triangles which have one angle of the one equal to one angle of the other, have to one another the ratio which is compounded of the ratios of their sides.

Then VI. 19 is an immediate consequence of this theorem. For let ABC and DEF be similar triangles, so that AB is to BC as DE is to EF; and therefore, alternately, AB is to DE as BC is to EF. Then, by the theorem, the triangle ABC has to the triangle DEF the ratio which is compounded of the ratios of AB to DE and of BC to EF, that is, the ratio which is compounded of the ratios of BC to EF and of BC to EF. And, from the definitions of duplicate ratio and of compound ratio, it follows that the ratio compounded of the ratios of BC to EF and of BU to EF is the duplicate ratio of BC to EF.

VI. 25. It will be easy for the student to exhibit in detail the process of shewing that BC and CF are in one straight line, and also LE and EM; the process is exactly the same as that in I. 45, by which it is shewn that KH and HM are in one straight line, and also FG and GL.

It seems that VI. 25 is out of place, since it separates propositions so closely connected as VI. 24 and VI. 26. We may enunciate VI. 25 in familiar language thus: to make a figure which shall have the form of one figure and the size of another.

VI. 26. This proposition is the converse of VI. 24; it might be extended to the case of two similar and similarly situated parallelograms which have a pair of angles vertically opposite.

We have omitted in the sixth Book Propositions 27, 28, 29, and the first solution which Euclid gives of Proposition 30, as they appear now to be never required, and have been condemned as useless by various modern commentators; see Austin, Walker, and Lardner. Some idea of the nature of these propositions may be obtained from the following statement of the problem proposed by Euclid in VI. 29. AB is a given straight line; it has to be produced through B to a point 0, and a parallelogram described on 40 subject to the following conditions; the parallelogram is to be equal to a given rectilineal figure, and the parallelogram on the base BO which can be cut off by a straight line through B is to be similar to a given parallelogram.

VI. 32. This proposition seems of no use. Moreover the enunciation is imperfect. For suppose ED to be produced

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 DC, 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. D occurs in the Meyáλn 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 Cyclopædia. 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 con tained 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

two convex solid figures are equal if they are contained by equal plane figures similarly arranged; see Catalan's Théorèmes et Problèmes de Géométrie Elémentaire. This result was first demonstrated by Cauchy, who turned his attention to the point at the request of Legendre and Malus; see the Journal de l'École Polytechnique, Cahier 16.

XI. Def. 26. The word tetrahedron is now often used to denote a solid bounded by any four triangular faces, that is, a pyramid on a triangular base; and when the tetrahedron is to be such as Euclid defines, it is called a regular tetrahedron. Two other definitions may conveniently be added.

A straight line is said to be parallel to a plane when they do not meet if produced.

The angle made by two straight lines which do not meet is the angle contained by two straight lines parallel to them, drawn through any point.

XI. 21. In XI. 21 the first case only is given in the original. In the second case a certain condition must be introduced, or the proposition will not be true; the polygon BCDEF must have no re-entrant angle. See note on I. 32.

The propositions in Euclid on Solid Geometry which are now not read, contain some very important results respecting the volumes of solids. We will state these results, as they are often of use; the demonstrations of them are now usually given as examples of the Integral Calculus.

We have already explained in the notes to the second Book how the area of a figure is measured by the number of square inches or square feet which it contains. In a similar manner the volume of a solid is measured by the number of cubic inches or cubic feet which it contains; a cubic inch is a cube in which each of the faces is a square inch, and a cubic foot is similarly defined.

The volume of a prism is found by multiplying the number of square inches in its base by the number of inches in its altitude; the volume is thus expressed in cubic inches. Or we may multiply the number of square feet in the base by the number of feet in the altitude; the volume is thus expressed in cubic feet. By the base of a prism is meant either of the two equal, similar, and parallel figures of XI. Definition 13; and the altitude of the prism is the perpendicular distance between these two planes.

The rule for the volume of a prism involves the fact that prisms on equal bases and between the same parallels are equal in volume.

A parallelepiped is a particular case of a prism. The volume of a pyramid is one third of the volume of a prism on the same base and having the same altitude.

For an account of what are called the five regular solids the student is referred to the chapter on Polyhedrons in the Treatise on Spherical Trigonometry.

THE TWELFTH BOOK.

Two propositions are given from the twelfth Book, as they are very important, and are required in the University Examinations. The Lemma is the first proposition of the tenth Book, and is required in the demonstration of the second proposition of the twelfth Book,

APPENDIX.

THIS Appendix consists of a collection of important propositions which will be found useful, both as affording geometrical exercises, and as exhibiting results which are often required in mathematical investigations. The student will have no difficulty in drawing for himself the requisite figures in the cases where they are not given.