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XX. The Base of a Cone is the circle, described by that side, containing the right angle, which revolves.

XXI. A Cylinder is a solid figure, described by the revolution of a rectangle about one of its sides, which remains fixed.

XXII. The Axis of a Cylinder is the fixed straight line about which the rectangle revolves.

XXIII. The Bases of a Cylinder are the circles, described by the two revolving opposite sides of the rectangle.

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

XXV. A Cube is a solid figure, contained by six equal squares.

XXVI. A Tetrahedron is a solid figure, contained by four equal and equilateral triangles.

XXVII. An Octahedron is a solid figure, contained by eight equal and equilateral triangles.

XXVIII. A Dodecahedron is a solid figure, contained by twelve equal pentagons, which are equilateral and equiangular.

XXIX. An Icosahedron is a solid figure, contained by twenty equal and equilateral triangles.

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

POSTULATE.

Let it be granted that a plane may be made to pass through any given straight line.

PROPOSITION I. THEOREM.

(Eucl. XI. 2.)

If two straight lines meet one another, a plane can be drawn to contain both; and every plane containing both must coincide with the aforesaid plane.

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Let the two st. lines AC, BC meet in C.

Then a plane can be drawn to contain AC and BC.

Let any plane EF be drawn to contain AC,

and let EF be turned about AC till it pass through B. Then B and C are points in the plane EF,

.. BC lies in the plane EF.

Post.

XI. Def. 1. Also, any plane containing AC and BC must coincide with EF. For let Q be any point in a plane containing AC and BC. Draw QMN in this plane to cut BC, AC in M and N. Then ·· M and N are points in the plane EF,

..Q is a point in the plane EF. XI. Def. 1. Similarly, any point in a plane containing AC, BC must lie in EF;

and .. any plane containing AC, BC must coincide with EF.

Q. E. D.

COR. I. Hence it follows that a plane is completely determined by the condition that it passes through two intersecting straight lines.

COR. II. A straight line and a point without the line determine

a plane.

D

A

Let AB be a straight line, and C a point without AB.
Draw the st. line CD to any point D in AB.
Then one plane can be drawn to contain AB and CD.

.. one....

.AB and C.

Again, any plane containing AB must contain D,

XI. 1.

.. any plane containing AB and C must contain CD also. But there is only one plane that can contain AB and CD, .. there is only one plane

Hence the plane is completely determined.

AB and C.

COR. III. Three points, not in the same straight line, determine a plane.

For let A, B, C be three such points (fig. Cor. 2).

Draw the straight line AB.

Then a plane, which contains A, B and C, must contain AB and C,

and a plane, which contains AB and C, must contain A, B, C. Now AB and C are contained by one plane, and one only,

.. A, B, C are contained by one plane, and one only. Hence the plane is completely determined.

COR. IV. Two parallel lines determine a plane.

Cor. 2.

For, by the definition of parallel lines, the two lines are in the same plane, and as only one plane can be drawn to contain one of the lines and any point in the other line, it follows that only one plane can be drawn to contain both lines.

PROPOSITION II. THEOREM. (Eucl. xI. 3.)

If two planes cut one another, their common section must be a straight line.

N

M

B

Let AB and CD be two planes that cut one another.

Then must their common section be a straight line. Let M and N be two points common to both planes. Draw the straight line MN.

Then ·.· M and N are common to both planes,

.. the st. line MN lies in both planes.

XI. Def. 1.

And no point, out of this line, can be common to both planes. For, if it be possible, let P be such a point.

But there can be but one plane common to the point P and the st. line MN. XI. 1, Cor. 2.

... P is not common to both planes.

Hence every point in the common section of the planes lies in the straight line MN.

Q. E. D.

Note.-The Propositions which follow are numbered as in

Euclid.

PROPOSITION IV. THEOREM.

If a straight line stand at right angles to each of two straight lines, at the point of their intersection, it must also be at right angles to the plane that passes through them.

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Let the st. line EF be 1 to each of the st. lines AB, CD, at E, the pt. of their intersection.

Then must EF be 1 to the plane passing through AB, CD.

Make AE, EB, CE, ED, all equal to one another, and through E, draw, in the plane in which AB, CD are, any st. line GEH, and join AD, CB.

Take any pt. F, in EF, and join FA, FG, FD, FC, FH, FB. Then in As AED, BEC,

·· AE=BE, and DE=CE, and ▲ AED = 2 BEC, I. 15. .. AD=BC, and ▲ DAE= ▲ CBE,

Then in As AEG, BEH,

I. 4.

LAEG = ▲ BEH, and ▲ GAE = ▲ HBE, and AE=BE,

.. GE=HE, and AG=BH.

Then in As AEF, BEF,

I. B. p. 17.

·.· AE=BE, and EF is common, and rt. ▲ AEF=rt. 2 BEF,

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