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.
\[\frac ab\sim \frac cd\Longleftrightarrow ad=cb.\]
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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.
\[\frac ab\sim \frac de\Longleftrightarrow ae=db,
\frac ac\sim \frac df\Longleftrightarrow af=dc.\]
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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).
Mentioned in:
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
Thank you to the contributors under CC BY-SA 4.0! 
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- @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
