XVI

Geometry

Geometry, like the rest of mathematics, is

abstract. In it the properties of the shapes and relative positions of things are studied. But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations. Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties. Of course it helps our imagination to look at particular examples of spheres and cones and triangles and squares. But the propositions do not merely apply to the actual figures printed in the book, but to any such figures.

Thus geometry, like algebra, is dominated by the ideas of "any" and "some" things. Also, in the same way it studies the interrelations of sets of things. For example, consider any two triangles $A B C$ and $D E F$ .

What relations must exist between some of

the parts of these triangles, in order that the triangles may be in all respects equal? This is one of the first investigations undertaken in all elementary geometries. It is a study 33 of a certain set of possible correlations between the two triangles. The answer is that the triangles are in all respects equal, if:–- Either, (a) Two sides of the one and the included angle are respectively equal to two sides of the other and the included angle:

Or, (b) Two angles of the one and the side joining them are respectively equal to two angles of the other and the side joining them:

Or, (c) Three sides of the one are respectively equal to three sides of the other.

This answer at once suggests a further enquiry. What is the nature of the correlation between the triangles, when the three angles of the one are respectively equal to the three angles of the other? This further investigation leads us on to the whole theory of similarity

(cf. XIII.), which is another type of correlation.

Again, to take another example, consider the internal structure of the triangle $A B C$ . Its sides and angles are inter-related–-the greater angle is opposite to the greater side, and the base angles of an isosceles triangle are equal. If we proceed to trigonometry this correlation receives a more exact determination in the familiar shape $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},$ $a^{2} = b^{2} + c^{2} - 2 b c \cos A$ , with two similar formulæ.

Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles; and between the three sides, namely, that the sum of the lengths of any two is greater than the length of the third.

Thus the true method to study geometry is to think of interesting simple figures, such as the triangle, the parallelogram, and the circle, and to investigate the correlations between their various parts. The geometer