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Dr Keill of Oxford had keenly espoused the claim of would cause the body to describe uniformly in the time Sir Isaac Newton to this invention, and bad engaged in i with the velocity which it generates in that time. a very acrimonious altercation with the celebrated John

Let this be resolved into n N, by which it defiects the body Bernoulli of Basle. Bernoulli had published in the Acta

into a curvilineal path, and mn, by which it retards the Eruditorum Lipsiæ an investigation of the law of forces,

ascent and accelerates the descent of the body along the by which a body moving in a resisting medium might tangent. The resistance of the air acts solely in retarddescribe any proposed curve, reducing the whole to the ing the motion, both in ascending and descending, and simplest geometry. This is perhaps the most elegant bas no deflective tendency. The whole action of graspecimen which he has given of his great talents. Dr Keill proposed to him the particular problem of the

vity then is to its accelerating or retarding tendency as

m N to mn, or (by similarity of triangles) as m M to trajectory and motion of a body moving through the air, as one of the most difficult. Bernoulli very soon

SY

and the whole retardation in 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

gy

the ascent will be rt. The same fluxionary symbol the law of density, provided only that they were regular, and capable of being expressed algebraically. Dr Brook will express the retardation during the descent, because Taylor, the celebrated author of the Method of Incre- in the descent the ordinates decrease, and y is a negaments, solved it at the same time, in the limited form tive quantity. in which it was proposed. Other authors since that

The diminution of velocity is - i. This is proportime have given other solutions. But they are all (as indeed they must be) the same in substance with Ber

tional to the retarding force and to the time of its action noulli's. Indeed they are all (Bernoulli's not excepted) the same with Newton's first approximations, modified jointly, and therefore - v=r+%Yxi; but the time by the steps introduced into the investigation of the

i is spiral motions mentioned above ; and we still think it

z divided by the velocity v; therefore most strange that Sir Isaac did not perceive that the

gy =r+

rx +gy variation of curvature, wbich he introduced in that in

Х

and
vestigation, made the whole difference between his ap-
proximations and the complete solution. This we shall

02

gy. Because m N is the deflection point out as we go along. And we now proceed to the Bernoulli's

problem itself, of which we shall give Bernoulli's solu- by gravity, it is as the force g and the square of the time tion, restricted to the case of uniform density and a re

jointly (the momentary action being held as uniform), sistance proportional to the square of the velocity. This solution is more simple and perspicuous than any

We have therefore m N, or — y=gi. (Observe that that has since appeared.

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 strictPROBLEM. To determine the trajectory, and all the circuinstances of the motion of a body projectedly equal to gi). a

; therefore

y= Fig. 7:

through the air from A (fig. 7.) in the direction AB,
and resisted in the duplicate ratio of the velocity.

and was and – 22 j = 57°. The fluxion of

ve ?.
Let the arch AM be put=%, the time of describing
it t, the abscissa AP=x, the ordinate PM=y. Let

this equation is q? — 2 vyö = 2g ; but, bethe velocity in the point M=v, and let MN=%, be

- % described in the moment i ; let r be the resistance of cause % : y = mM:mo,= mN:mn,=y: %, we bave the air, g the force of gravity, measured by the ve

% % = y y. Therefore 2 g y y = 28 % 2,= - v'y –

= = 3 3
locity which it will generate in a second ; and let a be
the height through which a heavy body must fall in va- 2 výv, and — 2 vůj = v*7 — 2 g ý ý, and -

y y
cuo to acquire the velocity which would render the
resistance of the air equal to its gravity: so that we have

g ý. But we have already – vė
va
= ; because, for any velocity w, and producing

2 y

and finally = height h, we have

8
zh

y

y Let M m touch the curve in M; draw the ordinate 2, or a y = x y, for the fluxionary equation of the m, and draw M 0, N n perpendicular to Np and Then we have MN =ž, and Mo=, also mo

75 is ultimately =y and M m is ultimately=MN or z.

