uniformly resisting medium approaches nearer
to these hyperbolas than to a parabola. That
line is certainly of the hyperbolic kind, but
about the vertex it is more distant from the
asymptotes, and in the parts remote from the
vertex draws nearer to them than these hy- M JL BD K
perbolas here described. The difference, however, is not so great between
the one and the other but that these latter may be commod^ously enough
used in practice instead of the former. And perhaps these may prove more
useful than an hyperbola that is more accurate, and at the same time more
compounded. They may be made use of, then, in this manner.
Complete the parallelogram XYGT, and the right line GT will touch
the hyperbola in G, and therefore the density of the medium in G is re-
GT 2
ciprocally as the tangent GT, and the velocity there as ^ -^=- ; and the
resistance is to the force of gravity as GT to
Therefore if a body projected from the
place A, in the direction of the right line
AH, describes the hyperbola AGK and
AH produced meets the asymptote NX in
H, arid AI drawr
ri parallel to it meets the
other asymptote MX in I
; the density of
the mediu.n in A will be reciprocally as
AH. and the velocity of the body as -J
AH1
. . . and the resis an^e there to the force
Al
2nn
n +2^- X GV.
. TT 2nn + 2n
of gravity rs AH to ZiTo
~ X AI. Her,ce the following rules a e
deduced.
RULE 1. If the density of the medium at A, and the velocity with which
the body is projected remain the same, and the angle NAH be changed ,
the lengths AH, AI, HX will remain. Therefore if those lengths, in any
276 THE MATHEMATICAL PRINCIPLES [BOOK II.
one case, are found, the hyperbola may afterwards be easily determined
from any given angle NAH.
RULE 2. If the angle NAH, and the density of the medium at A, re
main the same, and the velocity with which the body is projected be
changed, the length AH will continue the same ; and AI will be changed
in a duplicate ratio of the velocity reciprocally.
RULE 3. If the angle NAH, the velocity of the body at A, and the accelerative
gravity remain the same, and the proportion of the resistance at
A to the motive gravity be augmented in any ratio ; the proportion of AH
to A I will be augmented in the same ratio, the latus rectum of the above-
AH2
mentioned parabola remaining the same, and also the length AI proportional
to it
; and therefore AH will be diminished in the same ratio, and
AI will be diminished in the duplicate of that ratio. But the proportion
of the resistance to the weight is augmented, when either the specific grav
ity is made less, the magnitude remaining equal, or when the density of
the medium is made greater, or when, by diminishing the magnitude, the
resistance becomes diminished in a less ratio than the weight.
RULE 4. Because the density of the medium is greater near the vertex
of the hyperbola than it is in the place A, that a mean density may be
preserved, the ratio of the least of the tangents GT to the tangent AH
ought to be found, and the density in A augmented in a ratio a little
greater than that of half the sum of those tangents to the least of the
tangents GT.
RULE 5. If the lengths AH, AI are given, and the figure AGK is to be
described, produce HN to X, so that HX may be to AI as n -\- 1 to 1
; and
with the centre X, and the asymptotes MX, NX, describe an hyperbola
through the point A, such that AI may be to any of the lines VG as XV"
to xr.
RULE 6. By how much the greater the number n is, so much the more
accurate are these hyperbolas in the ascent of the body from A, and less
accurate in its descent to K ; and the contrary. The conic hyperbola
keeps a mean ratio between these, and is more simple than the rest. There
fore if the hyperbola be of this kind, and you are to find the point K,
where the projected body falls upon any right line AN passing through
the point A, let AN produced meet the asymptotes MX, NX in M and N,
and take NK equal to AM.
RULE 7. And hence appears an expeditious method of determining this
hyperbola from the phenomena. Let two similar and equal bodies be pro
jected with the same velocity, in different angles HAK, hAk, and let them
fall upon the plane of the horizon in K and k ; and note the proportion
of AK to AA". Let it be as d to e. Then erecting a perpendicular AI of
uny length, assume any how the length AH or Ah, and thence graphically,
SEC. II. OF NATURAL PHILOSOPHY. 27?
or by scale and compass, collect the lengths AK, A/>* (by Rule 6). If the
ratio of AK to A/.* bo the same with that of d to e, the length of AH was
rightly assumed. If not, take on the indefinite right line SM, the length
SM equal to the assumed AH ; and erect a perpendicular MN equal to the
AK d
difference -r-r of the ratios drawn into any given right line. By the
like method, from several assumed lengths AH, you may find several points
N ; and draw througli them all a regular curve NNXN, cutting tr.e right
line SMMM in X. Lastly, assume AH equal to the abscissa SX, and
thence find again the length AK ; and the lengths, which are to the as
sumed length AI, and this last AH, as the length AK known by experi
ment, to the length AK last found, will be the true lengths AI and AH,
which were to be found. But these being given, there will be given also
the resisting force of the medium in the place A, it being to the force of
gravity as AH to JAI. Let the density of the medium be increased by
Rule 4, and if the resisting force just found be increased in the same ratio,
it will become still more accurate.
