), yet in the majority of cases we should obtain no such result (i.e.,
). Hence, we should have to say that in certain cases the square root of a true number would be a true number, whereas in other cases it would correspond to nonsense. Once again, in order to obtain generality, we should have to incorporate these seemingly nonsensical magnitudes along with the true numbers, obtaining thereby the so-called irrational numbers. In a similar way negative numbers would have to be introduced in order to confer sense in all cases on the subtraction of one true number from another. Imaginary numbers would follow when we wished to generalise the significance of the square root of a number, whether positive or negative; then complex numbers, when we wished to confer generality on the significance of the addition of numbers, whether real or imaginary. A further generalisation would yield quaternions and hyper-complex numbers generally. Following still another line of generalisation, Cantor introduced transfinite numbers into mathematics.
A further illustration would be afforded by geometry. Thus, if in a plane we trace a straight line and a second-order curve (circle, ellipse, parabola or hyperbola), it may happen that the line intersects the curve in two points, but it may also happen that no intersection takes place. Generality can be re-established, however, provided we introduce imaginary points and consider them as true points. With this extended concept of a point we may say that the straight line always intersects the second-order curve in two true points, whether real or imaginary. Further generalisations require the introduction of ideal points, points at infinity, imaginary lines and angles.
Also, when we consider a second-order curve, it is necessary to state the positions of five of its points for the curve to be determined without ambiguity. But a circle is also a second-order curve; yet, in the case of a circle, when we state the positions of three of its points the circle is completely determined. Hence, our rule lacks generality and cannot be claimed to hold for all second-order curves, whether ellipses, parabolas, hyperbolas or circles. Once again, however, when imaginary points are introduced, this duality disappears and our rule becomes general. It is found that all circles, without exception, pass through the two same imaginary points called the circular points at infinity. When we take these two imaginary points into consideration we are able to state that all second-order curves, whether circles or ellipses, etc., are determined when five of their points are given. It is also interesting to note, as Cayley and Klein have shown, that when imaginary points, lines and angles are introduced, the various geometries (non-Euclidean and Euclidean) can be treated in terms of projective geometry. We see that in all cases mathematicians attempt to replace the word sometimes by the word always, the particular by the general, thereby revealing unity where diversity once held sway.
Let us mention yet another geometrical example. It is always possible to draw a triangle circumscribed to a circle, that is to say, one whose three sides are tangent to the circle. Also it is always possible to inscribe a triangle in a circle; which means that the three summits lie on the circle. Now, Euler noticed that if two circles were drawn at random, it was impossible in the majority of cases to draw a triangle which would be inscribed in one circle while circumscribed to the other. When, however, one such triangle could be found, an indefinite number of others could also be drawn.
Poncelet then gave a celebrated generalisation of this theorem, and Jacobi showed its intimate connection with elliptical functions. We may understand the nature of this generalisation as follows:
A circle is a particular instance of a conic, and a triangle a particular instance of a polygon; whence Poncelet extended Euler’s theorem, which deals with circles and triangles, to one dealing with conics and polygons. In this illustration, again, we witness the same tendency, progress towards generalisation.
Possibly it is in the theory of functions that the most beautiful examples will be found. Mathematical functions are magnitudes whose values vary with that attributed to the variable they contain. We can construct as many different types of functions as we please by annexing additional terms, and the functions thus constructed differ in their behaviour from one another.