Therefore tural sines, or for log. among the logarithmic sines,

In the next place, let a body, whose terminal velothe ascent city is u, be projected perpendicularly upwards, with of a body velocity V. It is required to determine the height to which it ascends, so as to have any remaining velocity, and the time of its ascent; as also the height and time in which its whole motion will be extinguished.

projected perpendicularly.

26

terminal velocity, or V=u; then v= If V=
689, v=487.

We have now
for both gravity and resistance act now in the same di-
rection, and retard the motion of the ascending body:
therefore (2+v2)

We must in the last place ascertain the relation of g(u2+v3) for the expression of f; the space and the time.

g(

Here £(u*+v3) ;=~

(art. FLUXIONS) is an arch whose tangent

i=—v, and i——

=

g

u

S= -v v, and s

ยย

+C, =

v v
u2+v2
·×L√u2+v2+

and t =

แข

g

with that with which

the ground

and the complete fluent will be s≈

u2 + V3
u2+v2
Let h be the greatest height to which the body will

rise.

Then sh when vo; and # + Va u

u

L

=

X λ
Mg

h=- X
g
We have

u

ent is tx (arc. tan.

u

tities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore ab

We learn from this expression of the time, that however great the velocity of projection, and the height, to which this body will rise, may be, the time of its mited. ascent is limited. It never can exceed the time of falling from the height a in vacuo in a greater proportion than that of a quadrantal arch to the radius, nearly the proportion of 8 to 5. A 24 pound iron ball cannot continue rising above 14 seconds, even if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than 12 seconds, let its velocity be ever so great.

In the preceding example of the whole ascent, e=0,

and

68

This time

compared

in bodies projected in air and

in vacuo.

Fig. 6.

Cor. 1. The time in which a body, projected in the air with any velocity V, will attain its greatest height, is to that in which it would attain its greatest height in vacuo, as the arch whose tangent expresses the velocity is to the tangent; for the time of the ascent in the air V

บ

is x arch; the time of the ascent in vacuo is

g

g

V

X tan.

g

Now

istan. and Vux tan. and
g g
It is evident, by inspecting fig. 6. that the arch AI
is to the tangent AG as the sector ICA to the tri-
angle GCA; therefore the time of attaining the great-
est height in the air is to that of attaining the greatest
height in vacuo (the velocities of projection being the
same), as the circular sector to the corresponding tri-
angle.

If therefore a body be projected upwards with the
terminal velocity, the time of its ascent will be to the
time of acquiring this velocity in vacuo as the area of a
circle to the area of the circumscribed square.

2

2. The height H to which a body will rise in a void,
is to the height h to which it would rise through the
air when projected with the same velocity V as M.V' to
u2 + V2
κ' Χλ
for the height to which it will rise in

rate notions of the air's resistance. Mr Robins's me-
thod with the pendulum is impracticable with great
shot; and the experiments which have been generally
resorted to for this purpose, viz. the ranges of shot and
shells on a horizontal plane, are so complicated in them-
subject to such irregularities, that they may be brought
selves, that the utmost mathematical skill is necessary
for making any inferences from them; and they are
to support almost any theory whatever on this subject.
But the perpendicular flights are affected by nothing
but the initial velocity and the resistance of the air;

and a considerable deviation from their intended direc

tion does not cause any sensible error in the conse-
quences which we may draw from them for our purpose.

70

71

But we must now proceed to the general problem, of obto determine the motion of a body projected in any di-lique prorection, and with any velocity. Our readers will be-jection. lieve beforehand that this must be a difficult subject, when they see the simplest cases of rectilineal motion abundantly abstruse: it is indeed so difficult, that Sir Isaac Newton has not given a solution of it, and has This prothought himself well employed in making several ap-blem not proximations, in which the fertility of his genius appears solved by in great lustre. In the tenth and subsequent proposi- Newton. tions of the second book of the Principia, he shows what state of density in the air will comport with the motion of a body in any curve whatever: and then, by applying this discovery to several curves which have some similarity to the path of a projectile, he finds one which is not very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations, that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes which he had committed in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which. a body will describe in our atmosphere. This is somewhat surprising. In prop. 14. &c. he shows how a body, actuated by a centripetal force, in a medium of a V density varying according to certain laws, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceedingly small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation, that mathematicians have adopted the other hypothesis.

and the height to which it rises in the air is 22 + V1

;

therefore H: h=

V1
:

2g

Mg

Mg

=V1: X 2λ
M

=M•V1 : u3×λ

WE have been thus particular in treating of the perpendicular ascents and descents of heavy bodies through the air, in order that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connection in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can deterNecessity mine with any confidence. On this account it were to of further be wished that the professional gentlemen would make experiments. many experiments on these motions. There is no way that promises so much for assisting us in forming accu

69

72

Soon after the publication of this second edition of Disputes the Principia, the dispute about the invention of the among fluxionary calculus became very violent, and the great British and promoters of that calculus upon the continent were in foreign the habit of proposing difficult problems to exercise the ticians. talents of the mathematician. Challenges of this kind frequently passed between the British and foreigners.

Dr

mathema

73

Bernoulli's

solution.

Fig. 7:

would cause the body to describe uniformly in the time
with the velocity which it generates in that time.
Let this be resolved into n N, by which it deflects the body
into a curvilineal path, and mn, by which it retards the
ascent and accelerates the descent of the body along the
tangent. The resistance of the air acts solely in retard-
ing the motion, both in ascending and descending, and
has no deflective tendency. The whole action of gra-
vity then is to its accelerating or retarding tendency as
m N to mn, or (by similarity of triangles) as m M to
mo. Orx:y=g:
8y
and the whole retardation in

2

gy

the ascent will be r+ The same fluxionary symbol

Dr Keill of Oxford had keenly espoused the claim of
Sir Isaac Newton to this invention, and had engaged in
a very acrimonious altercation with the celebrated John
Bernoulli of Basle. Bernoulli had published in the Acta
Eruditorum Lipsia an investigation of the law of forces,
by which a body moving in a resisting medium might
describe any proposed curve, reducing the whole to the
simplest geometry. This is perhaps the most elegant
specimen which he has given of his great talents.
Dr
Keill proposed to him the particular problem of the
trajectory and motion of a body moving through the
air, as one of the most difficult. Bernoulli very soon
solved the problem in a way much more general than it
had been proposed, viz. without any limitation either of
the law of resistance, the law of the centripetal force, or
the law of density, provided only that they were regular,
and capable of being expressed algebraically. Dr Brook
Taylor, the celebrated author of the Method of Incre-
ments, solved it at the same time, in the limited form,
in which it was proposed. Other authors since that
time have given other solutions. But they are all (as
indeed they must be) the same in substance with Ber-
noulli's. Indeed they are all (Bernoulli's not excepted)
the same with Newton's first approximations, modified
by the steps introduced into the investigation of the
spiral motions mentioned above; and we still think it
most strange that Sir Isaac did not perceive that the
variation of curvature, which he introduced in that inv
vestigation, made the whole difference between his ap-
proximations and the complete solution. This we shall
point out as we go along. And we now proceed to the
problem itself, of which we shall give Bernoulli's solu-
tion, restricted to the case of uniform density and a re-
sistance proportional to the square of the velocity.
This solution is more simple and perspicuous than any
that has since appeared.

PROBLEM. To determine the trajectory, and all the
circumstances of the motion of a body projected
through the air from A (fig. 7.) in the direction AB,
and resisted in the duplicate ratio of the velocity.
Let the arch AM be putz, the time of describing
it t, the abscissa AP=x, the ordinate PM=y. Let
the velocity in the point Me, and let MN=%, be
described in the moment t; let r be the resistance of
the air, g the force of gravity, measured by the ve-
locity which it will generate in a second; and let a be
the height through which a heavy body must fall in va-
cuo to acquire the velocity which would render the
resistance of the air equal to its gravity: so that we have
r=; because, for any velocity u, and producing

will express the retardation during the descent, because in the descent the ordinates decrease, and y is a negative quantity.

The diminution of velocity is . This is proportional to the retarding force and to the time of its action

jointly, and therefore-vr+xi; but the time

is as the space z divided by the velocity v; therefore
rx+gy
v=r+= x =

B

and

บ

gy. Because m N is the deflection

by gravity, it is as the force g and the square of the time
t jointly (the momentary action being held as uniform),
We have therefore m N, or -
-ÿ=g?. (Observe that
m N is in fact only the half of-y; but g being twice
the fall of a heavy body in a second, we have -y strict-
༧
ly equal to g). But = ; therefore

--

y=

=g. The fluxion of

gy; therefore

૪

2 a

2, or a y=zy, for the fluxionary equation of the