RULE 8. The lengths AH, HX being found ;
let there be now re
quired the position of the line AH, according to which a projectile thrown
with that given velocity shall fall upon any point K. At the joints A
and K, erect the lines AC, KF perpendicular to the horizon : whereof let
AC be drawn downwards, and be equal to AI or ^HX. With the asymp
totes AK, KF, describe an hyperbola, whose conjugate shall pass through
the point C ; and from the centre A, with the interval AH. describe a cir
cle cutting that hyperbola in the point H ; then the projectile thrown in
the direction of the right line AH will fall upon the point K. Q.E.I. For
the point H, because of the given length AH, must be somewhere in the
circumference of the described circle. Draw CH meeting AK and KF in
E and F: and because CH, MX are parallel, and AC, AI equal, AE will
be equal to AM, and therefore also equal to KN. But CE is to AE as
FH to KN. and therefore CE and FH are equal. Therefore the point H
falls upon the hyperbolic curve described with the asymptotes AK,.KF
whose conjugate passes through the point C ; and is therefore found in the
27S THE MATHEMATICAL PRINCIPLES [BOOK 11
common intersection of this hyperbolic
curve and the circumference of the de
scribed circle. Q.E.D. It is to be ob
served that this operation is the same,
whether the right line AKN be parallel to
the horizon, or inclined thereto in any an
gle : and that from two intersections H,
//., there arise two angles NAH, NAA ;
and that in mechanical practice it is suf
ficient once to describe a circle, then to
apply a ruler CH, of an indeterminate length, HO to the point C, that its
part PH, intercepted between the circle and the right line FK, may bo
equal to its part CE placed between the point C and the right line AK
What has been said of hyperbolas may be easily
applied to pir i >;>l.i3. For if a parabola be re
presented by XAGK, touched by a right line XV
in the vertex X, and the ordinates IA, YG be as
any powers XI", XV"
; of the abscissas XI, XV ;
draw XT, GT, AH, whereof let XT be parallel
to VG, and let GT, AH touch the parabola in
G and A : and a body projected from any place
A, in the direction of the right line AH, with a
due velocity, will describe this parabola, if the density of the medium in
each of the places G be reciprocally as the tangent GT. In that case the
velocity in G will be the same as would cause a body, moving in a nonresisting
space, to describe a conic parabola, having G for its vertex, VG
2GT2
produced downwards for its diameter, and -. for its latus
nn n X VG
rectum. And the resisting force in G will be to the force of gravity as GT to
2nti 2tt
~2~ VG. Therefore if NAK represent an horizontal line, and botli
the density of the medium at A, and the velocity with which the body is
projected, remaining the same, the angle NAH be any how altered, the
lengths AH, AI, HX will remain; and thence will be given the vertex X
of the parabola, and the position of the right line XI ; and by taking VG
to IA as XVn to XI", there will be given all the points G of the parabola,
through which the projectile will pass.
SEC. IILJ OF NATURAL PHILOSOPHY. 279
SECTION III.
Of the motions of bodies which are resisted partly in the ratio of the ve
locities, and partly in the duplicate of the same ratio.
PROPOSITION XI. THEOREM VIII.
If a body be resisted partly in the ratio and partly in the duplicate ratio
of its velocity, and moves in a similar medium by its innate force
only; and the times be taken in arithmetical progression; then
quantities reciprocally proportional to the velocities, increased by a cer
tain given quantity, will be in geometrical progression.
With the centre C, and the rectangular asymptotes ^
OADd and CH, describe an hyperbola BEe, and let
| \p
AB, DE, de. be parallel to the asymptote CH. In
|
the asymptote CD let A, G be given points ; and if
the time be expounded by the hyperbolic area ABED
uniformly increasing, I say, that the velocity may ~r
be expressed by the length DF, whose reciprocal
GD, together with the given line CG, compose the
length CD increasing in a geometrical progression.
