The Teaching of Geometry

by David Eugene Smith

Some one who had begun to read geometry with Euclid, when he had learned the first theorem, asked, "But what shall I get by learning these things?" Euclid called his slave and said, "Give him three obols, since he must make gain out of what he learns."

Whether true or not, the story expresses the sentiment that runs through Euclid's work, and not improbably we have here a bit of real biography,—practically all of the personal Euclid that has come down to us from the world's first great textbook maker. It is well that we read the story occasionally, and also such words as the following, recently uttered[4] by Sir Conan Doyle,—words bearing the same lesson, although upon a different theme:

In the present utilitarian age one frequently hears the question asked, "What is the use of it all?" as if every noble deed was not its own justification. As if every action which makes for self-denial, for hardihood, and for endurance was not in itself a most precious lesson to mankind. That people can be found to ask such a question shows how far materialism has gone, and how needful it is that we insist upon the value of all that is nobler and higher in life.

An American statesman and jurist, speaking upon a similar occasion[5], gave utterance to the same sentiments in these words:

When the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence.

There have not been wanting, however, in every age, those whose zeal is in inverse proportion to their experience, who were possessed with the idea that it is the duty of the schools to make geometry practical. We have them to-day, and the world had them yesterday, and the future shall see them as active as ever.

These people do good to the world, and their labors should always be welcome, for out of the myriad of suggestions that they make a few have value, and these are helpful both to the mathematician and the artisan. Not infrequently they have contributed material that serves to make geometry somewhat more interesting, but it must be confessed that most of their work is merely the threshing of old straw, like the work of those who follow the will-o'-the-wisp of the circle squarers. The medieval astrologers wished to make geometry more practical, and so they carried to a considerable length the study of the star polygon, a figure that they could use in their profession. The cathedral builders, as their

art progressed, found that architectural drawings were more exact if made with a single opening of the compasses, and it is probable that their influence led to the development of this phase of geometry in the Middle Ages as a practical application of the science. Later, and about the beginning of the sixteenth century, the revival of art, and particularly the great development of painting, led to the practical application of geometry to the study of perspective and of those curves[6] that occur most frequently in the graphic arts. The sixteenth and seventeenth centuries witnessed the publication of a large number of treatises on practical geometry, usually relating to the measuring of distances and partly answering the purposes of our present trigonometry. Such were the well-known treatises of Belli (1569), Cataneo (1567), and Bartoli (1589).[7]

The period of two centuries from about 1600 to about 1800 was quite as much given to experiments in the creation of a practical geometry as is the present time, and it was no doubt as much by way of protest against this false idea of the subject as a desire to improve upon Euclid that led the great French mathematician, Legendre, to publish his geometry in 1794,—a work that soon replaced Euclid in the schools of America.