Source — Euclid's Elements, encoded as an rrxiv paper

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VIII.18 Proposition 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.

Proof

For similar plane numbers

ab

and

c d

with

a : b = c : d

, the mean proportional is the geometric mean of

ab

and

c d

, which by VII.19 / VIII.2 equals

a d

(or

b c

, equal by VII.19).

lines 74–74 in main.tex