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OK, let's give it a whirl.

First, let's spread the lines out so we can drop a 6 and put in a 9.

6x6x6 6x6x9

We have found the first geometric mean between the cube of 6 and the cube of 9.

Then we will spread the lines again and drop another 6 and put in another 9 for the next line.

6x6x6 6x6x9 6x9x9

We have found the second geometric mean between the cubes.

And then in the last line we will drop all 6's and put in all 9's so we will have:

6x6x6 6x6x9 6x9x9 9x9x9

Now let's do our multiplication:

6x6x6=216 6x6x9=324 6x9x9=486 9x9x9=729

Take the four answers and divide the first into the second, the second into the third and the third into the fourth.

324 divided by 216=1.5 486 divided by 324=1.5 729 divided by 486=1.5 PATTERN?

Yes, we have the same multiplier that we had when we were finding the geometric mean between the squares, 6x6 and 9x9.

That's because 9 divided by 6 is 1.5. It works every time!

If we put our four numbers down in this series 216, 324, 486, 729 what does that suggest?

We have "two" geometric means between the cubes whereas with the squares we only had one geometric mean.

With the squares we could prove our answer by multiplying the two end terms and taking the square root.

If we multiply the two end terms of the cubes 216 times 729 we find that they equal the product of the two middle terms 324 times 486.

216x729=157464 324x486=157464

We can see from our box that the top row has three 6's and the bottom three 9's and the two middle rows have three 6's and three 9's and 6x6x6x9x9x9 is the same regardless of the order.

6x6x9x6x9x9 is equal to 6x6x6x9x9x9.

We found that the squares have one mean and the cubes two means.

PATTERN?

What if we wanted to find the geometric means between:

6x6x6x6 9x9x9x9

How many would we have?

Yes, you were a little quick for me that time. You had seen the

PATTERN. The number of means will be one less than the powers of the numbers. Since 6x6x6x6 and 9x9x9x9 are 6 to the fourth power and 9 to the fourth power than there must be three means.

We can check that by using the box method and dropping 6's and adding 9's:

6x6x6x6 6x6x6x9 6x6x9x9 6x9x9x9 9x9x9x9

The top line represents 6 to the fourth power and the bottom line represents 9 to the forth power and there are "three" lines in between which represent the geometric means so there must be "three"

geometric means between numbers to the "fourth" power.

You can work it out for yourself by the same method we used above with a different set of numbers.

In my example what will the multiplier be?

It's still 1.5 or 9 divided by 6.

Give it a try. Use some other numbers. It's good practice for finding PATTERNS.

I noted in my preface that our method would be simple,

observational arithmetic. No algebra involved. And you can see from the above I did just that. Once a PATTERN is established, the rest is easy.

In the above I used the "box" method, dropping one number and adding another. However, we could have just used the multiplier.

Say we were looking for the geometric means between 6 and 9 to the "fifth" power. We now know that we would have "four" geometric means since "four" is one less that the "fifth" power. Using the multiplier we would simply do this:

6x6x6x6x6=7776 7776x1.5=11664 11664x1.5=17496 17496x1.5=26244 26244x1.5=39366

39366x1.5=59049 which is 9x9x9x9x9

That works, but I like the box method better as it is more

graphic and lends itself to a better understanding of Gann's work.

In the preceding examples, I used the "natural" numbers. (The natural numbers were explained in Book IV). But the method for finding geometric means can be used for any numbers, even irrational numbers like the square root of 5 (which is involved in the golden mean) or pi. The same box method could be used.

Let's see how that would work. My computer does not have a square root or pi symbol, so I will used SR5 for the square root of 5 and PI for pi. Let's assume we want to find the geometric means of the cube of each.

We would make our "box" thus:

SR5xSR5xSR5 SR5xSR5xPI SR5XPIxPI PIxPIxPI

And the multiplier would be PI divided by SR5.

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In light of what you know now, reread the second page of Chapter 9, page 112 in the "old" commodity course (Section 10, Master Charts, page 3, Square of Nine in the "new" course) and see if you can see a PATTERN.

Gann talks about building the cube. He talks about the cube of 9 or 9x9x9. He also mentions 6 times the square of 9 or 6x81. We can always (and we should, to look for PATTERNS) break down our numbers into their component parts.

6x81 can also be read as 6x9x9 or 486. Check the work we did on the cube in Chapter 5 and see if you can find that number.

Yes, we can see that that number is one of the geometric means between the cube of 6 and the cube of 9.

As I mentioned earlier I always thought that the cube, 9x9x9, stood for a time period of days, months, etc. Maybe those were your thoughts, too.

