iiS
"The
Method of FLUXIONS,
Demonftration of the ConftruRion in Example 4.
no. Draw from
ce
CE, and
parallel to
at
an indefinitely fmall diflance
and the tangent of the Hyperbola ck t and let fall it,
KM
AP. Now perpendicular from the nature of the Hyperbola it will be AC A? :: to
:
AP AM, and therefore AC? GLq AC?: LE? (or AP V :: :
:
::
')
AP?
AM?
:
and
;
divlfim*
AG/
:
AL? (DE?) ::.AP?: AM? AP?(MK?) And invent, AG: ;
AP
DE
::
little
Area
:
MK.
DEed
is
But the
to the Tri-
DE
angle CKr, as the altitude as to -LAP. Wherefore to are all the contemporaneous
AG
And
to 4-AP.
is
to half the altitude
KM
in. Draw c*/ parallel and meeting the Curve
Then by
q.
litude
of the
Eq
::
(
Dp
x
P
HI
:
that
is,
AG
Demonjlration of the Conjlruftion in Example
and
;
all the moments of the Space PDE moments of the Space PKC, as therefore thofe whole Spaces are in the fame ratio.
AE
in
e,
infinitely near to and draw hi and
the Hypothefis
~Dd=
Triangles Ddp and DCP, (PD) HI, fo that Dp x HI
moment
CD,
6.
(Fig. in p. 115-)
fe meeting DCJ in p Eg, and from the fimiit
will
be
= Eg xCPj AC moment
D/>
:
(Dd)
and thence
(the Eg x AC. Wherefore
EF/e) :: fince PC and AC EyxAC :: CP are in the given ratio of the latus tranlverfum to the Jatus rectum of the Conic Section QD, and fince the moments HI//) and EFfe of the Areas HIKQ^and AEF are in that ratio, the Areas themfelves will be in the fame ratio. Q-^E. D.
E ? xCP
(the
HI/'/.)):
:
:
112. In this kind of demonilrations it is to be obferved, that I affume fuch quantities for equal, whofe ratio is that of equality And that is to be efteem'd a ratio of equality, which differs lefs from equality than by any unequal ratio that can be Thus :
affign'd.
in the laft
demon ftration
fuppos'd the rectangle E^xAC, or FE?/, to be equal to the fpace FEt/j becaufe (by realon of the difference lefs than them, or nothing in comparifon of Eqe infinitely them,) I
they