COROLLARY. If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn.

These two propositions and the corollary are all parts of one general proposition: If through a point a line is drawn cutting a circle, the product of the segments of the line is constant.

Ꮲ

If P is within the circle, then xx'= yy; if P is on the circle, then x and y become 0, and 0 ⋅ x'= 0 ⋅ y' = 0; if P is at P, then x and y, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if P is at P4, then y = y', and we have xx' y2, or x : y = y: x', as stated in the proposition.

We thus have an excellent example of the Principle of Continuity,

y

P

P

y/x

and classes are always interested to consider the result of letting P assume various positions. Among the pos sible cases is the one of two tangents from an external point, and the one where P is at the center of the circle.

Students should frequently be questioned as to the meaning of "product of lines." The Greeks always used "rectangle of lines," but it is entirely legitimate to speak of " product of lines," provided we define the expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter.

THEOREM. The square on the bisector of an angle of a triangle is equal to the product of the sides of this angle

diminished by the product of the segments made by the bisector upon the third side of the triangle.

This proposition enables us to compute the length of a `bisector of a triangle if the lengths of the sides are known.

THEOREM. In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side.

This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circumscribed circle in terms of the three sides.

For if we designate the sides by a, b, and c, as usual, and let CD d and PB = x, then

=

This is not available at this time, however, because the Pythagorean Theorem has not been proved.

These two propositions are merely special cases of the following general theorem, which may be given as an interesting exercise:

If ABC is an inscribed triangle, and through C there are drawn two straight lines CD, meeting AB in D, and CP, meeting the circle in P, with angles ACD and PCB equal, then AC BC will equal CD x CP.

Fig. 1 is the general case where D falls between A and B. If CP is a diameter, it reduces to the second figure given on page 249. If CP bisects ACB, we have Fig. 3, from which may be proved the proposition given at the foot of page 248. If D lies on BA produced, we have Fig. 2. If D lies on AB produced, we have Fig. 4. This general proposition is proved by showing that ADC and PBC are similar, exactly as in the second proposition given on page 249.

These theorems are usually followed by problems of construction, of which only one has great interest, namely, To divide a given line in extreme and mean ratio.

The purpose of this problem is to prepare for the construction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called "the golden section," and is probably "the section " mentioned by Proclus when he says that Eudoxus 'greatly added to the number of the theorems which

66

Plato originated regarding the section." The expression "golden section" is not old, however, and its origin is uncertain.

If a line AB is divided in golden section at P, we have
AB × PB = AP2.

about 0.6 a, and a

about 0.4 a, and a is

That is, x = therefore divided in about the ratio of 2:3.

There has been a great deal written upon the æsthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern.

CHAPTER XVII

THE LEADING PROPOSITIONS OF BOOK IV

Book IV treats of the area of polygons, and offers a large number of practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made. in this chapter to suggest practical problems to the teachers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important.

THEOREM. Two rectangles having equal altitudes are to each other as their bases.

Euclid's statement (Book VI, Proposition 1) was as follows: Triangles and parallelograms which are under the same height are to one another as their bases. Our plan of treating the two figures separately is manifestly better from the educational standpoint.

In the modern treatment by limits the proof is divided into two parts: first, for commensurable bases; and second, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geometry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important