Commentary on Proposition III.13¶
Back to Proposition III.13.
The difficulties which have been felt in regard to the proofs of this proposition need not trouble us now, because they have already been disposed of in the discussion of the more crucial propositions III. 11, 12.
Euclid’s proof of the first part of the proposition differs from Simson’s; and we will deal with Euclid’s first. On this Camerer remarks that it is assumed that the supposed second point of contact lies on the line of centres produced beyond the centre of the “outer” circle, whereas all that is proved in III. 11 is that the line of centres produced beyond the centre of the “inner” circle passes through a point of contact. But, by the same argument as that given on III. 11, we show that the circles cannot have a point of contact, or even any common point, outside the line of centres, because, if there were such a point, there would be a corresponding common point on the other side of the line, and the circles would have three common points. Hence the only hypothesis left is that the second point of contact may be on the line of centres but in the direction of the centre of the “outer” circle; and Euclid’s proof disposes of this hypothesis.
Heron (in an-Nairīzī, ed. Curtze, pp. 122—4), curiously enough, does not question Euclid’s assumption that the line of centres passes through both points of contact (if double contact is possible); but he devotes some space to proving that the centre of the “outer” circle must lie within the “inner” circle, a fact which he represents Euclid as asserting (“sicut dixit Euclides”), though there is no such assertion in our text. The proof of the fact is of course easy. If the line of centres passes through both points of contact, and the centre of the “outer” circle lies either on or outside the “inner” circle, the line of centres must cut the “inner” circle in three points in all: which is impossible, as Heron shows by the lemma, which he places here (and proves by I. 16), that a straight line cannot cut the circumference of a circle in more points than two.
Simson’s proof is as follows (there is no real need for giving two figures as he does).
“If it be possible, let the circle EBF touch the circle ABC in more points than one, and first on the inside, in the points B, D; join BD, and draw GH bisecting BD at right angles.
Therefore, because the points B, D are in the circumference of each of the circles, the straight line BD falls within each of them: And their centres are in the straight line GH which bisects BD at right angles:
Therefore GH passes through the point of contact [III. 11]; but it does not pass through it, because the points B, D are without the straight line GH: which is absurd.
Therefore one circle cannot touch another on the inside in more points than one.”
On this Camerer remarks that, unless III. 11 be more completely elucidated than it is by Euclid’s demonstration, which Simson has, it is not sufficiently clear that, besides the point of contact in which GH meets the circles, they cannot have another point of contact either (1) on GH or (2) outside it. Here again the latter supposition (2) is rendered impossible because in that case there would be a third common point on the opposite side of GH; and the former supposition (1) is that which Euclid’s proof destroys.
Simson retains Euclid’s proof of the second part of the proposition, though his own proof of the first part would apply to the second part also if a reference to III. 12 were substituted for the reference to III. 11. Euclid might also have proved the second part by the same method as that which he employs for the first part.