and 31 for a start

It works every time!

If we wanted to add another row to the side of the square of 7

we would start at 50 at the bottom which would be 5. This eighth row in the square of 52 would start at 365 which is also 5. We could add a ninth row to the square of 7 and it would end at 63.

Beginning a 10th row we would have 64 which is 1 and brings us back to where we stated in the first row since we cannot go beyond 9 without repeating. The ninth row of 52 would end at 468 and the 10th row would begin at 469 which is 4+6+9 or 19 or 1+9 which is 10 and 1+0 is 1.

It works every time!

I know. I said earlier I would not do any more reducing since you would understand by now how to do it.

But its does make it easier, doesn't it?

We could make an overlay of the square of 7 or 49 numbers

reduced to SDVs and place it over the various parts of the square of 52 and numbers working to the square of 7 would also be working to the square of 52.

C C ha h a pt p te e r r 9 9 -T - T he h e 2 24 4- -H Ho ou ur r S Sq qu ua ar re e

Looking at another important square, the 24-hour square which ends at 576 or 4 times 144, we can see that it has an SDV of 6 so we

could use a square of 15 or a square of 6 to overlay it.

Let's lay out a square of 6 SDV style and observe it. This time let's give more attention to the numbers going straight across instead of the diagonal.

6 3 9 6 3 9

5 2 8 5 2 8

4 1 7 4 1 7

3 9 6 3 9 6

2 8 5 2 8 5

1 7 4 1 7 4

Along the bottom we have 1, 7, 4, 1, 7, 4. In the next row we have 2, 8, 5, 2, 8, 5. In the third row the numbers are 3, 9, 6, 3, 9, 6. In the fourth row the numbers are 4, 1, 7, 4, 1, 7. In the fifth row they are 5, 2, 8, 5, 2, 8. In the sixth are 6, 3, 9, 6, 3, 9.

As you can see there are only three lines of different numbers, the other three being the same numbers, but in a different order.

We can see why repetition starts in the forth row from the left.

Any number in the first row will become the number in the fourth since we are adding three 6's to the number in the first row to get the one in the fourth. Three 6's are l8 which is a multiple of 9 and adding 9's does not change the value of the original number.

It would be the same if we were using a square of 24. We would be adding three 24's to the number in the first row to get the number in the fourth row. Three 24's equal 72, another multiple of 9, which still does not change the value of the original number.

With like reasoning we can see that the number in the fifth row is the same as in the second and the one in the sixth is the same as in the third, all for the same reason.

Now we will look at something a little different. Since this is a 24 hour and 15 degree chart or 15 days and 24 degree chart, let's convert the degrees to SDV. The first 15 degrees will convert to 6 and we will place it in front of 1, 7, 4, etc.

The second unit of 15 is 30 which becomes 3 and we place it in front of 2, 8, 5, etc.

The third unit of 15 is 45 and we will make that 9 and place it in front of 3, 9, 6, etc. If we kept converting the results would be the same since 60 is 6 and would be in the next line of 4, 1, 7, etc.

So for simplicity we could work with 3 lines corresponding to three numbers.

x 6 3 9 6 3 9

x 5 2 8 5 2 8

x 4 1 7 4 1 7

45 Deg (9) 3 9 6 3 9 6

30 Deg (3) 2 8 5 2 8 5

15 Deg (6) 1 7 4 1 7 4

The brief account Gann gives about this square will have you going around the circle which can be a little confusing. That's why I converted it to a square and took a small portion, the three lines, to work with.

A few numbers are given by Gann which on the surface are

confusing but after we make the conversion we can put them into an interesting perspective. By converting 44, 239, 311 and 344, we can see that they fall into three groups, 8, 5, 2, which fall on one of our three lines. No matter what angle they fall against, the angle will convert to a 3 such as 120.

The other two numbers, 67 and 436 are both 4's. When Gann subtracts 360 degrees from 436, he gets 76, which is 4 and just a reversal of 67. 67 plus 9 is 76. Subtracting the 360 degrees from 436 didn't change its value since 360 is a multiple of 9 as we saw earlier.

