392 G E 0 ill But we have also (C‘DAB‘. = -1, so that A and B are harmonic conjugates with regard to C and 1). The principal property of harmonic points, upon which almost all applications depend, is this, that their cross-ratio remains un- altered if we interchange the two points belonging to one pair, viz. : (ABCD)=(ABDC)=-=(BACD). _ For four harmonic points the six cross-ratios which are generally ditferciit become two and two equal : K=—1, 1-K-2, ’:=§ 1 1 K-1 0 l;_—1’1-—x —" Hence if we get four points whose cross-ratio is 2 or <5, then they are harmonic, but not arranged so that conyugates are paired. lt this is the case the cross-ratio= -1. ,5 19. If we equate any two of the above six values of the cross- ratios, we get either K=1, 0, oo, or K= — 1, 2, ='_., or else It becomes a root of the equation K2— K + 1 = O, that is, an imaginary cube root of — 1. In this case the six values become three aml three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are all-iniportant. § '20. From the definition of harmonic points, and by aid of §11, the following properties are easily deduced. If C andD are harmonic conjugates with regard to A and B, then one of thcm lies in, the other without AB; it is impossible to move from A to B without passing either through C‘ or through D ; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and I} are “separated” by U and D. For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B. If A and B are difl'erent points, and if C coincides with A or B, D does. But if A and B coineide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, “if of four harmonic conjugates two 4-oincidc, then a third coincides with them, and the fourth may be an 3/ point in the line.” If C is the middle point "between A and B, then D is the point at infinity for AC: CB= +1, hence AD :l)B must be equal to ~1. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the mirlillc point of AB. This important property gives a first example how metric pro- perties are connected with projective ones. § 21. Complete QuarIrilateral.—A figure formed by four lines in a plane is called a complete quadrilateral or, shorter, a foiir-side (fig. 7). The figure has six vcrticcs, that is, points where the sides meet, and three diagonals AB, EF, GH, which join opposite vertices. Similarly a figure formed by four points in a plane is called a four-point. It has six sides, which join the points, or vertices, and three diagonal points, where the sides meet. 'l‘lic three diagonals of a four-side out each other harmonically. If we project the points E, l", K, D from II to AB, we get (I-IFK D) = (ABCD), and if we project them from G, we get (EFKD)=(BACD); (ABCD)=(BACD), that is, the cross-ratio (ABCD) is equal to its reciprocal, hence = -1, as the four points are all ditlercnt. This gives the im- portant theorem :— Tm-:om~:.{. —In every four-side any diagonal is cut harmonically by the oth.er two. This allows the solution of the problem :— Pn0BLE.i.—To construct the harmonic conjugate D to a point C with regard to two gircn points A and B. Solution.—-Through A draw any two lines, and through C one cutting the former two in G and ll. Join these points to B, cutting the former two lines in E and F. The point I) where El" cuts AB will be the harmonic conjugate required. This remarkable construction requires nothing but the drawing of lines, and is tlicrt-fore. independent of measiircineiit. It follows, also, that all four-siiles u'hieh. hare tu'u 'rr:'rticr's at A and B, and Fig. 7- so that 1'] T R Y one diagonal passing th rough C‘, will each hate the third cliugonut‘ passing through ll. § 22. The theory of cross-ratios may be extended from points in a row to lines in a Hat pencil, and to planes in an axial pencil. We have seen (§ 13) that it" the lines which join four points A, ’._, C. l) to any point S be cut by any other line in A’. B’, C’, 1)’, then (_.-l’.'L‘|l) =(A'B'C’l)'). In other words, four lines in a fiat pencil are cut by every other line in four points whose (-ross-i'atio is constant. .D¢;finition.—By the cross-ratio of four rays in a Hat pencil is meant the cross-ratio of the four points in which the rays are cut by any line. If a, b, c, d be the lines, then this cross-ratio is denoted by (abcd). Dr_/z‘n.ition.—By the cross-ratio of four planes in an axial pencil is understood the cross-ratio of the four points in which any line cuts the planes, or, what is the saiue thing, the ('i'0.~s-l‘:ilio oi" the four rays in which any plane cuts the four planes. In order that this definition may have a meaning, it has to be proved that all lines cut the pencil in points which have the same cross-ratio. This is seen at once for two intersecting lines, as tl_. ir plaiie cuts the axial pencil in a Hat pencil, which is itself cut l-_v the two lines. The cross-ratio of the four points on one line is tllL‘l‘(‘f0l‘C} Cqillfll to thiit on the other, and equal to that of the foiir rays in t ie t at penci . If two non-intersecting lines p and q cut the four planes in A, 1B, C, ll alnd A’, B’, C’, D’, di'..i' a line 7' to meet both 1: and :1, ant let this ine cut the ilancs in A”,B",(,"’,l)”. Then All('l)) = (A’B’C'l)’), for each is eqiial to (A"B"L"'l)”). ( § 23. ’e may now also extend the notion of harmonic elements, viz. : Dc_/inz'tion.—l-'oiii' rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their cross-ratio equals — 1, that is, if they are cut by a line in four harmonic points. Harmonic pencils are constructed by aid of the theorem in § 21. which may now be stated thus :— In a four-sitlc tu'o stiles are harmonic conjugates ’lI't[]l- -rrgaril In the diagonal through their intersection and the line from this point to the point 'u'hrrc the other diagonals meet. Or thus: In a four- point the lines joining one diagonal point to the other /tI'0 are har- nionic con_ju_r/ates irilh rega‘r(l to the sides passing through the _/irs/. If we iinderstaiid by a “ iiic(lian line" of a triaii_-_;_le a line which joins a vertex to the middle point of the opposite side, and by a “median line” of a parallelogram a line joining middle points r~i' opposite sides, we get as special cases of the last theorem :— The diagonals anrl nzediaa lines of a parallelogram ‘own an harmonic pencil ; aml J /It a 'rcrtc:c of any lri'anglr', the two sirles, the median l inc, and tlu linc parallel to the base form an harmonic pencil. Takting the parallelogram a rectangle, or the triangle isosceles, we we :— TtllEORE.‘[.—-/l)l_7_[ two lines and the bisections Q/'lheirungl¢'s_fi:/‘m an harmo1'cpe1u'il. Or :— In an harmonic pencil, if two conjugate rays are perpendicular, [i~i:oJLc'riv i-;. then the other two are equally inclined to them ; and, Coni'crsel_v, if one ray bisects the angle between conjugate rags, it is prrpc-niliculur to its eonjztgatc. This connects perpendiciilarity and bisection of angles with projective properties. §24. We add a few theorems and problems which are easily proved or solved by aid of harmonics. An harmoiiic pencil is cut by a line parallel to one of its rays in three equidistant points. Through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point. Having given two parallel lines, to bisect on either any given segment without using a pair of compasses. llaving given in a line a segment and its middle point, to draw through any given point in the plane a line parallel to the given line. To (lrawa line which joins a given point to the intersection of two given lines which meet off the drawing paper (by aid of § 21). COItRESI‘0.'DE.'Cl~}. §25. Two rows, p and p’, which are one the projection of the other (as in fig. 5), stand in a definite relation to each other, characterized by the following properties. 1. To each point in either corresponrls one point in the other; that is, those points are said to correspond which are projections of one another. 2. The cross-ratio of any four points in one equals that of the co'rrespr.mling points in thc olher. 3. The lines joining corrrsponrling points all pass tln'ou_r/h the same point. If we suppose eorrcspoinliiig points marked, and the rows brought into any other position, then the lines joining corresponding points will no longer meet in a common point, and hence the thii'il_ of
the above properties will not hold any longer ; but we have still a