A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

1OOK VI.

129

Cor. 4.-If A, B, C, D be four points in the circumference of a circle, and E and F any two other points
also in the circumference, then the pencil (E. ABCD)
= (F. ABCD). This is evident, since the pencils have
equal angles.
Cor. 5.-If through the middle point 0 of any chord
AB of a circle two other chords CE and DF be drawn,
and if the lines ED and CF joining their extremities
intersect AB in G and H, then OG = OH.
Dem.-The pencil (E. ADCB) =(F. ADCB; therefore the anharmonic ratio of the points A, G, O, B = the
anharmonic ratio of the points A, O, HI, B; and since
AO = OB, OG = OH.
DEF.-The anharmonic ratio of the cyclic pencil
(E. ABCD) is called the anharmonic ratio of the four
cyclic points A, B, C, D.
Prop. 3.-The anharmonic ratio of four concyclic
points can be expressed in terms of the chords joining
these four points.
Dem. (see fig., Prop. 9,Section IV.)-The anharmonic
ratio of the pencil (0. ABCD) is AC. BD: AB. CD;
and this, by Prop. 9, Section IV. = A'C'. B'D': A'B'. C'D'; but the pencil (O. ABCD) = the pencil
(O. A'B'C'D') = the anharmonic ratio of the points
A', B', C', D'. Hence the Proposition is proved.
Cor. 1.-The six functions formed, as in Def. 1,
with the six chords joining the four concyclic points
A', B', C', D', are the six anharmonic ratios of these
points.
Cor. 2.-If two triangles CAB, C'A'B' be inscribed
in a circle, any two sides, viz., one from each triangle,
are divided equianharmonically by the four remaining
sides. For, let the sides be AB, A'B'; then the
pencils (C. A'BAB'), (C'. A'BAB') are equal (Cor. 4,
Prop. 2).
Prop. 4.-PAscAL's THEOREM.-If a hexagon be
inscribed in a circle, the intersections of opposite sides
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