502 
PRO 
PRO 
P R O 
And most of these expressions v/ill become much 
simpler if the first term is 0 instead of a. 
Geometrical Progression, is a series of quan- 
tities proceeding in the same continual ratio or 
proportion, either increasing or decreasing ; or 
it is a series of quantities that are continually 
proportional; or which increase by one common 
multiplier, or decrease by one common divisor ; 
which common multiplier or divisor is called 
the common ratio. As, 
increasing, 1, 2, 4, 8, 16, &c. 
decreasing, 81, 27, 9, 3, 1 , Sec . ; 
where the former progression increases continu- 
ally by the common multiplier 2, and the latter 
decreases by the common divisor 3. 
Or ascending, a, ra, r 2 a, fa, Sec. 
or descending, a, ~, &c : 
r r r 
where the first term is a , and common ratio r. 
1. Hence, the same principal properties ob- 
tain in a geometrical progression, as have been 
remarked of the arithmetical one, using only 
multiplication in the geometricals for addition 
in the arithmeticals, and division in the former 
for subtraction in the latter. So that, to con- 
struct a geometrical progression, from any 
given first term, and with a given common ra- 
tio ; multiply the 1st term continually by the 
common ratio for the rest of the terms, when 
the series is an ascending one ; or divide con- 
tinually by the common ratio, when it is a de- 
scending progression. 
2. In every geometrical progression, the pro- 
duct of the extreme terms is equal to the pro- 
duct of every pair of the intermediate terms- 
that are equidistant from the extremes, and also 
.equal to the square of the middle term when 
there is a middle one, or an uneven number of 
{he terms. 
Thus, 1, 2, 4, 8, 16, 
16 8 4 2 1 
tprod. 16 16 16 16 16. 
Also, a, ra, fa, ra, fa, 
fa fa fa ra a 
gression 1 -|- — — 1 _ _ _ Sec. where 
rr- 9 
z — 1> and r — 2, it is s or — - 
2— 1 
1 ~ 
5. The first or least term of a geometrical 
progression, is to the sum of all the terms, as 
the ratio minus 1, to the «th power of the ratio 
minus 1 ; that is, a \ j ” r — 1 * r n — 1. 
Other relations among the five quantities a, 
”, s, where 
a denotes the least term, 
z the greatest term, 
r the common ratio, 
« the number of terms, 
s the sum of the progression, 
are as below ; viz. 
rz — a r n — 1 r n — 1 
r~ 1 ~ r — 1 U ~ r — 1 
" ~\/ z n — n ~ {/ a n 
r n ~ ‘ 
And the other values of a, z, and r, are to be 
found from these equations, viz. 
( s z ) n ~ 1 z — (j — a)" ~ * a, 
f — r == 1 — — . 
prod, r'a 1 r'a 1 r'a 2 fa 2 r'a 2 . 
3. The last term of a geometrical progression, 
•is equal to the first term multiplied, or divided, 
by the ratio raised to the power whose exponent 
is less by 1 than the number of terms in the se- 
ries ; so z ar n ~ 1 , when the series is an as- 
o 
cending one, or z — — : — -, when it is a de- 
r" — * 
scending progression. 
4. As the sum of all the antecedents, or all 
the terms except the least, is to the sum of all 
the consequents, or all the terms except the 
greatest, so is 1 to r, the ratio. For, 
if * + ra + r 2 a+fa are all except the last, 
then ra -f- r 2 a -j- ra-f- r'a are all except the first ; 
where it is evident that the former is to the lat- 
ter as I to r, or the former multiplied by r gives 
,the latter. So that, z denoting the last term, a 
the first term, and r the ratio, also s the sum of 
all the terms ; then j — z [ s — a ; * 1 r , or 
s — a = s — z . r. And from this equation all 
the relations among the four quantities a, z, r, s 
are easily. derived ; such as, s = — — ; viz. 
r — 1 
multiply the greatest term by the ratio, subtract 
the least term from the product, then the re- 
niaiuder.divlded by 1 less than the ratio, will 
give the sum of the series. And if the least 
term a is 0, which happenswhen the descend- 
ing progression is. infinitely continued, then the 
sum is barely As -in the infinite pro- 
6 
r" r n 1 — .See Series. 
