# Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers

Similar plane and solid numbers are those having proportional sides.

### Modern Definitions

Two rectangles with the integer side lengths $$a,b$$ and $$c,d$$ are called similar, if the ratios of their sides are proportional numbers, i.e.

$\frac ab\sim \frac cd\Longleftrightarrow ad=cb.$ | :------------- |:-------------

Two cuboids with integer side lengths $$a,b,c$$ and $$d,e,f$$ are called similar, if the ratios of their respective sides are proportional numbers, i.e.

$\frac ab\sim \frac de\Longleftrightarrow ae=db, \frac ac\sim \frac df\Longleftrightarrow af=dc.$ | ### Notes

• Nowadays, the term similarity is related to geometrical figures rather than numbers.
• However, the numerical proportions of the sides of rectilinear figures are indeed a defining property of similar rectilinear figures, as it was already demonstrated by Euclid in the previous Book 6 for triangles (Prop 6.04 and Prop 6.05).

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Propositions: 19 20 21 22 23 24 25 26

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