ELL
[ 2?4 1
ELL
To find the Axis, "Parameter, and Semi-ordinate of of the Axis Hence, the Square of the Semi- Axis AC,
an Ellipsis. 1S equal t0 the Rectangle of C T, into PC.
Hence, alfo z°. The Parameter, Abfcifsj and Serai- ir°. The Rectangle under the Subtangent, and theDi-
ordinates, in an ffllit&Z betes given 5 the Axis is found by {knee of the Ordinate from the Centre, is equal to the
makino 1 b ■ v ■ ■ v * v 1 a x —■ y'~ = (lw — y 7 ) : x ~x : a. Difference of this Diftance, and the Square of the Tranf-
5 . . j . j . j_, . __j_ _ ^^ g emi _ Axis<
3 . The Axis AB, the Abfcifs A P, (Fig* 23.) and the 12 . In an ElliMis, the Square of the Semi-ordinate is to
Semi-ordinate P M being given* the Parameter A G is the Square of the Conjugate Semi-diameter 5 as the
thus found : Make A I ^^P M 5 and from A, thro' M, Rectangle under the Segments of the Diameter, to the
draw the right Line A L. In I ereft a Perpendicular LI: Square of the Semi-diameter. Confequently, the Relation
Then fincc &/ V P = PJVt :: AN: LI; L I = y 1 : x. Pro- of the Semi-ordinatcs to the Diameters, is the fame as
duce 'p M to 6, till P O = LI = y- : x and from B, to the Axes : And the Parameter of the Diameter, is a
thro' O draw a'right Line BG, In A creel a Perpen- third proportional to the Diameters.
dicular G A — ay* : (a# — x') : This will be the Para- meter A G.
4 . The Axis A B, and the Parameter AG being given, we can affign every Abfcifs, as B P, its Semi-ordinate PN - by drawing the Line GB to the Parameter A G, which is perpendicular to the Axis A B : Then, erecting a Perpendicular P N, make PL^PH. Laflly, on A L defcribc a Semi-circle.
2o find the Foci, conjugate Axis, Ratio of the Ordl- nates, Sec. of an Ellipsis.
i°. From B to L, (Fig. 22.) fet off half the Parameter; then will C L — i a — \ b. In the Centre C erect a Per- pendicular C K, meeting the Semi-circle defcribed on A L. Thus will CK=/ (4 a 1 — 4 ab) Therefore, making CF — CK; F will be the Focus. The latter Equation furnifhes us this Theorem.
If the Axis A B, be cut in the Focus F, the Rectangle under the Segments of the Axis A F, F B will be fub- quadruple of the Rectangle under the Parameter and the Axis. Sec Focus.
2 . The Parameter, and Axis A B given, the Conjugate Axis is eafily found; as being a mean proportional, be- tween the Axis and Parameter. Confequently, the Parameter is a third proportional to the greater and leifer Axis. Add, that the Square of half the Conjugate Axis, is equal to the Rectangle, under the Distance of the Focus from the Vertex, and its Complement to the Axis.
3 . In an Ellipfis, the Squares of the Semi-ordinates P M, and p m, c3V. are to each other as the Rectangles under the Segments of the Axis. Hence DC 1 : PM : = CB : : AP PB. Confequently DC » : C B 2 — P M * -. A P
Infinite Ellipses, are thofe defined by the Equation ay 111 -j- n = b# m (a — * n ) which fome call Ellipto'ides, if m be greater than 1, and n greater than 1. See Elliptoides.
In Refpeft of thefe Curves, the Ellipfis of the firft Kind is called the Apollonian Ellipfis.
Quadrature of the Ellipsis. See Quadrature.
Ellipsis, in Grammar, and Rhetoric, a figurative Way of fpeaking, wherein fomething is fupprefs'd, or left out in a Difcourfe, and fuppofed or underlfood. See Figure.
This chiefly happens, when, being under the Tranfport of a violent Paflion, a Man is not at Leifure to fay every Thing cut at Length; the Tongue being too flow to keep Pace with the rapid Motions of the Mind. So that on thefe Occafions we only bring forth broken, interrupted Words and Expreffions; which exprefs the Violence of a Paflion, better than any confiftent Difcourfe.
Fa. Soffu confidcrs the Ellipfis, as a Way of difguifing Sentences; by fuppreffing the Word which fliould make the particular Application, and leaving the whole in a Kind of ingenious Ambiguity. Sec Sentence.
Thus, the Trojans, in Virgil, being reduced by T'urnus to the laft Extremity, and ready to be deftroy'd, fpy JEneas coming to aflift them : Upon which the Poet fays, Spes addita fvfeitat Iras. Which Expreffion figni- fies either, in particular, that the Hope they conceive re- trieves and augments their Courage : Or in general, that the Hope of Affiftance at Hand naturally raifes Courage, and gives new Strength.
