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then DO: CO≈m: n, and DC: CO—m—n:n, and Therefore any velo

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DC-2 CO, or te

n

AC and BD are as AC and BD'. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC: therefore the velocities at all the other instants x, g, D, are properly represented by the corresponding ordinates. Hence,

1. Since the abscissæ of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC xe is to Acgf as the space described in the time Cx to the space described in the time C g (1st Theorem on varied motions).

2. The rectangle ACOF is to the area ACDB as the space formerly expressed by 2 a, or E to the space described in the resisting medium during the time CD: for AC being the velocity V, and OC the extinguishing time e, this rectangle is e V, or E, or 2 a, of our former disquisitions; and because all the rectangles, such as ACOF, BDOG, &c. are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had ev E, or 2 a.

3. Draw the tangent A; then, by the hyperbola Cx-CO: now Cx is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc A a of the curve, the first impulse of the uniform action of the resistance is the same with the first impulse of its varied action. By this the velocity AC is reduced to a c. If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle ACx.

This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity.

4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the assymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned.

5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion, is to the extinguishing time alone: for AC: BD=OD (or OC+CD): OC; or V: vee+i.

6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG are equal; and therefore AVGF and BVCD are equal, and VC: VA VG: VB; thereforete

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and ext

v

-V.

7. Any velocity is reduced in the proportion of m to n in the time e

m-n

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For, let AC BD≈m : n;

:

n

m-n

n

city is reduced to one half in the time in which the initial resistance would have extinguished it by its uniform

action.

mode of

Thus may the chief circumstances of this motion be Anethe determined by means of the hyperbola, the ordinates determinand abscissæ exhibiting the relations of the times and; ing this velocities, and the areas exhibiting the relations of both motion. to the spaces described. But we may render the conception of these circumstances infinitely more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML, parallel to OD, for its abscissa, and of such a nature, that if the ordinates to the hyperbola AC, ex, fg, BD, &c. be produced till they cut this curve in L, p, n, K, &c. and the abscissa in L, s, h, d, &c. the ordinates s p, h, n, d K, &c. may be proportional to the hyperbolic areas e AC, ƒA C g, Ac K. Let us examine what kind of curve this will be.

Make OC: 0 x=0x: 0g; then (Hamilton's Conics, IV. 14. Cor.), the areas AC x e, ex gfare equal: therefore drawing ps, n t perpendicular to OM, we shall have (by the assumed nature of the curve Lp K), Ms=st; and if the abscissa OD be divided into any number of small parts in geometrical progression (reekoning the commencement of them all from O), the a.is Viof this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, p s, nt, &c. in geometrical progression, and its abscisse in arithmetical progres

sion.

Also, let KN, MV touch the curve in K and L, and let OC be supposed to be to O c, as OD to O d, and therefore Ce to D d as OC to OD; and let these lines Cc, D d be indefinitely small; then (by the nature of the curve) Lo is equal to Kr: for the areas a AC c, b BD d are in this case equal. Also ko is to k r, as LM to KI, because c Cd D=CO: DO:

Therefore IN: IK≈r Krk
IK : ML=rk ;ol
ML: MV ol : 0 L
and IN: MN=rK: 0 L

That is, the subtangent IN, or MV, is of the same magnitude, or is a constant quantity in every part of the

curve.

Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate Kas the rectangle BDOG or ACOF to the parabolic arex BDCA.

For let fg hn be an ordinate very near to BDK; and let n cut the curve in a, and the ordinate Kl in 9; then we have

Kq: qn KI: IN, or
Dg:gn=DO: IN;

but BD: AC CO: DO;
therefore B D.Dg: AC.qn=CO: IN.

Therefore the sum of all the rectangles BD. Dg is to the sum of all the rectangles AC. qn, as CO to IN;

but

56

The whole

but the sum of the rectangles BD. Dg is the space
ACDB; and, because AC is given, the sum of the rect-
angles AC.qn is the rectangle of AC and the sum of
all the lines qn; that is, the rectangle of AC and RL :
therefore the ACDB: AC. RL=CO; IN, and
ACDBXIN=AC. CO. RL; and therefore IN: RL
=AC.CO: ACDB.

space

Hence it follows that QL expresses the area BVA, and in general, that the part of the line parallel to OM, which lies between the tangent KN and the curve LpK, expresses the corresponding area of the hyperbola which lies without the rectangle BDOG.

And now, by the help of this curve, we have an easy way of convincing and computing the motion of a body through the air. For the subtangent of our curve now represents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion.

