Does everyone agree as to what a square actually is? Is a square a rectangle? And how do oblongs fit into all this?

Here are some definitions

• an equal oblong; where an oblong is "longer than broad" - from the Penguin Crossword Dictionary
• an equilateral rectangle; where a rectangle is a "four sided plane rectilinear figure with four right angles, especially one with adjacent sides unequal" - from the Concise Oxford Dictionary
• a plane figure with all four sides equal and four right angles - from the Penguin English Dictionary
• a simple quadrilateral with all sides equal and all four angles right angles - from the Penguin Dictionary of Mathematics

These definitions are of the noun form of the word square. The word also has adjective and verb forms. The Concise Oxford Dictionary has an interesting definition of the verb form. According to them, to square means "to make rectangular". The Penguin English Dictionary defines the adjectival form as "having four equal sides and four right angles, or approximating to this shape".

Here lie potential and actual confusions. However, there is a clear concensus in the above definitions and they seem to imply that, in order to decide if a given shape is a square, we must measure its angles to see if they are all right angles, and measure its sides to see if they are all the same.

Bringing measurement to the issue sounds dangerous. What level of accuracy is required?

Surely there are not degrees of squareness.

We need to reframe the problem.