# 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.
4. Other

Related:

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Are kindergartners supposed to be steered from squares being rectangles?

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

Why do we have circles for ellipses, squares for rectangles but nothing for triangles?

What are/should kids (be) taught about the colour of the sun?

• Where did you find that phrase? Euclid's elements? Mar 22 '18 at 7:26
• Oblong is another word for a rectangle of length more than width. I for one would not make simple things more complicated than needed and expend time on it. Mar 22 '18 at 8:29
• @Arthur Yes! Updated. Thanks.
– BCLC
Mar 22 '18 at 9:54
• @Narasimham Relevance please? My question is why some texts give 'oblong' while others give 'rectangle' or even both words with one in round brackets.
– BCLC
Mar 22 '18 at 9:55
• collinsdictionary.com/dictionary/english/oblong It has a Latin origin, old English, refers to the figure when one dimension is larger than the other. It is meant to specifically exclude the square in its subset reference, which is not the case for the rectangle. About round brackets please refer to explanation of notation in the index pages. Mar 22 '18 at 14:01

In mathematics, the term "rectangle" includes squares.

"Oblong" is rarely used as a mathematical term.

– BCLC
Mar 22 '18 at 5:56
• I thought that was a complete answer. If you want a better answer be more specific in exactly what you are asking. "'that which is right-angled but not equilateral" is very awkward and and old fashioned and extremely vague wording. And as to why do some translate with one word and others to another? Because different people translate them. To make any sense of your question would be to answer what sense do mathematicians use the term. And the answer is: we use "rectangle" but it also includes squares. We don't use "oblong". I think Robert Israel answered perfectly. Mar 22 '18 at 6:03
• Robert Israel, @fleablood, I was supposed to ask about Euclid's elements which one would discover is the source of the quote upon googling, but I guess I should have been explicit. Edited question. Thanks, I guess. Any updates please?
– BCLC
Mar 22 '18 at 9:38

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.