Let the chords A B and C D intersect each other in the point E; then will the angle B E D or A E C be measured by half the sum of the arcs A C, B D. For from the point C draw C F parallel to A B. Because the chords A B and C F are parallel, the arcs A C, B F, are equal. Add each of these equals to B D, and we have B D plus A C equal to B D plus B F; that is, the sum of the arcs B D, A C, is equal to the arc F D. Because the chords A B, C F, are parallel, the opposite exterior and interior angles D E B, D C F, are equal. But D C F is an angle at the circumference, and is therefore measured by half the arc F D. Then the equal angle D E B must be measured by half of the arc F D, or its equal B D, plus A C. [Illustration] PROPOSITION XXV. THEOREM. DEMONSTRATION. We wish to prove that _The angle formed by two secants meeting without a circle is measured by half the difference of the intercepted arcs._ Let the secants A B, A C, intersect the circumference in the points D and E; then will the angle B A C be measured by half the difference between the arcs B C and D E. For from the point D draw the chord D F parallel to E C. Because A C and D F are parallel, the opposite exterior and interior angles B D F and B A C are equal. Because the chords D F, E C, are parallel, the arcs D E and F C are equal. If from the arc B C we take the arc D E, or its equal F C, we shall have left the arc B F; But the angle B D F, being at the circumference, is measured by half the arc B F: Then the equal of B D F, or B A C, must be measured by half the arc B F, or half the difference between the intercepted arcs B C and D E. APPENDIX. NOTE A.—To those teachers who think that the line should be derived from a surface, and the surface from a solid, the author would say, that, according to his experience, children apprehend the ideas conveyed by the terms _line_ and _surface_ as readily as they do any ideas whatever; and that, therefore, there seems to be no necessity for extraordinary care in this case to avoid giving wrong impressions. Still, if it be considered desirable in this manner to derive lines and surfaces, it will be apparent that all that can be done in the matter is to give such instruction only by way of a preliminary lesson. * * * * * NOTE B.—Crooked and curved lines are here treated of before straight lines, because the first two are defined by means of an affirmative property,—they _do_ change direction; while the last is defined by means of the absence of one,—they do _not_ change direction. It is easier for a child to comprehend what is than what is _not_. * * * * *