1 C and PD, cutting each other at K; bisect KC by a perpendicular meeting CD in O; and on O, with the radius OC, describe the quadrant CGQ. Through Q and A, draw QG, cutting the quadrant at G; then draw GO, cutting AB at M; make EL equal to EM, also EN equal to EO. From N, through M and L draw NH and NI; then M, L, N, O, are the four centres by which the four quarters of the ellipsis are drawn. It must be observed, that this is not a true ellipsis, but only an approximation to it; for it is impossible to draw a perfect ellipsis by means of compasses, which can only describe parts of circles. But the curve of an ellipsis differs essentially from that of a circle in every part; and no portions of circles put together, can ever form an ellipsis. But by this means, a figure may be drawn, which approaches nearly to an ellipsis, and therefore may be often substi tuted for it when a trammel cannot be had, or when the ellipsis is too small to be drawu by it. At the joining of the portions of circles in this operation, the defect is not perceivable; and the best way is not to join them quite, and to help the curve by hand. Prob. 29. An ellipsis, ACDB, being given, to find the transverse and conjugate axis. Draw any two parallel lines, AB and CD, cutting the ellipsis at the points A, B, C, D; bisect them in e and f. Through e and f, draw GH, cutting the ellipsis at G and H; bisect GH at I; and it will give the centre. Upon I, with any radius, describe a circle, cutting the ellipsis in the four points k, l, m, n; join k, l, and m, n; bisect k 1, or in n, at o or p. Through the points o, I, or I, p, draw QR, cutting the ellipsis at Q and R; then QR will be the transverse axis. Through I draw TS, parallel to k 1, cutting the ellipsis at T and S; and TS will be the conjugate axis. Prob. 30. To describe an ellipsis similar to a given one ADBC, to any given length IK, or to a given width ML. Let AB and CD be the two axes of the given ellipsis. Through the points of contact A,D,B,C, complete the rectangle GEHF; draw the diagonals EF and GH: they will pass through the centre at R. Through I and K draw PN and OQ parallel to CD, cutting the diagonals EF and GH, at P,N,Q,O. Join PO and NQ, cutting CD at L and M; then IK is the transverse, and ML the conjugate axis of an ellipsis, that will be similar to the given ellipsis ADBC, which may be described by some of the foregoing methods. Prob. 31. To describe a parabola. If a thread equal in length to BC, be fixt at C, the end of a square ABC, and the other end be fixt at F; and if the side AB of the square be moved along the line AD, and if the point E be always kept close to the edge BC of the square, keeping the string tight, the point or pin E will describe a curve EGIH, called a parabola. The focus of the parabola is the fixed point F, about which the string revolves. The directrix is the line AD, which the side of the square moves along. The axis is the line LK, drawn through the focus F, perpendicular to the directrix. The vertex is the point I, where the line LK cuts the curve. The latus rectum, or parameter, is the line GH passing throngh the focus F, at right-angles to the axis IK, and terminated by the curve. The diameter is any line MN, drawn parallel to the axis IK. A double ordinate is a right line RS, drawn parallel to a tangent at M, the extreme of the diameter MN, terminated by the curve. The abscissa is that part of a diameter contained between the curve and its ordinate, as MN. Prob. 32. To describe a parabola, by finding points in the curve; the axis AB, or any diameter being given, and a double ordinate CD. Through A draw EF parallel to CD; through C and D draw DF and CE parallel to AB, cutting EF at E and F. Divide BC and BD, each into any number of equal parts, as four; likewise divide CE and DF into the same number of equal parts. Through the points 1, 2, 3, &c. in CD, draw the lines 1 a, 2 b, 3 c, &c. parallel to CD; also through the points 1, 2, 3, in CE and DF, draw the lines 1 A, 2 A, 3A, cutting the parallel lines at the points a, b, c then the points a, b, c, are in the curve of the parabola. Prob. 33. To describe an hyperbola. If B and C are two fixed points, and a rule AB be made moveable about the point B, a string ADC being tied to the other end of the rule, and to the point C; and if the point A be moved round the centre B, towards G, the angle D of the string ADC, by keeping it always tight and close to the edge of the rule AB, will describe a curve DHG, called an hyperbola. If the end of the rule at B were made moveable about the point C, the string being tied from the end of the rule A to B, and a curve being described after the same manner, is called an opposite hyperbola. The foci are the two points B and C, about which the rule and string revolves. The transverse axis is the line IH terminated by the two curves passing through the foci, if continued. The centre is the point M, in the middle of the transverse axis IH. The conjugate axis is the line NO, passing through the centre M, and terminated by a circle from H, whose radius is MC, at N and O. A diameter is any line VW, drawn through the centre M, and terminated by the opposite curves. Conjugate diameter to another, is a line drawn through the centre, parallel to a tangent with either of the curves, at the extreme of the other diameter terminated by the curves. Abscissa is when any diameter is continued within the curve, terminated by a double ordinate and the curve; then the part within is called the abscissa. Double ordinate is a line drawn through any diameter parallel to its conjugate, and terminated by the curve. Parameter or latus rectum, is a line drawn through the focus, per pendicular to the transverse axis, and terminated by the curve. Prob. 34. To describe an hyperbola by finding points in the curve, having the diameter or axis AB, its abscissa BG, and double ordinate DC. Through G draw EF, parallel to CD; from C and D draw CE and DF, parallel to BG, cutting EF in E and F. Divide CD and BD, each into any number of equal parts, as four; through the points of division, 1, 2, 3, draw lines to A. Likewise divide EC and DF into the same number of equal parts, viz. four; from the divisions on CE and DF, draw lines to G; a curve being drawn through the intersec tions at G, a, b, &c. will be the hyperbola required. Remarks. In a circle, the half chord DC, is a mean proportion between the segments AD, DB of the diameter AB perpendicular to That is AD: DC :: DC: DB. it. 2. The chord AC is a mean proportional between AD and the d ameter AB. And the chord BC a mean proportional between DB and AB. That is, AD: AC:: AC: AB, and BD: BC:: BC: AB. 3. The angle ACB, in a semicircle, is always a right. 4. The square of the hypotenuse of a right-angled triangle, is equal to the squares of both the sides. That is, AC = AD2 + DC, and BC = BD2 + DC, and AB AC2 + BC2. 5. Triangles that have all the three angles of the one respectively equal to all the three of the other, are called equiangular triangles, or similar triangles. 6. In similar triangles, the like sides, or sides opposite the equal angles, are proportional. 7. The areas, or spaces, of similar triangles, are to each other, as the squares of their like sides. MENSURATION OF SUPERFICIES. Prob. 1. To find the area of a parallelogram: whether it be a square, a rectangle, a rhombus, or a rhomboid. Multiply the length by the breadth, or perpendicular height, and the product will be the area. Ex. 1. To find the area of a square, whose side is 6 inches, or 6 feet, &c. 6 6 36 Ansr 0018 |