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Doubling the temple stone

At the time of  Plato, the people hated the geometry and mathematics knowledge. It happened that an epidemic of a disease spread in Athens. The people asked their prophet for help, so he prayed for God and was ordered to double the size of the temple stone – a cubic figure. The people then brought another cube stone near the other one but the epidemic did not resolve. They asked again their prophet for help. He told them that they have just got another stone near the first stone and this is not doubling. He ordered them to consult Plato who had some knowledge in geometry. When they asked Plato about the problem, he agreed with them to solve it in condition they will start learning geometry and they agreed. How to get the doubling of this cubic stone so that the epidemic is left up?

Answer

The size or volume S of a cube is the product of the width, length and depth. In other words, if these were equal then the size would be to (width)³. In doubling the cube stone, it was meant to double each of these three dimensions i.e., the new temple stone would be (2 x width)³ = 8 x S. As the people brought 7 similar stones to the original stone and arranged them in a cubic shape, the epidemic was left. Since then, people studied geometry more seriously.

Plato Geometrical Numbers:

The cubic and the square structures below are constructed with exactly equal numbers of blocks. What is the number of blocks that can be used to construct a cube as well as a square?

Answer

The cube and the square are each made of 64 small blocks. It happen that 4 x 4 x4 = 8 x 8 = 64. In other words, numbers which when cubed give a square call for a solution of the problem. These are the numbers which can be written as squares. Another example other is 9. It is evident that 9³ = 27² = 729.

Next number is 16. (16)³ = 64 x 64 = 4096 

It is easy to prove that the cube of a number which can be written as square is equal to a square.

Let t be such a number. Since t is a square then t can be written as

t =(b x b).

Cube of t = t³ = [b x b]³ = (b³)².

In other words, cube of t = square of (cube of b).

t³ = [(square root of t)³]² 

For example, if t = 4, a square of 2, then cube of t can be written as 64 = 4³ = [2 x 2]³ = 8²

How to prove Pythagoras rule?

Pythagoras rule states that the hypotenuse c is related to the sides a and b of a right-angle triangle in the relation:

c² = a² + b².

Answer

The hypotenuse c is the longest side of a right angle triangle and it is opposing the right angle.

Using the triangle with sides (a, b, c), we can construct 2 identical squares in 2 different ways. See figure 1 and 2.

In figure 1, the square area S =  a² + b² + 2ab

In figure 2, the square area S = c²+ 2ab

a² + b² + 2ab = c²+ 2ab. Hence, c² = a² + b².