That is, GI: IH:: CG: GD :: DH: HE. For, through the points G, I, D, H, draw the diame ters GK, IL, DM, HN; as also the lines CI, EI, which are double ordinates to the diameters GK, HN, by cor. 1 theor. 16; therefore the diameters GK, DM, HN, bisect the lines CL, CE, LE; LMN hence км= CM CELE CL=CK or KL. But, by parallels, GI IH KL : LN, and also But the 3d terms as also the 4th terms second terms, in all the lines, are IH: CG: GD :: DH: HE. Therefore the first and proportional, namely GI THEOREM XXII. Q. E. D. If a Rectangle be described about a Parabola, having the same Base and Altitude; and a diagonal Line be drawn from the Vertex to the Extremity of the Base of the Parabola, forming a right-angled Triangle, of the same Base and Altitude also; then any Line or Ordinate drawn across the three Figures, perpendicular to the Axis, will be cut in Continual Proportion by the Sides of those Figures. ог that is theref. by Geom. th. 78, EF, EG, EH are proportionals, Q. E. D. THEOREM EF EH EG2: EH2, EF EG EG: EH. THEOREM XXIII. The Area or Space of a Parabola, is equal to Two-Thirds of its Circumscribing Parallelogram. FOR, conceive the space ADCGA to be composed of, or divided into, indefinitely small parts, by lines parallel to DC or AB, such as IG; which divide AD into like small and equal parts, the number or sum of which is expressed by the line AD. Then, by the parabola, that is, BC2: EG2 :: AB: AE, Hence it follows, that any one of these narrow parts, as 1G, is = DC AD2 X AI2; hence, AD and DC being given or constant quantities, it appears that the said parts 1G, &c. are proportional to A1, &c. or proportional to a series of square numbers, whose roots are in arithmetical progression, and the DC area ADCGA equal to drawn into the sum of such a series AD2 of arithmeticals, the number of which is expressed by AD. Now, by the remark at pag. 227 this vol. the sum of the squares of such a series of arithmeticals, is expressed by fn. n+1. 2n+1, where n denotes the number of them. In the present case, n represents an infinite number, and then the two factors n + 1, 2n + 1, become only n and 2n, omitting the 1 as inconsiderable in respect of the infinite number n: hence the expression above becomes barely Jn. n. 2n = }n3. To apply this to the case above: n will denote AD or BC; and the sum of all the A12's becomes AD3 or BC3; conse DCBD, which is the area of the exterior part ADCGA. That is, the said exterior part ADCGA, is of the parallelogram ABCD; and consequently the interior part ABCGA is { of the same parallelogram. Q. E. D. Corel. Corol. The part AFCGA, inclosed between the curve and the right line AFC, is of the same parallelogram, being the difference between ABCGA and the triangle ABCFA, that is bẹtween and of the parallelogram. THEOREM XXIV. The Solid Content of a Paraboloid (or Solid generated by the Rotation of a Parabola about its Axis), is equal to Half its Circumscribing Cylinder. LET ABC be a paraboloid, generated by the rotation of the parabola ac about its axis AD. Suppose the axis AD be divided into an infinite number of equal parts, through which let circular planes pass, as EFG, all those circles making up the whole solid paraboloid. Now if c = the number 3.1416, then 2c X FG is the circumference of the circle EFG whose radius is FG ; therefore cX FG2 is the area of that circle. H A But, by cor. theor. 