through D to a point F, such that DF is equal to DE; and join CF. Then the triangle CDF will satisfy all the conditions in Euclid's enunciation, as well as the triangle CDE; but CF and CB are not in one straight line. It should be stated that the bases must lie on corresponding sides of both the parallels; the bases CF and BC do not lie on corresponding sides of the parallels AB and BC, and so the triangle CDF would not fulfil all the conditions, and would therefore be excluded.
VI. 33. In VI. 33 Euclid implicitly gives up the restriction, which he seems to have adopted hitherto, that no angle is to be considered greater than two right angles. For in the demonstration the angle BGL may be any multiple whatever of the angle BGC, and so may be greater than any number of right angles.
VI. B, C, D. These propositions were introduced by Simson. The important proposition VI. B occurs in the Μεγάλη Σύνταξις of Ptolemy.
THE ELEVENTH BOOK.
In addition to the first six Books of the Elements it is usual to read part of the eleventh Book. For an account of the contents of the other Books of the Elements the student is referred to the article Eucleides in Dr Smith's Dictionary of Greek and Roman Biography, and to the article Irrational Quantities in the English Cyclopcaedia. We may state briefly that Books VII, VIII, IX treat on Arithmetic, Book X on Irrational Quantities, and Books XI, XII on Solid Geometry.
XI. Def. 10. This definition is omitted by Simson, and justly, because, as he shews, it is not true that solid figures contained by the same number of similar and equal plane figures are equal to one another. For, conceive two pyramids, which have their bases similar and equal, but have different altitudes. Suppose one of these bases applied exactly on the other; then if the vertices be put on opposite sides of the base a certain solid is formed, and if the vertices be put on the same side of the base another solid is formed. The two solids thus formed are contained by the same number of similar and equal plane figures, but they are not equal.
It will be observed that in this example one of the solids has a re-entrant solid angle; see page 264. It is however true that