Other possibilities can be seen in working off angles. The

number 436 is 16 degrees beyond the angle of 420 which is a 6, in the same way that 76 is 16 degrees beyond the angle of 60 which is a 6.

Subtracting 360 from 436 gives us 76 and subtracting 360 from 420 gives 60. This provides for the same degree in the circle and off the same angle.

The other angles can be worked the same way.

C C h h a a p p t t e e r r 1 1 0 0 - - T T h h e e P P r r o o p p e e r r t t i i e e s s o o f f " " 1 1 " "

Let's now turn our attention to a couple of other aspects of the

SDV system. Until now we have been dealing with 9 and have seen that it's properties in the SDV system is the same as zero in our usual numbering system.

In the usual numbering system the addition of zero does not

change the value of a number. Multiplication does change the value to zero. In an opposite manner the addition of "1" changes the value of a number but multiplication does not.

Is there a number in the SDV system that has the same properties as the one in our regular numbering system?

Yes. It is still 1 or numbers that add to 1. We can readily see

that because when we multiply a number by 10 such as 5 and get 50 its SDV value is still 5. But we can also do the same thing with 19, 28, 37, etc. because their SDV is 1. 19 times 5 is 95 and 9+5=14 and 1+4=5.

I'm sure you have seen by now that we can deal in certain ways with the SDV numbers other than 9 or 1. My discussion to this point has been the 9 as zero and the 1 as 1.

Let's add up a couple of big numbers and then get their SDV value. Let's take 194,536 and add it to 233,327. We get out our little hand held calculator and find that the answer is 427,863. I believe at this point I no longer have to show you how to simply add those six numbers to come up with the SDV of 3.

But there is an easier way of doing it!

Without the calculator we could have established the fact that the answer would be 3 by doing some mental calculations.

And doing mental calculations is the whole point of this book as you might have gathered from the first paragraph.

The key is this:

We could have reduced each number to SDV before adding. Since the addition of 9 never changes the value in our SDV system we can simply drop them or cast them out.

Looking at 194,536 we can drop the 9, drop the 4 and 5 since they add to 9 and drop the 3 and 6 since they add to 9. All we have remaining is 1.

Looking at 233,327, we can drop the three 3's and the 2 and 7 and have 2 as the single digit value.

So our first number has an SDV of 1 and our second an SDV of 2.

Add them together and we have 3, which was the answer we got when we ran the two large numbers through the calculator and then reduced the answer to its SDV.

But now you are probably ahead of me so before reading further why don't you give multiplication a try SDV style!

We can perform multiplication in the same way. We can take a couple of Gann numbers like 44 and 67 and multiply them:

44X67=2948 2+9+4+8=5

We could have first reduced our multipliers to single digits:

44=4+4=8

67=6+7=13=1+3=4 Then multiply them:

8x4=32=3+2=5

Let's look at subtraction. In our normal system if we subtract 7 from 5 we get -2. If we subtract any greater number from a lesser number we will always have a minus number.

But in our SDV system if we subtract 7 from 5 we get 7!

It took me quite a while to figure that out, long after I had

come up with the 9 as zero. But when I did figure it out it seemed so simple.

Want to give it a try before going on? I'll wait. I need a good rest!

Got it? Ok. Let's have a look at it.

To subtract a larger number from a smaller in the SDV system simply add 9 to the smaller and then subtract. Adding 9 to 5 gives 14 which does not change its value. Now subtract 7 from 14 and the answer is 7.

25 is a 7 and 32 is a 5 and 25 from 32 is 7, so 7 from 5 is 7.

We know that 7 minus 7 is 0 in our regular number system. In our SDV system we can change the first 7 to 16 by adding 9. Now subtract 7 and you are back to 9. Again we are using 9 as zero.

C C ha h a pt p te e r r 1 1 1 1 -S - S in i ng gl le e D D ig i g it i t o on n t th he e S S qu q ua a r r e e o of f N Ni in ne e

Going along the angles on a square chart like a 19x19 is pretty straight forward. However the charts where the numbers begin in the center such as the Square of Nine and the Square of Four can be a little confusing because of the way they are constructed.

As I said before I do not want to get into a long discussion of