When we add 8 to the 13 to get 21 we are simply adding another 8 to the pot. Dividing 8 into 21 we have two units of 8 with 5 left

over and 5 divided by 8 is still .625. We can add 8's to the pot until the cows come home and divide by 8 and still have a remainder of 5. It works every time.

In the third sequence of 19, 88, 107, we divide 88 by 19 and get four units of 19 which equal 76 with 12 left over. The remainder of 12 divided by 19 gives us the decimal fraction, .1635789. By adding 19 to 88 to get 107, we are simply adding 19 to the pot of 88 and getting five 19's or 95 with 12 left over. Need I tell you about the remainder of 12 and the cows again? It's still working!

Now for the second "amazing fact" about Fibonacci numbers that the experts like to point out to us. The fact that when you multiply 1.6l8 by .618 you get one or to be more exact .999924. The reason why the multiplication does not equal one is because the two Fibonacci ratios have been rounded off.

The two Fibonacci ratios are obtained by dividing the smaller Fibonacci number into the larger and then the larger into the smaller. Let's take the first part of our first example, 55 and 89.

When we divide 89 by 55 we get 1.6181818 as before and then we divide 55 by 89 and obtain .6179775. When we multiply the two results we get .9999999.

From our second example we can use 8 and 13. Dividing 8 into 13 we get 1.625 as before. Dividing 8 by 13 we get .6153846. Multiplying the two results we get .9999999.

Let's look at our non-Fibonacci example and divide 19 into 88.

We get 4.6315789. When we divide 88 into 19 we obtain .215909.

Multiplying the two results we get .9999995.

Let's pull a Dr. Watson and observe what we have done. In each case we divided a smaller number into a larger one and then the larger one into the smaller one and multiplied the results, getting a decimal number almost equal to one. The reason for it not being exactly one is the rounding off that has to be done.

So, let's express each of the numbers as fractions and then multiply. We can express 89 divided by 55 as 89 over 55 and 55 divided by 89 as 55 over 89. We multiply and get 4895 over 4895. Any number over itself or divided by itself is always one. The same procedure for 8 and 13 gets us 104 over 104 or one. Ditto for 19 and 88 which equals 1672 over 1672 or one. In our review of the two properties of the Fibonacci sequence we have really found nothing magic or amazing. It is simply how our arithmetic system works.

Interesting, but not amazing.

Now let's look at some of the other interesting properties of the Fibonacci sequence staring first with the Master numbers.

(That was the end of the article in Gann and Elliott Wave Magazine. The rest of this book continues that work.)

C C ha h a pt p te e r r 2 2 -T - T he h e M Ma as s te t e r r N N um u mb be e rs r s

This book is written with the assumption that the reader is

somewhat familiar with the Fibonacci series of numbers. Therefore no attempt is being made to explain its history, how the numbers are found in nature, the pyramids, etc. That has been done by a number of other writers.

But before we proceed I would like for you to close up this book (or step away from the computer if you are reading this on disk) and write down the series up to 144 before you continue reading.

I asked you to do that because of an important concept we must understand before we go on to the master numbers.

Like many of you I have seen the series expressed in two ways.

Like this:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

And like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

For a long time I thought the first way seemed to be just as

good as the second. The extra numeral "1" didn't seem to make any difference. It seemed to confuse the issue and was just in the way.

Wrong!

And if you wrote it that way, you should understand why it is wrong.

The Fibonacci sequence is also known as a summation series. That is, adding two numbers to get a third, adding the second to the third to get the next number, etc.

So, let's examine the series, working backwards.

We obtained 144 by adding 55 to 89; 89 by adding 34 to 55; 55 by adding 21 to 34; 21 by adding 8 to 13; 13 by adding 5 to 8; 5 by adding 3 to 2; and 3 by adding 1 to 2.

But how do we get the 2?

We can't do it in the first series because 0 plus 1 is 1. So we have to have 1, 1, in order to add to 2.

Now let's look at the master numbers.

I do not want to get into an argument here about what Gann

considered a master number. In one place he indicates it is 45.

Others believe it was 13, some think it was 17 and others 72 because of the references in Masonry. That discussion is for another time and place.

But here we will look at master numbers as understood by the numerologists.

