Page 10 - GALIET THE BEAUTIFUL INNATE: Meno & Theatetus Plato IV

Galiet & Galiet
To demonstrate his theory and the immortality of the soul, Socrates arrogantly leads the slave to discover that if he (the slave) draws a line connecting the mid points of two sides of a square and does this, four times connecting all four mid points, then he will produce another square inside the first square which is one half the size of the first square. In other words, Socrates shows the slave the internal square formed by the diagonals of the four squares of four areas juxtaposed in such a way that it forms a square of 4x4, which the slave accepts. To arrive at the demonstration, the slave would have to deduce according to Socrates’ plan, first, that the diagonal cuts the square in two equal parts; second, that the area of the square is four; third, that half of four is two; fourth, that the square formed by the four diagonals is formed by four halves, that is, by four times two. It appears that Socrates also attempts to induce reasoning through a rule of three: the one that has two halves measures four in area, then the internal one that has four halves (double the halves) must measure twice, that is, eight in area. Here, the slave seems to demonstrate that the rule of three is natural; however, he may have learned it at Meno’s house or it may be the case that he is completely lost and is simply saying so to please Socrates.
By using elemental algebra, Socrates’ demonstration is as follows: Any area of a square is defined as:
A = L2 (A = area of square)
(L = length of side of a square)
What Socrates does is lead the slave through a diagrammatic explanation of this formula. He does this by dividing squares in half starting at sides of one foot for a basic square to sides of four feet for a larger square. Socrates shows that by
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