Corol. 1. In like manner, if any other line p''q', paralles to cr or to pq, meet PHQ; since the rectangles PH'a, р'H'q′ are also in the same ratio of CR2 to cr2; therefore the rect. PHQ: рHq :: PH'Q : p'H'q. Also, if another line P'ha' be drawn parallel to PQ or CR; because the rectangles p'ho', p'hq' are still in the same ratio, therefore, in general, the rectangle PHQ: pHq :: P'ho' : p'hq'. That is, the rectangles of the parts of two parallel lines, are to one another, as the rectangles of the parts of two other parallel lines, any where intersecting the former. Corol. 2. And when any of the lines only touch the curve, instead of cutting it, the rectangles of such become squares, and the general property still attends them. If a Line be drawn through any Point of the Curves, Parallel to either of the Axes, and terminated at the Asymptotes; the Rectangle of its Segments, measured from that Point, will be equal to the Square of the Semi-axis to which it is.. parallel. For, draw AL parallel to ca, and aL to CA. by the parallels, and, by theor. 2, theref. by subtr. Then CA2: ca2 or AL2:: CD2: DH2; CA2: ca2:: CA2: DH2 DE2 or HEK. But the antecedents CA2, CA2 are equal, In like manner it is again, by the parallels, and, by theor 3, theref. by subtr. CA2: ca2 or AL2 :: CD2: DH2; CA2: ca2:: CA2 : De2 - DH2 or HeK. But the antecedents CA2, CA2 are the same, theref. the conseq. ca2, HeK must be equal. In like manner, by changing the axes, is hek or hek = CA2. Corol. 1. Because the rect. HEK = the rect. HеK. therefore EH: CH: eK : EK. And consequently HE is always greater than He. SCHOLIUM. It is evident that this proposition is general for any line oblique to the axis also, namely, that the rectangle of the segments of any line, cut by the curve, and terminated by the asymptotes, is equal to the square of the semi-diameter to which the line is parallel. Since the demonstration is drawn from properties that are common to all diameters. THEOREM XXVI (24). All the Rectangles are equal which are made of the Segments of any Parallel Lines cut by the Curve, and limited by the Asymptotes. For, each of the rectangles HEK or Hek is equal to the square of the parallel semi-diameter cs; and each of the rectangles hɛk or hek is equal to the square of the parallel semidiameter cr. And therefore the rectangles of the segments of all parallel lines are equal to one another. CI. Q. E. D. Corol. Corol. 1. The rectangle HEK being constantly the same, whether the point E is taken on the one side or the other of the point of contact 1 of the tangent parallel to HK, it follows that the parts HE, KE, of any line HK, are equal. And because the rectangle Heк is constant, whether the point e is taken in the one or the other of the opposite hyperbolas, it follows, that the parts He, Ke, are also equal. Corol. 2. And when HK comes into the position of the tangent DIL, the last corollary becomes IL ID, and IMIN, and LM DN. Hence also the diameter CIR bisects all the parallels to DL which are terminated by the asymptote, namely RH = RK. Corol. 3. From the proposition, and the last corollary, it follows that the constant rectangle HEK or EHE is = IL2. And the equal constant rect. Hek or eнe MLN or IM2 — IL2. Corol. 4. And hence IL = the parallel semi-diameter cs, For, the rect. EHE = IL2, IL CS. And so the asymptotes pass through the opposite angles of all the inscribed parallelograms. THEOREM XXVII (25). The Rectangle of any two Lines drawn from any Point in the Curve, Parallel to two given Lines, and Limited by the Asymptotes, is a Constant Quantity. That is, if AP, EG, DI be parallels, as also AQ, EK, DM parallels, then shall the rect. PAQ rect. GEK = rect. IDM. G K 72 For, For, produce KE, MD to the other asymptote at H, L. but theref. the rectangle : EK : EK DM: DM;" HEK GEK: LDM: IDM. But, by the last theor. the rect. HEK = LDM; and therefore the rect. GEK = IDM = PAQ THEOREM XXVIII (27). Q. E. D. Every Inscribed Triangle, formed by any Tangent and the two Intercepted Parts of the Asymptotes, is equal to a Constant Quantity; namely Double the Inscribed Parallelogram. G T That is, the triangle CTS 2 paral. GK. For, since the tangent Ts is bisected by the point of contact E, and EK is parallel to TC, and GE to CK; therefore CK, KS, GE are all equal, as are also CG, GT, KE. Consequently the triangle GTE = the triangle KES, and each equal to half the constant inscribed parallelogram GK. And therefore the whole triangle CTs, which is composed of the two smaller triangles and the parallelogram, is equal to double the constant inscribed parallelogram GK. Q. E. D. THEOREM XXIX (29). K If from the Point of Contact of any Tangent, and the two Intersections of the Curve with a Line parallel to the Tangent, three parallel Lines be drawn in any Direction, and terminated by either Asymptote; those three Lines shall be in continued Proportion. For, by the parallels, EI: IL :: DH: HM; and, by the same, theref. by compos. EI IL GK: KM; EI: IL:: DH, GK: HMK; but, by theor. 26, the rect. HMK = IL2; M and theref. the rect. or DH. GK = EI2, THEOREM XXX (30). Draw the semi-diameters CH, CIN, CK; For, because HK and all its parallels are bisected by CIN, therefore the triangle CNH = tri. CNK, and the segment INH seg. INK; consequently the sector CIH = sec. CIK. Corol. If the geometricals DH, EI, GK be parallel to the other asymptote, the spaces DHIE, EIKG will be equal; for they are equal to the equal sectors CHI, CIK. So that by taking any geometricals CD, CE, CG, &c, and drawing DH, EI, GK, &c, parallel to the other asymptote, as also the radii CH, CI, CK; then the sectors CHI, CIK, &c, or the spaces DHIE, EIKG, &c, or the spaces DHIE, DHKG, &c, will be in arithmetical progression. And therefore these sectors, or spaces, will be analogous to the logarithms of the lines or bases CD, CE, CG, &c; namely CHI or DHIE the log. of the ratio of CD to CE, or of CE to CG, &c; or of EI to DH, or of GK to EI, &c; and CHK or DHKG the log. of the ratio of CD to CG, &c, or of GK to DH, &c. SECTION III. OF THE PARABOLA. THEOREM XX (7). If an Ordinate be drawn to the Point of Contact of any Tangent, and another Ordinate produced to cut the Tangent; It will be, as the Difference of the Ordinates: Is to the Difference added to the external Part: : So is Double the first Ordinate: To the Sum of the Ordinates. |