I noted in Book I-"The Cycle of Mars" that Mars and Jupiter make conjunctions approximately every 27 months and a cycle of 27 times 27 months would be 729 since the cube of 9 is also the square of 27.

There is another planetary cycle that could be applied here, but we will not go down that path now.

The more I read that section of Gann the more I became convinced that the cube was not a "time period" at all but a "division" of time. In his discussion of the hexagon or six-sided figure he

mentions 30 years so I finally decided that maybe 30 years should be divided into 729 parts.

30 years is:

30x365.2422=10957.266 days

10957.266 days divided by 729=15.030543 days Interesting?

It should be.

15, among other things, is one of the divisions of the circle by Gann.

We will get back to that, but let's have another look at that geometric mean.

We saw that 9 divided by 6 is 1.5. We also know now from our work that 729 divided by 486 is 1.5. If we divide 20 into 30 it is also 1.5.

Gann says we can use 486 or 729. Let's see what he meant by that. If we divided 30 years by 486 what do we get?

10957.266 divided by 486 is 22.545814 or 22.5 days, another division of the circle. Gann says you can go over 22.5 days on your chart and reach 22.5 degrees in the circle.

And 22.5 divided by 15 is 1.5!

Now, let's reverse the process and divide 20 years by 729.

20x365.2422=7304.844

7304.844 divided by 729=10.020362 days and 10 divided into 15 is 1.5

All that is very interesting you say, but what does that have to do with the cycle of Venus?

Then again you might be way ahead of me.

You will recall that the heliocentric period of Venus was 225 days and if we take the square root we find it is 15!

So the 30-year period divided by the cube of 9 equals the square root of the cycle of Venus.

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Is there any other clues that the cube could refer to the cycle of Venus?

One could be the 15-day, 24-hour chart discussed on page 153 (in my copy) of the Gann material.

He notes that the time periods of 15 days equal 24 degrees.

Since Venus goes 1.6 degrees in one day then in 15 days it would go 24 degrees.

Gann ends this circular chart at 576 which is 24 squared, but it is also 4x144 as we saw earlier. We found earlier that 360 times 1.6 is 576 and we also found that 225 times 1.6 is 360.

What does that suggest? Let me put down the numbers like this:

225, 360, 576 Got it now?

That's correct!

360 is the geometric mean between the square of 15 and the

square of 24 and we can put down the numbers like this:

15x15=225 15x24=360 24x24=576

What else does it suggest from the three products above?

Remember how to verify a geometric mean?

That's right. We can multiply the two end terms and take their square root and the answer will be the middle term.

Ah, hah. You just saw it! 225 times 576 is 360x360 or the square of the degrees in a circle. 576 cycles of Venus equals 360x360!

This is just one of the examples of squaring the circle as you have noticed from your reading of Gann. It is the only one I will deal with here as the "square of the circle" is a study in itself.

There are also other explanations for the 15-degree, 24-hour chart and again this is not the place for a discussion of those as it would take us far afield and we want to keep to the discussion at hand.

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I am going to put down four number and let you have a look at them:

90, 144, 225, 360

Recognize them? The 90 is Gann's 90 degree angle, the 144 is the square of 12, the 225 is his 5/8 of a circle and square and also the cycle of Venus and the 360 is the degrees in a circle.

They have a special relationship and we will look at them a little bit later. But first some background.

In Book IV-"On the Square" I noted that the ancients recognized 10 different "means" and those 10 could be the basis of the tetraxes of Pythagorus. We looked at two those means in that book and I have given a recap in this one, but let's look at them once more.

The arithmetic mean is Gann's halfway point. The usual way used by most writers is to take two numbers like 6 and 12, subtract them to find their difference, take half of the difference and add to the smaller number to find the half-way point.

12 minus 6 is 6 and half of 6 is 3 and 3 added to 6 is 9 and 9 is the arithmetic mean between 6 and 12.

Gann's method was a lot simpler. He added the smaller to the

larger and divided by 2. Six plus 12 is 18 and 18 divided by 2 is 9.

Using simple numbers like this, the two methods don't really differ in ease of use, but when using bigger numbers Gann's method is easier.

The low on beans in 1932 was 44 and the high in 1948 was 436.

Gann added the two to get 480 and divided by 2 to get 240, the arithmetic mean, which in this case also ended up being the 2/3 point of the circle (240 is 2/3 of 360).

We also saw that the geometric mean was the middle number

between two other numbers which was arrived at by multiplying the two end numbers and taking the square root. We used squares to make the task simpler.

We found that the geometric mean between the square of 6 and square of 9 could be found thus:

6x6=36 6x9=54 9x9=81

We could prove the answer by multiplying the end terms and taking the square root to find the middle term.