Commentary on Proposition II.4

Back to Proposition II.4.

The editions of the Greek text which preceded that of E. F. August (Berlin, 1826—9) give a second proof of this proposition introduced by the usual word ἄλλως or “otherwise thus.” Heiberg follows August in omitting this proof, which is attributed to Theon, and which is indeed not worth reproducing, since it only differs from the genuine proof in that portion of it which proves that CGKB is a square. The proof that CGKB is equilateral is rather longer than Euclid’s, and the only interesting point to notice is that, whereas Euclid still, as in I. 46, seems to regard it as necessary to prove that all the angles of CGKB are right angles before he concludes that it is right-angled, Theon says simply “And it also has the angle CBK right; therefore CK is a square.” The shorter form indicates a legitimate abbreviation of the genuine proof; because there can be no need to repeat exactly that part of the proof of I. 46 which shows that all the angles of the figure there constructed are right when one is.

There is also in the Greek text a Porism which is undoubtedly interpolated: “From this it is manifest that in square areas the parallelograms about the diameter are squares.” Heiberg doubted its genuineness when preparing his edition, and conjectured that it too may have been added by Theon; but the matter is placed beyond doubt by a papyrus-fragment referred to already (see Heiberg, Paralipomena zu Euklid, in Hermes XXXVIII., 1903, p. 48) in which the Porism was evidently wanting. It is the only Porism in Book II., but does not correspond to Proclus’ remark (p. 304, 2) that “the Porism found in the second book belongs to a problem.” Heiberg regards these words as referring to the Porism to IV. 15, the correct reading having probably been not δευτέρῳ but δʹ, i.e. τετάρτῳ.

The semi-algebraical proof of this proposition is very easy, and is of course old enough, being found in Clavius and in most later editions. It proceeds thus.

By II. 2, the square on AB is equal to the sum of the rectangles AB, AC and AB, CB.

But, by II. 3, the rectangle AB, AC is equal to the sum of the square on AC and the rectangle AC, CB;

while, by II. 3, the rectangle AB, CB is equal to the sum of the square on BC and the rectangle AC, CB.

Therefore the square on AB is equal to the sum of the squares on AC, CB and twice the rectangle AC, CB.

The figure of the proposition also helps to visualise, in the orthodox manner, the proof of the theorem deduced above from II. 1—3, viz. that the difference of the squares on two given straight lines is equal to the rectangle contained by the sum and the difference of the lines.

For, if the lines be AB, BC respectively, the shorter of the lines being measured along BA, the figure shows that

the square AE is equal to the sum of the square CK and the rectangles AF, FK;

that is, the square on AB is equal to the sum of the square on BC and the rectangles AB, AC and AC, BC.

But the rectangles AB, AC and BC, AC are, by II. 1, together equal to the rectangle contained by AC and the sum of AB, BC,

i.e. to the rectangle contained by the sum and difference of AB, BC.

Whence the result follows as before.

The proposition II. 4 can also be extended to the case where a straight line is divided into any number of segments; for the figure will show in like manner that the square on the whole line is equal to the sum of the squares on all the parts together with twice the rectangles contained by every pair of the parts.