G.319 similar plane numbers

VIII.18

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.
similar_plane_numbers(A, B, C, D)
given A, B be two similar plane numbers, let the numbers C, D be the sides of A, E, F of B
prove A also has to B the ratio duplicate of that which C has to E or D to F
A B G C D E F
given A, B be two similar plane numbers, let the numbers C, D be the sides of A, E, F of B
∵ similar_plane_solid_defassert similar plane numbers are those which have their sides proportional
assert as C is to D, so is E to F
assert I say then that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or D to F, that is, of that which the corresponding side has to the corresponding side
assert Now since, as C is to D, so is E to F,
∵ numbers_proportional_proportionalassert alternately, as C is to E, so is D to F
assert since A is plane, and C, D are its sides,
assert D by multiplying C has made A
assert For the same reason also
assert E by multiplying F has made B
assert Now let D by multiplying E make G
assert D by multiplying C has made A, and by multiplying E has made G,
∵ number_multiplying_numbersassert as C is to E, so is A to G
assert But, as C is to E, so is D to F
assert also, as D is to F, so is A to G
assert E by multiplying D has made G, and by multiplying F has made B,
∵ number_multiplying_numbersassert as D is to F, so is G to B
assert It was also proved that, as D is to F, so is A to G
assert also, as A is to G, so is G to B
assert A, G, B are in continued proportion
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 the corresponding side has to the corresponding side, that is, of that which C has to E or D to F
assert A, G, B are in continued proportion,
∵ magnitudes_proportional_said_defassert A has to B the ratio duplicate of that which it has to G
assert as A is to G, so is C to E, and so is D to F
∴ A also has to B the ratio duplicate of that which C has to E or D to F

Dependency Graph

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Required for

G.321, G.326, G.327, G.328, G.329, G.330, G.331, G.333, G.334, G.335, G.339, 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