APPENDIX

I. MODERN GEOMETRIC CONCEPTS

840. Modern Geometry. In recent times many new geometric ideas have been invented, and some of them developed into important new branches of geometry. Thus, the idea of symmetry (see Art. 484, etc.) is a modern geometric concept. A few other of these modern concepts and methods will be briefly mentioned, but their thorough consideration lies beyond the scope of this book.

841. Projective Geometry. The idea of projections (see Art. 345) has been developed in comparatively recent times into an important branch of mathematics with many practical applications, as in engineering, architecture, construction of maps, etc.

842. Principle of Continuity. By this principle two or more theorems are made special cases of a single more general theorem. An important aid in obtaining continuity among geometric principles is the application of the concept of negative quantity to geometric magnitudes.

Thus, a negative line is a line opposite in direction to a given line taken as positive.

B.

For example, if OA is +, OB is —.

A

same line rotating in the opposite direction forms the negative angle AOB'. Similarly, positive and negative arcs are formed.

In like manner, if P and P' are on opposite sides of the line AB and the area PAB is taken as positive, the area P'AB will be negative.

As an illustration of the law of continuity, we may take the theorem that the sum of the triangles formed by drawing lines from a point to the vertices of a polygon equals the area of the polygon.

B

P

B

Applying this to the quadrilateral ABCD, if the point P falls within the quadrilateral, ▲PAB+^PBC+^PCD +APAD=ABCD (Ax. 6).

Δ

Also, if the point falls without the quadrilateral at F", APAB+▲ P'BC+▲ P'CD + ▲ P'AD = ABCD, since APAD is a negative area, and hence is to be subtracted from the sum of the other three triangles.

843. The Principle of Reciprocity, or Duality, is a principle of relation between two theorems by which each theorem is convertible into the other by causing the words for the same two geometric objects in each theorem to exchange places.

Thus, of theorems VI and VII, Book I, either may be converted into the other by replacing the word "sides" by "angles," and "angles" by "sides." Hence these are termed reciprocal theorems.

The following are other instances of reciprocal geometric properties:

The reciprocal of a theorem is not necessarily true. Thus, two parallel straight lines determine a plane, but two parallel planes do not determine a line.

However, by the use of the principle of reciprocity, geometrical properties, not otherwise obvious, are frequently suggested.

844. Principle of Homology. Just as the law of reciprocity indicates relations between one set of geometric concepts (as lines) and another set of geometric concepts (as points), so the law of homology indicates relations between a set of geometric concepts and a set of concepts outside of geometry: as a set of algebraic concepts, for instance.

Thus, if a and b are numbers, by algebra (a+b) (a−b) = a2 — b2.

Also, if a and b are segments of a line, the rectangle (a+b)(ab) is equivalent to the difference between the squares a2 and b2.

By means of this principle, truths which would be overlooked or difficult to prove in one department of thought

are made obvious by observing the corresponding truth in another department of thought.

Thus, if a and b are line segments, the theorem (a+b)? +(ab)2=2(a2+b2) is not immediately obvious in geometry, but becomes so by observing the like relation between the algebraic numbers a and b.

845. Non-Euclidean Geometry. Hyperspace. By varying the properties of space, as these are ordinarily stated, different kinds of space may be conceived of, each having its own geometric laws and properties. Thus, space, as we ordinarily conceive it, has three dimensions, but it is possible to conceive of space as having four or more dimensions. To mention a single property of four dimensional space, in such a space it would be possible, by simple pressure, to turn a sphere, as an orange, inside out without breaking its surface.

As an aid toward conceiving how this is possible, consider a plane in which one circle lies inside another. No matter how these circles are moved about in the plane, it is impossible to shift the inner circle so as to place it outside the other without breaking the circumference of the outer circle. But, if we are allowed to use the third dimension of space, it is a simple matter to lift the inner circle up out of the plane and set it down outside the larger circle.

Similarly if, in space of three dimensions, we have one spherical shell inside a larger shell, it is impossible to place the smaller shell outside the larger without breaking the larger. But if the use of a fourth dimension be allowed, -that is, the use of another dimension of freedom of motion,-it is possible to place the inner shell outside the larger without breaking the latter.

846. Curved Spaces. By varying the geometric axioms of space (see Art. 47), different kinds of space may be conceived of. Thus, we may conceive of space such that through a given point one line may be drawn parallel to a given line (that is ordinary, or Euclidean space); or such

that through a given point no line can be drawn parallel to a given line (spherical space); or such that through a given point more than one line can be drawn parallel to a given line (pseudo-spherical space).

These different kinds of space differ in many of their properties. For example, in the first of them the sum of the angles of a triangle equals two right angles; in the second, it is greater; in the third, it is less.

These different kinds of space, however, have many properties in common. Thus, in all of them every point in the perpendicular bisector of a line is equidistant from the extremities of the line.

EXERCISES. GROUP 85

Ex. 1. Show by the use of zero and negative arcs that the principles of Arts. 257, 263, 258, 264, 265, are particular cases of the general theorem that the angle included between two lines which cut or touch a circle is measured by one-half the sum of the intercepted arcs.

Ex. 2. Show that the principles of Arts. 354 and 358 are particular cases of the theorem that, if two lines are drawn from or through a point to meet a circumference, the product of the segments of one line equals the product of the segments of the other line.

Ex. 3. Show by the use of negative angles that theorem XXXVIII, Book I, is true for a quadrilateral of the form ABCD. [BCD is a negative angle; the angle at the vertex D is the reflex angle ADC.]

Ex. 4. What is the reciprocal of the statement that two intersecting straight lines determine a plane?

B

Ex. 5. What is the reciprocal of the statement that three planes perpendicular to each other determine three straight lines perpendicular to each other?

1