I do not want to get into all of the theories of numerology

either (there are a number of books on that subject which can be read by those who are interested), so we are only going to look at the master numbers.

The master numbers in numerology are the double digits of 1 to 9 or 11, 22, 33, 44, 55, 66, 77, 88 and 99.

Some of those will be recognized by Gann students.

They are:

22, the number of Robert Gordon of "The Tunnel Thru the Air"

when his name is reduced to a number according to the numerology system.

33, the highest degree in Masonry and also a Gann square among other things.

44, the cash low on soybeans.

55, a number in the original Fibonacci system and a triangular

number. (Triangular numbers were discussed in Book VI of this series.

66, a number mentioned several times by Gann and also a triangular number.

77, a number well hidden in "The Tunnel Thru the Air."

88, the cycle of the planet Mercury.

For our present purposes we will divide the master numbers into two halves. Instead of calling this number, 11, eleven we will call it "one placed beside of one" as in the original Fibonacci series.

So let's pick out a couple of master numbers and see what we can discover.

We will start with 44 or 4 and 4 or 4 beside of 4 and add them in a summation series just as we do with the original Fibonacci series.

We add 4 to 4 and get 8; 4 to 8 to get l2, etc. as follows:

4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356.

Let's pause for a second and divide 356 by 220.

We get 1.6181818!

Let's select another number, 77, and use the same procedure obtaining:

7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623.

Let's pause once again to check some of our work and divide 385 into 623.

We get 1.6181818!

Before going on I would like for you to do the other six master

numbers in the same way, carry out the numbers to the 11th term and then divide the last term by the next to last term.

Ok. You have done them.

PATTERN?

Now you are probably doing a double take and saying, "What's going on here? This can't be true. I was told that only the original Fibonacci system worked like that."

"Sure," the experts say, "You can do summation series, but they will never work to the Fibonacci ratios."

Wrong!

We just showed that they can. Let's observe what we did.

Instead of thinking of 7 and 4 as just 7 and 4, we can think of them as units. In the same way that we added one unit of 1 to one unit of 1 and got two units of 1, we added a unit of 7 to a unit of 7 and got two units of 7 or 14.

When we added 4 to 4 we added one unit of 4 to another unit of 4 to obtain 8 or two units of 4. Adding 4 to 8 we get 12, or three units of 4, etc.

Think about that concept with those master numbers you have just counted.

Let's take the kids to the park and make some observations. The kids want to seesaw so we pick out one. One kid weighs more than the other so we have to adjust the seesaw to make it balance.

One kid weighs 55 pounds and the other 89. Never mind the weight of the seesaw itself, we move it until we get the two kids weights adjusted to it.

We can say that one kid is 1.6181818 heavier than the other

because this is what we find when we divide 55 into 89. That is their ratio, one to the other.

Let's see if we can let some other children ride on the seesaw

along with the first two kids, which we will call kid 55 and kid 89, without having to rebalance the seesaw.

We pick out a couple of 10 pound kids and put one on each side of the seesaw, but it doesn't balance.

Why? The 10 pounds added to kid 55 makes 65 pounds and the 10 pounds added to kid 89 makes 99 pounds. When we divide 99 by 65 we get 1.5230769. We upset the ratio and therefore lost the balance.

So we take off all the kids and replace 55 with a 45 pounder and kid 89 with a 79 pounder. Still out of balance. When we divide 79 by 45 we get 1.7555555.

Again we upset the ratio and put the seesaw out of balance.

So we take off those two kids and put kid 55 and kid 89 back on each end of the board. To this we add another 55 pounder to kid 55 and an 89-pounder to kid 89 and find that they balance.

We can check that by dividing 110 into 178. We obtain 1.6181818.

Why?

When we added the 10 pound to each side we were adding like amounts which do not work. In the last instance we added a multiple of each side. We could keep adding 55 to one side and a like amount of multiples of 89 to the other and obtain the same ratio.

C C h h a a p p t t e e r r 3 3 - - H H o o w w t t o o H H o o l l d d a a R R a a t t i i o o

In chapter two one might think we ought to have added the 89-pounder to number 55 and the 55-89-pounder to number 89 in which we would have had 144 pounds on each side. But you must remember that we started with a board which had to be adjusted so that the number 55 was on the longer end and 89 on the shorter end and our object was to maintain a proportion or ratio rather than equality.

