Example:-Prove that √2 × √√√3 = √√6.

2 is specified as a number which lies between a and a', where a2 < 2, a^2 > 2, and a' a can be made as small as we please. 3 is specified as a number which lies between band b', where b2 < 3, b'2 > 3, and b' b can be made as small as we please. By Def. III., √2 × √3 always lies between ab and a'b'. But a2b26, and a'2'2 > 6.

Hence √2 x 3 always lies between two rational numbers whose difference can be made as small as we please, such that the square of the smaller number is always less than 6, and the square of the larger number greater than 6. But this is the specification of No6.

385. THEOREM. The area of a rectangle is given by the product of the measures of its sides, when the sides are incommensurable with the unit of length.

A

b Bb

In rect. ABCD, let AB and AD contain X and Y units of length respectively, where X and Y are irrational. Let B lie between two points b and b' in AB, and let D lie between two points d and d' in AD, such that Ab, Ab', Ad, Ad', contain x, x', y, y' units of length, where xx, y < y', and these numbers are all rational.

D

FIG. 514.

Complete the rects. Abcd, Ab'c'd'. Then it follows from the figure that rect. Abcd < rect. ABCD <rect. Ab'c'd'.

Now the difference between x and x' and the difference between y and y' can be made as small as we please. Also X lies between x and x', and Y lies between y and y'. Hence XY is specified as the number which always lies between xy and x'y'. .

(i)

Again measure of area Abcd is xy, and measure of area Ab'c'd' is x'y'. Also area ABCD always lies between area Abcd and area Ab'c'd'. Hence

measure of area ABCD always lies bet. xy and x'y'. . (ii) From (i) and (ii) it follows that

XY.

386. THEOREM.-If two transversals intersect a series of parallel lines, and if the ratio of two intercepts (AB, CD) formed on the one transversal is irrational, it is equal to the ratio of the two corresponding intercepts (EF, GH) on the other.

AB

EF

G

H

FIG. 515.

Thus and are equal, for they lie between the

CD

GH

same two rational numbers whose difference ultimately vanishes.

387. Irrational Exponent. We have now completed the theory of irrational numbers, except for the case when the irrational is used as an index. As this case is not required in the present volume it will be omitted. The method of treatment is essentially the same as before, though the details of the discussion are more intricate.

EXERCISES CXXXIV.

1. How would you define A × b, where A is irrational and b rational? 2. How would you define a ÷ B, where a is rational and B irrational ? Prove from your definition that (a B) x B is equal to a.

3. How would you define A' where A is irrational and b integral ? Prove that Ab x Ac = Ab+c.

3

4. Prove that p÷q=where p and q are rational, and 3/p and 3/4 are irrational.

5. Prove that triangles of equal altitude are proportional to their bases when these are incommensurable.

6. Prove that in equal circles the arcs and areas of sectors are proportional to their angles when these are not commensurable.

CHAPTER XXVII.

STRAIGHT LINES AND PLANES.

388. The Plane. The earlier portions of this work have been devoted almost entirely to Plane Geometry, i.e. to the consideration of geometrical figures lying entirely in one plane. The remainder of the work will be devoted to the elements of Solid Geometry, i. e. to the consideration of geometrical figures which do not lie in one plane.

The following definition of a plane has already been given in § 47

DEFINITION. A plane surface (or plane) is a surface such that the straight line which joins any two points upon it lies entirely on the surface.

Examples of plane surfaces are the face of a wall, the flat top of a table, the surface of a garden lawn, etc. Fig. 516 represents two plane surfaces, say the surface of a rectangular lawn, A, and the face of a rectangular wall, B, along one side of the lawn: thus A represents a horizontal (i. e. level) plane, and B represents a vertical (i. e. upright) plane.

A complete straight line is infinite in length, but in a figure it is represented by any convenient portion of itself, which portion may be produced if desired. In the same way a complete plane is infinite in length and breadth, but in a figure it is represented by any convenient portion of itself (usually a rectangle) which portion may be produced in length or breadth or both,

if desired. For example, in Fig. 516 the complete plane represented by the rectangle B extends in all directions above, below, to the right of, and to the left of the rectangle B.

Notice that though A and B in Fig. 516 represent rectangles they are not actually drawn as rectangles.

A

B

FIG. 516.

They are irregular quadrilaterals - approximately parallelograms. If a geometrical figure lies in one plane it can be drawn in correct shape on a flat piece of paper; but if a geometrical figure does not lie in one plane it can only be represented on paper by a "picture" of the figure, i. e. by a diagram drawn in perspective.

Thus in the diagrams of solid geometry a complete plane is represented by some convenient portion of itself (usually a rectangle) drawn in perspective.

389. DEFINITIONS.

(1) A straight line is said to be parallel to a plane if the complete straight line and the complete plane have no point in common.

Illustrations.-Any straight line drawn on the flat top of a table standing in a room is parallel to the plane of the floor and also to that of the ceiling; any straight line drawn on one wall of a room is parallel to the opposite wall, etc.

(2) Two planes are said to be parallel if when completed they have no point in common.

Illustrations.-Two opposite walls in a room are parallel planes; the flat top of a table standing in a room is parallel to the floor, etc.

(3) A straight line is said to be perpendicular to a plane if it is perpendicular to every line in the plane which meets it. (See Fig. 517.)

A straight line perpendicular to a plane is called a normal to the plane.

FIG. 517.

Illustrations.-A flagstaff standing on level ground should be perpendicular to any straight line on the ground drawn from its foot, i.e. the flagstaff should be perpendicular to the plane of the ground; in machinery the axle of a wheel should be perpendicular to every spoke of the wheel, and therefore to the plane of the wheel; etc.

390. AXIOMS.—(1) Any point on a complete straight line divides it into two parts.

(2) Any complete straight line on a complete plane divides it into two parts.

(3) Any complete plane divides space into two parts. For example, in Fig. 516, the plane A, if produced indefinitely in all directions, divides space into two parts, or regions, one above the plane

and one below it.

(4) A half-plane, rotated about its bounding line, sweeps through all space.

Thus in Fig. 516 the complete line XY divides the complete plane A into two halves. If either half of this complete plane makes one complete revolution about the line XY it sweeps through every point in space.

(5) Two intersecting surfaces meet (in general) in a line (or lines), not in a point or isolated points.

Thus in Fig. 518 the two planes AA, BB meet in the line XY.