If we put this into the form of a proportion, we Relation 74 Action of Lastly, let us suppose x to be a constant quantity, the

have a : % =y: y. Now this evidently establishes a regravity in elementary ordinates being supposed equidistant.

lation between the length of the curve and its variation a given The action of gravity during the time i may be

i of curvature ; and between the curve itself and its evo- its rasistime. measured by m N, which is half the space which it luta, which are the very circumstances introduced by tion of

3

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q

19=föVi+p+C, C being the constant quantity perpendicular to the asymptote, and BC parallelerefore
aqfivi
miting conditions of the case. Now :=
fiviteite. Thereforeiz

Newton into his investigation of the spiral motions. And 2d, We get x by the area of another curve whose

1 the equation is evidently an equation connected

abscissa is p, and the ordinate is y

9 y

get y by the area of a third curve whose abwith the logarithmic curve and the logarithmic spiral. But we must endeavour to reduce it to a lower order of scissa is p, and the ordinate is

9. fluxions, before we can establish a relation between %, x',

The problem of the trajectory is therefore complete

ly solved, because we have determined the ordinate, abLet p express the ratio of ý to x, that is, let p be=

scissa, and arch of the curve for any given position of 76

its tangent. It now only remains to compute the magor p x=y. It is evident that this expresses the nitudes of these ordinates and abscissæ, or to draw them pute the

magnitude

by a geometrical construction. But in this consists the of the ordiinclination of the tangent at M to the horizon, and that p is the tangent of this inclination, radius being unity: lengths of * and y, can neither be computed nor exhi-abscissa.

difficulty. The areas of these curves, which express the nate and Or it may be considered merely as a number, multiply- bited geometrically, by any accurate method yet discoing x, so as to make it =y. We now have yü = p***, vered, and we must content ourselves with approximaand since z•= x3 + y, we have ji = ** + pez", =

tions. These render the description of the trajectory i tp x *, and =3 vitp.

exceedingly difficult and tedious, so that little advantage

bas as yet been derived from the knowledge we have got Moreover, because we have supposed the abscissa x

of its properties. It will however greatly assist our conto increase uniformly, and therefore å to be constant, ception of the subject to proceed some length in this we have y=x P, and

v=x p.
Now let

construction ; for it must be acknowledged that very

few distinct notions accompany a mere algebraic opera

P ratio of p to é, that is, make = q, or q * = 'p. tion, especially if in any degree complicated, which we

confess is the case in the present question.

Let B m NR (fig. 8.) be an equilateral hyperbola, of Plate This gives us * q=p, and x* q=* P, =y.

which B is the vertex, BA the semitransverse axis, CCCCXLII. By these substitutions our former equation a y=%y

which we shall assume for the unity of length. Let AV changes to a zog=inīt p*l* p, or a y=p

be the semiconjugate axis =BA, = unity, and AS the

assymptote, bisecting the right angle BAV. Let PN, vit p*, and, taking the fluent on both sides, we have pn be two ordinates to the conjugate axis, exceedingly

near to each other. Join BP, AN, and draw BB, N,

, to AP required for completing the fluent according to the li

is to .
PN: =BA? + AP. Now since BA=1, if we make
, and
9 9 AP=p of our formulæ, PN is Vi+p®, and Ppis=
ар

, and the area BAPNB:
+1+c

BAPNB=f; i Vi+pT: That is to
I+P+
Also, since y=p3, =pP, we have y=

PP
y=

say, the number f; Vi+po(foritisa number) has
,

p
9
app

the same proportion to unity of number that the area

BAPNB has to BCVA, the unit of surface. This Pri+pel+C

area consists of two parts, the triangle APN, and the op vitp Also :=: Vitpl=

hyperbolic sector ABN. APN = ; AP X PN, =

pi+p?, and the hyperbolic sector ABN=BN »B,
for
Vi+*+C which is equivalent to the hyperbolic logarithm of the

number represented by A , when A B is unity. ThereThe values of x, y, %, give us

fore it is equal to the logarithm of p +di+p.

P
ap
p+

1 +

\pi+p* + hyperbolic logarithm pt vi+po.

Now let AMD be another curve, such that its ordi

pp

nates V m, PD, &c. may be proportional to the areas itp*l+C

AB m V, ABNP and may have the same proportion

to AB, the unity of length, which these areas have to i

ABCV, the unity of surface. Then VM : VC=

V m BA : VCBA, and PD : Pd=PNBA : VCBA, for Vi+P+C The process therefore of describing the trajectory is, ist, with reference to a linear unit, as the areas to the to find q in terms of p by the area of the curve whose hyperbola represented it in reference to a superficial abscissa is p and the ordinate is vitp.

unit.

