Commentary on Proposition I.15

Back to Proposition I.15.

This theorem, according to Eudemus, was first discovered by Thales, but found its scientific demonstration in Euclid (Proclus, p. 299, 3—6).

Proclus gives a converse theorem which may be stated thus. If a straight line is met at one and the same point intermediate in its length by two other straight lines on different sides of it and such as to make the vertical angles equal, the latter straight lines are in a straight line with one another. The proof need not be given, since it is almost self-evident, whether (1) it is direct, by means of I. 13, 14, or (2) indirect, by reductio ad absurdum depending on I. 15.

The balance of MS. authority seems to be against the genuineness of this Porism, but Proclus and Psellus both have it. The word is not here used, as it is in the title of Euclid’s lost Porisms, to signify a particular class of independent propositions which Proclus describes as being in some sort intermediate between theorems and problems (requiring us, not to bring a thing into existence, but to find something which we know to exist). Porism has here (and wherever the term is used in the Elements) its second meaning; it is what we call a corollary, i.e. an incidental result springing from the proof of a theorem or the solution of a problem, a result not directly sought but appearing as it were by chance without any additional labour, and constituting, as Proclus says, a sort of windfall (ἕρμαιον) and bonus (κέρδος). These Porisms appear in both the geometrical and arithmetical Books of the Elements, and may either result from theorems or problems. Here the Porism is geometrical, and springs out of a theorem; VII. 2 affords an instance of an arithmetical Porism. As an instance of a Porism to a problem Proclus cites “that which is found in the second Book” (τὸ ἐν τῷ δευτέρῳ βιβλίῳ κείμενον); but as to this see notes on II. 4 and IV. 15.

The present Porism, says Proclus, formed the basis of “that paradoxical theorem which proves that only the following three (regular) polygons can fill up the whole space surrounding one point, the equilateral triangle, the square, and the equilateral and equiangular hexagon.” We can in fact place round a point in this manner six equilateral triangles, three regular hexagons, or four squares. “But only the angles of these regular figures, to the number specified, can make up four right angles: a theorem due to the Pythagoreans.”

Proclus further adds that it results from the Porism that, if any number of straight lines intersect one another at one point, the sum of all the angles so formed will still be equal to four right angles. This is of course what is generally given in the text-books as Corollary 2.