The Teaching of Geometry

by David Eugene Smith

7. Angle of a Segment. An angle of a segment is that contained by a straight line and a circumference of a circle.

This term has entirely dropped out of geometry, and few teachers would know what it meant if they should hear it used. Proclus called such angles "mixed."

8. Angle in a Segment. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.

Such an involved definition would not be usable to-day. Moreover, the words "circumference of the segment" would not be used.

9. And when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.

10. Sector. A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.

There is no reason for such an extended definition, our modern phraseology being both more exact (as seen in the above use of "circumference" for "arc") and more intelligible. The Greek word for "sector" is "knife" (tomeus), "sector" being the Latin translation. A sector is supposed to resemble a shoemaker's knife, and hence the significance of the term. Euclid followed this by a definition of similar sectors, a term now generally abandoned as unnecessary.

It will be noticed that Euclid did not use or define the word "polygon." He uses "rectilinear figure" instead. Polygon may be defined to be a bounding line, as a circle is now defined, or as the space inclosed by a broken line, or as a figure formed by a broken line, thus including both the limited plane and its boundary. It is not of any great consequence geometrically which of these ideas is adopted, so that the usual definition of a portion of a plane bounded by a broken line may be taken as sufficient for elementary purposes. It is proper to call attention, however, to the fact that we may have cross polygons of various types, and that the line that "bounds" the polygon must be continuous, as the definition states. That is, in the second of these figures the shaded portion is not considered a polygon. Such special cases are not liable to arise, but if questions relating to them are suggested, the teacher should be prepared to answer them. If suggested to a class, a note of this kind should come out only incidentally as a bit of interest, and should not occupy much time nor be unduly emphasized.

It may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure, and that the modern idea, which is met in higher mathematics, leads to a modification of earlier concepts. For example, here is a quadrilateral with one of its diagonals, BD, outside the figure. Furthermore, if we consider a quadrilateral as a figure formed by four intersecting lines, AC, CF, BE, and EA, it is apparent that this general quadrilateral has six vertices, A, B, C, D, E, F, and three diagonals, AD, BF, and CE. Such broader ideas of geometry form the basis of what is called modern elementary geometry.