HYPERBOLA. 



cond axis, and is bisected by the trans- 

 verse axis, the right lines drawn through 

 the centre, and the extremities of the pa- 

 rallel line, are called asymptotes. 1 1. The 

 right line drawn through the centre of 

 the hyperbola, parallel to the tangent, 

 and equal to the segment of the tangent 

 between the asymptotes, and which is 

 bisected in the centre, is called the se- 

 cond diameter of that which is drawn 

 through the point of contact. 12. A third 

 proportional to two diameters, one of 

 which is transverse, the other second to 

 it, is called the latus rectum, or parame- 

 ter of that diameter, which is the tirst of 

 the three proportionals. And, 13. Lastly, 

 fig. 9. If upon two right lines A a, B b, 

 mutually bisecting each other at right an- 

 gles, the opposite hyperbolas AG, a .-, 

 are described; and if upon the same right 

 lines there are described two other op- 

 posite hyperbolas, BK, b h, of which the 

 transverse axis, B 6, is the second axis of 

 the two first and the second axis of the 

 two last, A a, is the transverse axis of 

 the two first ; these four are called con- 

 jugated hyperbolas, and their asymptotes 

 shall be common. 



Prop. 1. (fig. 6.) The square of the 

 half of the second axis is equal to the 

 rectangle contained by the right lines be- 

 tween the foci and the vertexes of the ' 

 transverse axis. 



Let A a be the transverse axis, C the 

 centre, E and F the foci, and B b the se- 

 cond axis, whiph is evidently bisected in 

 the centre C, from the definition : let A B 

 be joined : then since (by def. 6.) AB and 

 CF are equal ; the squares of A.C and CB, 

 together, will be equal to the square of 

 CF, that is, (6. 2.) to the square of AC 

 and the rectangle AF a together; where- 

 fore, taking away the square of AC, which 

 is common, the square ofCB will be equal 

 to the rectangle AF a. 



Prop. II. If from any point G (fig. 7 and 

 8;) of the hyperbola, aright line GD is 

 drawn at right angles to the transverse 

 axis, A a, and if from the same point 

 there is drawn the right line GF to the 

 focus nearest to that point ; the half of 

 the transverse axis CA will be to the 

 distance of the focus' from the centre, 

 viz. CF, as the distance of the perpen- 

 dicular CD is to the sum of the half of 

 the transverse axis, and the right line 

 drawn to the focus. 



Let GE be drawn to the other focus, 

 and on the axis a A produced, let there 

 be set off AH equal GF ; then, with the 

 centre G, and the distance GF, describe 

 > circle cutting the axis a A in K and F, 



and the right line EG in the points L and 

 M : then since EF is double CF, and FK 

 double FD, EK shall be also double CD ; 

 and since EL or A a is double CA, and 

 LM double GE or AH, EM shall also 

 be double CH ; but because of the circle, 

 EL or A a : EF : : EK : EM ; and taking 

 their halves, it will be as CA: CF: : CD : 

 CH. 



Prop. III. (fig. 7 and 8.) the same things 

 being supposed, if from A, the extremity 

 of the transverse axis nearest to the point 

 G, there is set oil' a right line AH on the 

 axis produced, equal to the distance of 

 the point G from the focus F, nearest to 

 the said extremity ; the square of the per- 

 pendicular GD shall be equal to the ex- 

 cess of the rectangle EHF, contained 

 under the segments between H ^the ex- 

 tremity of the right line AH) and the 

 foci, above the rectangle AD a, con- 

 tained under the segments cut off be- 

 tween the perpendicular and the extremi- 

 ties of the axis. 



For since the right line CJI is any how 

 cut in A, the squares of CA and CH 

 together will be equal to twice the rect- 

 angle ACH, and the square of AH, 

 (7. 2.) i. e. because CA, CF, CD, CH, 

 are proportionals to twice the rectangle 

 FCD, and to the square of AH or GF ; 

 that is, to twice the rectangle of FCD 

 and the squares of- FD and DG, that is, 

 to the squares of FC, CD, and DG, (7 2.) 

 wherefore the two squares of CA and 

 CH are equal to three squares of FC, 

 CD and DG ; and taking away the 

 squares of CA and CF from both sides, 

 the remaining rectangle EHF, will be 

 equal to the remaining rectangle AD o, 

 and to the square of DG (6. 2.) 



Prop. IV. (fig. 7 and 8.) If from any 

 point G of the hyperbola, there is drawn 

 a right line parallel to the second axis B b t 

 meeting the transverse axis A a in D ; the 

 square of the transverse axis shall be to 

 the square of the second axis, as the rect- 

 angle contained under the segments of 

 the transverse axis, between the parallel 

 and its extremities, to the square of the 

 parallel. 



Prop. V. (fig. 8.) If from any point 

 G of the hyperbola there is drawn a right 

 line parallel to the transverse axis A a 

 meeting the second axis in N; the square 

 of the second axis shall be, to the square 

 of the transverse, as the sum of the 

 squares of the half of the second axis 

 and its segment, between the centre and 

 the right line, to the square of the line 

 itself; that is, CB 1 : CA 1 : : CB J H- 

 G D 1 : C A a -f- the rectangle AD a ; 



