Conclusiones

natural numbers.

But how could it be made? I couldn't simply add .5 to the triangle of 4. The triangle of 4 is 10 and 10 plus .5 is 10.5 and I thought the answer must be 12.375.

It should have been obvious. Put this work aside and see if you can figure it out.

Got the answer already? Yes, you are quicker than I am. As I noted before I'm a little dense and it takes me several whacks to figure something out.

Let's see now. We can take 4.5 and multiply it times the next number and divide by 2. (That's the way we made the triangular numbers from the natural numbers.)

But what is the next number?

It took a little experimentation until I came across the obvious.

I multiplied 4.5x5.5 and divided by 2 and got 12.375.

So the "next" number we use is simply 1 added to the number (the root) for which we are looking for the triangle.

That's really what we did when we were making the triangles from the natural numbers. The next number was simply 1 plus our original number.

If we wanted to find the triangle of 5.5 we would multiply it times 5.5+1 or 6.5 and then divide by 2.

Why would we want to know about the unnatural triangles? As I noted in Book IV-"On the Square" very little of Gann's material ended up on natural numbers, squares or otherwise. Most ended in fractions and the more we know about those fractions, maybe the closer we will be to the answers about his work.

So let's make a short list of the triangles of numbers which end with the fraction of one-half or .5. Where do we start from?

From .5 itself.

As in our earlier work the first set of numbers will be the

numbers (roots) for which we are looking to find the triangle and the second will be the answers.

.5 .375 1.5 1.875 2.5 4.375 3.5 7.875 4.5 12.375

5.5 17.875 6.5 24.375 7.5 31.875 8.5 40.375 9.5 49.875

Can we prove our work? Are we on the right track?

Do these numbers follow the same PATTERNS as the ones with the natural numbers? Let's check.

You will recall from our work with the natural numbers that the triangles were made simply by adding 1 to 2 to 3, etc.

But actually we were adding the "triangle" of 1, which is 1, to the natural numbers in order.

So here we do not add .5 to 1.5. We add the triangle of .5 to 1.5. So we add:

.375+1.5=1.875

and to 1.875 we add 2.5 and get 4.375.

It works!

Let's give it the acid test by using some other characteristics of the natural triangles.

You will recall that with the natural triangles we could add the triangle of one number to the triangle of the next and get the square of the next:

The triangle of 13 is 91 and the triangle of 14 is 105.

91+105=196 which is the square of 14.

Let's try it with our unnatural numbers.

Using the bottom of the ladder again we add .375 to 1.875 which is 2.25. Taking the square root we find it is 1.5. So the triangle of .5 plus the triangle of 1.5 equals the square of 1.5.

PATTERN made!

Let's check some of the other qualities the natural triangles have.

Remember that we took three successive triangular numbers,

multiplied the two end terms and added the middle term and the answer was the square of the middle term.

Let's try that.

.375, 1.875, 4.375

.375x4.375=1.l64065 1.64065+1.875=3.515625

And the square root of 3.515625 is 1.875 PATTERN made!

We also found that we could take any natural triangular number, multiply it by 8 and add 1 and we would end up with a square, actually an odd square every time.

So, let's apply that characteristic:

8x.375=3

3+1=4, the square of 2.

PATTERN made and we see that when we multiply 8 times the triangular number which have one-half as a fraction in its root and add 1 we end up with the "even" squares.

Try a few and prove it to yourself.

Another characteristic of the natural triangular numbers is that

when we multiply any by 9 and add 1 we get another triangular number.

So, let's try that. Again from the bottom of the ladder.

.375x9=3.373.375+1=4.375 That's the triangle of 2.5

Remember now we could find out what triangle we would be making with this method. We multiplied the root of the original triangle by 3 and added 1

The root of .375 is .5. And 3x.5 is 1.5 and when we add 1 we get 2.5.

PATTERN made!

Try a few and prove it to yourself.

Even the 72 times a triangular number plus 9 works.

.375x72=27

27+9=36, the square of 6.

1.875x72=135 135+9=144

Note that the square roots of 36 and 144 have a difference of 6.

What do you think the next square will be?

4.375x72=315

315+9=324, the square of 18.

Try making some triangular numbers with other unnatural numbers like the triangle of 4.25, 5.25, etc. You would start with the

triangle of .25 if you wanted to make a list and the first would be .25x1.25=.3125. Then divide by two to get your first triangular number. Take if from there. Then check with some of the characteristics found above.

C C ha h a pt p te e r r 1 1 2 2 -M - Mo or re e E E vi v id de e nc n c e e

Can we find some numbers in the Gann material which are triangles of unnatural numbers?

I have not studied them all and an index of Gann numbers will be along later, but let's see if we can find at least one.

Let's find the triangle number of 22.5:

22.5x23.5=525.75

525.75 divided by 2=264.375 Let's check another one:

21.5x22.5=483.75

483.75 divided by 2=241.875

Now check your answers in the private papers "Time Periods and Price Resistance," the table of 1/64th of the circle.

C C ha h a pt p te e r r 1 1 3 3 -G - Ga a nn n n' 's s H He e xa x ag go o n n a a nd n d t th he e A An nc c ie i e nt n t H He e xa x ag go o n n

Gann's hexagon and the hexagon we made, the ancient 6-sided figure, must not be confused. They are made in two different ways.

The ancient is made by adding numbers which are four units apart.

Gann's is made by placing 6 around 1 and 12 around that, and 18 around that, etc.

The octagon of the ancients and Gann's cycle of 8 or Square of Nine (also called an octagon by some writers) should not be confused either. All the "cycle of" work based on Gann"s method and the ancient way of making sided figures are two different things.

C C ha h a pt p te e r r 1 1 4 4 -T - T he h e T T r r ia i an ng gl le e o of f t th he e Z Zo od di ia ac c

The study of the triangles presents many interesting observations about other things.

When you are going through other material you will be able to spot them a little better.

Case in point in the Time-Life book "Visions and Prophecies" in the series "Myteries of the Unknown" on page 122 is a discussion of the tarot cards. It did not surprise me that there were 78 cards in the tarot deck.

Why 78? That number is the triangle of 12. There are 12 signs of the zodiac, 12 months in the year, 12 tribes of Israel, etc.

You can form a triangle with them by placing 1 card at the top, 2 under that, etc. The last row will have 12 and the 78 cards will form a large triangle.

And of course that is why the numbers we have been looking at are called triangular. Each row or number counts one more than the one before it.

Check the drawing in the Time-Life Book series, Mysteries of the Unknown, "Ancient Wisdom and Secret Sects," page 147. Look at the triangle in the middle and the 10 bars inside of it and notice how the triangle points to the fourth sign of the zodiac signifying the number 10.

At the top there are some figures which look like faces and some points of what appear to be crowns. Count the points and then count the ribbons that flow from the neck.

I'll let you make the connection. As I said before, why should I have all the fun.

If I had never learned about triangles, this drawing would have