Corol. 3. If two parallels, CL, NO, be cut by two diameters, EM, GI; the rectangles of the parts of the parallels, will be as the segments of the respective diameters. For theref, by equal. EK EM CK. KL NM. MO, Corol. 4. When the parallels come into the position of the tangent at r, their two extremities, or points in the curve unite in the point of contact P; and the rectangle of the parts becomes the square of the tangent, and the same properties still follow them. If two Parallels intersect any other two Parallels; the Rectangles of the Segments will be respectively Proportional. That is, CK. KL: DK. KE :: GI. IH: N1. 10. For, by cor. 3 theor. 23, PK: QICK. KLGI.IH; and by the same, PK: QI: DK: KE. NI. 10; theref. by equal. CK. KL: DK : KE :: GI. IH: N1. 10. Corol. When one of the pairs of intersecting lines comes into the position of their parallel tangents, meeting and limiting each other, the rectangles of their segments become the squares of their respective tangents. So that the constant ratio of the rectangles, is that of the square of their parallel tangents, namely, CK. KL: DK. KE :: tang. parallel to CL : tang. parallel to DE. THEOREM, XXI. if there be Three Tangents intersecting each other; their Segments will be in the same Proportion. That That is, GI IH :: CG: GD: : DH : HE. For, through the points , I, D. H, draw the diameters 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 H LMN CLLE LN or NE, the diameters GK, DM, HN, and also But the 3d terms LE = CLCK OF KL. as also the 4th terms LN, KM, NE. Therefore the first and second terms, in all the lines, are proportional, namely G1 1H:: CG GD: DH HE. Q. E, D, THEOREM XXII 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. EF EH that is EG2: EH2, theref. by Geom. th. 78, EF, EG, EH are proportionals, THEOREM XXIII. The Area or Space of a Parabola, is equal to Two-Thirds of its Circumscribing Parallelogram. That is, the space ABCGA = ABCD; or the space ADCGA = ABCD. 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, BC: EG:: AB: AE, : Hence it follows, that any one of these narrow parts, as IG, is = DC AD2 × A12; hence, AD and DC being given or constant quantities, it appears that the said parts IG, &c. are proportional to A12, &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 n.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, 2+ 1, become only n and 25, omitting the I as inconsiderable in respect of the infinite number n: hence the expression above becomes barely In.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 DC quently the sum of all the X A12's, is AD. DC= - AD2 BD, which is the area of the exterior part ADCGA. That is, the said exterior part ADCGA, is gram ABCD; and consequently the interior of the same parallelogram. of the parallelo part ABCGA is Q. E D, 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 between 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 31416, then 2c X FG is the circumference of the circle EFG whose radius is FG; therefore CX FG is the area of that circle. H B E T Ꭰ But, by cor. theor. 1, Parabola, / X AF = FG2, where f denotes the parameter of the parabola; consequently pc XAF 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 O 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 ADX 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 fc × AD2, or c X X AD2, is the sum of all the circular sections, or the content of the paraboloid. But, by the parabola, : DC :: DC : AD, or f = DC2 AD ; con sequently XxX AD2 becomes c X AD X DC for the solid content of the paraboloid. But cx AD X DC2 is equal, 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. But theref. AF2 = DF X (AD + AF), pcX DFX (AD + AF) = the frust. BEGC. But, by the parab. X AD = DC theref. eX DF X (DC2 + FG2) and X AF = 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; c = IH its conjugate; AK, any absciss, from the extremity of the diam. y DK the correspondent ordinate. Then, theor. 2, AB2: Hr2:: AK. KB: DK2, that is, de2 :: x (d—x): y", hence d2 y, c2 (dx-x2); or dye (dx-x2), 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, if ʼn denote the parameter, c2d 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. |