■ r Z 3 z 
PROHIBITION, is a writ properly issu- 
ing only out of the court of king’s bench, 
being the king’s prerogative writ ; but, for 
the furtherance of justice, it may now also 
he- had in some cases out of the court of 
chancery, common pleas, or exchequer, di- 
rected to the judge and parties of a suit in 
an inferior court, commanding them to cease 
from the prosecution thereof, upon a sugges- 
tion, that either the cases originally, or some 
collateral matter arising therein, does not 
belong to that jurisdiction, but the cogniz- 
ance of some other court. 3 Black. 1 12. 
Upon the court being satisfied that the 
matter alleged by the suggestion is sufficient, 
the writ ot prohibition immediately issues ; 
commanding the judge not to hold, and the 
party not to prosecute, the plea. And if 
either the judge or party shall proceed after 
such prohibition, an attachment may be 
had against them for the contempt, by the 
court that awarded it, and an action will lie 
against, them to repair the party in damages. 
3 Black. 113. 
PROJEC TILE, or Project, in mechanics, is 
any body which, being put into a violent mo- 
tion by an external force impressed upon it, is 
dismissed from the agent, and left to pursue its 
course ; such as a stone thrown out of the hand 
or a sling, an arrow from a bow, a ball from a 
gun, &c. 
PROJECTILES, the science of the motion 
velocity, flight, range, Ac. of a projectile put 
into violent motion by some external cause, a» 
the force of gunpowder, Sec. This is the foun- 
dation of gunnery, under which article mav he 
found all that relates peculiarly to that branch. 
All bodies, being indifferent as to motion or 
rest, will necessarily continue the state they are 
put into, except so far as they are hindered, and 
forced to change it by some new cause. Hence, 
a projectile, put in motion, must continue eter- 
nally to move on in the same right line, and with 
the same uniform or constant velocity, were it 
to meet with no resistance from the medium, 
por had any force of gravity to encounter. * 
In the first case, the theory of projectile* 
. ^ vou * (1 be very simple indeed ; for there would 
be nothing more to do, than to compute the 
space passed over in a given time by a mven 
constant velocity : or either of these, 'from the 
other two being given, 
® ut by the constant action of gravity, the 
piojectile is continually deflected more and more 
from its right-lined course, and that with an ac- 
celerated velocity : which, being combined with 
its projectile impulse, causes the body to move 
in a curvilineal path, with a variable motion, 
which path is the curve of a parabola, as will 
be proved below ; and the determination of the 
range, time of flight, angle of projection, and 
variable velocity, constitutes what is usually 
meant by the doctrine of projectiles, in the com- 
mon acceptation of the word. 
What is said above, however, is to be under- 
stood of projectiles moving in a non-resisting 
medium ; for when the resistance of the air is 
also considered, which is enormously great, and 
which very much impedes the first projectile ve- 
locity, the path deviates greatly from the para- 
bola, and the determination of the circumstances 
of its motion becomes one of the most complex 
and difficult problems in nature. 
In the first place, therefore, it will be proper 
to consider the common doctrine of projectiles, 
or that on the parabolic, theory, or as depending 
only on the nature of gravity and the projectile 
motion, as abstracted from the resistance of the 
medium. 
Little more than 200 years ago, philosophers 
took the line described bv a body projected 
horizontally, such as a buftet out of a cannon 
while the force of the powder greatly exceeded 
the weight of the bullet, to be a right line, after 
which they allowed it became a curve. Nicholas 
I artagha was the first who perceived the mis- 
take, maintaining that the path of the bullet 
was a curved line through the whole of its ex- 
tent. But it was Galileo who first determined 
what particular curve it is that a projectile de- 
scribes ; shewing that the path of a bullet pro- 
jected horizontally from an eminence, was a 
parabola the vertex of which is the point 
where the bullet quits the cannon. And the 
same is proved generally, in the 2d article fol- 
lowing, when the projection is made in any di- 
rection whatever, viz. that the curve is always 
a parabola, supposing the body moves in a non- 
resisting medium. 
The Laws of the Motion of Projectiles. 
I. If a heavy body is projected perpendicu- 
larly, it will continue to ascend or descend per- 
pendicularly ; because both the projecting and 
the gravitating force are found in the same line 
ot direction. 
parallel to the horizon, or in any oblique . 
reaction ; it will, by this motion in conjuncti 
with tire action of gravity, describe the cur 
line of a parabola, (Fig. i.) 
For, let the body he projected from A, in t 
direction AD, with any uniform velocity • th 
in anv equal portions of time it would by tl 
impulse alone, describe the. equal spaces / 
