Commentary on Proposition IV.10¶
Back to Proposition IV.10.
There is every reason to conclude that the connexion of the triangle constructed in this proposition with the regular pentagon, and the construction of the triangle itself, were the discovery of the Pythagoreans. In the first place the Scholium IV. No. 2 (Heiberg, Vol. V. p. 273) says “this Book is the discovery of the Pythagoreans.” Secondly, the summary in Proclus (p. 65, 20) says that Pythagoras discovered “the construction of the cosmic figures,” by which must be understood the five regular solids. Thirdly, Iamblichus (Vit. Pyth. c. 18, s. 88) quotes a story of Hippasus, “that he was one of the Pythagoreans but, owing to his being the first to publish and write down (the construction of) the sphere arising from the twelve pentagons (τὴν ἐκ τῶν δώδεκα πενταγώνων), perished by shipwreck for his impiety, having got credit for the discovery all the same, whereas everything belonged to HIM (ἐκείνου τοῦ ἀνδρός), for it is thus that they refer to Pythagoras, and they do not call him by his name.” Cantor has ($\text{I}_3$, pp. 176 sqq.) collected notices which help us to form an idea how the discovery of the Euclidean construction for a regular pentagon may have been arrived at by the Pythagoreans.
Plato puts into the mouth of Timaeus a description of the formation from right-angled triangles of the figures which are the faces of the first four regular solids. The face of the cube is the square which is formed from isosceles right-angled triangles by placing four of these triangles contiguously so that the four right angles are in contact at the centre. The equilateral triangle, however, which is the form of the faces of the tetrahedron, the octahedron and the icosahedron, cannot be constructed from isosceles right-angled triangles, but is constructed from a particular scalene right-angled triangle which Timaeus (54 A, B) regards as the most beautiful of all scalene right-angled triangles, namely that in which the square on one of the sides about the right angle is three times the square on the other.
This is, of course, the triangle forming half of an equilateral triangle bisected by the perpendicular from one angular point on the opposite side. The Platonic Timaeus does not construct his equilateral triangle from two such triangles but from six, by placing the latter contiguously round a point so that the hypotenuses and the smaller of the sides about the right angles respectively adjoin, and all of them meet at the common centre, as shown in the figure (Timaeus 54 D, E.). The probability that this exposition was Pythagorean is confirmed by the independent testimony of Proclus (pp. 304—5), who attributes to the Pythagoreans the theorem that six equilateral triangles, or three hexagons, or four squares, placed contiguously with one angular point of each at a common point, will just fill up the four right angles round that point, and that no other regular polygons in any numbers have this property.
How then would it be proposed to split up into triangles, or to make up out of triangles, the face of the remaining solid, the dodecahedron? It would easily be seen that the pentagon could not be constructed by means of the two right-angled triangles which were used for constructing the square and the equilateral triangle respectively. But attempts would naturally be made to split up the pentagon into elementary triangles, and traces of such attempts are actually forthcoming. Plutarch has in two passages spoken of the division of the faces of the dodecahedron into triangles, remarking in one place (Quaest. Platon. V. 1) that each of the twelve faces is made up of 30 elemen-
tary scalene triangles, so that, taken together, they give 360 such triangles, and in another (De defectu oraculorum, c. 33) that the elementary triangle of the dodecahedron must be different from that of the tetrahedron, octahedron and icosahedron. Another writer of the 2nd cent., Alcinous, has, in his introduction to the study of Plato (De doctrina Platonis, c. 11), spoken similarly of the 360 elements which are produced when every one of the pentagons is divided into 5 isosceles triangles, and each of the latter into 6 scalene triangles. Now, if we proceed to draw lines in a pentagon separating it into this number of small triangles as shown in the above figure, the figure which stands out most prominently in the mass of lines is the “star-pentagon,” as drawn separately, which then (if the consecutive corners be joined) suggests the drawing, as part of a pentagon, of a triangle of a definite character. Now we are expressly told by Lucian and the scholiast to the $Clouds$ of Aristophanes (see Bretschneider, pp. 85—86) that the triple interwoven triangle, the pentagram (τὸ τριπλοῦν τρίγωνον, τὸ δι’ ἀλλήλων, τὸ πεντάγραμμον), was used by the Pythagoreans as a symbol of recognition between the members of the same school (συμβόλῳ πρὸς τοὺς ὁμοδόξους ἐχρῶντο), and was called by them Health. There seems to be therefore no room for doubt that the construction of a pentagon by means of an isosceles triangle having each of its base angles double of the vertical angle was due to the Pythagoreans.
The construction of this triangle depends upon II. 11, or the problem of dividing a straight line so that the rectangle contained by the whole and one of the parts is equal to the square on the other part. This problem of course appears again in Eucl. VI. 30 as the problem of cutting a given straight line in extreme and mean ratio, i.e. the problem of the golden section, which is no doubt “the section” referred to in the passage of the summary given by Proclus (p. 67, 6) which says that Eudoxus “greatly added to the number of the theorems which Plato originated regarding the section.” This idea that Plato began the study of the “golden section” as a subject in itself is not in the least inconsistent with the supposition that the problem of Eucl. II. 11 was solved by the Pythagoreans. The very fact that Euclid places it among other propositions which are clearly Pythagorean in origin is significant, as is also the fact that its solution is effected by “applying to a straight line a rectangle equal to a given square and exceeding by a square,” while Proclus says plainly (p. 419, 15) that, according to Eudemus, “the application of areas, their exceeding and their falling short, are ancient and discoveries of the Muse of the Pythagoreans.”
We may suppose the construction of IV. 10 to have been arrived at by analysis somewhat as follows (Todhunter’s Euclid, p. 325).
Suppose the problem solved, i.e. let ABD be an isosceles triangle having each of its base angles double of the vertical angle.
Bisect the angle ADB by the straight line DC meeting AB in C. [I. 9]
Therefore the angle BDC is equal to the angle BAD; and the angle CDA is also equal to the angle BAD,
Again, since, in the triangles BCD, BDA, the angle BDC is equal to the angle BAD, and the angle B is common, the third angle BCD is equal to the third angle BDA, and therefore to the angle DBC.
Now, if a circle be described about the triangle ACD [IV. 5], since the angle BDC is equal to the angle in the segment CAD, BD must touch the circle [by the converse of III. 32 easily proved from it by reductio ad absurdum].
Hence [III. 36] the square on BD and therefore the square on CD, or AC, is equal to the rectangle AB, BC.
Thus the problem is reduced to that of cutting AB at C so that the rectangle AB, BC is equal to the square on AC. [II. 11] When this is done, we have only to draw a circle with centre A and radius AB and place in it a chord BD equal in length to AC. [IV. 1]
Since each of the angles ABD, ADB is double of the angle BAD, the latter is equal to one-fifth of the sum of all three, i.e. is one-fifth of two right angles, or two-fifths of a right angle, and each of the base angles is four-fifths of a right angle.
If we bisect the angle BAD, we obtain an angle equal to one-fifth of a right angle, so that the proposition enables us to divide a right angle into five equal parts.
It will be observed that BD is the side of a regular decagon inscribed in the larger circle.
Proclus, as remarked above (Vol. I. p. 130), gives IV. 10 as an instance in which two of the six formal divisions of a proposition, the setting-out and the “definition,” are left out, and explains that they are unnecessary because there is no datum in the enunciation. This is however no more than formally true, because Euclid does begin his proposition by setting out “any straight line AB,” and he constructs an isosceles triangle having AB for one of its equal sides, i.e. he does practically imply a datum in the enunciation, and a corresponding setting-out and “definition” in the proposition itself.