5 70 RESISTANCE. 
and gradual, continually from the smallest (o 
the highest velocities ; and that tire increased 
real resistance no where rises higher than to 
about double of that which Newton’s theory 
gives it. 
The subject of the resistance of the air, J 
as begun by Robins, has been prosecuted by ; 
Dr. Hutton, to a very great extent and va- j 
riety, both with the whirling-machine, and i 
with cannon-balls of all sizes, from lib to j 
61b. weight, as well as with figures of many [ 
other different shapes, 
and hind part of them, 
all varieties of angles 
path or motion of tiie 
he has 'obtained the re 
for all velocities, from 
second ; together with 
a nee to the same body 
ties, and for different s 
loci tv, and also for all 
both on the fore part 
and with planes set at 
of inclination to- the 
same ; from all which 
al resistance to bodies 
1 up to 2 000 feet per 
the law of the' resist- 
for all different veloci- 
izes with the same ve- 
angles of inclination.. 
RESISTANCES OF DIFFERENT' BODIES. 
Velocity 
per 
second. 
Small 
hemis. 
Large hemisph. 
Cone. 
Resistance 
as the 
power of 
the velocity 
flat 
side. 
round 
side. 
Cylinder. 
Whol 
flat side. 
vertex. 
base. 
globe 
feet. 
oz. 
OZ. 
OZ. 
OZ. 
OZ. 
OZ. 
OZ. 
3 
.028 
.051 
.020 
.028 
.064 
.050 
.027 
4 
.048 
.096 
.039 
.048 
.109 
.090 
.047 
5 
.072 
.148 
.063 
.071 
.162 
. 143 
.068 
6 
.103 
.211 
.092 
.098 
.225 
.205 
.094 
7 
.141 
.284 
.123 
.129 
. 298 
.278 
.125 
8 
.184 
.368 
. 160 
.168 
.382 
. 360 
.162 
9 
.233 
.464 
. 199 
.211 
.478 
.456 
.205 
10 
.287 
.573 
.242 
.260 
.587 
.565 
.255 
11 
.349 
.698 
.292 
.315 
.712 
.688 
.310 
2.05" 
12 
.418 
. 836 
. 347 
. 37 6 
.850 
.826 
.370 
2.042 
13 
. 492 
.988 
.409 
.440 
1 .000 
. 979 
.435 
2 . 9SGT 
14 
.573 
1.154 
.478 
.512 
1.166 
1 . 145 
.505 
2.031 
15 
.661 
1 . 336 
.552 
.569 
1 .346 
1 .327 
-581 
2.031 
16 
.754 
1 . 538 
. 634 
. 673 
1 . 546 
1.526 
.663 
2.033 
17 
.853 
1 . 757 
• 722 
.762 
1 . 763 
1 .745 
• 752 
2.038 
18 
.959 
1 • 998 
.818 
. 858 
2.002 
1 . 986 
.848 
2.044 
19 
1 .073 
2.258 
.922 
.959 
2.260 
2.246 
• 949 
2.047 
20 
1.196 
2.542 
1.033 
1 .069 
2 . 540 
2.528 
1 .057 
2.051 
Mean 
proper. 
Nos. 
140 
288 
119 
126 
291 
285 
124 
2.040. 
1 
2 
3 
4 
5 
6 
7 
8. 
9 
diff erent distances, and so discovered by experi- 
ment the quantity of velocity lost, when passing 
through those distances, or spaces of air, with 
the several known degrees of celerity. For hav- 
ing thus known, the velocity lost or destroyed, 
in passing over a certain space, in a certain 
time, (which time is very nearly equal to the 
quotient of the space divided by the medium 
velocity between the greatest and least, or be- 
tween the velocity at the mouth of the gun and 
that at the pendulum) ; that is, knowing the 
velocity v, tiie space s, and time t ; the resisting 
force is thence easily known, being equal to 
i’v!) 
or — , where b denotes the weight of the 
ball, and V the medium velocity above-men- 
tioned. 'I he balls employed upon this occasion 
by Mr. Robins, were leaden ones, of -j— of a 
pound weight, and £ of an inch diameter ; and 
to the medium velocity of 
1600 feet, the resisfance was 11 lb., 
1065 feet, ------ it was 2A; 
b 1 
but by fire theory of Newton, before laid down, 
the former of these should be only 4^ lb., and 
the latter 21b.; so that, in the former case the 
real resistance is more than double of that by 
the theory, being increased as 9 to 22 ; and in 
the lesser velocity the increase is from 2 to 24 -, 
or as 5 to 7 only. 
Euler has shewn, that the com- 
mon doctrine of resistance answers pretty 
well when the motion is not very swift, but 
in swift motions it gives tiie resistance less 
than it ought to be, on two accounts: 1. 