If the Poet had added a Word, and faid, Ollis /pes addita fufdtat Iras, the PafTage had been exprcfsly affected to the firft Senfe 5 and it had ceafed to be a Sentence, and commenced the Application of a Sentence
S IB. That is, the Square of the left Axis is to the Square The Su £ preffion of that Word mVkcs a Sentence in Form. of the greater; as the Square or the Semi-ordinate, to the g^ e Sentence
Rectangle under the Segments of the Axis.
4. . The right Line F D, (Fig. 24.) drawn from the Focus F, to the Extremity of the Conjugate Semi-Axis; is equal to half the tranfverfe Axis A C.
Hence, the Conjugate Axes being given, the Foci are eafily determined. For, hiffecting the greater Axis A B in C; from C erect, a Perpendicular C D, equal to the Con- jugate Semi- Axis. Then, from D, with the Interval C A, the Foci F and f are determined.
"To defcribe an Ellipsis.
5 . The Sum of two right Lines F M, and fm drawn from each Focus of an Elliffis, F and f to the fame Point of the Periphery M, being equal to the greater Axis A B : The Conjugate Axes of an Ellipfis being given, the Ellipfis is eafily defcribed. For, determining the Foci F and f, as already directed; and fixing two Nails therein, and about thefe Nails tying a Thread F M f, equal to the Length of the greater Axis A B : The Thread being ffretch'd, and a Style, or Pin, applied
This, that excellent Critick looks on as one of the
FinefTes of the Latin Tongue; wherein it had vaftly the Advantage of rhe Modern Tongues. v Traite du 'Foeme Epique. Page 4.66, $$c.
ELLIPTIC, or ELLIPTICAL, fomething that belongs to an Ellipfis. See Ellipsis.
Kepler firft maintain'd, that the Orbits of the Planets are not Circular, but Elliptical; which Hypothetic was" afterwards adhered to by M. 'Bouillaud. Mr. Flamfiead, Sir Ifaac Newton, Monf. Caffini, and others, of the later Aftronomers, have confirm'd the fame : So that this, which was once by Way of Contempt call'd the Elliptic Hypothefis, is now the prevailing Doctrine. See Orbit.
Dr. Ifaac Ne-wton demonstrates, that if any Body revolve round another in an Elliptic Orbit, its centrifugal Forces, or Gravities, will be in a duplicate Ratio; or as the Squares of its Diflances from the Umbilicus, or Focus. See Cen- tripetal.
Serlio, Haftman, &c. demonftrated that the belt. Form
its Extent, the Duct or Sweep of the Style or Thread for ^es, or Vaults, is the Elliptical. See Arch. about the Nails will defcribe snEM0s. Elliptic Space, is the Area, contained within the Ci
6°. The Reftangle under the Segments of the Conjugate cum f eren ce, or Curve of the Ellipfis.
Axis, is to the Square of its Semi-ordinate, as the Square of the Conjugate Axis, to the Square of the greater Axis. Hence, the Co-ordinates to the Conjugate Axis, have the fame Relation, as there is between the Co-ordinates to the greater Axis. Confequently, the Parameter of the Conju- gate Axis, is a third proportional to the Conjugate Axis, and the greater Axis.
T"o determine the Subtangent P T, (Fig. 25.) and Sub- normal PR in an Ellipsis.
7 , As the firft Axis, is to the Parameter; Co is the
'Tis demonftrated, 1°. That trie Elliptic Space is to a Circle defcribed on the tranfverfe Axis 5 as the Conjugate Diameter is to the tranfverfe Axis.
z°. That the Elliptic Space is a mean proportional be- tween two Circles defcribed on the tranfverfe and conjugate Axes. See Circle.
Elliptic Conoid, is the fame with the Spheroid. See Spheroid.
Elliptic Specula, or Mirrors. See Mirror.
ELLIPTICAL Compajfes, is an Inftrument, made ufually
Diltance of the Semi-ordinate from the Centre, to the Sub- j n grafs; for the drawing any Ellipfis, or Oval, at one Re- normal. See Subnormal.
8 Q . The Rectangle under the Segments of the Axis, is equal to the Rectangle, under the Diitance of the Semi- ordinate from the Centre, into the Subtangent. ' See Sub- tangent.
9 . As the Diftance of the Semi-ordinate from the Centre, is to half of the Axis; fo is the Abfcifs to the Portion
of the- Subtangent intercepted between the Vertex of the defined by the Equation ay 11 Ellipfis and the Tangent. 7^ > 1 and n > 1.
j io°. The Rectangle under the fubtangent P T, into the Of this there are feveral Kinds or Degrees -. As the Cubical Abfcifs A P, is equal to the Reclangle under the Segments Ellittoid, wherein ay J = hx* (a — x), A Biquadratic or
in-
volution of an Index. See Compasses.
Elliptical 2)ial, is an Inffrument, ufually of Brafs. with a Joint to fold together, and the Gnomen to fall flat; for the Sake of the Pocket.
By it are found the Meridian, Hour of the Day, Riling and Setting of the Sun, i$c. See Dial.
ELLIPTOIDES, an Infinite Ellipfis, i. e. an Ellipfis bjc m (a— x) n , wherein