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Again, in every ordinate make PD:P)=P): PO, will describe similar trajectories if the velocities are in and thus we obtain a reciprocal to PD, or to the subduplicate ratio of the diameters. This we shall

find to be of considerable practical importance. But let or equivalent to

This
P 1+pa

us now proceed to determine the velocity in the differ- relocini de

ent points of the trajectory; and the time of describing diferen? will evidently be ---, and POo p will be

and the area

its several portions. ар

gx contained between the lines AF, AW, and the curve Recollect, therefore, that vis and that je GEOH, and cut off by the ordinate PO, will represent

Y

= **1+p* and j = xp. This gives viz-gxi +8". Lastly, make PO ; PQ=AV: AP, =1:7; and

But p=qx. Therefore v=-5x1+pa

-g
y
then PQqp will represent, and the area ALEQP

9
agitp

and v=M

-git pa will represent

P itpl+C

9 But we must here observe, that the fluents expressed

vad-g/i+p
by these different areas require what is called the cor-

itp
rection to accommodate them to the circumstances of fi Vī+p/+C
the case. It is not indifferent from what ordinate we
begin to reckon the areas. This depends on the initial Also i was found =
direction of the projectile, and that point of the abscis-
sa AP must be taken for the commencement of all the ė Vitpa
areas which gives a value of p suited to the initial di-

now substitute for v its value rection. Thus, if the projection has been made from

9v Fig. 7. A (fig. 7.) at an elevation of 45°, the ratio of the just found, we obtain i

and t =

P
fluxions x and y is that of equality; and therefore the
Fig. 8.
point E of fig. 8. where the two curves intersect and

pva
bave a common ordinate, evidently corresponds to this
condition. The ordinate EV passes through V, so that

Vitpitc
AV or p=AB, =1, = tangent 45°, as the case re-
quires. The values of x and of y corresponding to any
other point of the trajectory, such as that which has AP
for the tangent of the apgle which it makes with the

p
horizon, are now to be bad by computing the areas
VEOP, VEQP.
Another curve might have been added, of which the modating these formulæ, which appear abundantly sim- of accos-

The greatest difficulty still remains, viz. the accom

Difficulty ordinates would exhibit the fluxions of the arch of the ple, to the particular cases. It would seem at first modating

apitpa trajectory x =

and of which the area sight, that all trajectories are similar; since the ratio of the fornisf,i

Vi+P+C
p

the fluxions of the ordinate and abscissa corresponding to da to para would exhibit the arch itself. And this would bave the same in them all : but a due attention to what bas

any particular angle of inclination to the borizon seems pitpa

been hitherto said on the subject will show us that we been very easy, for it is 2 za

have as yet only been able to ascertain the velocity in f, PVI++C

the point of the trajectory, which has a certain inclinawhich is evidently the fluxion of the byperbolic loga- tion to the horizon, indicated by the quantity P, and the

time (reckoned from some assigned beginning) when the

projectile is in that point. Å vi+p, and we have already got a. It is only in-,

To obtain absolute measures of these quantities, the creasing PO in the ratio of BA to BP.

term of commencement must be fixed upon. This will be Consequen- And thus we have brought the investigation of this expressed by the constant quantity C, which is assumed ces of

problem to considerable length, having ascertained the for completing the fluent of ; i+p?, which is the the form form of the trajectory. This is surely done when the basis of the whole construction. We there found q =

ratio of the arch, absciss, and ordinate, and the position jectory of its tangent, is determined in every point. But it is

. still very far from a solution, and much remains to be

This fluent is in general q = done before we can make any practical application of it. The only general consequence that we can the premises is, that in every case where the resistance

quantity C is to in any point bears the same proportion to the force of gravity, the trajectory will be similar. Therefore, two be accommodated to some circumstances of the case. balls, of the same density, projected in the same direction, Different authors have selected different circumstances. 2

Euler,

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Euler, in his Commentary on Robins, and in a disserta

va

р Euler's me-tion in the Memoirs of the Academy of Berlin publish- a X

and the thod the

P simplest.

ed in 1753, takes the vertex of the curve for the begin-
ning of his abscissa and ordinate. This is the simplest
method of any, for C must then be so chosen that the height producing the velocity at N=*a(1+pe).

n-P
whole fluent

may

vanish when p=0, which is the case in the vertex of the curve, where the tangent is paral

Hence we learn by the bye, that in no part of the Remarklel to the horizon. We shall adopt this method.

ascending branch can the inclination of the tangent be able pro-, Therefore, let AP (fig. 9.) =x, PM=y, AM=%.