For let the areola DEec/ be the least given increment of the time, and
Dd will be reciprocally as DE, and therefore directly as CD. Therefore
the decrement of ^TR, which (by Lem. II, Book II) is ^ no , will be also as
D
tf
CD CG + GD 1 CG
GO* r GD2
~
fc
1S>aS GD +
GJD 2
* * nerefore tne timc
uniformly increasing by the addition of the given particles EDcfe, it fol
lows that r decreases in the same ratio with the velocity. For the de
crement of the velocity is as the resistance, that is (by the supposition), as
the sum of two quantities, whereof one is as the velocity, and the other as
the square of the velocity ; and the decrement of ~~ is as the sum of the
1 C^(^ 1
quantities ~-^=r
and
pfp,>
whereof the first is ^^r itself, and the last
i i
is a* /-TFT; therefore T^-R is as tne velocity, the decrements of both
- CilJ
being analogous. And if the quantity GD reciprocally proportional to
T, be augmented by the given quantity CG ; the sum CD, the time
ABED uniformly increasing, will increase !n a geometrical progression.
Q.E.D.
THE MATHEMATICAL PRINCIPLES [BOOK II
COR. 1. Therefore, if, having the points A and G given, the time bo
expounded by the hyperbolic area ABED, the velocity may be expounded
by -r the reciprocal of GD.
COR. 2. And by taking GA to GD as the reciprocal of the velocity at
the beginning to the reciprocal of the velocity at the end of any time
ABED, the point G will be found. And that point being found the ve
locity may be found from any other time given.
PROPOSITION XII. THEOREM IX.
The same things being supposed, I say, that if the spaces described are
taken in arithmetical progression, the velocities augmented by a cer
tain given quantity will be in geometrical progression.
In the asymptote CD let there be given the
point R, and, erecting the perpendicular RS
meeting the hyperbola in S, let the space de
scribed be expounded by the hyperbolic area
I
RSED ; and the velocity will be as the length
J GD, which, together with the given line CG,
**
composes a length CD decreasing in a geo
metrical progression, while the space RSED increases in an arithmetical
[(regression.
For, because the incre nent EDde of the space is given, the lineola DC?,
which is the decrement of GD, will be reciprocally as ED, and therefore
directly as CD ; that is, as the sum of the same GD and the given length
CG. But the decrement of the velocity, in a time reciprocally propor
tional thereto, in which the given particle of space D^/eE is described, is
as the resistance and the time conjunctly, that is. directly as the sum of
two quantities, whereof one is as the velocity, the other as the square of
the velocity, and inversely as the veh city ; and therefore directly as the
sum of two quantities, one of which is given, the other is- as the velocity.
Therefore the decrement both of the velocity and the line GD is as a given
quantity and a decreasing quantity conjunctly; and, because the decre
ments are analogous, the decreasing quantities will always be analogous ;
viz., the velocity, and the line GD. U.E.D.
COR. 1. If the velocity be expounded by the length GD, the space de
scribed will be as the hyperbolic area DESR.
COR. 2. And if the point be assumed any how, the point G will be
found, by taking GR to GD as the velocity at the beginning to the velo
city after any space RSED is described. The point G being given, the
space is given from the given velocity : and the contrary.
Cotw 3. Whence since (by Prop. XI) the velocity is given from the given
SEC. Ilt.1 Or NATURAL PHILOSOPHY. 281
time, and (by this Prop.) the space is given from the given velocity ; the
space will be given from the given time : and the contrary.
PROPOSITION XKI. THEOREM X.
Supposing that a body attracted downwards by an uniform gravity as
cends or descends in a right line; and that the same is resisted
partly in the ratio of its velocity, and partly in the duplicate ratio
thereof: I say, that, if right lines parallel to the diameters of a circle
and an hyperbola, be drawn through the ends of the, conjugate diame
ters, and the velocities be as some segments of those parallels drawn
from a given point, the times will be as the sectors of the, areas cut
off by right lines drawnfrom the centre to the ends of the segments ;
and the contrary.
CASE 1. Suppose first that the body is ascending,
and from the centre D, with any semi-diameter DB,
describe a quadrant BETF of a circle, and through
the end B of the semi-diameter DB draw the indefi
nite line BAP, parallel to the semi-diameter DF. In
chat line let there be given the point A, and take the
segment AP proportional to the velocity. And since
one part of the resistance is as the velocity, and
another part as the square of the velocity, let the
whole resistance be as AP 2
-f 2BAP. Join DA, DP, cutting the circle
in E and T, and let the gravity be expounded by DA2
, so that the gravity
shall be to the resistance in P as DA2 to AP2+2BAP ; and the time of the
whole ascent will be as the sector EDT of the circle.
For draw DVQ,, cutting off the moment PQ, of the velocity AP, and the
moment DTV of the sector DET answering to a given moment of time ;
and that decrement PQ, of the velocity will be as the sum of the forces of
gravity DA2 and of resistance AP 2 + 2BAP, that is (by Prop. XII
BookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ:
is as DP2
, and the area DTV, which is to the area DPQ, as DT2 to DP 2
, it
as the given quantity DT2
. Therefore the area EDT decreases uniformly
according to the rate of the future time, by subduction of given particles DTV,
and is therefore proportional to the time of the whole ascent. Q..E.D.
