you can make a PATTERN. Hint: Look at the differences in numbers.
Did you get it?
Let's look at them.
The difference in 4 and 8 is 4 The difference in 8 and 14 is 6 The difference in 14 and 22 is 8 The difference in 22 and 32 is 10
Now without even looking at the chart do you know what the next number on that 45-degree angle will be?
If you said 44 you would be correct.
Here again you saw that the "variable of 2" or the gain of two was at work again. The differences in our example were 4, 6, 8, 10 and the next difference would be 12 and 32 plus 12 is 44. And the next number would be 44 plus 14 or 58.
Now let's look at the numbers running up on the 45-degree line parallel with those on the line of 4 that begin at 9.
9, 15, 23, 33
Can you see the PATTERN here without the hint? What would be the next number on the line?
That's right. It's 45.
The difference in 9 and 15 is 6 The difference in 15 and 23 is 8 The difference in 23 and 33 is 10
The next difference would be 12 and 33 plus 12 is 45. That old
"variable of two" again. Now do a couple for yourself. Try 16, 25 and 36.
Now go to 1, 2, 5 and 10 and check them on the 45-degree angle.
Going up from 2 on the diagonal you will notice that each number is the geometric mean between successive squares.
I noted above that there is an interesting relationship between the usual square and the Square of Nine chart so let's get to it.
Look at the 45 degree angle again that contains the numbers 4, 8, 14, 22, 32.
Now let's multiply 4x8 and get 32. Look at the location of each of those three numbers, 4, 8 and 32 on the usual square. They are on the same 45-degree angle. You will find that when you multiply any two numbers on a 45-degree angle, the answer will also be on the same angle. Take the 5 numbers above and use any two to multiply and then extend the series on out and you will find the answer.
Now try it with some of the numbers on the other 45-degree
angles.
Let's look at the Square of Nine chart. Locate 4, 8 and 32.
Notice anything?
We find that 4 and 32 are on a 45-degree angle and that 32 is 2 units away on that angle. If we go the 90-degree route we can go up two units to 34 and the over two units to 32.
Let's try another, multiplying 8 times 14 to get 112. Locate 8, 14 and 112 on the Square of Nine chart. Notice anything?
We find that 8 and 112 are on a 45-degree angle and that 112 is 4 units away from 8 or if we take the 90 degree route we can do down four units to 116 and then four units over to 112.
PATTERN?
In both examples above the answer is on the 45-degree angle with the smaller number.
4x8=32 and 4 and 32 are on the 45-degree angle. 8x14 is 112 and 8 and 112 are on the 45-degree angle.
The second part of the PATTERN is that the distance on the 45-degree angle from the small number to the answer is one-half the small number. The distance between 4 and 32 is half of 4 or two units. The distance between 8 and 112 is 4 units or one-half of eight. And when we go the 90 degree route the distance is the small number itself.
Now try 14 and 22 for yourself.
Before you multiply to get the answer try to find the answer on the Square of Nine chart.
Let's try some from the 9 row on the regular square. We multiply 9 times 15 and get 135. Check those three numbers on the Square of Nine chart. Then check 15 times 23.
Move over to the other side of the regular square and check 5 times 11 and 11 times 19.
PATTERN?
Yes, we have a PATTERN, but not like the one with the other numbers.
You might have noticed that when we were doing 4, 8, 14 and 16, 24, etc. we were using even numbers and now with 9, 15 and 5, 11, etc. we are using odd numbers.
When we use 9 and 15 and 5 and 11, the answer is found on the
45-degree lines running from the larger number. With the even numbers the answer ran on the same 45-degree line as the smaller number.
With the even numbers the answer was half the distance of the smaller number. If the small number was four then the distance was two.
It takes a little while to find the PATTERN with the odd numbers. Give it a try before looking below.
With 9 and 15 the distance is four. With 11 and 19 the distance is five. Got it now?
The distance is half of one less than the smaller number. 9 minus 1 is 8 and half is 4. 11 minus 1 is 10 and half is 5.
If we use the 90-degree method explained earlier we find that the distance is 1 less than the smaller number. With 9 it would be 8 and with 11 it would be 10.
Try 21x31 and check your answer on the Square of Nine chart.
In our discussion of the geometric means between squares we saw that the multiplication of any two numbers made a geometric mean between the squares of the two numbers so all of our answers in the discussion above are geometric means between squares.
When you multiplied 21x31 you found the geometric mean between the square 21x21 and the square 31x31.
You might wonder if a number can serve as a geometric mean between more than just two squares. Think about it a minute and you can probably answer the question yourself.
Let's look once again at the number 30 from our discussion of the successive squares. We found that 30 was the geometric mean between the square of 5 and the square of 6. But 30 has factors other than 5 and 6.
30 is also the product of 2x15 and 3x10 so it can be a geometric mean between those squares like so:
2x2=4 2x15=30 15x15=225 3x3=9 3x10=30 10x10=100
In the first example we would have the series, 4, 30, 225 and the second 9, 30, 100.
And checking our answers we can see that 4 times 225 is 900 and 9 times 100 is 900 and the square root of 900 is 30 so we have a geometric series.
Of course the multiplier for each would be different. The
multiplier for the first would be 30 divided by 4 and for the second would be 30 divided by 9.
This sets up another interesting question you might ask. Can a square itself be a geometric mean between two other squares?
C C ha h a pt p te e r r 1 1 3 3 -T - T he h e S Sq qu ua ar re e a as s a a G Ge eo om me e tr t ri ic c M Me e a a n n
The answer again is yes. Let's look at a familiar square, 144, which is the square of 12, but it too has many other factors. I will not mention all of them, just a couple to give you the idea.
9x16 is 144. Of course both 9 and 16 are squares, but that is not the point here.
144, the square of 12, is the geometric mean between the square of 9 and the square of 16 as shown below:
9x9=81 9x16=144 16x16=256
8x18 is also 144 so we could have:
8x8=64 8x18=144 18x18=324
If we multiply 81 times 256 or 64 times 324 we will find that they equal 144x144.
Do a few for yourself, 2x72, 3x48, 4x36, etc. to prove to
yourself that a square itself can also be a geometric mean between two other squares.
There are other observations to be made about the squares of the Square of Nine chart, but they must be made in connection with the single digit numbering system, the triangle numbers and the TELEOIS.
They will be dealt with in their proper place.
I believe that now when you look at the Square of Nine chart you will be looking at it with a different frame of mind than before.
If I have only taught you one thing in this whole material, I