has in his mind not a detached proposition, but a figure with its various parts mutually inter-dependent. Just as in algebra, he generalizes the triangle into the polygon, and the side into

the conic section. Or, pursuing a converse route, he classifies triangles according as they are equilateral, isosceles, or scalene, and polygons according to their number of sides, and conic sections according as they are hyperbolas, ellipses, or parabolas.

The preceding examples illustrate how the fundamental ideas of geometry are exactly the same as those of algebra; except that algebra deals with numbers and geometry with lines, angles, areas, and other geometrical entities. This fundamental identity is one of the reasons why so many geometrical truths can be put into an algebraic dress. Thus if $A$ , $B$ , and $C$ are the numbers of degrees respectively in the angles of the triangle $A B C$ , the correlation between the angles is represented by the equation $A + B + C = 180 °;$ and if $a$ , $b$ , $c$ are the number of feet respectively in the three sides, the correlation between the sides is represented by $a < b + c$ , $b < c + a$ , $c < a + b$ . Also the trigonometrical formulæ quoted above are other examples of the same

fact. Thus the notion of the variable and the correlation of variables is just as essential in geometry as it is in algebra.

But the parallelism between geometry and algebra can be pushed still further, owing to the fact that lengths, areas, volumes, and

angles are all measurable; so that, for example, the size of any length can be determined by the number (not necessarily integral) of times which it contains some arbitrarily known unit, and similarly for areas, volumes, and angles. The trigonometrical formulæ, given above, are examples of this fact. But it receives its crowning application in analytical geometry. This great subject is often misnamed as Analytical Conic Sections, thereby

fixing attention on merely one of its subdivisions. It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body.

Though the mathematical procedures in geometry and algebra are in essence identical and intertwined in their development, there is necessarily a fundamental distinction between the properties of space and the properties of number–-in fact all the essential difference between space and number. The "spaciness" of space and the "numerosity" of number are essentially different things, and must be directly apprehended. None of the applications of algebra to geometry or of geometry to algebra go any step on the road to obliterate this vital distinction.

One very marked difference between space and number is that the former seems to be so much less abstract and fundamental than the

latter. The number of the archangels can be counted just because they are things. When we once know that their names are Raphael, Gabriel, and Michael, and that these distinct names represent distinct beings, we know without further question that there are three of them. All the subtleties in the world about the nature of angelic existences cannot alter this fact, granting the premisses.

But we are still quite in the dark as to their relation to space. Do they exist in space at all? Perhaps it is equally nonsense to say that they are here, or there, or anywhere, or everywhere. Their existence may simply have no relation to localities in space. Accordingly, while numbers must apply to all things, space need not do so.