such thať P shall be greater than n; and that if we sup-perty of the Put the quantity C which is introduced into the fluent pose P equal to n in any point of the curve, the velo

trajectory.

city in that point will be infinite. That is to say, there equal to . It is plain that n must be a number ; for is a certain assignable elevation

of the tangent which a

cannot be exceeded in a curve which has this velocity it must be homologous with PVī + p, which is a in the vertex. The best way for forming a conception number. For brevity's sake let us express the fluent of of this circumstance in the nature of the curve, is to Pit pe by the single letter P; and thus we shall invert the motion, and suppose an accelerating force,

р
have r=aX

equal and opposite to the resistance, to act on the body
PP
%=ax

in conjunction with gravity. It must describe the same

curve, and this branch ANC must have an assymptote
it pa
And ud =

LO, which has this limiting position of the tangent.
Lag (1 +p')

Now the
n+P

For, as the body descends in this curve, its velocity
height h necessary for communicating any velocity v is this accelerating force, and yet the tangent never ap-

increases to infinity by the joint action of gravity and
h
Lag (1+p)
-fu(1 + p)

And lastly,

proaches so near the perpendicular position as to make
2g(n+P)
28

P=n.
n+P

This remarkable property of the curve was
known to Newton, as appears by his approximations,

which all lead him to curves of a hyperbolic form, ha-
Ngo
n+P

ving one assymptote inclined to the horizon. Indeed
These fluents, being all taken 30 as to vanish at the it is pretty obvious: For the resistance increasing faster
vertex, where the computation commences, and where than the velocity, there is no velocity of projection so
p is = 0 (the tangent being parallel to the horizon), we great but that the curve will come to deviate so from
sa

the tangent, that in a finite time it will become paobtain in this case h=

2h

rallel to the horizon. Were the resistance proportional

to the velocity, then an infinite velocity would produce Hence we see that the circumstance which modifies a rectilineal motion, or rather a deflection from it less all the curves, distinguishing them from each other, is than any that can be assigned. the velocity (or rather its square) in the highest point We now see that the particular form and magnitude on what of the curve. For h being determined for any body of this trajectory depends on two circumstances, a and its form and whose terminal velocity is u, n is also determined ; and n. a affects chiefly the magnitude. Another circum-magnitude this is the modifying circumstance. Considering it geo- stance might indeed be taken in, viz. the diminution of depends. metrically, it is the area which must be cut off from the accelerating force of gravity by the statical effect of the area DMAP of fig. 8. in order to determine the the air's gravity. But, as we have already observed, this ordinates of the other curves.

· is too trifling to be attended to in military projectiles. We must further remark, that the values now given

y relate only to that part of the area where the body is

was made equal to . Therefore the ra

p
descending from the vertex. This is evident ; for, in
order that y may increase as we recede from the vertex,

dius of curvature, determined by the ordinary me-
its fluxion must be taken in the opposite sense to what

*(1+p*) (1+p) it was in our investigation. There we supposed y to in thods, is

, and, because

is * Simpson crease as the body ascended, and then to diminish du

P

P

Flurions, ring the descent; and therefore the fluxion of y was first

ý 6s, &c.

nip for the descending branch of the curve, the
positive and then negative.

+P
,
the sign of P; for if we consider y as decreasing during

n+P
the ascent, we must consider

in the ascending branch at N, it is a Xi+p* x Vi+po
as expressing
9
and

1-P
On both sides, therefore, when the velocity is infinitely

9
P

must be great, and P by this means supposed to equal or exceed

n, the radius of curvature is also infinitely great. We taken negatively. Therefore, in the ascending branch, also see that the two branches are unlike each other, and we have AQ or X (increasing as we recede from A) that when p is the same in both, that is, when the tan

gent is equally inclined to the horizon, the radius of pp

curvature, the ordinate, the absciss, and the arch, are nap

all greater in the ascending branch. This is pretty obVol. XVII. Part II.

a
-
n

and n=

2n'

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vious.

n

n+Q

n

u

a

"fivit

а

ag

of n.

vious. For as the resistance acts entirely in diminishing

n+P

and Am
the velocity, and does not affect the deflection occasioned qits position at m; then AM=a x log.
by gravity, it must allow gravity to incurvate the path
so much the more (with the same inclination of its line

= ax log. nt, and therefore M m is = a x log.
=

m a
of action) as the velocity is more diminished. The

n+Q curvature, therefore, in those points which have the same inclination of the tangent, is greatest in the dea n+P. Thus we can find the values of a great numscending branch, and the motion is swiftest in the ber of small portions and the inclination of the tanascending branch. It is otherwise in a void, where both gents at their extremities. Then to each of these porsides are alike. Here u becomes infinite, or there is no tions we can assign its proportion of the abscissa and terminal velocity; and n also becomes infinite, being ordinate, without having recourse to the values of x and y.

