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are constructed between one plane and one point above it at which they meet.

13. A 'prism' is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others are parallelograms.

14. A 'sphere' is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed.

15. The ‘axis of a sphere' is the fixed straight line about which the semicircle revolves.

16. The 'centre of a sphere' is the same with that of the semicircle.

17. The ‘diameter of a sphere' is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

18. A 'cone' is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.

If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled cone; and if greater, an acuteangled cone.

19. The 'axis of a cone' is the fixed straight line about which the triangle revolves.

20. The 'base of a cone' is the circle described by that side containing the right angle which revolves.

21. A 'cylinder' is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed.

22. The ‘axis of a cylinder' is the fixed straight line about which the parallelogram revolves.

23. The 'bases of a cylinder' are the circles described by the two revolving opposite sides of the parallelogram.

24. 'Similar cones and cylinders' are those which have their axes and the diameters of their bases proportionals.

25. A 'cube' is a solid figure contained by six equal squares.

26. A tetrahedron' is a solid figure contained by four equal and equilateral triangles.

27. An 'octahedron' is a solid figure contained by eight equal and equilateral triangles.

28. A 'dodecahedron' is a solid figure contained by twelve equal pentagons which are equilateral and equiangular.

29. An icosahedron' is a solid figure contained by twenty equal and equilateral triangles.

A. A 'parallelopiped' is a solid figure contained by six quadrilateral figures, of which every opposite two are parallel.

The following, though not given by Euclid, are important (Legendre's Éléments de Géométrie) :—

DEF.-A straight line and a plane are said to be parallel to one another when they do not meet however far they are produced.

PROP. 1.—If a straight line AB without a plane MN be parallel to a straight line CD in that plane, it is parallel to the plane.

For, since it lies in pl. ABDC, if it met pl. MN it would

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meet it at some point in the common section CD, which is contrary to the hypothesis.

PROP. 2.—If a straight line AB is parallel to a plane MN, and a plane passing through AB cut the plane MN, the common section CD is parallel to AB.

For if AB met CD it would meet pl. MN, in which CD lies, which is contrary to the hypothesis.

PROP. 3.--If from a point C in a plane MN, which is parallel to a straight line AB, a straight line be drawn parallel to AB, it will lie in the plane MN.

For the plane BAC cuts pl. MN in a st. line CD || to AB by Prop. 2, and only one || to AB can be drawn from a point. [I. 30 and XI. 9.

PROP. 4.-If a straight line AB is parallel to two planes CDE, CDF, which cut one another, it is parallel to the common section CD.

For the to AB through C must lie in both planes by Prop. 3;

.. it is their common section CD.

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Printed by T. and A. CONSTABLE, Printers to Her Majesty, at the Edinburgh University Press.

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