In Book 1 of Euclid’s Elements, Definition 17 reads (Fitzpatrick 6):
| Greek | English |
|---|---|
| ιζʹ. Διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ περατουμένη ἐφ ̓ ἑκάτερα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου περιφερείας, ἥτις καὶ δίχα τέμνει τὸν κύκλον. | 17. And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each direction by the circumference of the circle. (And) any such (straight-line) also cuts the circle in half. |
Like the previous definition, the first part of Definition 17 simply establishes the correct terminology to be used when we are talking about circles: Any straight line segment that passes through the centre of a circle and whose two endpoints are on the circumference of the circle is called a diameter.
If there were nothing more to this definition, there would be nothing more to say. But Euclid adds an important property of the diameter: it bisects the circle (δίχα τέμνει τὸν κύκλον [dikha temnei ton kuklon] = cuts the circle in two). This addition has led to much discussion.
In his translation of Euclid’s Elements, Richard Fitzpatrick adds the following footnote to Definition 17:
This should really be counted as a postulate, rather than as part of a definition. (Fitzpatrick 7 fn)
He is referring specifically to the last part of Definition 17, which says that a diameter cuts the circle in half. David Joyce concurs:
That a diameter “also bisects the circle” should not be part of the definition, but either assumed as a postulate or proved as a proposition. It depends on the fact that circles are drawn on planes, and planes have constant curvature. The analogous figure on a surface of nonconstant curvature does not have this property. For such figures the two “semicircles” on either side of a “diameter” need not be equal. (David Joyce)
Euclid never proves this fact as a proposition (theorem). It should therefore be added to his five postulates, which are listed after the twenty-three definitions of Book 1. The final clause of Definition 17 is authentic and can be found in all extant manuscripts of the Elements (Heiberg 4-5). Some later editors, however, took it upon themselves to “correct” Euclid by omitting these troublesome words, as Heath notes:
The last words, literally “which (straight line) also bisects the circle,” are omitted by Simson and the editors who followed him. But they are necessary even though they do not “belong to the definition” but only express a property of the diameter as defined. For, without this explanation, Euclid would not have been justified in describing as a semi-circle a portion of a circle bounded by a diameter and the circumference cut off by it. (Heath 185)
In his Commentary on the First Book of Euclid’s Elements, Proclus tells us that the early Greek philosopher Thales of Miletus was the first to point out that the diameter bisects the circle (Morrow 124). Proclus gives a demonstration of this fact:
This is a proof by contradiction. One “half” of the circumference of the circle is folded over the diameter so that it falls on the other “half”. In linear algebra, we call this type of geometric transformation a reflection in a line. If the reflected half fails to coincide with the other half, then some of the radii of the reflected semicircle will no longer be as long as the radii of the other semicircle. But this is impossible, as all the radii were equal in length before the reflection (Definition 15), and reflection in a line is an isometry: it preserves length, area and measure of angle. Therefore, the reflected half must coincide precisely with the other half, demonstrating that they are indeed equal halves of the circumference. Note that this proof requires the assumption that reflection in a line preserves length.
In a footnote, the translator Glenn R Morrow cites the philosopher of mathematics Ian Mueller of the University of Chicago to the effect that this is probably Thales’ proof:
Presumably Proclus gives Thales’ proof. Euclid’s incorporating the assertion in a definition where it certainly doesn’t belong is probably due to a desire to avoid proofs in which a geometric object is moved. In the same way Euclid probably states Post[ulate] IV to avoid a proof in which motion is used, like the one given by Proclus at 188.20ff. (Morrow 125)
In the Elements, Euclid does use the rigid motion of a geometrical object in some of his proofs. For example, in Proposition 1:4, the SAS [Side-Angle-Side] Theorem, he proves that if two sides of one triangle are equal in length to two of the sides of another triangle, and the angles between the two sides are also equal, then the two triangles are similar.
He does this by “applying” or “fitting” (εφαρμόζω
) one triangle to the other. In linear algebra, such a transformation involves a translation and a rotation, which are rigid motions. Both of these transformations are isometries, though Euclid never proves this or includes this assumption among his Postulates.
In Ancient Greek διάμετρος [diametros] means literally “measured through”:
In Euclid’s Elements, and in the writings of many of his contemporaries, the word διάμετρος was also used to denote the diagonal of a parallelogram. See for example Elements Proposition 1:34. In Proposition 10:91, the same word is used to denote the diagonal of a square. In Proposition 11:28, however, Euclid uses the word διαγωνίους [diagōnious]—the accusative plural of διαγωνίος—to denote the diagonals of two opposing faces of a parallelepiped.
Diameter was the regular word in Euclid and elsewhere for the diameter of a square, and also of a parallelogram; diagonal (διαγωνίος) was a later term, defined by Heron (Def. 67] as the straight line drawn from an angle to an angle. (Heath 185)
Heron—also known as Hero of Alexandria—was a mathematician and engineer who flourished in the middle of the 1st century of the Common Era. Many of his writings have survived, including his Definitions. This work, which some have attributed to the 3rd-century mathematician Diophantus of Alexandria, comprises a list of 133 definitions of geometrical terms preparatory to a study of Euclid’s Elements. Definition 67 defines diagonal (διαγωνίος) as a straight line drawn from one angle to another (Heibig 1976:46-47).
Unfortunately, Heron’s Commentary on Euclid’s Elements survives only in small extracts preserved by Proclus and the Persian mathematician Al-Nayrizi.
And that’s a good place to stop.
