The Teaching of Geometry

by David Eugene Smith

a solid,"—definitions still in use and not without their value. For it must not be assumed that scientific priority is necessarily priority in fact; a child knows of "solid" before he knows of "point," so that it may be a very good way to explain, if not to define, by beginning with solid, passing thence to surface, thence to line, and thence to point.

The first definition of point of which Proclus could learn is attributed by him to the Pythagoreans, namely, "a monad having position," the early form of our present popular definition of a point as "position without magnitude." Plato defined it as "the beginning of a line," thus presupposing the definition of "line"; and, strangely enough, he anticipated by two thousand years Cavalieri, the Italian geometer, by speaking of points as "indivisible lines." To Aristotle, who protested against Plato's definitions, is due the definition of a point as "something indivisible but having position."

Euclid's definition is essentially that of Aristotle, and is followed by most modern textbook writers, except as to its omission of the reference to position. It has been criticized as being negative, "which has no part"; but it is generally admitted that a negative definition is admissible in the case of the most elementary concepts. For example, "blind" must be defined in terms of a negation.

At present not much attention is given to the definition of "point," since the term is not used as the basis of a proof, but every effort is made to have the concept clear. It is the custom to start from a small solid, conceive it to decrease in size, and think of the point as the limit to which it is approaching, using these terms in their usual sense without further explanation.

2. Line. A line is breadthless length. This is usually modified in modern textbooks by saying that "a line is that which has length without breadth or thickness," a statement that is better understood by beginners. Euclid's definition is thought to be due to Plato, and is only one of many definitions that have been suggested. The Pythagoreans having spoken of the point as a monad naturally were led to speak of the line as dyadic, or related to two. Proclus speaks of another definition, "magnitude in one dimension," and he gives an excellent illustration of line as "the edge of a shadow," thus making it real but not material. Aristotle speaks of a line as a magnitude "divisible in one way only," as contrasted with a surface which is divisible in two ways, and with a solid which is divisible in three ways. Proclus also gives another definition as the "flux of a point," which is sometimes rendered as the path of a moving point. Aristotle had suggested the idea when he wrote, "They say that a line by its motion produces a surface, and a point by its motion a line."

Euclid did not deem it necessary to attempt a classification of lines, contenting himself with defining only a straight line and a circle, and these are really the only lines needed in elementary geometry. His commentators, however, made the attempt. For example. Heron (first century A.D.) probably followed his definition of line by this classification:

		Straight
Lines				Circular circumferences
		Not straight		Spiral shaped
				Curved (generally)

Proclus relates that both Plato and Aristotle divided lines into "straight," "circular," and "a mixture of the two," a statement which is not quite exact, but which shows the origin of a classification not infrequently found in recent textbooks. Geminus (ca. 50 B.C.) is said by Proclus to have given two classifications, of which one will suffice for our purposes:

		Composite (broken line forming an angle)
Lines				Forming a figure, or determinate. (Circle, ellipse, cissoid.)
		Incomposite		Not forming a figure, or indeterminate and extending without a limit. (Straight line, parabola, hyperbola, conchoid.)

Of course his view of the cissoid, the curve represented by the equation y²(a + x) = (a - x)³, is not the modern view.