Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/425

Rh GEOM 1 1 ] 1 (_1/—m’x) — (E+—;4)(_i/—7na:)+.‘2m,’-—2m=0, which by what precedes is the equation of a line through the point . . . . 1 1 - - Q. Substitutiiig herein for A + 1-, — + —— their foregoing values. ‘ .r 2-2 (133 :1‘, PL.-.'E A.’ALYTICAI..j the equation becomes — (A —l~ l‘mi)(y — -m’.c) + (A + Bm’)(y — ma‘) +-m’ — m= O ; that is, (m — 7n’)(A:v+ By+ C) -= O; or finally it is ..::+l3y+C=O, showing that the point Q lies in a Iiiie the position of which is independent of the particular lines H..’, OIBB’ used in the construction. It is proper to notice that there is no correspondence to each other of the points A, A’ and I}, ii’; the grouping might as well have been A, A’ and P»',B ; and it thence appears that the line A)‘.-i—B_l/+C=0 just obtained is in fact the line joining the point Q i'itli the point It ivliicli is the iiiteisection of AB and A’l5’. 10. The equation .A.c+ B3/+ C = O of a line contains in appearance 3, but really only 2 constants (for one of the constants can be divided out), and a line depends accord- ingly upon :3 paraineters, or can be made to satisfy 2 condi- tions. Siinilarly, the equation (a, I), c, f, _r/, }{Q}r, 3/, 1)2=0 of a conic contains really 5 constants, and the equation (-.=)(..-, 3/, 1)“ = 0 of a cubic contains really 9 constants. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely deterinined. There is, liow- ever, a reinarkable exception. Considering two given cubic curves 8 :0, S’ = 0, these intersect in 9 points, and through these 9 points we have the whole series of ciibics S —l.-."=0, where in is an arbitrary constant : I: may be (lt‘t'..‘I‘ll)lIlL'.‘.l so that the cubic shall pass through a given tenth point (/.'=S0+S'O, if the coordinates are (re, 3/”), and S0, 8'0 denote the correspondiiig values of S, The resulting curve F-S'(,— S'b'0=0 may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (I0, 3/0); and from th- equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection. The applications of this theorem are very numerous; for instance, we derive from it Pascal’s theorem of the inscribed liu-xa_«_;oii. Consider a hexagon inscribed in a eoiiie. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the l’-asealiaii points, the conic and this line con- stitute a cubic passing through 8 of the 9 points of inter- section, and it therefore passes through the .remaining point of iiitersection—tliat is, the third Pascaliaii point; and sin ‘e obviously this does not lie on the conic, it must li -. on the liiie——tliat is, we have the theorem that the three l’a.s-e:ili:iii points (or points of intersection of the pairs of opposite sides) lie on a line. J[tfrz'rrtl T/icor_//. 11. The foundation of the metrical theory consists in the simple the Jl'Ull1 that if a finite line PQ (fig. 10) be projected upon any other line 00' by lines perpendicular to 00', then the length of the projection P13’ is equal to the la-iigth of PQ, into the cosine of its inclination to P7’; oi-, whit is the same thing, that the pei-peiidicular dis- tallce l"Q' of any two parallel lines is equal to the inclined distance 1’Q into the cosine of the inclination. It at once f-illo ‘s that the algebraical sum of the projections of the sides of a closed polygon upon any line is = 0; or, re- versing the signs of certain sides, and coiisideriiig the ETRY 411 polygon as Consistillg Of two broken lines, each extend- ing from the same initial to the same terminal point, the sum of the projections of the lines of the first set upon any line is equal to the sum of the projections of the lines of the second set. Observe that if any line be perpen- dicular to the line on which the projection is made, then its projection is = 0. Thus, if we have a right- _ angled triangle PQlt (fig. 11), F1{:‘'- 10- where QII, 1-} l’, QP arc={, -:7, p respectively, and whereof the base- 0 .7‘ Fig. 11. angle is=a, then projecting successively on the three sides, We have {=pc%a,n=p§nmp={Cwa+n§na; and we thence obtain f=§+f;m§w%m%=L And again, by projecting on a line Qml, inclined at the angle a’ to QR, we have p cos (a—a')=§ eosa'+-:7 sin a’; and by substituting for g, -,7 their foregoing values, cos (a— a’) =eos a cos a'+ sin a sin a’. It is to be remarked that, assuming only the theory of similar triangles, we have herein a proof of Euclid, Book I., Prop 47 ; in fact, the same as is given Book 'I., Prop. 31 ; and also a proof of the trigoiioinetrical formula for cos (a. — a.’). The formulzc for cos(a.+ 0.’) and sin (a :1: a’) could be obtained in the same manner. Draw PT at right angles to Q.i.',, and suppose QT‘, TP=§1, -:71 re- spcctively, so that we have now the quadrilateral (_llIl’TQ, or, what is the same thing, the two broken lines Qlll’ and (3'l'l’, each exteiid- ing from Q to 1’. Projectiiig on the four sides successively, we have 3 = 5, cos a’ — -:7, sin a’, 17 = 3, sin a'+'n, cos a’, g,= g cos a’+-q sin a’, -q,= —-5 sin a'+'i7 cos a’, where the third equation is that previously written p cos (a- a')={ e0sa+n sin a. E«1ua1‘z'ons of I21}/lit Line and Circle.—Ti'((22s)‘?ii'iizaﬂoiz of C0o)'(l1'im1‘cs. 12. The required formula) are really contained in the foregoing results. For, in ﬁg. 11, supposing that the axis of :2‘ is parallel to QR, and taking (1, b for the co- ordinates of Q, and (.r, 3/) for those of P, then we have 5, 7;=.c — u, 3/ — Ii respectively ; and therefore a'—a=p eosa, 3/—b=p sin a, P‘ =(-v-a)"'+(y—br. Writing the first two of these in the form x—a y—b ('3: = T( =P): os a sin a we may regard Q as a ﬁxed point, but P as a point moving in the direction (3 to 1‘, so that a reniaiiis constant, and then, oni_itting the equation (=p), we have a relation between the coordinates 3:, 3/ of the point I’ thus inoviiig in a right line_,—tliat is, we have the equation of the line through the given point ((1, 1)) at a given