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.

TWO FIGURES DIRECTLY SIMILAR.

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first figure, it will be its own homologue in the second
figure.
DEP.-FThe point 0 is called a double point of the two
figures; it is also called their centre of similitude.
Prop. 4.-Being given two polygons directly similar,
it is required to find their double point.
Let AB, A'B' be two homologous sides of the figures;
C their point of intersection. Through the two triads
of points A, A', C; B, B', C describe two circles interA
secting again in the point O: 0 will be the point required. For it is evident that the triangles OAB, OA'B'
are similar, and that either may be turned round the
point 0, so that the two bases AB, A'B' will be
parallel.
Observation.-The foregoing construction must be modified
when the homologous sides of the two figures are consecutive
- - I.^ ^~~~~

sides BA, AC of a triangle. In this case, upon the lines BA,