In Book 1 of Euclid’s Elements, Definition 22 reads (Fitzpatrick 6):
| Greek | English |
|---|---|
| κβʹ. Τὼν δὲ τετραπλεύρων σχημάτων τετράγωνον μέν ἐστιν, ὃ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτερόμηκες δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλευρον δέ, ῥόμβος δέ, ὃ ἰσόπλευρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοειδὲς δὲ τὸ τὰς ἀπεναντίον πλευράς τε καὶ γωνίας ἴσας ἀλλήλαις ἔχον, ὃ οὔτε ἰσόπλευρόν ἐστιν οὔτε ὀρθογώνιον· τὰ δὲ παρὰ ταῦτα τετράπλευρα τραπέζια καλείσθω. | 22. And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one another which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia. |
This is the longest of the twenty-three definitions in Book 1 of the Elements. Like Definitions 20 and 21, this one simply establishes the correct terminology to be used when describing certain types of rectilinear figures. Three types of trilaterals (triangles) were defined in terms of the relative lengths of their sides, and three types in terms of their angles.
Definition 22 defines five types of quadrilateral in terms of both side-length and angle. It first divides all quadrilaterals into two broad groups:
Those in which opposite sides and opposite angles are equal to one another. These will be further subdivided into two subgroups (right-angled and not right-angled), and each subgroup will be further subdivided into two sub-subgroups (equilateral and not equilateral).
Those in which opposite sides and opposite angles are not equal to one another. These are all grouped together as trapezia.
| English | Greek | Opposite Sides and Angles | Side-Lengths | Angles |
|---|---|---|---|---|
| Square | τετράγωνον | equal | equilateral | right-angled |
| Rectangle | ἑτερόμηκες | equal | not equilateral | right-angled |
| Rhombus | ῥόμβος | equal | equilateral | not right-angled |
| Rhomboid | ῥομβοειδὲς | equal | not equilateral | not right-angled |
| Trapezium | τραπέζιον | not equal | not equilateral | not right-angled |
For Euclid, right-angled means that all four angles are right-angles (Joyce).
Euclid defines the rectangle as a quadrilateral that is right-angled but not equilateral. Today, we define the rectangle more simply as a right-angled quadrilateral, the square being a special type of rectangle. Euclid, however, specifically excludes squares. This is clear from the technical term he uses for rectangles in this definition: ἑτερόμηκες [heteromēkes], which literally means with sides of uneven length, oblong (Liddell & Scott 590). Richard Fitzpatrick, whose translation of the Elements is my preferred text, translates ἑτερόμηκες as rectangle, but Thomas Little Heath uses oblong:
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong 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. (Heath 188)
Having defined the rectangle, Euclid never again uses the term ἑτερόμηκες in the Elements. Whenever he wishes to refer to a rectangle, he uses the undefined term ὀρθογώνιον [orthogōnion], which is the Greek equivalent of rectangle. Is this because everything he has to say about these rectangles also applies to squares? ὀρθογώνιον literally means right-angled, so it must include squares as well as oblong rectangles.
Euclid uses the term trapezion [τραπέζιον], plural trapezia [τραπέζια], to designate those quadrilaterals in which the opposite sides and the opposite angles are not equal to one another. Today, geometers in the English-speaking word have two technical terms similar to Euclid’s that both go back to the Ancient Greek:
trapezium
trapezoid
The first of these is just a Latinate form of the Greek τραπέζιον, and the second an Anglicized form of the Greek τραπέζοειδὲς [trapezoeides], which is also derived from τραπέζιον and means trapezion-like. In English, however, these words do not necessarily mean the same thing as Euclid’s τραπέζιον. And to confuse matters further, their meanings differ from country to country.
In the English-speaking world outside the US and Canada, trapezoid designates a quadrilateral with no sides parallel, while trapezium designates a quadrilateral with two and only two sides parallel.
In the United States of America and Canada, these definitions have been swapped: trapezium, which is now only rarely encountered, designates a quadrilateral with no sides parallel. A trapezoid is a quadrilateral with two and only two sides parallel. According to Wikipedia, the transposition of these two terms was the result of an error in Charles Hutton’s mathematical dictionary (Hutton 610):
This is borne out by the Oxford English Dictionary, which prefaces its definition of trapezoid with a lengthy note on the subject:
Euclid’s term is more general and covers both types. For Euclid, a trapezium is any quadrilateral that is not a square, rectangle, rhombus, or rhomboid.