Let's go back and look at our series of 4's and 7's. When we divided 4 into 356 we get 89 and when we divide the next to the last term, 220 by 4, we get 55.

In the 7 series we divide 7 into 623 and get 89 and when we divide 7 into the next to the last term, 385, we get 55.

PATTERN?

Using the master numbers you worked with you should come up with the same results if you did it right and made no mistakes.

So, in summary we can say that any series of two equal numbers, such as 8 and 8, when added together in a summation series will always equal multiples of the Fibonacci series or vice versa.

Another way to look at is to say that when two numbers, such as 50 and 100 that have a proportional value, that is, one-half to one, will not change that proportion when both are multiplied by the same number.

If we multiply 50 by 4 and 100 by 4 we get 200 and 400. 200 is still one-half of 400.

When we added 55 to 89 to each side of the original 55 and 89 on the seesaw, we were doing the same thing as multiplying each number by two. Multiplying both sides by two did not change the ratio or proportion. Ditto if we multiplied each side by any other number.

So there are two ways of changing the original Fibonacci series into a new group of numbers which will retain the same relationship or ratios.

We can put down 4 and 4 or any other number and add them in a summation series.

Or we can simply multiply the original series by a given number such as 4x1, 4x1, 4x2, 4x3, 4x5, etc.

Now, we can set up a table which will show the relationship between the original Fibonacci numbers and the master numbers.

They can be listed down the page or across the page. We will

also have a column showing the ratio between each number which will correspond to the ratio between any given set of master numbers.

We can see that the ratio of 1 and 1 is the same for 2 and 2, 3 and 3, etc.

The ratio between 8 and 13 is l.625. It is the same for 2 times 8 and 2 times 13; 4 times 8 and 4 times 13; 9 times 8 and 9 times 13, etc.

1 2 3 4 5 6 7 8 9 Ratio

1 2 3 4 5 6 7 8 9 1

2 4 6 8 10 12 14 16 18 2

3 6 9 12 15 18 21 24 27 1.5

5 10 15 20 25 30 35 40 45 1.666

8 16 24 32 40 48 56 64 72 1.600

13 26 39 52 65 78 91 104 117 1.625

21 42 63 84 105 126 147 168 189 1.615 34 68 102 136 170 204 238 272 306 1.619 55 110 165 220 275 330 385 440 495 1.617

C C ha h a pt p te e r r 4 4 -L - Lo oo ok k in i ng g f fo or r G Ge e o o me m e tr t r ic i c M Me ea an n

One writer once said they thought that Gann used Fibonacci, but didn't know how he did it.

In the above discussion we have shown one way that he could have done it.

Gann's work also was concerned with squares, so let's explore that a little and see what relationship we can find between the squares and the Fibonacci numbers.

We will make a little detour here in order to grasp an idea.

It was in search of a possible "geometric mean" in the Fibonacci series that led me to the discoveries on the master numbers.

The mathematicians in the crowd will see this as elementary, but us Watsons will have to make some observations.

Don't let the mathematical term "geometric mean" scare you. We will go through it easily so it can be grasped.

A mean is a number between a smaller and larger number that has a certain relationship to the smaller and larger numbers.

There are arithmetic means, geometric means, harmonic means and 7 other means known to the ancients.

I discussed the arithmetic and geometric means in Book IV-"On

the Square" and showed how to find the geometric means on the Square of Nine chart.

I discussed the harmonic means in Book V-"The Cycle of Venus."

I do not want to go into all the means here as that would take us far afield and I want to stick to the discussion at hand.

But, I will do a little recap on the geometric means.

A geometric progression is one that grows by a fixed multiplier and has a geometric mean. The mean can be found by multiplying each end of the progression and taking the square root of the answer. The answer will be the middle term of the series and is a check to see if you are doing it right.

For example, the geometric progression most understood by the public is the one that involves the story of the hired hand who told the farmer that he would work for a penny the first day if the farmer agreed to double that each day for a month.

As the story goes, by the end of the month the hired hand would be making over $1 million a day.

In document The PATTERNS of GANN (Page 194-200)