Because in quick motions, the air does not 
hd up the space behind the body fast enough 
to press on the hinder parts, to counterbalance 
tiie weight of tiie atmosphere on the fore 
part. 2. Phe density of tiie air before tiie 
bail being increased by the quick motion, 
will press more strongly on the fore part, and 
so will resist more than lighter air in its na- 
tural state. He has shewn that Mr. Robins 
has restrained his rule to velocities not ex- 
ceeding i67o feet per second; whereas had 
he extended it to greater velocities, the re- 
sult must have been. erroneous ; and he gives 
another formula himself, and deduces con- 
clusions differing from those of Mr. Robins. 
Mr. Robins having proved that, in very 
great changes of velocity, the resistance does 
not accurately .follow the duplicate ratio of 
the velocity, lays down two positions, which 
he thought might be of some service in the 
practice of artillery, till a more complete and 
accurate theory of resistance, and the changes 
of its augmentation, may be obtained. The 
first of these is, that till the velocity of the 
projectile surpasses 1 100 or 1200 feet in a se- 
cond, the resistance may be esteemed to lie 
in the duplicate ratio of the velocity; and the 
second is, that when tiie velocity exceeds 
1 100 or 1200 feet, then the absolute quantity 
oi the resistance will be near 3 times as great 
as it should be by a comparison with the 
smaller velocities. Upon these principles he 
proceeds in approximating fo the actual 
ranges of pieces with small angles of eleva- 
tions* viz. such as do not ekceecl 8° or 1 O’, 
which lie sets down in a table, compared with 
their corresponding potential ranges. But 
we shall see presently that these positions are 
• i -' r l5«th without foundation ; that there is no 
such thing as a sudden or abrupt change in 
tiie law ot resistance, from the square of the 
velocity to one that gives a quantity three 
Hines as much; but that the change is slow 
In this table are contained the resistances 
to several forms of bodies, when moved with 
several degrees of velocity, from three feet 
per second to twenty. The names of the 
bodies are at the tops of the columns, as also 
which end went foremost through the air ; 
the different velocities are in the lirst column, 
and the resistances on the same line, in their 
several columns, in avoirdupois ounces and 
decimal parts. So on the first line are con- 
tained the resistances when tiie bodies move 
with a velocity of three feet in a second, viz. 
in the second column for tiie small hemi- ! 
sphere, of 4’ inches diameter, its resistance 1 
.028 ounces when (he flat side went fore- 
most ; in the third and fourth columns the 
resistances to a larger hemisphere, first with 
the flat side, and next the round side fore- 
most ; the diameter of this, as well as all the 
following figures, being 6£ inches, and there- 
fore the area of the great circle = 32 square 
inches, or 2 . of a square foot ; then in- the 
5th and 6th columns are the resistances to a 
cone, first its vertex, and then its base fore- 
most, the altitude of the cone being 6| inches, 
tiie same as the diameter of its base ; in the 
seventh column the resistance to the end of 
the cylinder, and in the eighth, that against 
the whole globe or sphere. All the numbers 
shew the real weights which are equal to the 
resistances ; and at the bottoms of the co- 
lumns are placed proportional numbers, 
which shew the mean proportions of the re- 
sistances of all the figures to one another. 
with any velocity. Lastly, in the ninth ce<- 
lumn are placed the exponents of the power 
of the velocity which the resistances in the 
eighth column bear to eacli other, viz. which 
that of the ten-feet velocity bears to each of 
the following ones, the medium of allot them 
being as the 2.04 power of the velocity, that 
is, very little above the square or second 
power of tire velocity, so far as the velocities 
in this table extend. 
From this table the following inferences 
are easily deduced: 
1. That the resistance is nearly in the same 
proportion as the surfaces ; a small increase 
only taking place in the greater surfaces, and 
for the greater velocities. Thus, by coin- 
paring together the numbers in the second 
and third columns, for the bases of the two 
hemispheres, the areas of which bases are in 
the proportion of 17f to 32,. or 5 to & very 
nearly, it appears that the numbers in those 
two columns, expressing the resistances, are 
nearly as 1 to 2 or 5 to 10, as far as the ve- 
locity of 1 2 feet ; but after that, the resist- 
ances on the greater surface increase gradu- 
ally more and more above that proportion. 
2. The resistance to the same surface 
with different velocities, is, in these slow 
motions, nearly as the square of the velocity ; 
but gradually increases more and more above 
that proportion as the velocity increases. 
This is manifest from all the columns : and 
the index of the power of the velocity is set 
down in tbe ninth column, for the resistance* 
3 
