Commentary on Proposition III.35¶
Back to Proposition III.35.
In addition to the two cases in Euclid’s text, Simson (following Campanus) gives two intermediate cases, namely (1) that in which one chord passes through the centre and bisects the other which does not pass through the centre at right angles, and (2) that in which one passes through the centre and cuts the other which does not pass through the centre but not at right angles. Simson then reduces Euclid’s second case, the most general one, to the second of the two intermediate cases by drawing the diameter through E. His note is as follows: “As the 25th and 33rd propositions are divided into more cases, so this 35th is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted them because they are easy; as he has given the case which by far is the easiest of them all, viz. that in which both the straight lines pass through the centre: And in the following proposition he separately demonstrates the case in which the straight line passes through the centre, and that in which it does not pass through the centre: So that it seems Theon, or some other, has thought them too long to insert: But cases that require different demonstrations should not be left out in the Elements, as was before taken notice of: These cases are in the translation from the Arabic and are now put into the text.” Notwithstanding the ingenuity of the argument based on the separate mention by Euclid of the simplest case of all, I think the conclusion that Euclid himself gave four cases is unsafe; in fact, in giving the simplest and most difficult cases only, he seems to be following quite consistently his habit of avoiding too great multiplicity of cases, while not ignoring their existence.
The deduction from the next proposition (III. 36) which Simson, following Clavius and others, gives as a corollary to it, namely that, If from any point without a circle there be drawn two straight lines cutting it, the rectangles contained by the whole lines and the parts of them without the circle are equal to one another, can of course be combined with III. 35 in one enunciation.
As remarked by Todhunter, a large portion of the proofs of III. 35, 36 amounts to proving the proposition, If any point be taken on the base, or the base produced, of an isosceles triangle, the rectangle contained by the segments of the base (i.e. the respective distances of the ends of the base from the point) is equal to the difference between the square on the straight line joining the point to the vertex and the square on one of the equal sides of the triangle. This is of course an immediate consequence of I. 47 combined with II. 5 or II. 6.
The converse of III. 35 and Simson’s corollary to III. 36 may be stated thus. If two straight lines AB, CD, produced if necessary, intersect at O, and if the rectangle AO, OB be equal to the rectangle CO, OD, the circumference of a circle will pass through the four points A, B, C, D. The proof is indirect. We describe a circle through three of the points, as A, B, C (by the method used in Euclid’s proofs of III. 9, 10), and then we prove, by the aid of III. 35 and the corollary to III. 36, that the circle cannot but pass through D also.