Commentary on Proposition III.11¶
Back to Proposition III.11.
Again August and Heiberg give in an Appendix the additional or alternative proof, which however shows little or no variation from the genuine proof and can therefore well be dispensed with.
The genuine proof is beset with difficulties in consequence of what it tacitly assumes in the figure, on the ground, probably, of its being obvious to the eye. Camerer has set out these difficulties in a most careful note, the heads of which may be given as follows:
He observes, first, that the straight line joining the centres, when produced, must necessarily (though this is not stated by Euclid) be produced in the direction of the centre of the circle which touches the other internally. (For brevity, I shall call this circle the “inner circle,” though I shall imply nothing by that term except that it is the circle which touches the other on the inner side of the latter, and therefore that, in accordance with the definition of touching, points on it in the immediate neighbourhood of the point of contact are necessarily within the circle which it touches.) Camerer then proceeds by the following steps.
The two circles, touching at the given point, cannot intersect at any point. For, since points on the “inner” in the immediate neighbourhood of the point of contact are within the “outer” circle, the inner circle, if it intersects the other anywhere, must pass outside it and then return. This is only possible (a) if it passes out at one point and returns at another point, or (b) if it passes out and returns through one and the same point. (a) is impossible because it would require two circles to have three common points; (b) would require that the inner circle should have a node at the point where it passes outside the other, and this is proved to be impossible by drawing any radius cutting both loops.
Since the circles cannot intersect, one must be entirely within the other.
Therefore the outer circle must be greater than the inner, and the radius of the outer greater than that of the inner.
Now, if F be the centre of the greater and G of the inner circle, and if FG produced beyond G does not pass through A, the given point of contact, then there are three possible hypotheses.
(a) A may lie on GF produced beyond F.
(b) A may lie outside the line FG altogether, in which case FG produced beyond G must, in consequence of result 2 above, either
meet the circles in a point common to both, or
meet the circles in two points, of which that which is on the inner circle is nearer to G than the other is.
(a) is then proved to be impossible by means of the fact that the radius of the inner circle is less than the radius of the outer.
(b) (ii) is Euclid’s case; and his proof holds equally of (b) (i), the hypothesis, namely, that D and H in the figure coincide.
Thus all alternative hypotheses are successively shown to be impossible, and the proposition is completely established.
I think, however, that this procedure may be somewhat shortened in the following manner.
In order to make Euclid’s proof absolutely conclusive we have only (1) to take care to produce FG beyond G, the centre of the “inner” circle, and then (2) to prove that the point in which FG so produced meets the “inner” circle is not further from G than is the point in which it meets the other circle. Euclid’s proof is equally valid whether the first point is nearer to G than the second or the first point and the second coincide.
If FG produced beyond G does not pass through A, there are two
conceivable hypotheses: (a) A may lie on GF produced beyond F, or (b) A may be outside FG produced either way. In either case, if FG produced meets the “inner” circle in D and the other in H, and if GD is greater than GH, then the “inner” circle must cut the “outer” circle at some point between A and D, say X.
But if two circles have a common point X lying on one side of the line of centres, they must have another corresponding point on the other side of the line of centres. This is clear from III. 7, 8; for the point is determined by drawing from F and G, on the opposite side to that where X is, straight lines FY, GY making with FD angles equal to the angles DFX, DGX respectively.
Hence the two circles will have at least three points common: which is impossible.
Therefore GD cannot be greater than GH; accordingly GD must be either equal to, or less than, GH, and Euclid’s proof is valid.
The particular hypothesis in which FG is supposed to be in the same straight line with A but G is on the side of F away from A is easily disposed of, and would in any case have been left to the reader by Euclid.
For GD is either equal to or less than GH.
Therefore GD is less than FH, and therefore less than FA.
But GD is equal to GA, and therefore greater than FA: which is impossible.
Subject to the same preliminary investigation as that required by Euclid’s proof, the proposition can also be proved directly from III. 7.
For, by III. 7, GH is the shortest straight line that can be drawn from G to the circle with centre F;
This proposition is the crucial one as regards circles which touch internally; and, when it is once established, the relative position of the circles can be completely elucidated by means of it and the propositions which have preceded it. Thus, in the annexed figure, if F be the centre of the outer circle and G the centre of the inner, and if any radius FQ of the outer circle meet the two circles in Q, P respectively, it follows, from III. 7, III. 8, or the corresponding theorem with reference to a point on the circumference, that FA is the maximum straight line from F to the circumference of the inner circle, FP is less than FA, and FP diminishes in length as FQ moves round from FA until FP reaches its minimum length FB. Hence the circles do not meet at any other point than A, and the distance PQ cut off between them on any radius FQ of the outer circle becomes greater and greater as FQ moves round from FA to FC and is a maximum when FQ coincides with FC, after which it diminishes again on the other side of FC.
The same consideration gives the partial converse of III. 11 which forms the 6th lemma of Pappus to the first book of the $Tactiones$ of Apollonius (Pappus, VII. p. 826). This is to the effect that, if AB, AC are in one straight line, and on one side of A, the circles described on AB, AC as diameters touch (internally at the point A). Pappus concludes this from the fact that the circles have a common tangent at A; but the truth of it is clear from the fact that FP diminishes as FQ moves away from FA on either side; whence the circles meet at A but do not cut one another.
Pappus’ 5th lemma (VII. p. 824) is another partial converse, namely that, given two circles touching internally at A, and a line ABC drawn from A cutting both, then, if the centre of the outer circle lies on ABC, so does the centre of the inner. Pappus himself proves this, by means of the common tangent to the circles at A, in two ways. (1) The tangent is at right angles to AC and therefore to AB: therefore the centre of the inner circle lies on AB. (2) By III. 32, the angles in the alternate segments of both circles are right angles, so that ABC is a diameter of both.