The Teaching of Geometry

by David Eugene Smith

If part of the curve APB is known, take P as the mid-point. Then stretch the tape from A to B and draw PM perpendicular to it. Then swing the length AM about P, and PM about B, until they meet at L, and stretch the length AB along PL to Q. This fixes the point Q. In the same way fix the point C. Points on the curve can thus be fixed as near together as we wish. The chords AB, PQ, BC, and so on, are equal and are equally distant from the center.

Theorem. A line perpendicular to a radius at its extremity is tangent to the circle.

The enunciation of this proposition by Euclid is very interesting. It is as follows:

The straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further, the angle of the semicircle is greater and the remaining angle less than any acute rectilineal angle.

The first assertion is practically that of tangency,—"will fall outside the circle." The second one states, substantially, that there is only one such tangent, or, as we say in modern mathematics, the tangent is unique. The third statement relates to the angle formed by the diameter and the circumference,—a mixed angle, as Proclus called it, and a kind of angle no longer used in elementary geometry. The fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity. This gives rise to a difficulty that seems to have puzzled many of Euclid's commentators, and that will interest a pupil: As the circle diminishes this angle apparently increases, while as the circle increases the angle decreases, and yet the angle is always stated to be zero. Vieta (1540-1603), who did much to improve the science of algebra, attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time. He said that a circle was a polygon of an infinite number of sides (which it cannot be, by definition), and that, a tangent simply coincided with one of the sides, and therefore made no angle with it; and this view was also held by Galileo (1564-1642), the great physicist and mathematician who first stated the law of the pendulum.

Theorem. Parallel lines intercept equal arcs on a circle.

The converse of this proposition has an interesting application in outdoor work.

Suppose we wish to run a line through P parallel to a given line AB. With any convenient point O as a center, and OP as a radius, describe a circle cutting AB in X and Y. Draw PX. Then with Y as a center and PX as a radius draw an arc cutting the circle in Q. Then run the line from P to Q. PQ is parallel to AB by the converse of the above theorem, which is easily shown to be true for this figure.