But it remains to make another observation on the reduced to curve L pK, which will save us all the trouble of graphical operations, and reduce the whole to a very cal compu- simple arithmetical computation. It is of such a na

a simple arithmeti

ation.

ture, that when MI is considered as the abscissa, and is
divided into a number of equal parts, and ordinates are
drawn from the points of division, the ordinates are a
series of lines in geometrical progression, or are conti-
nual proportionals. Whatever is the ratio between the
first and second ordinate, there is the same between the
second and third, between the third and fourth, and so
on; therefore the number of parts into which the ab-
scissa is divided is the number of these equal ratios
which is contained in the ratio of the first ordinate to
the last For this reason, this curve has got the name
of the logistic or logarithmic curve; and it is of immense
use in the modern mathematics, giving us the solution
of many problems in the most simple and expeditious
manner, on which the genius of the ancient mathema-
ticians had been exercised in vain. Few of our readers
are ignorant, that the numbers called logarithms are of
equal utility in arithmetical operations, enabling us
not only to solve common arithmetical problems with
astonishing dispatch, but also to solve others which are
quite inaccessible in any other way. Logarithms are no-
thing more than the numerical measures of the abscissa
of this curve, corresponding to ordinates, which are
measured on the same or any other scale by the natural
numbers; that is, if ML be divided into equal parts,
and from the points of division lines be drawn parallel to
MI, cutting the curve Lp K, and from the points of
intersection ordinates be drawn to MI, these will divide
MI into portions, which are in the same proportion to
the ordinates that the logarithms bear to their natural
numbers.

In constructing this curve we were limited to no par-
ticular length of the line LR, which represented the
and all that we had to take care of was,
space ACDB;
that when OC, Ox, O g were taken in geometrical pro-
gression, M s, Mt should be in arithmetical progression.
The abscisse having ordinates equal tops, nt, &c.might
have been twice as long, as is shown in the dotted curve
which is drawn through L. All the lines which serve to
measure the hyperbolic spaces would then have been
doubled. But NI would also bave been doubled, and

our proportions would have still held good; because this
subtangent is the scale of measurement of our figure, as
E or 2a is the scale of measurement for the motions.

Since then we have tables of logarithms calculated
for every number, we may make use of them instead
of this geometrical figure, which still requires consi-
derable trouble to suit it to every case. There are two
sets of logarithmic tables in common use. One is call-
It is
ed a table of hyperbolic or natural logarithms.
suited to such a curve as is drawn in the figure, where
the subtangent is equal to that ordinate which cor-
responds to the side O of the square λ O inserted be-
tween the hyperbola and its assymptotes. This square
is the unit of surface, by which the hyperbolic areas
are expressed; its side is the unit of length, by which
the lines belonging to the hyperbola are expressed; v
is=1, or the unit of numbers to which the logarithms
are suited, and then IN is also 1. Now the square
Ox being unity, the area BACI will be some number;
O being also unity, OD is some number: Call it x.
Then, by the nature of the hyperbola, OB: 0 %=
: DB: That is, x: 1=1:- so that DB is
Now calling Ddx, the area BD db, which is the
fluxion (ultimately) of the hyperbolic area, is Now
in the curve Lp K, MI has the same ratio to NI that
BACD has to λO: Therefore, if there be a scale of
which NI is the unit, the number on this scale cor-
responding to MI has the same ratio to 1 which the
number measuring BACD has to I; and I, which
corresponds to BD db, is the fluxion (ultimately) of
MI: Therefore, if MI be called the logarithm of x,

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I

x

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I

1

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is properly represented by the fluxion of MI. In short the line MI is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as

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viz. the fluxion

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E, for a ball of cast-iron one inch diameter, and if it spaces, showing the motion during each successive se

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cond; the fourth column is the velocity at the end of the time t; and the last column is the differences of ve locity, showing its diminution in each successive second. We see that at the distance of 1000 yards the velocity is reduced to one half, and at the distance of less than a mile it is reduced to one-third.

II. It may be required to determine the distance at which the initial velocity V is reduced to any other quantity v. This question is solved in the very same manner, by substituting the logarithms of V and u for those of e+t and e; for AC: BD=OD: OC, and AC OD V e+t OC'

therefore log.log. or log.

-

log.

e

Thus it is required to determine the distance in which the velocity 1780 of a 24 pound ball (which is the medium velocity of such a ball discharged with 16 pounds of powder) will be reduced to 1500.

