Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

(3) Pappus gave a pretty proof of 1. 5. This proof has, I think, been wrongly understood; on this point see my note on the proposition.

(4) On 1. 47 Proclus says¹: “As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more, like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result.” We shall see what Heron's addition consisted of; what Pappus may have added we do not know, unless it was something on the lines of his extension of 1. 47 found in the Synagoge (IV. p. 176, ed. Hultsch).

We may fairly conclude, with van Pesch², that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus:

1. We are reminded of the curvilineal angle which is equal to but not a right angle by the note on 1. 32 to the effect that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles “equal to” right angles in the sense described by Pappus on Post. 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on 1. 4, 23 to curvilineal angles of which certain moon-shaped angles (μηνοειδεῖς) are shown to be “equal to” rectilineal angles savour of Pappus.

2. On 1. 9 Proclus says³ that “Others, starting from the Archimedean spirals, divided any given rectilineal angle in any given ratio.” We cannot but compare this with Pappus iv. p. 286, where the spiral is so used; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus.

3. The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subject⁴. Now on I. 35 Proclus speaks⁵ about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on 1. 37 about triangles⁶. Compare⁷ also his note on 1. 4. These notes may have been taken from Pappus.