169 which is the difference in the cube of and the cube of since 28 is the triangle of Those numbers I have firmly in mind as well as the triangle of number up to about 25 After that it gets a little hazy in my mind 372 So I found an easier way of doing it Say I wanted to find the difference in the cubes of 144 and 145 First I would have to find the triangle of 144 and then multiply the answer by and add But let's look at that easier way We know that when we want to find the difference in the cubes of and we would have to find the triangle of by multiplying times and divide by to get 28 Then we would multiply 28 times and add like we did above to get 169 We could also say 7x8x6 divided by two and then plus But why not divide the by and get Then we could simply say 7x8x3 and then plus 7x8 is 56 and when we multiply by we get 168 and when we add our we get 169 which is the difference in the cube of and the cube of So back to our difference in the cubes of 144 and 145 We would simply say 144x145x3 and then add our That number certainly gets around! Appendix 16 NUMBERS BOTH TRIANGULAR AND SQUARE There are some numbers that are both triangular and square The number 36 is both the triangle of and the square of The number 1225 is both the triangle of 49 and the square of 35 I was trying to figure out what other numbers would be triangular and square and how I would find those numbers With a little playing around I finally figured it out Can you it? Can you tell me the next number that would be both triangular and square? We know that to make a triangular number we multiply a number by the next and divide by We also know that a square times a square is a square Can you take it from there? Let's look at our first number, 36, which is the triangle of and the square of We know that to make the triangle of we can multiply times and divide by and get 36 But there is something else we could We could take half of 373 which is and multiply by and get 36 Do you have it now? When we divided by and got we got a square (2x2) and when we multiplied by (3x3) we got 36 Got it now? Think of as a double square and when we divide by two we get a square So what we need are two successive numbers in which one is a square (in this case 9) and the number before or after it is a double square (in this case 8) The 1225 above is both triangular and square It is the triangle of 49 and the square of 35 To make the triangle of 49 we would multiply by 50 and divided by We see that 50 is a double square (the double of 25 which is 5x5) and we get the square of 35 because 7x7x5x5 is 35x35 The next number that would be both triangular and square I figured in my head one night I was looking for a square that had a double square before it or after it We have already done the square of so let's work up from there I will put down some squares and we will look at the number next to it both before and after to see if it is a double square We can divide those numbers by and take the square root to see if we have the square of a natural number 8x8=64 (63, 64, 65) No double square 9x9=81 (80, 81, 82) No double square 10x10=100 (99, 100, 101) No double square 11x11=121 (120,121,122) No double square 12x12=144 (143, 144, 145) No double square 13x13=169 (168, 169, 170) No double square 14x14=196 (195, 196, 197) No double square 15x15=225 (224, 225, 226) No double square 16x16=256 (255, 256, 257) No double square 17x17=289 (288, 289, 290) We have a double square! Do you see it? Yes from all of your Gann reading you should know that 288 is two squares of 144 So if we found the triangle of 288 we would have a triangular number which is also a square So multiply 144 times 279 and we would have 12x17x12x17 and the answer will be the square of 204 and the triangle of 288 374 The numbers 12 and 17 are intriguing since if we had a square of 12, its diagonal would almost be 17 Appendix 17 FINDING THE TRIANGLE OF DOUBLE OF A NUMBER (Discovered Dec 24, 1998) When reading "Square of the Magi," the author had 28 plus 36 plus 36 plus 36 Couldn't figure out why he didn't stop at just adding 28 and 36 to arrive at square of But then it dawned on me He carried it out until he had the triangle of 16 which is 2x8 Thought I had something new But then I saw that when 28 is added to 36 we get the square of and I knew