Ex. 4. The tranfverfe diameter is 3, the conjugate 1, and the ordinate, required the abfciffas. PROBLEM V. TO Anf. and 20. The conjugate, ordinate, and abscissa being given, to find the tranfverfe. RULE. Find the fquare root of the difference of the fquares of the femi-conjugate and ordinate, and, according as the greater or lefs obfciffa is propofed, add this root to or fubtract it from the femi-conjugate. Then use the following proportion As the fquare of the ordinate Is to the product of the conjugate and abfciffa, So is the fum or difference, found as above, To the tranfverfe. EXAMPLE. I. The conjugate CD is 20, the ordinate Fx is 8, and the abscissa Ax 12, required the tranfverse AB. 102=100 82= 64 36(6 root of the differences of the fquare of the fe 36 10 femi-conjugate. 16 fum. As 64 12X 20 :: 16 64: 240 :: 16 16 64)3840(60 the tranfverfe. [mi conj. and ord. 384 Ex. 2. The conjugate 30, the ordinate 12, and the abfciffa 18, required the tranfverfe. Anf. 90. Ex. 3. The conjugate 35, the ordinate 14, and the abfciffa 84, required the tranfverfe. 4 Anf. 105. Ex. 4. The conjugate 1, the abfciffa 20, and the ordinate , required the transverse. Anf. 3. PROBLEM VI. The tranfverfe, ordinate, and abfciffa being given, to find the con jugate. RULE. As the fquare root of the products of the two abfciffas Is to the ordinate, So is the tranfverfe To the conjugate. EXAMPLE I. The tranfverfe AB is 60, the ordinate Fx 8, and the abfciffa 12, required the conjugate. Ex. 2. The transverse 90, the ordinate 12, and the abscissa 18, required the conjugate diameter. Anf. 30. Ex. 3. The tranfverfe 105, the ordinate 14, and the abfciffa 84, it is required to find the conjugate. Anf. 35 Anf. 1. Ev, 4. The tranfverfe 3, the ordinate, and the abscissa 2, required the conjugate. PROBLEM VII. The abfciffa, tranfverfe, and conjugate being given, to find the or dinate. RULE. As the tranfverfe Is to the conjugate, So is the fquare root of the product of the two abfciffas EXAMPLE I, The tranfverfe AB 60, the abfciffas 12, 48, and the conju gate 20, required the ordinate. Ex. 2. The tranfverfe 90, the abfciffa 18, and the conjugate 30, required the ordinate. Anf. 12. Ex. 3. The tranfverfe 105, the abfciffa 84, and the conjugate 35, required the ordinate. Anf. 14. Ex. 4. The tranfverfe 36, the abfciffa 28, and the conjugate 12, required the ordinate. Anf. 4.8., PROBLEM VIII. To find the area of an elliptic segment, whofe bafe is parallel to either of the axis. RULE. Divide the height of the fegment by that axe of the ellipfe of which it is a part, and find, in the table of circular fegments, an area, whofe verfed fine fhall be equal to this quotient. Then multiply the area fo found, and the two axes continually, and the last product will give the area of the fegment required. EXAMPLE I. Required the area of the elliptic fegment ECF, whose height is GC 20, and the axes CD and AB 70 and 50. Ex. 2. Required the area of an elliptic, fegment, cut off parallel to the conjugate, at the distance of 18 from the centre, the axis being 60 and 20. Anf. 134.1876. Ex. 3. Required the area of an elliptical fegment, cut off parallel to the transverse, whose height is 6, the diameters being 30 and 20. Anf. 118 9008. Ex. 4. Required the area of an elliptical fegment, cut off parallel to the tranfverfe, whofe height is 10, the diameters being 70 and 50. A f. 391.3829. PROBLEM IX. To defcribe a parabola, the abfciffa and ordinate to the axle being gi ven. RULE. Bifect the given ordinate BA in G, join VG, and draw GD at right angles to VG, meeting the axis in D, and make VO, OF, each equal to BD, and F will be the focus of the parabola." Take any number of points, x, x, &c. in the axis, and through thefe points draw double ordinates of an indefinite length. Then with the radii VF, Vx, &c. and centre F, defcribe the arches c, c, &c. and through all the points of interfection the curve may be drawn. Note. The line cFc is called the parameter. PROBLEM X. Any three of the four following particulars being given, viz. any two ordinates and their two abfciffas, to find the fourth. |