G.309 square numbers there

VIII.11

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.
square_numbers_there(A, B, C, D)
given A, B be square numbers, let C be the side of A, D of B
prove between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D
A B C D E
given A, B be square numbers, let C be the side of A, D of B
assert Let C by multiplying D make E
assert A is a square and C is its side,
assert C by multiplying itself has made A
assert For the same reason also
assert D by multiplying itself has made B
assert then C by multiplying the numbers C, D has made A, E,
∵ number_multiplying_numbersassert as C is to D, so is A to E
assert For the same reason also,
∵ numbers_multiplying_numberassert As C is to D, so is E to B
assert also, as A is to E, so is E to B
assert between A, B there is one mean proportional number
assert I say next that A also has to B the ratio duplicate of that which C has to D
assert A, E, B are three numbers in proportion,
∵ magnitudes_proportional_said_defassert A has to B the ratio duplicate of that which A has to E
assert But, as A is to E, so is C to D
∴ A has to B the ratio duplicate of that which C has to D

Dependency Graph

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Required for

G.314, G.315, G.316, G.317, G.381, G.383, G.403, G.404, G.405, G.406, G.407, G.408, G.409, G.413, G.414, G.415, G.419, G.420, G.423, G.425, G.427, G.429, G.431, G.433, G.437, G.438, G.439, G.440, G.443, G.444, G.445, G.448, G.449, G.450, G.451, G.452, G.458, G.459, G.460, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.480, G.481, G.482, G.486, G.487, G.488, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.584, G.589, G.597, G.600, G.601