Here d is 5.68, and therefore the logarithm of 2 ad is

by hyperbolic logarithms S=2ad X log.

e+t

e

+3.78671

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+8.87116

v

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Let the ball be a 12 pounder, and the initial velocity Log. 0.43429

be 1600 feet, and the time 20 seconds. We must first

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This reduction will be produced in about of a se-
cond.

+3.03236
III. Another question may be to determine the time
+0.65321 which a ball, beginning to move with a certain veloci
-3.20415 ty, employs in passing over a given space, and the dimi
nution of velocity which it sustains from the resistance
0.48145 of the air.

We may proceed thus:
2ad: S=0,43429 : log. +t, =

1.36229

0.48145

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=t. Then to log.

add log. e, and we obtain log. e+t, and e+t; from which if we take e we have t. Then to find v, say e+t:e= =V : V.

riment

Mr

We shall conclude these examples by applying this app +3.68557 last rule to Mr Robins's experiment on a musket bullet of of an inch in diameter, which had its velocity re +9.9449° duced from 1670 to 1425 by passing through 100 feet see 9.63778 of air. This we do in order to discover the resistance which it sustained, and compare it with the resistance to r a velocity of 1 foot per second.

OC=AC: BD, or e+t: eV: v. 23",03: 3′′,03=1600: 2101, v.

3.99269

The ball has therefore gone 3278 yards, and its velocity is reduced from 1600 to 210.

It may be agreeable to the reader to see the gradual progress of the ball during some seconds of its motion.

T. S. Diff. V. Diff.

We must first ascertain the first term of our analogy." The ball was of lead, and therefore 2a must be multiplied by d and by m, which expresses the ratio of the density of lead to that of cast iron. dis 0.75, and m is

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and 2adm=1274.2

3" 3336

744

4" 4080

690

645

804 114 86

5" 4725

604

569

67

537

6" 5294 The first column is the time of the motion, the second is the space described, the third is the differences of the

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59 ecapita stion.

62
remains t, o".062, or of a second, for the time

1000

of passage. Now, to find the remaining velocity, say
825.763=1670: 1544, =v.

But in Mr Robins's experiment the remaining velo-
city was only 1425, the ball having lost 245; whereas
by this computation it should have lost only 126. It
appears, therefore, that the resistance is double of what
it would have been if the resistance increased in the
duplicate proportion of the velocity. Mr Robins says
it is nearly triple, But he supposes the resistance to
slow motions much smaller than his own experiment, so
often mentioned, fully warrants.

The time e, in which the resistance of the air would
extinguish the velocity is o".763. Gravity, or the
1670
weight of the bullet, would have done it in
32

or 52";
therefore the resistance is 52 times, or nearly 68 times
0.763

its weight, by this theory, or 5.97 pounds. If we cal-
culate from Mr Robins's experiment, we must say log.
0.43429100: e V, which will be 630.23, and

V

v

e=

630.23
52
"3774, and gives 138 for the
1670
0.3774
proportion of the resistance to the weight, and makes
the resistance 12.07 pounds, fully double of the other.
It is to be observed that with this velocity, which
greatly exceeds that with which the air can rush into
a void, there must be a statical pressure of the atmo-
sphere equal to 6 pounds. This will make up the dif-
ference, and allows us to conclude that the resistance
arising solely from the motion communicated to the air
follows very nearly the duplicate proportion of the ve-
locity.

The next experiment, with a velocity of 1690 feet,
gives a resistance equal to 157 times the weight of the
bullet, and this bears a much greater proportion to the
former than 1690' does to 1670', which shows, that
although these experiments clearly demonstrate a pro-
digious augmentation of resistance, yet they are by no
means susceptible of the precision which is necessary
for discovering the law of this augmentation, or for a
good foundation of practical rules; and it is still great-
ly to be wished that a more accurate mode of investi
gation could be discovered.

Thus we have explained, in great detail, the principles and the process of calculation for the simple case of the motion of projectiles through the air. The learned reader will think that we have been unreasonably prolix, and that the whole might have been comprised in less room by taking the algebraic method. We acknowledge that it might have been done even in a few lines.

followed the proportion of the hyperbolic areas, we
shewed the nature of another curve, where lines could
be found which increase in the very same manner as the
path of the projectile increases; so that a point describ-
ing the abscissa MI of this curve moves precisely as
the projectile does. Then, discovering that this line is
the same with the line of logarithms on a Gunter's
scale, we shewed how the logarithm of a number really
represents the path or space described by the projectile.

Having thus, we hope, enabled the reader to con-
ceive distinctly the quantities employed, we shall leave
the geometrical method, and prosecute the rest of the
subject in a more compendious manner.

