centres. Accordingly in his own demonstration Euclid confines himself to the right-hand figure; and he shews that this case cannot exist, because the straight line BD would be a diameter of both circles, and would therefore be bisected at two different points; which is absurd.
Euclid might have used a similar method for the second part of the proposition; for as there cannot be a point of contact out of the straight line joining the centres, it is obviously impossible that there can be a second point of contact when the circles touch externally. It is easy to see this; but Euclid preferred a method in which there is more formal reasoning.
We may observe that Euclid's mode of dealing with the contact of circles has often been censured by commentators, but apparently not always with good reason. For example, Walker gives another demonstration of III. 13; and says that Euclid's is worth nothing, and that Simson fails; for it is not proved that two circles which touch cannot have any arc common to both circumferences. But it is shewn in III. 10 that this is imposthe case of circles which cut. See the note on III. 10,
III. 17. It is obvious from the construction in III. 17 that two straight lines can be drawn from a given external point to touch a given circle; and these two straight lines are equal in length and equally inclined to the straight line which joins the given external point with the centre of the given circle.
After reading III. 31 the student will see that the problem in III. 17 may be solved in another way, as follows: describe a circle on AE as diameter; then the points of intersection of this circle with the given circle will be the points of contact of the two straight lines which can be drawn from A to touch the given circle.
III. 18. It does not appear that III. 18 adds anything to what we have already obtained in III. 16. For in III. 16 it is shewn, that there is only one straight line which touches a given circle at a given point, and that the angle between this straight line and the radius drawn to the point of contact is a right angle.
III. 20. There are two assumptions in the demonstration of III. 20. Suppose that A is double of B and C double of D; then in the first part it is assumed that the sum of A and C is double of the sum of B and D, and in the second part it is as-
