408 
are swimming, fying, crawling, climbing, 
leaping, running, walking, &c. The con- 
sideration of these diversified methods of 
progression involves the theory of the mo- 
tion of bodies in general, of the lever, the 
pulley, the centre of gravity, specific gravity, 
and the resistance of fluids, &c.; and, as 
we shall have occasion for constant refer- 
ence to the mechanical principles connected 
with these subjects, they will be first dis- 
cussed; but for the convenience of those 
who are unacquainted with the algebraic 
method of computation and analysis, the 
latter will generally be separated from the 
text. 
Fundamental Axioms. — First, every body 
continues ina state of rest or of uniform mo- 
tion in a right line until a change is effected by 
the agency of some mechanical force. Secondly, 
any change effected in the quiescence or mo- 
tion of a body is in the direction of the force 
impressed, and is proportional to it in quantity. 
Thirdly, reaction is always equal and con- 
trary to action, or the mutual actions of two 
bodies upon each other are always equal and 
in opposite directions. 
Thus if M (fig. 203) be a particle of matter 
free to move in any direction, and if the lines 
MA, MB, represent the intensity of two forces 
Fig. 203. 
Bhi=+ sseceeee eee 
M B 
acting on itin the direction MC, the particle M 
will move towards C by the combined action 
of the two forces, and it will require a force 
in the direction of CM, equal to MA+ 
MB to keep it ina state of rest; but if MA 
and MB (fig. 204) represent the intensities 
and directions of two forces acting on the par- 
ticle M in opposite directions, if MA be 
Fig. 204. 
A M 
MOTION. 
see gon of forces. 
which RN is the diagonal, — 
and consequently into an — 
indefinite number of such 
pairs.* This construction is — 
called the parallelogram of — 
forces. daa 
The resultant of any 
number of forces meeting — 
in acommon point may be — 
ascertained thus: first, let 
the resultant of any two 
forces be found as b . 
and substitute this one force 
for the two components pro- 
ducing it; then combine — 
this new force with one of 
the remaining forces, and continue this process 
until all the forces are reduced toa single foree, 
which is the resultant sought. The following 
geometrical solution will render the subject 
more apparent: let P, P’, P’, &e. (jig. 
represent a number of forces meeting in the 
common point A, and let A P, AP’, A P” be - 
proportional to these forces vely > 
through P draw PR equal and to 
AP’, and through R draw RR’ equal and 
parallel to AP”, and through R/ draw R’ R4 
equal and parallel to A P”’; join AR’, which 
represents the resultant of the four forces AP, — 
AP’, AP”, AP”. A similar opera- — 
tion will serve for any number of forces. 
Fig. 205. 
a 
This figure is denominated the poly- 
If the directions of 
three forces are rectangular, and in dif- 
ferent planes, the resultant may be L 
as follows: let PC, PC’, PC”, (fig. 207) be — 
the intensities and directions of three forces, — 
complete the lelopiped BD; then the — 
forces PC, PC’ have Pr for an eepen 
therefore P r may be substituted for teks ‘ 
forces; and by compounding the forces PC”,~ 
Pr, we get PR the diagonal of the parallelo- 
piped BDr for an equivalent to the three 
forces. This construction is called the 
parallelopipedon of forces.+ “7 
An equilibrium cannot subsist be- 
* 
greater than MB, the particle M will be 
moved towards A by the difference of these 
two forces, and it will require a farce equal to 
that difference to keep it at rest, 
The composition and resolution of forces — 
In the composition of forces it is proposed ta 
find the resultant, arising from any number or 
system of forces acting upon a given point. The 
resolution of forces, which is the converse of 
the former process, consists in discovering what 
forces acting in given directions would com- 
bine to produce a given resultant: Thus, if there 
be two forces F F* (fig. 205), whose directions 
and magnitudes are represented by F N, F’N, 
and if FR, F’R be drawn respectively parallel 
to FN, FN, then by the composition of forces 
we find the magnitude and direction of the 
equivalent or resultant of these two forces to 
be RN, and conversely it may be resolved 
into a pair of forces as RF, RF’ represented by 
the adjacent sides of any parallelogram, of 
~B tween any two forces acting a 
int of matter, if the lines represet t- 
ing the directions of the forees be inclined” 
to each other at any angle; but a third force 
equivalent to their resultant and jn an oppo- 
® Vide Gregory, vol. iii. ch. ii. p. 23. 
t From the same construction the resultant ¢ 
system of forces may be found, dis in diffe 
ier by the method of rectangular co-ordin 
etP Cuz, PC =y, PC’=z, (fig. 2 
by the 47 Euc, Lib. I. we have Pr? = PC*4> 
Cr? = x?-+y?, and PR? = Pr?+4rR? oe 
x? -+- y?-+- 2%, whence the resultant PR 
J/x?-+y?+2*. The position of the resultant 
is thus determined ; let r, r’,r” denote the mm 
known angles formed by the direction of the re~ 
sultant with each of the co-ordinates, and R cos. 
r, Ros. r’, R cos. r” willrepresent the equivalents — 
of the resultant in the several directions of theaxes, 
hence we have Ros. r = x, Reos. r’ 
K cos r” =z, .°- 
«ee 
} 5 
Cos. r= 7’ cos. 1 = Co. ae Re 
