Proof
For similar plane numbers and with , the
mean proportional is the geometric mean of and , which by
VII.19 / VIII.2 equals (or , equal by VII.19).
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Depends on (3)
- VII.19Proposition VII.19If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced…
- VIII.5Proposition VIII.5Plane numbers have to one another the ratio compounded of the ratios of their sides.
- VII.21Definition VII.21Similar plane and solid numbers are those which have their sides proportional.
Required by (dependents) (4)
- VIII.19Proposition VIII.19Between two similar solid numbers there fall two mean proportional numbers, and the solid number has to the solid…
- VIII.20Proposition VIII.20If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.
- VIII.26Proposition VIII.26Similar plane numbers have to one another the ratio which a square number has to a square number.
- IX.1Proposition IX.1If two similar plane numbers by multiplying one another make some number, the product will be square.
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