Geometry

Typically one learns arithmetic and algebra by experiment or by being told how to do it. Geometry, however, is taught logically. Its ideas are established by means of "proof." One starts with definitions, postulates, and primitive terms; then proves his or her way through the course.

The reason for this goes back to the forenamed Greeks, and in particular to Euclid. Twenty-three hundred years ago he wrote a beautiful book called the Elements. This book contains no exercises, no experiments, no applications, no questions—just proofs, the proof of one proposition after another.

For centuries the Elements was the basic text in geometry. Heath, in his 1925 translation of the Elements, quotes De Morgan: "There never has been...a system of geometry worthy of the name, which has any material departures...from the plan laid down by Euclid." Nowadays the Elements has been replaced with texts which do have exercises, problems, and applications, but the emphasis on proof remains. Even the most obvious fact, such as the fact that the opposite sides of a parallelogram are equal, is supposed to go unnoticed, or at least unused, until it has been proved. Whether or not this makes sense, the reader will have to decide for himself or herself, but sensible or not, proof is and will probably continue to be a dominant component of a course in geometry.

Proofs can vary in formality. They can be as formal as the two-column proofs used in text-books in which each statement is identified as an assumption, a definition, or the consequence of a previously proved property; they can be informal with much left for the reader to fill in; or they can be almost devoid of explanation, as in the ingenious proof of the Pythagorean theorem given by the Hindu mathematician Bhaskara in the twelfth century. His proof consisted of a single word "behold" and a drawing.