For the portion of absciss corresponding to the arch Mm, 2h

whose middle point is inclined to the horizon in the It is therefore in the quantity P, or

angle b, will be Mm X cosine b, and the corresponding 1+p,

portion of the ordinate will be Mm X sin. b. Then we that the difference between the trajectory in a veid and

obtain the velocity in each part of the curre by the

a in a resisting medium consists; it is this

quantity which equation h= ax 1 +po; or, more directly the velocity expresses the accumulated change of the ratio of the

n+p increments of the ordinate and absciss. In vacuo the

1+p second increment of the ordinate is constant when the

v at M will be =

Lastly, divide the first increment of the abscissa is so, and the whole

Vn+P increment of the ordinate is as itp. And this diffe- length of the little arch by this, and the quotient will be rence is so much the greater as P is greater in respect the time of describing Mm very nearly. Add all these

P is nothing at the vertex, and increases along together, and we obtain the whole time of describing with the angle MTP, and when this is a right angle, the arch AM, but a little too great, because the moP is infinite. The trajectory in a resisting medium tion in the small arch is not perfectly uniform. The will come therefore to deviate infinitely from a para- error, however, may be as small as we please, because bola, and may even deviate farther from it than the

we may make the arch as small as we please ; and for parabola deviates from a straight line. That is, the di- greater accuracy, it will be proper to take the p by stance of the body in a given moment from that point which we compute the velocity, a medium between of its parabolic path where it would have been in a the P for the beginning and that for the cod of the void, is greater than the distance between that point of arch. the parabola from the point of the straight line where This is the method followed by Euler, who was one Euler's me

$4 it would have been, independent of the action of gra- of the most expert analysts, if not the very first, in Eu-thod previty. This must bappen wheneverthe resistance is great- rope. It is not the most elegant, and the methods of ferred. er than the weight of the body, which is generally the some other authors, who approximate directly tothe areas case in the beginning of the trajectory in military pro- of the curves which determine the values of x and y, have jectiles; and this (were it now necessary) is enough to a more scientific appearance; but they are not ultimateshow the inutility of the parabolic theory,

ly very different: For, in some methods, these areas are Although we have no method of 'describing this 83

taken piecemeal, as Euler takes the arch; and by the Several

trajectory, which would be received by the ancient methods of others, who give the value of the areas by properties geometers, we may ascertain several properties of it, Newton's method of describing a curve of the parabolic of it asoer- which will assist us in the solution of the problem. In kind through any number of given points, the ordinates tained. particular, we can assign the absolute length of any part

of these curves, which express x and y, must be taken of it by means of the logistic curve. For because P

singly, which amounts to the same thing, with the P

great disadvantage of a much more complicated calculus, as any one may see by comparing the expressions of x and y with the expressions of %.

As to those me fore %, which was = a X

+C,or=ax

thods which approximate directly to the areas or va1+pl

lues of x and y by an infinite series, they all, without exception, involve us in most complicated expressions,

with coefficients of sines and tangents, and ambiguous n+ po may be expressed by logarithms; or = a

signs, and engage us in a calculation almost endless.

And we know of no series which converges fast enough x hyp. log. of since at the vertex A, where x to give us tolerable accuracy, without such a number of

terms as is sufficient to deter any person from the atmust be = 0, P is also = 0.

tempt. The calculation of the arches is very modeBeing able, in this way, to ascertain the length AM rate, so that a person tolerably versant in arithmetical of the curve (counted from the vertex), corresponding operations may compute au arch with its velocity and

( to any inclination p of the tangent at its extremity M, time in about five minutes. We have therefore no bewe can ascertain the length of any portion of it, such sitation in preferring this method of Euler's to all that as Mm, by first finding the length of the part A m, and we bave seen, and therefore proceed to determine some then of the part AM. This we do more expeditiously other circumstances which render its application more thus. Let p express the position of the tangent in M, and general.

If

=fi Vi+e“, we have p Vi+P = ř, and there

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