1, Parabola p X AF = FG3, where P denotes the parameter of the parabola; consequently pc X AF will also express the same circular section EG, and therefore pc x the sum of all the AF's will be the sum of all those circular sections, or the whole content of the solid paraboloid. But all the AF's form an arithmetical progression, beginning at 0 or nothing, and having the greatest term and the sum of all the terms each expressed by the whole axis AD. And since the sum of all the terms of such a progression, is equal to AD X AD or AD2 half the product of the greatest term and the number of terms; therefore AD2 is equal to the sum of all the AF's, and consequently pc X AD3, or c XpX AD2, is the sum of all the circular sections, or the content of the paraboliod. But, by the parabola, p: DC :: DC: AD, X DC2 for the sequentlyc XpX AD2 becomes c X AD solid content of the paraboloid. But c X AD to the cylinder BCIH; consequently the paraboloid is the half of its circumscribing cylinder. Q. E. D. THEOREM THEOREM XXV. The Solidity of the Frustum BEGC of the Paraboloid, is equal to a Cylinder whose Height is DF, and its Base Half the Sum of the two Circular Bases EG, BC. FOR, by the last theor. and, by the same, theref. the diff. But theref AD2 pc X AD2 = the solid ABC, pc X AF2 = the solid AEG, pc X (AD2-AF2)=the frust. BEGC, AF2 = DF X(AD + AF), pc X DFX (AD + AF) = the frust BEGC. But by the parab. p X AD DC2. and p X AF = FG2; theref.cXDF X (DC2 + FG2) = the frust BEGC. Q. E. D. ON THE CONIC SECTIONS AS EXPRESSED BY ALGEBRAIC EQUATIONS, CALLED THE EQUATIONS OF THE CURVE. Let d denote AB, the transverse, or any diameter; x = AK, any absciss, from the extremity of the diam. Then, theor. 2, AB2: HI? :: AK. KB: DK3, that is, de c2: x (d-x): y, bence d2 y2 = c2 (dx—x3), or dy = c(dx-xa), the equation of the curve. And from these equations, any one of the four letters or quantities, d, c. x, y, may easily be found, by the reduction of equations, when the other three are given. Or, ifp denote the parameter, = c d by its definition ; then, by cor. th. 2, d: p: : x (d—x): y2, or dy2=p(dx—x2), which is another form of the equation of the curve. Otherwise. Otherwise. Or, if d = AC the semiaxis; c = CH the semiconjugate ; pcd the semiparameter; x = ck the absciss counted from the centre; and y = DK the ordinate as before = Then is AK d-x, and Kв=d+x, and AK. KB = (d-x) x (d + r) = d2 r2. Then, by th. 2, d2: c2:: d2x2: y3, and d2 y2 = c2 (d2 — x2), or dy = c √ (d2 — x2), the equation of the curve. Or, dp d2 x2 y2, and dy2 = = p(d2x2), another form of the equation to the curve; from which any one of the quantities may be found, when the rest are given. 2. For the Hyperbola. Because the general property of the opposite hyperbolas, with respect to their abscisses and ordinates, is the same as that of the ellipse, therefore the process here is the very same as in the former case for the ellipse; and the equation to the curve must come out the same also, with sometimes only the change of the sign of a letter or term from + to, or from -to +, because here the abscisses lie beyond or without the transverse diameter, whereas they lie between or upon them in the ellipse. Thus, making the same notation for the whole diameter, conjugate, absciss, and ordinate, as at first in the ellipse; then, the one absciss AK being x, the other BK will be d+x, which in the ellipse was d-x; so the sign of x must be changed in the general property and equation, by which it becomes d2: c2:: r (d+x): y2; hence day2 = c2 (dx + x2) and dy = c√ (dx + x2), the equation of the curve. Or, using p the parameter as before, it is, d: p :: x (d+x): y3, or dy2 = p (dx + x2), another form of the equation to the curve. 1= Otherwise, by using the same letters d, c. p, for the halves of the diameters and parameter, and for the absciss CK counted from the centre; then is AK = x--d, and вK = x+d, and the property d2 c2: (x−d) × (x + d) : y2, gives d2 y2 = c2 (x2 - d2), or dy=c(x2-d2), where the signs of d2 and 2 are changed from what they were in the ellipse. Or again, using the semiparameter, dp :: x2 d2 : y3, and dy2 = P(x2d2) the equation of the curve. But for the conjngate hyperbola, as in the figure to theorem 3, the signs of both 2 and d2 will be positive; for the property in that theorem being CA2: ca2:: CD2 + CA2 : De2, it |