60

We are, in the next place, to consider the perpendi- Of the percular ascents and descents of heavy projectiles, where pendicular the resistance of the air is combined with the action of ascents of heavy projectiles. gravity and we shall begin with the descents.

Let u, as before, be the terminal velocity, and g the
accelerating power of gravity: When the body moves
and
with the velocity u, the resistance is equal to gi
in every other velocity v, we must have ua: v2 = g :
g va
=r, for the resistance to that velocity. In the
u
descent the body is urged by gravity g, and opposed
by the resistance therefore the remaining acce-

9

g v
u2

:

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its hyperbolic logarithm. Therefore S-
L√u2—v2+C. Where I means the hyperbolic lo-
garithm of the quantity annexed to it, and ▲ may be
used to express its common logarithm. (See article
FLUXIONS).

The constant quantity C for completing the fluent
is determined from this consideration, that the space
described is o, when the velocity is o: therefore C-
u3
-XL √uo, and C = XL √
and the

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g

g

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u2 u—

But we have observed, and our observation has
been confirmed by persons well versed in such subjects,
that in all cases where the fluxionary process introduces
the fluxion of a logarithm, there is a great want of di- complete fluent S =
stinct ideas to accompany the hand and eye. The so-
lution comes out by a sort of magic or legerdemain, we
cannot tell either how or why. We therefore thought
it our duty to furnish the reader with distinct conceptions
of the things and quantities treated of. For this reason,
after showing, in Sir Isaac Newton's manner, how the
spaces described in the retarded motion of a projectile
VOL. XVII. Part II.

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= ·x L

g

Χλ

0.43429 g
or (putting M for 0.43429, the modulus or subtangent

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This equation establishes the relation between the space fallen through, and the velocity acquired by the gS fall. 2g S

u

We obtain by it =L

u2

=L.

v

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and

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is 14850, and 323,62
689,34

or, which is still more conveni- 0,46947, which is the sine of 28°. The logarithmic
secant of this arch is 0,05407. Now M or 0,43429:
MX 2g S
0,05407=14850: 1848, the height wanted.

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u2

-*

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ent for us,
that is, equal to
the logarithm of a certain number: therefore having
found the natural number corresponding to the fraction
MX 2gS
mic secants.
consider it as a logarithm, and take out the
garithmic sine is
Add to this the log. of u

2. Required the velocity acquired by the body by falling 1848 feet. Say, 14850 1848 = 0,43429: 0,05407. Look for this number among the logarithIt will be found at 28°, of which the lo

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n is equal to

u1 u2-va

number corresponding to it: call this n. Then, since we have nu—n v3 — u3, and nu2—u3— n v3, or n v2 = ua × n—v, and v3 u2 xn―I

=

n

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is the logarithm of 323,62, the velocity required. We may observe, from these solutions, that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expres sed by the sine of an arch of which the terminal velocity is the radius. We cannot help taking notice here Erronect of a very strange assertion of Mr Muller, late professor assertion of of mathematics and director of the royal academy at Mr Maler, Woolwich. He maintains, in his Treatise on Gunnery, his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the air with a greater velocity than this; and he makes this a fundamental principle, on which he establishes a theory of motion in a resisting medium, which he asserts with great confidence to be the only just theory; saying, that all the investigations of Bernoulli, Euler, Robins, Simpson, and others, are erroneous. We use this strong expression, because, in his criticisms on the works of those celebrated mathematicians, he lays aside good manners, and taxes them not only with ignorance, but with dishonesty, saying, for instance, that it required no small dexterity in Robins to confirm by his experiments a theory founded on false principles; and that 5.67688 Thomas Simpson, in attempting to conceal his obliga2.83844 tions to him for some valuable propositions, by chan1.50515 ging their form, had ignorantly fallen into gross errors. 3.26670 Nothing can be more palpably absurd than this asserA blown bladder will have but a +1.44396 tion of Mr Muller. +5.26670 small terminal velocity; and when moving with this 5.67688 velocity, or one very near it, there can be no doubt that it will be made to move much swifter by a smart stroke. Were the assertion true, it would be impossible for a portion of air to be put into motion through the rest, for its terminal velocity is nothing. Yet this author makes this assertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth Robins and others are grossly mistaken, when they give of a cannon with a greater velocity than this; and that them velocities three or four times greater, and resistances which are 10 or 20 times greater than is possible; The process of this solution suggests a very perspicu- and by thus compensating his small velocities by still perspicu-maller resistances, he confirms his theory by many ex

We shall take an example of a ball whose terminal velocity is 689 feet, and ascertain its velocity after a fall of 1848 feet. Here,

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