Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 6, CHI-Crystallization

536
Of the therefore,
Hyperbola.
CONIC SECTIONS.
DH : PH :: DL : p L,
and by alternation
c DH : DL :: PH :/.L>
therefpre, becaufe of the parallel lines PH, ED, p L,
EP : E/> :: PK : p K.
Take CGzrCE, then PG—E/>, and by compohticn
EG : EP :: Vp : PK,
and taking the halves of the antecedents
CE : EP :: CP : PK j
hence, by divifion, CE : CP :: CP : CK.
Pan in,
Of the
Hypetboh,
Cor.. I. The re&angle contained by PE and E
is equal to the re&angle contained by KE and CE.
For CP*=KC CE=EC*—KE EC (2. 2. E.y
alfo CP*=EC*—PE’E/t (6. 2. E.)
therefore EC*—KE'ECrrEC*—-PE‘E/>,
and KE-EC=PE-E/>.
COR. 2. The refliangle contained by PK and
is equal to the re&angle contained by KE and KC.
For KC*=rCP*—^PK*K/> (5. 2. E.)
alfo KC*=EC-KC—EK-KC=CP*—EK-KC (3. 2« E. and by the prop.)
therefore CP*—PK K/>=CP*—EK-KC.
and PK-K/>=EK'KC.
Prop. X.
If a tangent to an hyperbola meet the conjugate
axis, and from the point of contadt a perpendi¬
cular be drawn to that axis, the femiaxis will
be a mean proportional between the fegments
of the axis intercepted between the centre and
the perpendicular, and between the centre and
the tangent.
pic, £T> Let DH, a tangent to the hyperbola at D, meet
the conjugate axis B in H, and let DG be perpen¬
dicular to that axis, then
CG ; CB :: CB : CH.
Let DPI meet the tranfverfe axis in K, draw DE
perpendicular to that axis, draw DF, Dy to the foci,
and defcribe a circle about the triangle Djf F j the
conjugate axis Avill evidently pafs through the centre
of the circle, and becaufe the angle FDf is bife£ted
by the tangent DK, the line DK will pafs through
one extremity of the diameter $ therefore the circle
paffes through H. Draw DL to the other extremity
„ of the diameter. The triangles LGD, KCPI, are IP
miiar, for each is hmilar to the right-angled triangle
LDH, therefore,
LG : GD (=CE) :: CK : CH 5
hence LG’CH—CE'CKzr (by laft prop.) CA*.
Now LC.CH=CF* (35. 3. E.)
therefore LC.CH— LG-CH=:‘CF*—CAJ,
that is, CG-CH=CB* (Def. 7.)
wherefore CG : CB :: CB : CH.
Definition.
XL If through A, one of the vertices of the tranf¬
verfe axis, a ftraight line HA h be drawn, equal and
parallel to B £ the conjugate axis, and bifeCted at A
by the tranfverfe axis, the ftraight lines CPIM, C h m
drawn through the centre, and the extremities of that
parallel, are called Afymptotes.
Cor. 1. The afymptotes of two oppofite hyperbo¬
las are common to both. Thrbugh a, the other ex¬
tremity of the axis, draw H' a h\ parallel to B b, and
meeting the afymptotes of the hyperbola DAD in H'
tod k'. Becaufe « C is equal to AC, a H' is equal to
2
A h, or to BC 5 alfo a h! is equal to AH, or to BC ;
hence, by the definition, CH' and C /? are afymp¬
totes of the oppofite hyperbola dad.
Cor. 2* The afymptotes are diagonals of a refl-
angle formed by drawing perpendiculars to the axes
at their vertices. For the lines AH, CB, a H' being
equal and parallel, the points H, B, H' are in a
ftraight line palling through B parallel to A 0 ; the
fame is true of the points h, b, h1.
Prop. XL
The afymptotes do not meet the hyperbola; and
if from any point in the curve a ftraight line be
drawn parallel to the conjugate axis, and ter¬
minated by the afymptotes, the re&angle con¬
tained by its fegments from that point is equal
to the fquare of half that axis.
Through D any point in the hyperbola draw a Fig. 53.
ftraight line parallel to the conjugate axis, meeting
the tranfverfe axis in E, and the afymptotes in M
and m ; the points M and fn Ihall be without the hy--
perbola, and the rectangle MD’D m is equal to the
fquare of BC.
Draw DG perpendicular to B £ the conjugate axis,
let a tangent to the curve at D meet the tranfverfe
axis in K, and the conjugate axis in L, and let a per¬
pendicular at the vertex A meet the afymptote in H.
Becaufe DK is a tangent, and DE an ordinate to the
axis, CA is a mean proportional between CK and
CE (9.), and therefore
CK : CE :: CA* : CE2 (2 cor. 20. 6. E.)
But CK : CE :: LC : LG,
and CA* : CE* :: AH* : EM*;
therefore LC : LG :: AH* : EM*.
Again, CB being a mean proportional between CL
and CG (10.)
LC : CG :: CB* : CG*,
and therefore
LC : LG :: CB* :: CB*+CG*, or CB5+ED*;
wherefore AH* : EM* :: CB*: CB*-j-ED*;
Now AH*=CE* (Def. 11.)
therefore EM*z^CB*-j-ED*,
confequently EM* is greater than ED*, and EM
greater