Commentary on Proposition III.20¶
Back to Proposition III.20.
The early editors, Tartaglia, Commandinus, Peletarius, Clavius and others, gave the extension of this proposition to the case where the segment is less than a semicircle, and where accordingly the “angle” corresponding to Euclid’s “angle at the centre” is greater than two right angles. The convenience of the extension is obvious, and the proof of it is the same as the first part of Euclid’s proof. By means of the extension III. 21 is demonstrated without making two cases; III. 22 will follow immediately from the fact that the sum of the “angles at the centre” for two segments making up a whole circle is equal to four right angles; also III. 31 follows immediately from the extended proposition.
But all the editors referred to were forestalled in this matter by Heron, as we now learn from the commentary of an-Nairīzī (ed. Curtze, p. 131 sqq.). Heron gives the extension of Euclid’s proposition which, he says, it had been left for him to make, but which is necessary in order that the caviller may not be able to say that the next proposition (about the equality of the angles in any segment) is not established generally, i.e. in the case of a segment less than a semicircle as well as in the case of a segment greater than a semicircle, inasmuch as III. 20, as given by Euclid, only enables us to prove it in the latter case. Heron’s enunciation is important as showing how he describes what we should now call an “angle” greater than two right angles. (The language of Gherard’s translation is, in other respects, a little obscure; but the meaning is made clear by what follows.)
“The angle,” Heron says, “which is at the centre of any circle is double of the angle which is at the circumference of it when one arc is the base of both angles; and the remaining angles which are at the centre, and fill up the four right angles, are double of the angle at the circumference of the arc which is subtended by the [original] angle which is at the centre.”
Thus the “angle greater than two right angles” is for Heron the sum of certain “angles” in the Euclidean sense of angles less than two right angles. The particular method of splitting up which Heron adopts will be seen from his proof, which is in substance as follows.
Let CDB be an angle at the centre, CAB that at the circumference. Produce BD, CD to F, G; take any point E on BC, and join BE, EC, ED.
Then any angle in the segment BAC is half of the angle BDC; and the sum of the angles BDG, GDF, FDC is double of any angle in the segment BEC.
Proof. Since CD is equal to ED,
Therefore the exterior angle GDE is equal to twice the angle DEC. Similarly the exterior angle FDE is equal to twice the angle DEB. By addition, the angles GDE, FDE are double of the angle BEC.
But
therefore the sum of the angles BDG, GDF, FDC is double of the angle BEC.
And Euclid has proved the first part of the proposition, namely that the angle BDC is double of the angle BAC.
Now, says Heron, BAC is any angle in the segment BAC, and therefore any angle in the segment BAC is half of the angle BDC. Therefore all the angles in the segment BAC are equal.
Again, BEC is any angle in the segment BEC and is equal to half the sum of the angles BDG, GDF, FDC. Therefore all the angles in the segment BEC are equal. Hence III. 21 is proved generally.
Lastly, says Heron, since the sum of the angles BDG, GDF, FDC is double of the angle BEC, and the angle BDC is double of the angle BAC, therefore, by addition, the sum of four right angles is double of the sum of the angles BAC, BEC.
Hence the angles BAC, BEC are together equal to two right angles, and III. 22 is proved.
The above notes of Heron show conclusively, if proof were wanted, that Euclid had no idea of III. 20 applying in terms (either as a matter of enunciation or proof) to the case where the angle at the circumference, or the angle in the segment, is obtuse. He would not have recognised the “angle” greater than two right angles or the so-called “straight angle” as being an angle at all. This is indeed clear from his definition of an angle as the inclination κ.τ.ἑ., and from the language used by other later Greek mathematicians where there would be an opportunity for introducing the extension. Thus Proclus’ notion of a “four-sided triangle” (cf. the note above on the definition of a triangle) shows that he did not count a re-entrant angle as an angle, and Zenodorus’ application to the same figure of the word “hollow-angled” shows that in that case it was the exterior angle only which he would have called an angle. Further it would have been inconvenient to have introduced at the beginning of the $Elements$ an “angle” equal to or greater than two right angles, because other definitions, e.g. that of a right angle, would have needed a qualification. If an “angle” might be equal to two right angles, one straight line in a straight line with another would have satisfied Euclid’s definition of a right angle. This is noticed by Dodgson (p. 160), but it is practically brought out by Proclus on I. 13. “For he did not merely say that ‘any straight line standing on a straight line either makes two right angles or angles equal to two right angles’ but ‘if it make angles.’
If it stand on the straight line at its extremity and make one angle, is it possible for this to be equal to two right angles? It is of course impossible; for every rectilineal angle is less than two right angles, as every solid angle is less than four right angles (p. 292, 13—20).” [It is true that it has been generally held that the meaning of “angle” is tacitly extended in VI. 33, but there is no real ground for this view. See the note on the proposition.]
It will be observed that, following his usual habit, Euclid omits the demonstration of the case which some editors, e.g. Clavius, have thought it necessary to give separately, the case namely where one of the lines forming the angle in the segment passes through the centre. Euclid’s proof gives so obviously the means of proving this that it is properly left out.
Todhunter observes, what Clavius had also remarked, that there are two assumptions in the proof of III. 20, namely that, if $A$ is double of $B$ and $C$ double of $D$, then the sum, or difference, of $A$ and $C$ is equal to double the sum, or difference, of $B$ and $D$ respectively, the assumptions being particular cases of V. 1 and V. 5. But of course it is easy to satisfy ourselves of the correctness of the assumption without any recourse to Book V.