ELL 



the circle, where the squares of the ordi- 

 nates are equal to such rectangles, as be- 

 ing mean proportionals between the seg- 

 ments of the diameter. In the same man- 

 ner, the ordinates to any diameter what- 

 ever are as the rectangles under the seg- 

 ments of that diameter. 



As to the other principal properties of 

 the ellipsis, they may be reduced to the 

 following propositions. 1. If from any 

 point M in an ellipsis, two right lines, 

 M F, M/, (fig. 1,) be drawn to the foci F, 

 /, the sum of these two lines will be equal 

 to the transverse axis A B. This is evi- 

 dent from the manner of describing an 

 ellipsis. 2. The square of half the lesser 

 axis is equal to the rectangle under the 

 segments of the greater axis, contained 

 between the foci and its verticles ; that 

 is, DC 1 = AF XFB = A/'x/B. 3. 

 Every diameter is bisected in the centre 

 C. 4. The transverse axis is the great- 

 est, and the conjugate axis the least, of 

 all diameters. 5. Two diameters, one of 

 which is parallal to the tangent in the 

 vertex of the other, are conjugate diame- 

 ters ; and, vice versa, a right line drawn 

 through the vertex of any diameter, pa- 

 rallel to its conjugate diameter, touches 

 the ellipsis in that vertex. 6. If four tan- 

 gents be drawn through the vertices of 

 two conjugate diameters, the parallelo- 

 gram contained under them will be equal 

 to the parallelogram contained under 

 tangents drawn through the vertices of 

 any other two conjugate diameters. 7. If 

 aright line, touching an ellipsis, meet 

 two conjugate diameters produced, the 

 rectangle under the segments of the tan- 

 gent, between the point of contact and 

 these diameters, will be equal to the 

 square of the semi-diameter, which is 

 conjugate to that passing through the 

 point of contact. 8. In every ellipsis, the 

 sum of the squares of any two conjugate 

 diameters is equal to the sum of the 

 squares of the two axes. 9. In every 

 ellipsis, the angles F GI, /G H, (fig. 1), 

 made by the tangent II I, and the lines 

 FG, /G, drawn from the foci to the 

 point of contact, are equal to each other. 

 10. The area of an ellipsis is to the area 

 of a circumscribed circle, as the lesser 

 axis is to the greater, and vice versa with 

 respect to an inscribed circle ; so that it 

 is a mean proportional between two cir- 

 cles, having the transverse and conjugate 

 axes for their diameters. This holds 

 equally true of all the other correspond- 

 ing- parts belonging to an ellipsis. 



The curve of any ellipsis may be obtain- 

 ed by tj*e following series. Suppose the 



ELL 



semi-transverse axis C B = r, the semi- 

 conjugate axis C D= c .andthe semi-ordi- 

 nate = a; then the length of the curve 



TU r> , r* a3 4 r 1 c 1 tf r a* 

 M o = a H -4- 



40 c 8 



8 c 4 r* gi -\- r 6 a"i 4 c 1 . _ 



n2~c^ , &c. And 



if the species of the ellipsis be determin- 

 ed, this series will be more simple : for if 



+ 2048r* 



2r, then M B = 



113a7 3419 a? 



458752^ +75497472r 8 , 



- Thls 



series will serve for an hyperbola, by 

 making the even parts of all the terms 

 affirmative, and the third, fifth, seventh, 

 &c. terms negative. 



The periphery of an ellipsis, according 

 to Mr. Simpson, is to that of a circle, whose 

 diameter is equal to the transverse axis 



?<.* ir d $ dt 



of the ellipsis, as 1- _____ 



3. 3 . 5</3 2 . 3 . 5 . 5 . 7 d+ 



2.2.4.4.6.6~~ 2 .2 .4.4 .6.6. 8.8 

 8cc. is to 1 , where d is equal to the differ- 

 ence of the squares of the axis applied to 

 the square of the transverse axis. 



ELLIPSIS, in grammar, a figure of syn- 

 tax, wherein one or 1 more words are not 

 expressed ; and from this deficiency it has 

 got the name ellipsis. 



ELLIPSIS, in rhetoric, a figure nearly 

 allied to preterition, when the orator, 

 through transport of passion, passes over 

 many things, which, had he been cool, 

 ought to have been mentioned. In prete- 

 rition, the omission is designed ; which, 

 in the ellipsis, is owing to the vehemence 

 of the speaker's passion, and his tongue 

 not being able to keep pace with 'the 

 emotion of his mind. 



ELLIPTIC, or ELLIPTICAL, some- 

 thing belonging to an ellipsis. Thus we 

 meet with elliptical compasses, elliptic 

 conoid, elliptic space, elliptic stairs, &c. 

 The elliptic space is the area contained 

 within the curve of the ellipsis, which 

 is to that of a circle described on the 

 transverse axis, as the conjugate dia- 

 meter is to the transverse axis ; or it is a 

 mean proportional between two circles, 

 described on the conjugate and trans- 

 verse axis. 



ELLIPTO1DES, in geometry, a name 

 used by some to denote infinite ellipses, 



m-\-n 



defined by the equation ag =bxm 

 n. 



