302 / 1042
e o
N
( 268 )
e o
N
flightH in degree thao thofe annexed to the fenfe of ju·
/lice: which is wifely ordered, becaufe duty to othm is
/lill more eí1"eotial to fociety, than duty 10 ourfelves; for
fociety could nOl fubil a moment, were individuals not
proteéled ftom the headilrong and turbulent paffiolt
of their neighbours.
CONI, a/lrong towo of Piedmoot in lt¡Jy, Gtuated up'
00
the river Stura, thihy two miles {outh· of TuriD,
in
,0
30'
E. loog. and
44° 2S'
N.lat.
e o
N 1
e
S E
e
T
ION
s.
C
ON1C SECTIONS are curve Iines formed by [he
imneélions of a cone and plane.
. Ir
a cone be cut by aplane through the venex, tbe
feélioowill be a triangle ABC, PlateLXVII 6g.
l.
Ir
a eone be CUt by aplane parallel to ita bafe, the
{eélion will be a circle. If it be cut by a plane DEF,
fig.
1.
in fueh a direllion, tbat tbe fide AC of a triangle
paffing lhrough the venex, sod having ilS bafe .BC pero
pendicular 10 EF, may be paraJleI to DP, me feélion is
a
parabola ; ir it be cut by aplane DR, fig.
2.
meeting
AC, the feélion il an ellipfe; and if it be CUl by a plane
DMO, 6g.
3.
which would meet ·AC extended beyond
A,
il is an byperbola.
If
aoy Jioe HG, fig.
l.
bedrawn in a parabola perpen–
dicular to DP, lhe fq uareof HG will be 10 the
fqu~re
of
EP, as DG to DP; for let LHK be a feélioo parallcl to
the bafe, aod thmfore a .circle, the reélangle 'LGK,
will be equal,to lhe fquare of HG, and tbe reélangle
BPC equal to the fq eare of EP ; lherefore lhefe fquares
will be to each omer as lheir refuogJes; !hat is, as
BP to LG, that is DP to DG.
Defcriptiol/ 01 Conie Se{
fionJ.on'
¡¡
P¡all~.
PAR A BOL A.
" Let AB, 6g.
4,
be aoy right line, and C any point
.. without it, and DKF a ruler, which let be pJaced
.. in fame plane io which the right lioe and poiot are,
.. in fuch a manner that one fide of it, as DK, be applied
.. to the right line AB, and me olher fide
KF
coincide
.. with the point C¡ and at
F,
the extremity of lhe fide
" KF, lel be fixed one end of the thread FNC, wbofe
" lenglb il equal to
KF,
and lhe olher eXlremily of it
" at lhe POiOl C, aod Jel patt of lhe lh read, as FG, be
" brought clofe
lO
the fide KF by a fmalJ pin G; lhen
" lel lhe fqum OKF be moved from
B
lowards
A,
fo
" lhal all lhe while ilS fide OK be applied clofe to lhe
.. line BA, and in lhe mfan lime lhe lhread beiftg ex.
.. lended wdl always be applitd to the fide KF, being
.. tlOpl fro01 going from it by means of lh( fmall pin;
.. and by lhe mOlioo
of
l~e
fmall pio N thm will be de•
.. fcribed a ccnaio curve, which is called a
¡,mi para.
..
bola.
..
A~n
if lhe fqllm be brought to itl 6dl given pofi.
" tion, '01110 lhe fame mrnner h. moved along lhe "ne
.. AS,
(ron,
R
lO'Nards H,
lh~
other fel. ¡.parabola \ViII
.. b: defcribo","
The Jine
AH
i,
~,lIed
lhc direélrix ;
t,
lhe foeu l : any
line perpendicular 10
AH,
a diammr; lhe point wltere
il meta lhe curve, its vertex; and four limes lhe di.
/lance of me vertex from tbe direélrix,
itl
latus reélum,
or pararneter•
E L L 1 P S E.
.. If
aoy I'qo poiots, as A and B, 6g.
S.
be takeo iD
.. any plane, aod io,themart hxed me exmmilies of a
.. thread, whofe length is greater lhan the dinaoce be·
.. tweeo the poina, aod lhe tbread exteoded by me
lo'
.. of a fmall pin C, and ir the pin be moved round
fro~
.. aoy point unlil il returo to the place from whence 11
.. begao to mOfe, the lhread being eXlended during me
" whole time of the revolulioo, the 6gure wbich lbe
.. fmall pio by tbis revolution defcribes is c¡Jled an
El·
" lip/t."
The poinls AB are ealled tbe foci; D, lhe
centre
I
EF, tbe lranfverfe axis; GH, me Jeí1"er
axis;
and
aoy
o–
lher !ine palliog througb D, adiameler.
H
y
P E R
BOL
A.
"If tothe pointA, 6g.
6,
io anyplane, one end of ¡he
" rule AB be pJace'J, io fueh a manner, tbal
a~ou:
lbat
.. point, as a center, il may freely move; aod If ¡
Ol.he.. other end
B,
of the rule
AS,
be fixed the,wremllY
" of me thread BOC, whofe lenglh is fmaller lbaD
\he
" rule AB, and the olher end of lhe thread beiog fixed
.. in ¡he point C, coincidiog Wilh lhe fide
of
lhe
r~lc
.. AB, which is in lhe fame plane wilh lbe
gtren
pOlOl
" A; aod lel par! of the thread, as BD,. be
bro~gbt
.. clofe to the fide of lhe rule AB, by meaDS of a fmall
.. pin D · thenlel¡he rule be
mO~'ed
abOUl the poinl
A,
.. from
é
towards T, ¡he thread alllhe whiJe heing ex,
" leoded, and lhe remaioing part coiocidiog
wit~
the
.. fide of lhe ,rule beiog /lopt from goiog from It by
.. meaos of lhe fmall piD, and by
lhe
motioo. of l,he fmall
" pin D, a certaio figure is defcribed whlch
11
called
.. the
/tm;·/v'perhola."
The olher femi.hyperbola isdefcribed in tbe fame way,
aod the oppofile HKF, by fixing lheruler tO C, ,nd lhe
thread to A, and defcribing il in me fame manoer. A
aodCareealled foci ; the poinl G, which bifetlsAC, me
cenler
I
KE, the lrafverfe axis; alioe drawo through lhe
teOler meeliog lhe hyperbolas, a tranfvn fe diamcler¡ ,
line drawn lbrough lhe center, perpeodicular10lhetranf·
verfe axis, and cut off by lhe cirele MN, whofe cenler
i, E, and radius cquaJ 10 CG, is C4l1ed tbe fecood
axis,
d
Jf a line be drawo lbrough ¡he vwex E, equal ao
par,dlellO lhe fecond axis GP and GObe joioed, ¡hey are
called aí1"ymplolcs, Aoy Iioe drolVn lhrough the center,
001 meeling lhe hypcl'bol.s, aod equaJ io kDgtb 10 the
parl