that a square plus two triangles of its root equals the triangle of a number that is twice as big Example: Thd square of plus 10 plus 10 equals 36 which is triangle of We could say triangle of plus square of plus triangle of But we can find some interesting numbers if we keep on adding triangles Starting with 15 (the triangle of 5) and adding 21 (triangle of 6) we get 15 36 (the square of 6) 57 78 (the triangle of 12 naturally) 99 120 141 162 183 204 225 (the square of 15) 246 267 (where Gann stopped in weeks on the bean chart) 288 (double square of 144) 309 330 351 372 393 414 435 (just one cent short of his high on beans) When I saw the 15 on top of this list and the 225 (the square of 15) down the list I thought I saw a germ of a PATTERN 375 I saw that when I subtracted 15 from 225 and got 210 and divided it by 21 I got 10 and that is the triangle of Hmmmmmmmmmmmmmmmmm But after I gave it a little thought I found that we had already discovered that Remember when we looked at some of the properties of triangles we found that if we took three successive triangular numbers like 10, 15, 21, we could multiply the two end terms and add the middle term and would have the square of the middle term So that is what was happening above I took 15 and after adding 21 for a total of 10 times I came to the square of 15 Let's extend our list and see what else we can find 15 36 (the square of 6) 57 78 (the triangle of 12 naturally) 99 120 141 162 183 204 225 (the square of 15) 246 267 (where Gann stopped in weeks on the bean chart) 288 (double square of 144) 309 330 351 372 393 414 435 (just one cent short of his high on beans) 456 477 498 519 540 561 582 603 624 645 666 the triangle of 36 (the square of 6) Note that we started by adding 15 and 21 and got the square of or 36 and here we have arrived at 666 which is the triangle of 36 Note that when we subtract 36 from 666 we get 330 and when we divide 330 by 21 we get 30 PATTERN? Yes, we see that we started with the triangle of (15) and the triangle of (21) and we know that 5x6 is 30 376 We saw earlier how triangles of squares can be made, but now we have a new one For the triangle of the square of (49) which is 1225 we could say 6x7=42 and 42 times 28 is 1176 Add 49 and we get 1225 Why 6x7 or 42 Remember that the triangle of and the triangle of when added equals the square of So taking the square of (49) and adding 42 times the triangle of (28) gives us 1225, the triangle of the square of Appendix 18 THE CUBES AND THE TRIANGLE OF THE SQUARE We saw earlier that the cubes are contained in the squares of the triangle For instance if we take the triangle of which is 28 and square 28 the answer will include all the cubes up through the cube of The cube is also connected to the triangle of the square I discovered this in one of my usual PATTERN searches Gann had talked about a change in cycles when you come to 325 I knew that 325 was the triangle of a square, the triangle of 25 which is the square of I was curious about how the cube of might be involved with 325, so I subtracted 125 from 325 and got 200 200 is a double square, two times 10x10 And I knew that 10 was the triangle of and is one unit less than Looks like there is a PATTERN there so let's check it out If we are right then: Double square of triangle of plus cube of will equal 136 which is the triangle of the square of The triangle of is The square of is 36 and the double square of is 72 The cube of is 64 and 64 plus 72 is 136 And 136 is the triangle of the square of which is 16 PATTERN made! I still don't know what Gann meant by the change of cycles at 325, but I still believe it has to with the fact that it is a triangle of a square 377 ... according to weeks 2-1 5-1 920 to 1 2-2 8-1 932 -6 72 weeks or 24x28 or 12x56 2-1 5-1 920 to 1 2-2 0-1 939 -1 036 weeks or 37x28 2-1 5-1 920 to 4-2 1-1 952 1680 weeks or 60x28 1 2-2 8-1 932 to 4-2 1-1 952 -1 008 weeks or... 4 -The Cycle of the Moon We have learned what the ancient writer meant about the 28-year cycle of the sun, or the cycle of the days of the week and the day of 31 the month Now let us look for the. .. to the 267-week chart Chapter 3 -The Cycle of the Sun First, let''s look at the cycle of the sun and why it is called a 28-year cycle It takes 28 years for the day of the week and the day of the