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, which 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 AC 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 of CB will be equal to the rectangle AF a.
Prop. IL If from any point G (fig. 7 and 8.) of the hyperbola, a right line GD is drawn at right angles to the transverse axis, A a, and if from the same point there is drawn the right title GP 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, 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 All equal GF ; then, with the centre G, and the distance GF, describe a 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 All, 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.111. (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 off a right line All 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 11 ex tremity of the right line All) 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 CH 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 All, (7. 2.) i. e. because CA, CF, CD, CH, are proportionals to twice the rectangle FCD, and to the square of All, or OF ; that is, to twice the rectangle of FCD and the sqUares of FD and DO, that is, to the squares of FC, CD, and DO, (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 FAD`, will he equal to the remaining rectangle All a, and to the square of DO (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, 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' : CA' • : CEP+ G : C A' + the rectangle AD a ; that is, as C B'±C N' is to C D= or G N'.
Prop. VI. (fig. 9.) It is another pro perty of the hyperbola, that the asymp totes, D d, E e, do never absolutely meet with the curve. See ASYMPTOTE.
Prop. VII. If through any point F (fig. 9.) of the hyperbola, there is drawn a right line 1 F L parallel to the second axis, and meeting the asymptotes in I and L; the rectangle contained under the right lines which are intercepted be tween the asymptotes and the hyper bola, is equal to the square of the half of the second axis, that is, C B' = I F L = I H L.