EXAMPLE 1

Let the line PFHQ, be a semi-circle described upon the PQ, to find the density of the medium that shall make a projec diameter move tile in that line. PQ Bisect the diameter CD, o ; then DI 2 in A and ; AD AQ, 2 or AQ, n ; AC, call = nn 2 aa a 2ao ; CH, oo, or e ; ec.

and 2ao and the root being extracted by our method, will give DI e aaoo ao 3 a 3 o 3

&C Here put nn f r ee + aa and
~e~~~2e
2e?
~2?
oo ;
ao
W
oo
DI
will become
=
*
>
nnoo
ao
e
2e
e
Such
3
anno 3
,
2e 5
&c.
manner
series I distinguish into successive terms after this
that the
:
I call
term in which the infinitely small quantity o is not found
the second, in which that quantity is of one dimension only the third, in
which it arises to two dimensions the fourth, in which it is of three and
first
;
;
;
;
so
And
ad infinitum.
the
the length of the ordinate
quantity
between
o.
CH
The
and
DN
;
position of the tangent
o,
or
a
to
e.
CH,
term, which here is e, will always denote
standing at the beginning of the indefinite
second term, which here
that
pleting the parallelogram
to
first
The
is,
the lineola
HC DM;
HN
;
as,
is
,
MN
will denote the difference
which
is
cut off by
com
and therefore always determines the
in this case,
third term, which here
by taking
is
-,
MN to HM
CM?
as
will represent the
G
li
neola IN, which lies between the tangent and the curve and therefore
determines the angle of contact IHN, or the curvature which the curve line


of a finite magnitude, it will be expressed
follow in wfinitu:.:i.
But if
by the third term, together with those that
has in H.
IN
If that lineola
that lineola be diminished
is
in.
infini-
tnm, the terms following become in
and
finitely less than the third term,
therefore
be
may
The
neglected.
fourth term determines the variation
of the curvature
of the
tion
the
;
the varia
fifth,
variation
and
;
on.
so
Whence, by the way, appears no
B~C~D~E~
~K
con-p~"
temptible use of these series in the solution of problems that depend upon
tangents, and the curvature of curves.
ao
Now
series
compare the
P
Qo
and ~
,
2e 5
the
-
and for
medium
will
^
^
So 3
1
come out
as
;
that
is,
ee
that
is
as that length of the tangent
AF
semi-diameter
Q,
e
(because
HT,
3a
to
2>/,
that
is,
as
?
n
PQ
SAC
-,
e,
^-^
and the density
;
is
which
standing perpendicularly on
will be to the gravity as
and S put
II
or -
H
1
put
lie
~Yj,
OH.
Ze*
&c., and for P,
+ QQ
&c., with the
^~
3
Ze
e
Roo
anno 3
77/700
series e
is
:
given),
as
oi
- or
e
terminated at the
and the resistance
to the diameter
PQ
of
Therefore if the body goes
the circle; and the velocity will be as i/ CH.
from the place F, with a due velocity, in the direction of a line parallel to
H
H
is as the
PQ, and the density of the medium in each of the places
length
is to the force
of the tangent HT, and the resistance also in any place
of a
of gravity as SAC to PQ, that body will describe the quadrant
FHQ
Q.E.I.
circle.
But
if
the
same body should
line perpendicular to
circle
Then
AC or a on the contrary side of the centre A
a for + a.
be changed, and we must put
must
sign
PFQ, we must
and therefore
its
go-frorn the place P, in the direction of a
begin to move in an arc of the semi
PQ, and should
take
the density of the
;
medium would come
out as
.
But nature
does not admit of a negative density, that is, a density which accelerates
the motion of bodies; and therefore it cannot naturally come to pass that
a body by ascending from P should describe the quadrant
of a circle.
PF
To
produce such an
effect, a body ought to be accelerated by an impelling
and
not
medium,
impeded by a resisting one.
be a parabola, having its axis