The Teaching of Geometry

by David Eugene Smith

2. Tangent. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.

Teachers who prefer to use "circumference" instead of "circle" for the line should notice how often such phrases as "cut the circle" and "intersecting circle" are used,—phrases that signify nothing unless "circle" is taken to mean the line. So Aristotle uses an expression meaning that the locus of a certain point is a circle, and he speaks of a circle as passing through "all the angles." Our word "touch" is from the Latin tangere, from which comes "tangent," and also "tag," an old touching game.

3. Tangent Circles. Circles are said to touch one another which, meeting one another, do not cut one another.

The definition has not been looked upon as entirely satisfactory, even aside from its unfortunate phraseology. It is not certain, for instance, whether Euclid meant that the circles could not cut at some other point than that of tangency. Furthermore, no distinction is made between external and internal contact, although both forms are used in the propositions. Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's.

4. Distance. In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn to them from the center are equal.

It is now customary to define "distance" from a point to a line as the length of the perpendicular from the point to the line, and to do this in Book I. In higher mathematics it is found that distance is not a satisfactory term to use, but the objections to it have no particular significance in elementary geometry.

5. Greater Distance. And that straight line is said to be at a greater distance on which the greater perpendicular falls.

Such a definition is not thought essential at the present time.

6. Segment. A segment of a circle is the figure contained by a straight line and the circumference of a circle.

The word "segment" is from the Latin root sect, meaning "cut." So we have "sector" (a cutter), "section" (a cut), "intersect," and so on. The word is not limited to a circle; we have long spoken of a spherical segment, and it is common to-day to speak of a line segment, to which some would apply a new name "sect." There is little confusion in the matter, however, for the context shows what kind of a segment is to be understood, so that the word "sect" is rather pedantic than important. It will be noticed that Euclid here uses "circumference" to mean "arc."