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THEOREM XLIV.

269. If a figure has two axes of symmetry perpendicular to each other, then their intersection is a center of symmetry.

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For, if x and y be two axes at right angles, then to a point A will correspond a point A' with regard to x as axis.

To these will correspond points A, and A,' with regard to y as axis. These points A, and A,, will correspond to each other with regard to x. To see this, let us first fold over along y; then A falls on A1, and A' on A,'.

If we now, without folding back, fold over along x, A, and with it A1, will fall on A', which coincides with 4,'.

At the same time OA and OA,' coincide, so that the angles AOx and A'' Ox' are equal, where x' denotes the continuation of x beyond O. It follows, that AOA'' are in a line, and that the sect AA, is bisected at O, or O is a center of symmetry for AA,, and similarly for A, and A.

BOOK II.

RECTANGLES.

270. A Continuous Aggregate is an assemblage in which two adjacent parts have the same boundary.

271. A Discrete or Discontinuous Aggregate is one in which two adjacent parts have different boundaries.

A pile of cannon balls is a discrete aggregate. We know that any adjacent two could be painted different colors, and so they have direct independent boundaries.

Our fingers are a discontinuous aggregate.

272. All counting belongs first to the fingers.

273. There is implied and bound up in the word "number" the assumption that a group of things comes ultimately to the same finger, in whatever order they are counted.

This proposition is involved in the meaning of the phrase "distinct things."

Any one and any other of them make two. If they are attached to two of my fingers in a certain order, they can also be attached to the same fingers in the other order. Thus, one order of a group of three distinct things can be changed into any other order while using the same fingers, and so on with a group of four, etc.

274. By generalizing the use of the fingers in counting, man has made for himself a counting apparatus, which each one carries around in his mind. This counting apparatus, the natural series of numbers, was made from a discrete aggregate, and so will only correspond exactly to discrete aggregates.

275. In a row of shot, we can find between any two, only a finite number of others, and sometimes none at all.

Just so in regard to any two numbers. A row of six shot can be divided into two equal parts; but the half, which is three, we cannot divide into two equal parts: and so in a series of numbers.

276. But in 136 we have shown how any sect whatever may be bisected, and the bisection point is the boundary of both parts. So a line is not a discrete aggregate of points. It is something totally different in kind from the natural series of numbers.

277. The science of numbers is founded on the hypothesis of the distinctness of things The science of space is founded on the entirely different hypothesis of continuity.

278. Numbers are essentially discontinuous, and therefore unsuited to express the relations of continuous magnitudes.

279. In arithmetic we are taught to add and multiply numbers we will now show how the laws for the addition and multiplication of these discrete aggregates are applicable to sects, which are continuous aggregates.

THE COMMUTATIVE LAW FOR ADDITION.

280. In a sum of numbers we may change the order in which the numbers are added.

If x and y represent numbers, this law is expressed by the equation

x + y = y + x.

It depends entirely on the interchangeability of any pair of the units of numeration.

281. The sum of two sects is the sect obtained by placing them on the same line so as not to overlap, with one end point in common.

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Thus, the sum of the sects a and b means the sect AC, which can be divided into two parts,

AB = a, and BC= b.

282. The commutative law holds for the summation of sects.

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for AC revolved through a straight angle may be superimposed upon C'A', and will coincide point for point.

The more general case, where three or more sects are added, follows from a repetition of the above.

Thus, the commutative law for addition in geometry depends entirely on the possibility of motion without deformation.

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