In the Elements, square is the only one of the five technical terms defined in Definition 22 that Euclid actually uses. But there is a type of quadrilateral that he refers to many times throughout the Elements, though he never actually defines it. This is the parallelogram. If we add this to our table of quadrilaterals, we get:
| English | Greek | Opposite Sides and Angles | Side-Lengths | Angles |
|---|---|---|---|---|
| Square | τετράγωνον | equal | equilateral | right-angled |
| Rectangle | ἑτερόμηκες | equal | not equilateral | right-angled |
| Rhombus | ῥόμβος | equal | equilateral | not right-angled |
| Rhomboid | ῥομβοειδὲς | equal | not equilateral | not right-angled |
| Parallelogram | παραλληλόγραμμον | equal | equal or not equal | not right-angled |
| Trapezium | τραπέζιον | not equal | not equilateral | not right-angled |
In other words, for Euclid a parallelogram is either a rhombus or a rhomboid. Today, we also include squares and rectangles in the definition of parallelogram, but Euclid treats these as separate.
In his Commentary on the First Book of Euclid’s Elements, Proclus introduced the modern distinction between quadrilaterals with two and only two sides parallel and quadrilaterals with no sides parallel. As we have seen, Euclid referred to both types indiscriminately as trapezia, but Proclus restricted this term to quadrilaterals with two and only two sides parallel. For the other type of Euclidean trapezia, he used the term τραπέζοειδὲς, trapezoid. He further subdivided trapezia into isosceles trapezia and scalene trapezia:
Proclus attributes this sevenfold classification to Posidonius of Apamea, a polymath who flourished in the first half of the 1st century BCE. If these seven types are to cover all possible quadrilaterals, an isosceles trapezium must be defined as a trapezium in which the two non-parallel sides are equal, and a scalene trapezium as one in which the two non-parallel sides are not equal. In other words, for Posidonius a scalene trapezium could have two adjacent sides equal in length. Today, most textbooks define a scalene trapezium (or, in the US and Canada, a scalene trapezoid) as a trapezium with no sides equal. The quotation from Proclus above makes it clear that for him scalene only referred to the non-parallel sides.
Critics of the Elements have pointed out that two of Euclid’s definitions in Definition 22 are superfluous. The square can be defined as an equilateral quadrilateral with at least one right angle. It is not necessary to require that all four angles be right angles: if the four sides are of equal length and one of the angles is a right angle, then the other three angles will also necessarily be right angles. Euclid actually demonstrates this in Proposition 1:46:
If it be objected, as it was by Todhunter for example, that the definition of a square assumes more than is necessary, since it is sufficient that, being equilateral, it should have one right angle, the answer is that, as in other cases, the superfluity does not matter from Euclid’s point of view; on the contrary, the more of the essential attributes of a thing that could be included in its definition the better, provided that the existence of the thing defined and its possession of all those attributes is proved before the definition is actually used; and Euclid does this in the case of the square by construction in 1.46, making no use of the definition before that proposition. (Heath 189)
Similarly, Euclid’s definition of the rhomboid requires that the opposites sides and the opposite angles be equal. But if the opposite sides are equal, then the opposite angles will necessarily be equal:
It is true that the definition of a rhomboid says more than is necessary in describing it as having its opposite sides and angles equal to one another. The answer to the objection is the same as the answer to the similar objection to the definition of a square. (Heath 189)
Most modern translations of Euclid’s Elements are based on Johan Ludvig Heiberg’s critical edition of 1883-88. In this edition, Euclid’s definitions of the five different types of quadrilaterals comprise Definition 22. But in earlier editions of the work, these definitions were given separately as Definitions 30-34. This division is not something that occurred in recent centuries. Even Proclus, whose commentary was written around 450 CE, refers to these as Definitions 30-34. This emendation was probably introduced by Theon of Alexandria, whose edited the Elements in the late 4th century CE.
François Peyrard’s French edition of 1814 reproduces Theon’s version (Peyrard 4), as does Ernst Ferdinand August in his edition of 1826 (August 2-3). Heiberg emended these to reflect what is believed to be Euclid’s original text, noting that:
Def. 21 uulgo in 3, def. 22 in 5 diuidunt [Definition 21 is commonly divided into three definitions, and Definition 22 into five]. (Heiberg 8)
It is, however, possible that Euclid only defined the square, as that is the only one of these technical terms that he goes on to use in the Elements:
The only figure defined here that Euclid actually uses is the square. The other names of figures may have been common at the time of Euclid’s writing, or they may have been left over from earlier authors’ versions of the Elements, or they may have been added later. (Joyce)
And that’s a good place to stop.
