ELL

367

ELL

the ft< e Un der side of the diameter a b, will also be in

Tp Ve

t ? et h°d may be very easily applied in perspective: byImpend* re P resentat * on °f a diameter of an original circletav e pillar, and that of another parallel to the picture, weSllip S i ^'ameter and double ordinate : the diameter of the’°r representation, is the diameter of the circle per-fiii(]i n u ar to the intersecting line of its plane ; thus it is onlytfj* ^' e con J u g a te diameter, and drawing as above.

Sibfy 13— y'/ie axes a b and c d of an ellipsis beingL r ’ 0 describe its representation by means of circular arcs .—Bp parallel and equal to e c; bisect it at 1; draw 1 c,^Cui a ’ eu tting each other at k ; bisect it C, by a perpen-the q feting o d at o; on o, with the radius o c, describeI'Wfq rant C g Q; through a and Q draw Q G, cutting the8 it* 1 ** at G » draw GO, cutting A b at M ; make e l equaland an ^ E N et l ua l to e o; from o, through m and l, draws “^ e from o, complete the arc gk; from N, with the

*te u j equal to the distance n d, describe the opposite

the e ’ ror Q m with the radius M G, describe the arc G h, attile 8a m remit 7 °f the longer axis ; and lastly, from E, with"lea A e ra dius, equal to l b, describe the opposite arc k i;Pass t}, GcKBlI > H a is the representation required, made to^^° u gh eight points in the curve.

" e f 14.— To find the representation of an ellipsis, by^Ve (d c i r cular arcs, passing through twelve points in the* 8 a-flJ 1 rnore accurate method than the former ) the axes! J U 0 t , c D being given. —Draw A 3 parallel to e c ; divide it

’ato ^ ee equal parts ; draw 2 c and 1 c ; divide A e also

Bp 66 e< l Ua l parts, and through the points 1, 2, draw d qP e 8(Ji c cut ting the former in Q and p. Bisect c P by a per-*8 a( . ar > meeting CD, produced at s ; and join p s, cutting'jl'aty p * > make e w equal to e x, and e u equal to e s ;|a<!ul ar X S ’ °W s, k xu, and l w u. Bisect p Q by a perpen-ae ra ^. rQe eting p s at p, and draw z F parallel to a b. With*4, * Us fq, describe the arc QZ, cutting fz at z ; join^ttjq produce it to meet the arc at Q z, at y: join y f,Val t 0 A B at v ; make x i equal to x f, also vv h, and w g,f ’ 1 1 ,-. . F ; makeET equal ev, and draw ivb, him, andpq t e ®> with the centre s, and distance s c, describe the0f iti e p ’ ^ith the centre u and distance u d, equal to the>, d ’ Scribe the opposite arc it L; with the centres I, f,t j t 11 he the arcs kb,pq,ko, and lm ; and, lastly,8 c entres v and t, describe the arcs Q e and m n, at

th

®*tr,

"8ti,

J r, p Passing through the twelve points A, Q, p, c, o, n,» ^his J, K ’ B > as required.

h e e thod differs so little from a true ellipsis, that itsed in preference to any instrument for describing

e mity of the longer axis ; and A c b d is the repre-

‘Vcupp r __.__ „

'Nts 0 f S +1 °^ v ery large arches of bridges, and in finding the

re ?ch ,

^ 8 Of tl --O- -.. D"~>- O .

° w • stones, as was the practice of the celebrated

- 4 rt otf le

-"ihe ^ lneer Berronet.l^ Uni ;r method of describing an ellipsis, by means of an

_^ ’ constructed upon the principle of the oval turning

3 We have never seen any investigation of this5nvi’ r , U P° n simple principles, the Author offers the>^ a nic ;°r, ttle public, which has only been given in his>b] Exercises.

6 • l ° n -—If there be any plane figure, and two inflex-

hi ‘Hraifyl, • 11| C1C UO IUI y piuuc w»m v hijioa.-

tb h e 8 fi lines at right angles to each other ; and if theC tw ° infl ^ to an axis at r 'ght angles thereto ; and if\ he gQ hoxible lines be made to coincide with the plane,^11 c ^? 0vea hle on its surface, that one of them, whichm p 0 ; ^ P le primary line, may always pass through twoiit e * s int ln tlle plane, and through the point where the3 be m a ., 6rsecte d by the axis; and if the other transversee to pass or slide along a given point, which is not

attached to the plane, but would remain stationary, even ^though the plane were in motion ; and if a secondary planebe fixed to the inflexible lines parallel to the primary plane;then if the axis be carried round while the point in thetransverse line is at rest, the primary plane will also becarried round, and every point in it will describe the circum-ference of a circle : the secondary plane will likewise becarried round, and perform its revolutions in the same timeas the primary plane and the axis, but being immoveablyfixed to the rectangular lines, they will cause it to have botha progressive and retrogressive motion, in the direction ofthe primary line, in each revolution ; and, lastly, if anotherpoint at rest be held to the surface of the secondary planewhile in motion, it will either describe an ellipsis, a circle, or ;a straight line. Hence the describing point will always beat the same distance from the centre, or point where the axisintersects the primary plane.

The eccentricity of the ellipsis, or the difference of the |axis, will be double the distance between the stationary point iin the transverse line and the axis.

Instead of the stationary point, a circle may be placed with !its centre in this point, and its plane perpendicular to the jaxis, and instead of the inflexible line moving backward and jforward upon two fixed points in the plane, the diametrically jopposite parts of the circumference may always touch a pairof parallel lines on the revolving plane.

Illustrations.—Figure 15. Let AB and ef, No. 1, 2, 3,

4, 5, 6, 7, 8, be the two inflexible lines, intersecting each i

other in i, at right angles, and let c d be the two fixed points. ;

Let A b be denominated the primary line, and E f the secon-dary line, and let the lines a b and e f at right angles, takenas a whole, be called a transverse ; also, let C represent a pri-mary point, and let the describing point be taken at G, in theline drawn through c d produced ; now, in all positions ofthe chuck, the primary line ab is always upon the point o,and e f upon d. Having premised this in general, suppose,before the machine begins to start, that e f, No. 1, thesecondary line, coincides with e g, and the point G with o,o being in the plane of the figure to be described ; thenbecause a b always passes through c, the points i and c willbe coincident, a b being then at right angles to E f. Let usnow suppose the motion to commence, and let it perform aneighth part of a revolution, as at No. 2, the describing pointg, still remaining in the same position with respect to cand d, viz., in the right line CDG; then the point o will beat a distance from the point g, and a part, g o, of the curvewill be described by the fixed point G, and the point i will beabove the line CDG; now let the motion proceed, anddescribe another eighth, as at No. 3, then, the point o beingalways in the line e f produced, e f will be at right anglesto the fixed line CDG, and A b coincident with cdg, and thepoint which was last at G, will now be at i. In like manner,when another eighth has been performed, as at No. 4, thepoint o has performed three-eighths of a revolution, the point1 is in a line drawn from the point c, perpendicular to thefixed line cdg, and the point 2, which was at g, in No. 3,is situated between 1 and G. In this manner, by continuingthe motion, the whole curve will be generated. No. 5 showsthe curve, when half a revolution has been described; No. 6,five-eighths ; No. 7, six-eighths, or three-quarters ; and No. 8,seven-eighths. !

Here it may be proper to observe, that the angles per- iformed by the revolution of the machine, are very different jfrom the corresponding angles, formed by lines drawn fromthe centre of the ellipsis to the describing point, and to theextremity of the curve at its commencement.

From what has been said, it is easy to conceive, that the