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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 ori. ginal. In the second case a certain condition must be introduced, or the proposition will not be true; the polygon BCDEP 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.
This Appendix consists of a collection of important prcpositions which will be found usefuu, 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.
1. The sum of the squares on the sides of a triangle is equal to twice the square on half the base, together with twice the square on the straight line which joins the vertex to the middle point of the base.
Let ABC be a triangle; and let D be the middle point of the base AB. Draw CE perpendicular to the base
meeting it at E; then E may be either in AB or in AB produced.
First, let E coincide with D; then the proposition follows immediately from I. 47.
Next, let E not coincide with D; then of the two angles ADC and BDC, one must be obtuse and one acute. Suppose the angle ADC obtuse. Then, by II. 12, the square on AC is equal to the squares on AD, DC, together with twice the rectangle AD, DE; and, by II. 13, the square on BC together with twice the rectangle BD, DE is equal to the squares on BD, DC. Therefore, by Axiom 2, the squares on AC, BC, together with twice the rectangle BD, DE are equal to the squares on AD, DB, and twice the square on DC, together with twice the rectangle AD, DE. But AD is equal to DB. Therefore the squares on AC, BC are equal to twice the squares on AD, DC.
2. If two chords intersect within a circle, the angle which they include is measured by half the sum of the intercepted arcs.
Let the chords AB and CD of a circle intersect at E; join AD. Tho angle AEC is equal to the angles ADE, and DAE, by I. 32; that is, to the angles standing on the arcs AC and BD. Thus the angle AEC is equal to an angle at the circumference of the circle standing on the sum of the arcs AC and BD; and is therefore equal to an angle at the centre of the circle standing on half the sum of these arcs.
Similarly the angle CEB is measured by half the sum of the arcs CB and AD.
3. If two chords produced intersect without a circle, the angle which they include is measured by half the difference of the intercepted arcs.
Let the chords AB and CD of a circle, produced, intersect at E; join AD.
The angle ADC is equal to the angles EAD and AED, by I. 32. Thus the angle AEC is equal to the difference of the angles ADC and BAD; that is, to an angle at the circumference of the circle standing on an arc which is the
difference of AC and BD; and is therefore equal to an angle at the centre of the circle standing on half the difference of these arcs.