Definition: 7.17: Cuboidal Number, Solid Number
And when three numbers multiplying one another make some (other number) then the (number so) created is (called) solid, and its sides (are) the numbers which multiply one another.
Modern Formulation
A cuboidal number \(h\) is the product of three positive integers \(a,b\) and \(c\):
\[h=a\cdot b\cdot c.\]
Notes
- Instead of Euclid's term solid number", the term cuboidal is used more often.
- From a geometrical point of view, a cuboidal number can be interpreted as the volume of a solid rectangular cuboid, i.e. a solid figure with rectangular faces.
- This formula, of course, holds not only for integer lengths but also for figures with side lengths represented by general real numbers. However, in the 7th Book of "The Elements", Euclid deals with positive integers only.
Mentioned in:
Definitions: 1 2
Propositions: 3
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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
