Hence, when C and D are distinct points, the anharmonic ratio of the parts into which C and D divide AB cannot be positive unity. 5. Let AC.BD AC AD BD And since C and D are now one external and one internal (2), they divide the segment AB in the same ratio internally and externally, disregarding sign. Such division of a line segment is called harmonic. (208°, Cor. 1) Harmonic division and harmonic ratio have been long employed, and from being only a special case of the more general ratio, this latter was named "anharmonic" by Chasles, "who was the first to perceive its utility and to apply it extensively in Geometry." 300°. Def.-When we consider AB and CD as being two segments of the same line we say that CD divides AB, and that AB divides CD. Now the anharmonic ratios in which CD divides AB are And the anharmonic ratios in which AB divides CD are But the anharmonic ratios of these sets are equal each to each in both sign and magnitude. .. the anharmonic ratios in which CD divides AB are the same as those in which AB divides CD. Or, any two segments of a common line divide each other equianharmonically. 301°. Four points A, B, C, D taken on a line determine six segments AB, AC, AD, BC, BD, and CD. These may be arranged in three groups of two each, so that in each group one segment may be considered as dividing the others, viz., AB, CD; BC, AD; CA, BD. Each group gives two anharmonic ratios, reciprocals of one another; and thus the anharmonic ratios determined by a range of four points, taken in all their possible relations, are six in number, of which three are reciprocals of the other three. These six ratios are not independent, for, besides the reciprocal relations mentioned, they are connected by three relations which enable us to find all of them when any one is given. Denote AC. BD by P, BA.CD by Q, CB. AD by R. Then P, Q, R are the anharmonic ratios of the groups ABCD, BCAD, and CABD, each taken in the same order. But in any range of four (233°) we have AB. CD+BC.AD+CA.BD=0. And dividing this expression by each of its terms in succession, we obtain Q + 1 = R + 1 = P + I. I P I I R From the symmetry of these relations we infer that any general properties belonging to one couple of anharmonic ratios, consisting of any ratio and its reciprocal, belong equally to all. Hence the properties of only one ratio need be studied. The symbolic expression {ABCD} denotes any one of the anharmonic ratios, and may be made to give all of them by reading the constituent letters in all possible orders. Except in the case of harmonic ratio, or in other special cases, we shall read the symbol in the one order of alternating the letters in the numerator and grouping the extremes and means in the denominator. Thus It is scarcely necessary to say that whatever order may be adopted in reading the symbol, the same order must be employed for each when comparing two symbols. 302°. Theorem.-Any two constituents of the anharmonic` symbol may be interchanged if the remaining two are interchanged also, without affecting the value of the symbol. Proof.- {ABCD}=AC.BD : AD. BC. Interchange any two as A and C, and also interchange the remaining two B and D. Then {CDAB} = CA. DB: CB. DA =AC.BD: AD. BC. Similarly it is proved that {ABCD} = {BADC} = {CDAB} = {DCBA}. q.e.d. 303°. If interchanging the first two letters, or the last two, without interchanging the remaining letters, does not alter the value of the ratio, it is harmonic. But the positive value must be rejected (299 ̊, 4), and the negative value gives the condition of harmonic division. 304°. Let ABCD be any range of four and O any point not on its axis. The anharmonic ratio of the pencil O. ABCD corresponding to any given ratio of the range is the same function of the sines of the angles as the given ratio is of the corresponding segments. sin AOC. sin BOD AC. BD sin AOD. sin BOC AD. BC; D corresponds to or, symbolically, O{ABCD} corresponds to {ABCD}. To prove that the corresponding anharmonic ratios of the range and pencil are equal. = АС_ДАОС_OA . OC sin AOC_OA sin AOC OB. OC sin BOC OB sin BOC and, with necessary formal variations, the anharmonic ratio of a range may be changed for that of the corresponding pencil, and vice versa, whenever required to be done. Cor. 1. Two angles with a common vertex divide each other equianharmonically. (300°) Cor. 2. If the anharmonic ratio of a pencil is + I, two rays coincide, and if - 1, the pencil is harmonic. (299°, 4, 5) Cor. 3. A given range determines an equianharmonic pencil at every vertex, and a given pencil determines an equianharmonic range on every transversal. Cor. 4. Since the sine of an angle is the same as the sine of its supplement (214°, 1), any ray may be rotated through a straight angle or reversed in direction without affecting the ratio. Corollaries 2, 3, and 4 are of special importance. 305. Theorem.-If three pairs of corresponding rays of two equianharmonic pencils intersect collinearly, the fourth pair intersect upon the line of collinearity. Proof.-Let O{ABCD}=O'{ABCD'}, and let the pairs of corresponding rays OA and O'A, OB and O'B, OC and O'C intersect in the three collinear points A, B, and C. Let the fourth corresponding rays meet the axis of ABC in D and D' respectively. Then R B {ABCD} = ABCD'}, (304°) which is possible only when D and D' coincide. .. the fourth intersection is upon the axis of A, B, and C, and the four intersections are collinear. q.e.d. Cor. If two of the corresponding rays as OC and O′′C become one line, these rays may be considered as intersecting at all points on this line, and however A and B are situated three corresponding pairs of rays necessarily intersect collinearly. .. when two equianharmonic pencils have a pair of corresponding rays in common, the remaining rays intersect collinearly. 306°. Theorem.—If two equianharmonic ranges have three pairs of corresponding points in perspective, the fourth points are in the same perspective. Proof {ABCD} = {A'B'C'D'}, and A and A', B and B', and C and C' are in perspective at O. Now O{ABCD} = 0 {A'B'C'D'}, and we have two equianharmonic pencils of which three pairs of correspond ing rays meet collinearly at A, B, and C. Therefore OD' and OD meet at D, or D and D' are in perspective at O. Cor. If two of the corresponding points, as C and C", become coincident, these two points are in perspective at every centre, and hence three corresponding pairs of points are necessarily in perspective. .. when two equianharmonic ranges have a pair of corresponding points coincident, the remaining pairs of corresponding points are in perspective. |