CASE 2. If the velocity in the ascent
of the body be expounded by the length
AP as before, and the resistance be made
as AP2
-f- 2BAP,and if the force of grav
ity be less than can be expressed by DA2
;
take BD of such a length, that AB2
BD 2 maybe proportional to the gravity,
and let DF be perpendicular and equal
F O
S2 THE MATHEMATICAL PRINCIPLES [BOOK ll.
to DB, and through the vertex F describe the hyperbola FTVE, whose con
jugate semi -diameters are DB and DF; and which cuts DA in E, and DP,
DQ in T and V ; and the time of the whole ascent will be as the hyper
bolic sector TDE.
For the decrement PQ of the velocity, produced in a given particle of
time, is as the sum of the resistance AP2 -f 2BAP and of the gravity
AB2 BD2
, that is, as BP 2 BD 2
. But the area DTV is to the area
DPQ as DT2 to DP 2
; and, therefore, if GT be drawn perpendicular to
DF. as GT2 or GD 2 DF2 to BD 2
, and as GD2 to BP 2
, and, by di
vision, as DF2 to BP 2 BD 2
. Therefore since the area DPQ is as PQ,
that is, as BP 2 BD 2
, the area DTV will be as the given quantity DF 2
.
Therefore the area EDT decreases uniformly in each of the equal particles
of time, by the subduction of so many given particles DTV, and therefore
is proportional to the time. Q.E.D.
CASE 3. Let AP be the velocity in the descent of
""" the body, and AP 2 + 2BAP the force of resistance,
and BD 2 AB 2 the force of gravity, the angle DBA
being a right one. And if with the centre D, and the
principal vertex B, there be described a rectangular
hyperbola BETV cutting DA, DP, and DQ produced
in E, T, and V : the sector DET of this hyperbola will
D be as the whole time of descent.
For the increment PQ of the velocity, and the area DPQ proportional
to it, is as the excess of the gravity above the resistance, that is, as
m)2
?_ AB 2 _2BAP AP2 or BD 2 BP 2
. And the area DTV
is to the area DPQ as DT 3 to DP 2
; and therefore as GT2 or GD" -
BD 2 to BP 2
, and as GD 2 to BD 2
, and, by division, as BD 2 to BD2 -
BP2
. Therefore since the ami DPQ is as BD2 BP2
, the area DTV
will be as the given quantity BD 2
. Therefore the area EDT increases
uniformly in the several equal particles of time by the addition of as
many given particles DTV, and therefore is proportional to the time of
the descent. Q.E.D.
Con. If with the centre D and the semi-diameter DA there be drawn
through the vertex A an arc A/ similar to the arc ET, and similarly subtendino^
the angle A DT, the velocity AP will be to the velocity which the
body in the time EDT, in a non-resisting space, can lose in its ascent, or
acquire in its descent, as the area of the triangle DAP to the area of the
Bector DA/ ; and therefore is given from the time given. For the velocity
ir a non-resistin^ medium is proportional to the time, and therefore to this
sector : in a resisting medium, it is as the triangle ; and in both mediums,
where it is least, it approaches to the ratio of equality, as the sector and
triangle do
SEC. III.] OF NATURAL PHILOSOPHY. 283
SCHOLIUM.
One may demonstrate also that case in the ascent of the body, where the
force of gravity is less than can be expressed by DA2 or AB 2 + BD 2
, and
greater than can be expressed by AB 2 DB 2
, and must be expressed by
AB2
. But I hasten to other things
PROPOSITION XIV. THEOREM XL
The same things being supposed, 1 say, that the space described in the
ascent or descent is as the difference of the area by which the time is
expressed, and of some other area which is augmented or diminished
in an arithmetical progression ; if the forces compounded of the re
sistance and the gravity be taken, in a geometrical progression.
Take AC (in these three figures) proportional to the gravity, and AK
to the resistance ; but take them on the same side of the point A, if the
\*
"1
\
B A K QP
body is descending, otherwise on the contrary. Erect A b, which make to
DB as DB 2 to 4BAC : and to the rectangular asymptotes CK, CH, de
scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area
A/AK will be augmented or diminished in an arithmetical progression,
while the forces CK are taken in a geometrical progression. I say, there
fore, that the distance of the body from its greatest altitude is as the excess
of the area A6NK above the area DET.
For since AK is as the resistance, that is, as AP 2 X 2BAP ; assume
any given quantity Z, and put AK equal to then (by Lem,
