# Whether Euclid considered squares to be rectangles

When I look up

'that which is right-angled but not equilateral'

there are translations that show the word before the above phrase to 'oblong', some that show 'rectangle' and some that show both with one term in brackets (1 2 3).

Why is this? Guesses:

1. Translation error
2. Euclid didn't consider squares to be rectangles.
3. Euclid made a mistake.
4. Other

Related:

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In what curricula are “rectangles” defined so as to exclude squares?

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• So is this a question about Euclid's writings or just about the quality of translations? – Carl Witthoft Mar 22 '18 at 11:25
• @CarlWitthoft My question is why some translations use 'oblong' while others use 'rectangle' – BCLC Mar 23 '18 at 7:58

The word in Euclid is "ἑτερομήκης". Whether you translate it "rectangle" or "oblong" depends on what you think those two words mean. It is clear from Euclid that "ἑτερομήκης" does not include squares. If the translator thinks rectangles do not include squares, then he can use "rectangle" in the translation. But if the translator thinks squares are rectangles, then he may use a different translation, such as "oblong".

So my answer is (2), with the proviso that we use "ἑτερομήκης", and not the English word "rectangle".

[Note: word ἑτερομήκης added from Mauro's answer.]

Definition 22.

Of quadrilateral (τετράπλευρος) figures, a square (τετράγωνος) is that which is both equilateral and right-angled; an oblong (ἑτερομήκης: with sides of uneven length) that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

And see Heath's commentary, page 188:

Tetragonon was already a square with the Pythagoreans, and it is so most commonly in Aristotle; but in De anima, II.3, 414b31 it seems to be a quadrilateral, and in Metaph., 1054b2, "equal and equiangular tetragona," it cannot be anything else but quadrilateral if "equiangular" is to have any sense. Though, by introducing tetrapleuron for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of tetragonon in much later writers.