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VI. Axial and Central Symmetry.

265. If two figures coincide, every point A in the one coincides with a point A' in the other. These points are said to correspond.

Hence to every point in one of two congruent figures there corresponds one, and only one, point in the other; those points being called “corresponding" which coincide if one of the two figures is superimposed upon the other. Hence, calling those parts corresponding which coincide if the whole figures are made to coincide, it follows, that corresponding parts of congruent figures are themselves congruent.

SYMMETRY WITH REGARD TO AN Axis.

266. If we start with two figures in the position of coincidence, and take in the common plane any line x, we may turn the plane of one figure about this line x until its plane, after half a revolution, coincides again with the plane of the other figure.

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The two figures themselves will then have distinct positions in the same plane; but they will have this property, that they can be made to coincide by folding the plane over along the

line x.

Two figures in the same plane which have this property are said to be symmetrical with regard to the line x as an axis of symmetry.

SYMMETRY WITH REGARD TO A CENTER.

267. If we take in the common plane of two coincident figures any point X, we may turn the one figure about this point so that its plane slides over the plane of the other figure without ever separating from it.

Let this turning be continued until one line to X, and therefore the whole moving figure, has been turned through a straight angle about X.

Then the two congruent figures still lie in the same plane, and have such positions that one can be made to coincide with the other by turning it in the plane through a straight angle about the fixed point X.

Two figures which have this property are said to be symmetrical with regard to the point X as center of symmetry.

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268. Any single figure has axial symmetry when it can be divided by an axis into two figures symmetrical with respect to that axis, or has central symmetry when it has a center such that

every line drawn through it cuts the figure in two points symmetrical with respect to this center.

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 A;'.

At the same time OA and OA;' coincide, so that the angles A Ox 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 0, 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.

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