The Teaching of Geometry

by David Eugene Smith

The Principle of Limits is now usually given as follows: "If, while approaching their respective limits, two variables are always equal, their limits are equal." This was expressed by D'Alembert in the eighteenth century as "Magnitudes which are the limits of equal magnitudes are equal," or this in substance. It would easily be possible to elaborate this theory, proving, for example, that if x approaches y as its limit, then ax approaches ay as its limit, and x/a approaches y/a as its limit, and so on. Very much of this theory, however, wearies a pupil so that the entire meaning of the subject is lost, and at best the treatment in elementary geometry is not rigorous. It is another case of having to sacrifice a strictly scientific treatment to the educational abilities of the pupil. Teachers wishing to find a scientific treatment of the subject should consult a good work on the calculus.

Theorem. In the same circle or in equal circles two central angles have the same ratio as their intercepted arcs.

This is usually proved first for the commensurable case and then for the incommensurable one. The latter is rarely understood by all of the class, and it may very properly be required only of those who show some aptitude in geometry. It is better to have the others understand fully the commensurable case and see the nature of its applications, possibly reading the incommensurable proof with the teacher, than to stumble about in the darkness of the incommensurable case and never reach the goal. In Euclid there was no distinction between the two because his definition of ratio covered both; but, as we shall see in Book III, this definition is too difficult for our pupils. Theon of Alexandria (fourth century A.D.), the father of the Hypatia who is the heroine of Kingsley's well-known novel, wrote a commentary on Euclid, and he adds that sectors also have the same ratio as the arcs, a fact very easily proved. In propositions of this type, referring to the same circle or to equal circles, it is not worth while to ask pupils to take up both cases, the proof for either being obviously a proof for the other.

Many writers state this proposition so that it reads that "central angles are measured by their intercepted arcs." This, of course, is not literally true, since we can measure anything only by some thing, of the same kind. Thus we measure a volume by finding how many times it contains another volume which we take as a unit, and we measure a length by taking some other length as a unit; but we cannot measure a given length in quarts nor a given weight in feet, and it is equally impossible to measure an arc by an angle, and vice versa. Nevertheless it is often found convenient to define some brief expression that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus we define "area of a circle," even when we use "circle" to mean a line; and so we may define the expression "central angles are measured by their intercepted arcs" to mean that central angles have the same numerical measure as these arcs. This is done by most writers, and is legitimate as explaining an abbreviated expression.

Theorem. An inscribed angle is measured by half the intercepted arc.

In Euclid this proposition is combined with the preceding one in his Book VI, Proposition 33. Such a procedure is not adapted to the needs of students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle standing on the same arc. Since Euclid never considered an angle greater than 180°, his inscribed angle was necessarily less than a right angle. The first one who is known to have given the general case, taking the central angle as being also greater than 180°, was Heron of Alexandria, probably of the first century A.D.[68] In this he was followed by various later commentators, including Tartaglia and Clavius in the sixteenth century.

One of the many interesting exercises that may be derived from this theorem is seen in the case of the "horizontal danger angle" observed by ships.

If some dangerous rocks lie off the shore, and L and L' are two lighthouses, the angle A is determined by observation, so that A will lie on a circle inclosing the dangerous area. Angle A is called the "horizontal danger angle." Ships passing in sight of the two lighthouses L and L' must keep out far enough so that the angle L'SL shall be less than angle A.

To this proposition there are several important corollaries, including the following: