CON 
PB — MP^ = AC^ — CP^ — MP' = 
AC^ — MC^ z= AC^— CD^ — CO" = 
C F2 _ C O" : and, in the hyperbola, A C" 
4- CD" = CF"; and AP X PB + MP" 
_ pc^ — AC" + MP" = MC" — 
AC" = CO" — CD" — CA" = CO" — 
G F". Tlierefore, the last analogy becomes, 
A C" : C F" :: A C" =F CP" : CF" T CO" 
Consequently, A C" : C P :: C P" : C O" 
19. 5. E. 
And, AC:CF:;,CP:CO. 
PROP. XXX. 
Fig. 44 and 46. If M be a point in an el- 
lipse or hyperbola, and M F and M/" be 
drawn to the foci ; then, in the ellipse, the 
sum of M F and M/ is equal to the trans- 
verse axis; and, in the hyperbola the dif- 
ference of M F and M.f is equal to tlie 
transverse axis. 
Draw MP perpendicular to the trans- 
verse axis, and take CO as in the last pro- 
position. And, because 
AC:CF::CP :CO, Pr. 29. 
Therefore, ACxCO = FCx CP; 
and4ACxCO=4CFxFO. But be- 
cause AB and F/are bisected m C, there- 
fore 4 A C X CO = B O" - A O", 8.2. E. 
and 4 F C X CP = P/^ - “ 
MF", 47. 1. E; therefore BO — AU — 
f M P. 
^ Again, MF" + M/" = /P + JP' + 
gMP" = 2FC" + 2CP" + 2MP = 
2PC"-+-2MC" = 2FC"4:2CD + 
SCO" = 2AC" + 2CO" = BO" + 
And, because B O" + A O" == /M" + 
M F", and BO" — AO" =/M — M F ; 
therefore, by adding the equals, 2 B O _ 
2fM"- and, by substracting the equals, 
2 pz’ = g A O". Therefore /M = B O, 
and F M = A O ; whence the proposition 
is manifest. 
PROP. XXXI. 
44 , 45, and 46. A straight line drawn 
froin*any point in a conic section to a focus 
has to a perpendicular drawn to the corres- 
ponding directrix, a ratio that is constantly 
th^e same wherever the point is assumed 
in the curve; and, in the ellipse, the con- 
stant ratio is a ratio of minority (or of a less 
magnitude to a greater); m the hyperbola 
the constant ratio is a ratio of majority (or 
of a greater magnitude to a less) ; and, m 
the parabola tlie constant ratio is a ratio 
of equality. . 
Let M (fig. 44 and 46) be a point in an 
ellipse or hyperbola, and draw M F to a 
focus, and MK perpendicular to the direc- 
CON 
trix H G, which corresponds to that focus, 
draw M P perpendicular to the tiansverse 
axis, and take C O as in Prop. 29. Then 
AC;CF::CP:CO,Pr.29. 
Invertendo, CF:QA::CO:CP 
Therefore, CF: CA :;FO i AP, 19. 5.E. 
But, CF:CA::AF:AG,Def.XX. 
Therefore, CF:CA::AO:GP,12.5.E. 
But, as has been shewn in the demonstra- 
tion of the last proposition, A O = M F, and 
G P = M K ; therefore 
_CF:CA::MF:MK. 
But the ratio of CF to C A is a constant 
ratio; and it is a ratio of minority in the 
ellipse, and a ratio of majority in the hy- 
perbola. 
Fig.45. In the parabola, (^A = AF,and 
4AF X AP = MP", Def. 1.; biit4AF 
A P = G P" — P F", 8. 2. E ; therefore 
MP" = GP" — PF"; and MP"-f PF", 
or M F" = GP", or M K". Therefore M F 
= MK. 
CONIFERiE, in botany, the name of 
one of the orders of Linnaeus’s fragments of 
a natural method, consisting of plants whose 
female flowers, placed at a distance from 
the male either on the same or distinct 
roots, are formed into a cone. Of this or- 
der are the- Abies, Cypressus, &c. 
All the coniferaj yield a resin which ren. 
ders most of them evergreen. The fruit in 
all is biennial, being produced in the spring, 
but not ripening and dropping its seeds un- 
til spring after. The coniferae compose 
also one of the natural orders of Jussieu. 
CONIUM, ill botany, a genus of the 
Pentandria Monogynia class and order. 
Natural order of Umbellataa. Essential 
character : partial involucre halved, three- • 
leaved ; fruit nearly globular, five-streaked, 
notched on each side. There are five spe- 
cies of which C. maculatum, common hem- 
lock, is obviously distinguished by its large 
and spotted stalk ; by the dark-green shin- 
ing leaves ; and particularly by their disa- 
greeable smell when bruised. Tlie root is 
biennial, resembling that of a small pars- 
uep. The stem is from four to six feet high, 
hollow, and covered wdtli a blueish powder, 
which easily wipes off. The leaves which 
"row near the bottom of the plant ai e 
about two feet in length. Calyx entire; 
corolla wliite, outer petals largest; seeds 
brownish, resembling those of aniseed. 
CONJUGATE diavieter, or axis oj an 
ellipsis, the shortest of the two diameters, 
or that bisecting the transverse axis. 
Conjugate hyperbolas. See Conic Sec- 
tions. 
