ELL

365

ELL

l l! U'all' V ^ be re P reser| te<l by points, and whatever arrises arep ar - to the elevation, they will be represented by linesPr t to ^ ie original. See Design.

^IZABETHAN ARCHITECTURE. See Tudor

ju^ABETHAN ARCHITECTURE.el} TECTURE -

f 0fm ^ I PS IS, or Ellipse, in geometry, a conic sectionStirfjw b y . cutting a cone entirely through the curvedtr ar Ce ’ neither parallel to the base, nor making a subcon-tha/ Secti °n;_ so that the ellipsis, like the circle, is a curveSe e t f eturns into itself, and completely encloses a space.

q le definitions under the word Cone .el]i D ^. e .°^ tlle principal and most useful properties of thedi am 18 ls ) that the rectangle under the two segments of as acQ G eter . * s as the square of the ordinate. In the circle, theof r& tio obtains, but the rectangle under the two segmentsb e diameter becomes equal to the square of the ordinate:ies Cr °^ ef ^ L —The two axes of an ellipsis being given to

e the

curve.

\^ tll °d 1. —Figure 1. Let AC be the greater axis, b g thefajjj ’ cut ting each other in the centre h ; and if with theit A H, or h c, from the point b, an arc e e be described,take CUt A c > at e and f, the foci ; fix two pins at e and f ;i, a thread equal in length to A c, and fix one end of it totovp tbe other to f ; then keeping a pencil at the point d,

such point forward in the same direction, so that the

Parts j.

U E an d e f may continue to be stretched during the_ » until the describent d come to the point whence itm °ve.

l °d II.— Figure 2 . Find the foci e and f, as before ;] ,, en E and f, take any point, 1 ; with the radii A 1, 1 c,® oontres e and f, describe arcs cutting each other atls ° at H ; then g and h are points in the curve; in thefltl 'l'th Qanner ’ the same radii, from the centres f and e,Join* o lnt ersections i and k. In like manner, if any other\ i,\ 2 > be taken ‘

bi

, if ' oe taken between e and f ; four other points,e j.’ \°> will be obtained, and thus as many more as willP 0 J* u ' s *te for drawing the curve by hand.

1) j 8 the construction of the ellipsis, e d + Dr ( Figuree< lual to ec + CF, = ef + 2fc; also ed + df =

fa + ae = ef+2ae; therefore ef + 2fc = ef +2 a e : consequently ae is equal tore; hence e d + df =e f + 2 a e = e f + 2 f C : that is, the sum of the two linesdrawn from the foci, to any point in the curve, is equal to thetransverse axis.

Method III.— Figure 3. To describe an ellipsis with theellipsograph, or trammel, as it is called by workmen, the axesA b and c d being given in position, bisecting each other in thecentre e. —In any piece of material, contained between anytwo parallel planes, cut two grooves, at right angles to eachother, in one of the planes ; then provide a rod with threepins, or points, so that at least two may be moveable, and in astraight line with the third : let h f g be the rod, with thepoints f and g moveable ; making h G equal to the greatersemi-axis ; then placing the grooves over the axes, andputting the points f and G in the two grooves, move thepoint h round, keeping the point f upon the greater axis A b,and the point G upon the lesser axis c d, until the describenth come to the point where the motion commenced ; and thefigure so described, will be an ellipsis. The trammel, usedby artificers, consists of two rulers, with a groove in each, sofixed that both grooves may be in the same plane, and atright angles to each other, and that the opposite sides of thecross may be in a plane parallel to those of the grooves. Therod above, is a bar with two moveable cursors, the fixed endis made to hold a pencil, and each of the other two an ironpoint, made to fill the groove, but capable of sliding freely.

Method IV.— Figure 4. Given one of the axis, a b, and anordinate, c d, to describe the ellipsis. —Bisect ab at i for thecentre ; through i draw E f, parallel to C D ; with the distancei a, or i b, from the point d, describe an arc, cutting i f at G :draw the straight line d g ii, cutting i a at h ; then if h d gbe conceived to be an inflexible line or rod, the points H, g, d,remaining at the same distance in respect to each other ; andif the point n be moved in the axis A b, and the point g inthe axis e f, while the describent d, is carried round thecentre, i, until A come to the point whence it began tomove; a curve d b e a f, will be described, which will be anellipsis.

Demonstration.—Figure 5.

l c,

L f, or B k : f G, or D BTherefore, bk ! : db'

BK 1 : B K 2 -DB 1

But B K 2 - DB 2 = BK + BDXBK

and cd ! = cl' — dl’= ab 2

or A b : c d ;

AB 2 :CD*j

ab' :ab ! — cd*.■ BD = DH x D K,- L D 2 .

J| Jt ^

’Jetho j 6 .demonstration which accompanies the following

aod,

ftfet/! fluite general, for every two c\ of „ —Figure 6. A diameter, K

Consequently, bk 2 : dh X dk : : a b 2 : d l which is a most principal property of the ellipsis

parallel to d l ; draw dec, and K b, at right angles to a a ,from l, with the distance k b, describe an arc, cutting I aat f ; draw ifc; through the points c and i, draw M n; thenif the point c be moved in m n, and the point f in A A, thepoint l will describe the curve of an ellipsis.

diameters.

, uj ~ . g _ __, „ h, and an ordinate,

an e ttipsis being given, to describe the curve by a con-m °tion —Bisect K h at I, and through i, draw AI A,

Demonstration.

For the triangles lfg, and LCD are similar . • .

L P

: p a :

: l c

: CD ;

But because fl = bk, fg = de, and l c = a i,

K B

; D E :

: A i

: C D ;

Again, by similar triangles, I K b, and ide , .

K B

: d e :

: k i

: di;

Therefore, by equality of ratios

K I

: d i :

: A i

: C d ;

By duplication .

KI 2

: D i 2 :

: Ai 2

: CD 2 ;

By division . . . . . . k i 2

K I 2 -

— d i 2 :

: A i 2

: ai 2 —

But KI 2 — D I 2 = (k I + D i) X (KI — D i) = D H X DK jand CD 2 = LC 1 — L D 2 = A I 2 - ld’.

Therefore by substituting dh X d k, for k i 2 — d i 2 , and a i 2 — l d 2 , for c d 2 ;

In the last analogy we have .k i 2 : d h X d k : : a i* : L O^,

a well-known property of the ellipsis.

5 A