Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/296

272 the boundary of a circle, and accordingly we use the word arc. Euclid himself uses circumference both for the whole boundary and for a portion of it.

Ill, I. In the construction, DC is said to be produced to E; this assumes that D is within the circle, which Euclid demonstrates in III. 1.

III. 3. This consists of two parts, each of which is the converse of the other; and the whole proposition is the converse of the corollary in III. 1.

III. 5 and III. 6 should have been taken together. They amount to this, if the circumferences of two circles meet at a point they cannot have the same centre, so that circles which have the same centre and one point in their circumferences common, must coincide altogether. It would seem as if Euclid had made three cases, one in which the circles cut, one in which they touch internally, and one in which they touch externally, and had then omitted the last case as evident.

III. 7, III. 8. It is observed by Professor De Morgan that in III. 7 it is assumed that the angle FEB is greater than the angle FEC, the hypothesis being only that the angle DFB is greater than the angle DFC; and that in III. 8 it is assumed that K falls within the triangle DLM, and E without the triangle DMF. He intimates that these assumptions may be established by means of the following two propositions which may be given in order after I.

The perpendicular is the shortest straight line which can he drawn from a given point to a given straight line; and of others that which is nearer to the perpendicular is less than the more remote, and the converse; and not more than two equal straight lines can he drawn from the given point to the given straight line, one on each side of the perpendicular.

Every straight line draWn from the vertex of a triangle to the base is less than the greater of the two sides, or than either of them if they he equal.

The following proposition is analogous to III. 7 and III. 8.

If any point he taken on the circumference of a circle, of all the straight lines touch can he drawn from it to the circumference, the greatest is that in which the centre is; and of any others, that which is nearer to the straight line which passes through the centre is always greater than one more remote; and from the same points there can, be drawn to the circumference two straight lines, and