Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/297

Rh only two, which are equal to one another, one on each side of the greatest line.

The first two parts of this proposition are contained in III. 15; all three parts might be demonstrated in the manner of III. 7, and they should be demonstrated, for the third part is really required, as we shall see in the note on III. 10.

III. 9. The point E might be supposed to fall within the angle ADC. It cannot then be shewn that DC is greater than DBF and DB greater than DA, but only that either DC or DA is less than DB; this however is sufficient for establisiting the proposition,

Euclid has given two demonstrations of III. 9, of which Simson has chosen the second. Euclid's other demonstration is as follows. Join D with the middle point of the straight line AB; then it may be shewn that this straight line is at right angles to AB; and therefore the centre of the circle must lie in this straight line, by III. 1, Corollary. In the same manner it may be shewn that the centre of the circle must lie in the straight line which joins D with the middle point of the straight line BC. The centre of the circle must therefore be at D, because two straight lines cannot have more than one common point.

III. 10. Euclid has given two demonstrations of III. 10, of which Simson has chosen the second. Euclid's first demonstration resembles his first demonstration of III. 9. He shews that the centre of each circle is on the straight line which joins K with the middle point of the straight line BG, and also on the straight line which joins K with the middle point of the straight line BH; therefore K must be the centre of each circle.

The demonstration which Simson has chosen requires some additions to make it complete. For the point K might be supposed to fall without the circle DEF, or on its circumference, or within it; and of these three suppositions Euclid only considers the last. If the point K be supposed to fall without the circle DEF we obtain a contradiction of III. 8; which is absurd. If the point K be supposed to fall on the circumference of the circle DEF we obtain a contradiction of the proposition which we have enunciated at the end of the note on III. 7 and III. 8; which is absurd.

What is demonstrated in III. 10 is that the circumferences of two circles cannot have more than two common points; there is