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 x BC will equal CD x CP._ [Illustration: FIG. 1] [Illustration: FIG. 2] [Illustration: FIG. 3] [Illustration: FIG. 4] 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 [L]_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 [triangles]_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 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_ x _PB_ = (_AP_)^2. Therefore, if _AB_ = _a_, and _AP_ = _x_, we have _a_(_a_ - _x_) = _x_^2, or _x_^2 + _ax_ - _a_^2 = 0; whence _x_ = - _a_/2 +- _a_/2[sqrt]5 = _a_(1.118 - 0.5) = 0.618_a_, the other root representing the external point. That is, _x_ = about 0.6_a_, and _a_ - _x_ = about 0.4_a_, and _a_ is therefore divided in about the ratio of 2 : 3. There has been a great deal written upon the aesthetic 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. FOOTNOTES: [76] For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited. [77] These two and several which follow are from Stark, loc. cit. [78] The author has a beautiful ivory specimen of the Sixteenth century. CHAPTER XVII THE LEADING PROPOSITIONS OF BOOK IV