M ECH 
acquired ; for, if the velocity of the fluid he only equal to 
the velocity of the vane or fail, it juft keeps up with it, 
but has no effect either to advance or retard its motions. 
It is, at the fame time, of the utrnoft importance to have 
regard to the dircBion in which the force is imprefled, in 
order to determine the change of motion produced by it. 
It would be very erroneous to fuppofe that the accelera¬ 
tion of the motion of a fltip, in the direction in which fhe 
fails, is proportional to the force imprefled when it acts 
obliquely upon the fail, or when the pofition of the fail is 
oblique to the direftion in which the fliip moves. The 
change of her motion is fir ft to be eft in: a ted in the direc¬ 
tion of the force imprefled, and thence, by a proper ap¬ 
plication of mechanical and geometrical principles, the 
change of the motion of the fltip in her own direction is 
to be derived. When gravity, or any centripetal force, 
afts upon a body moving with a direction oblique to the 
right line drawn from it to the centre, the change of its 
motion is not proportional to the whole centripetal force 
which afts upon it, but to that part only, which, after a 
jult rel'olution of the force, is found to aft in the direc¬ 
tion of its motion. It appears from thefe inftances, of 
how’ extenfive an ufe thefe general laws are in the doftrine 
of motion. 
The third general law of motion is, “ That a Eli on and re- 
aElion are equal with oppofite directions, and are to be efli- 
mated always in the fame right line.” Body not only never 
changes its (fate of itfelf, but refills, by its inertia , again ft 
every aftion that produces a change in its motion. When 
two bodies meet, each endeavours to perfevere in its Hate, 
and refills any change; and, becaufe the change which is 
produced in either may be equally meafured by the aftion 
which it exerts upon the other, or by the refinance which 
it meets with from it, it follow s that the changes produced 
in the motions of each are equal, but are made in con¬ 
trary directions. The one acquires no new force but 
what the other lofes in the fame dire&ion ; nor does this 
laft lofe any force but what the other acquires ; and hence, 
though by their collifions motion pafles from the one to 
the other, yet the fum of their motions, ellimated in a 
given direction, is preferred the fame, and is unalterable 
by their mutual aftion upon each other. In collecting 
this fum, motions that have contrary directions are to be 
affeCted with contrary flgns; amotion eaftward is con-- 
frary to a motion weltward ; fo that, if the motions are 
lammed up as having a weftern direction, a motion eaft¬ 
ward is to be confidered as negative, or to be fubdufted 
t fVom the relt. In this manner, this law ferves to render 
the fir It law more general, and to extend it to any number 
of bodies ; for as, by the firft law, a body perfeveres in 
its ftate of relt, or of uniform rectilinear motion, till fome 
external influence affeCt it; fo it follows from this law, 
“ That the fum of the motions of any number of bodies, 
ellimated in a given direction, perfeveres the fame in their 
mutual aCtions and collifions, till fome external influence 
difturb them.” 
Sir Ifaac Newton’s firft corollary, from the laws of mo¬ 
tion, is, that, when a body is aCted upon by two forces at 
the fame time, it will defcribe the diagonal, by the motion 
refulting from their compofition, in the fame time that it 
would defcribe the fides of the parallelogram by thofe forces 
aCting feparately. Let the body A, Plate I. fig. i. have a 
motion in the direction A B, reprefen ted by the right 
line A B ; at the fame time let another motion be com¬ 
municated to it in the direction A D, reprefented by the 
right line A D ; complete the parallelogram ABCD; 
and the body will proceed in the diagonal A C, and de¬ 
fcribe it in the fame time that it would have defcribed the 
fide A B by the firfi motion, or the fide A D by the fecond. 
To underftand the demonftration of this corollary, we mull 
‘premife this obvious principle, that, when a body is aCted 
upon by a motion or power parallel to a right line given 
in pofition, this power or motion has no effeft to caufe 
the body to approach towards that right line or recede 
from it, but to move in a line parallel to that right line 
A N I C S. 627 
only ; *s appears from the fecond law of motion. There¬ 
fore, A D being parallel to B C, the motion in the direc¬ 
tion A D has no effeCt in promoting or retarding the ap¬ 
proach of the body A towards the line B C; confequently 
it will arrive at this line B C in the fame time as if the 
firft motion A B only had been imprefled upon it. In 
like manner, becaufe A B is parallel to DC, the motion A B 
has no effect in promoting or retarding the approach of 
the body A towards the line D C ; confequently it will 
arrive at the line D C in the fame time as if the motion A D 
only had been impreffed upon it. Therefore the body A 
will arrive at both the lines B C and D C in the fame time 
that, by the firft motion alone, it would have defcribed A B, 
or, by the fecond alone, it would have defcribed AD. 
But it can arrive at both the lines B C and DC no other 
way than by coming to their interfeftion C ; therefore, 
when the two motions A B and A D are imprefled upon 
it at once, it moves from A to C, and defcribes the dia¬ 
gonal A C, in the fame time that, by thefe motions aft- 
ing feparately, it would nave defcribed the fides A B 
and A D. 
Becaufe this corollary is of very extenfive ufe, it may 
be worth while to illu(Irate it farther. Suppofe the fpace 
' E F G H, fig. 2. to be carried uniformly forward in the 
direction A B, and with a velocity reprefented by A B. 
Let a motion in the direction A D, and meafured by the 
right line A D, be impreffed upon the body A in the 
fpace E F G H. To thofe who are in this fpace, the body 
A will appear to move in the right line A D; but its real 
or abfolute motion will be in the diagonal A C of the pa¬ 
rallelogram ABCD: and it will defcribe A C in the 
fame time that the fpace by its uniform motion, or any 
point of it, is carried over a right line equal to A B, or 
that the body A, by its motion acrofs the fpace, defcribes 
A D. For it is manifeft that the line A D, in confe- 
quence of the motion of the fpace7is carried into the fitua- 
tion B C, and the point D to C ; fo that the body A 
really moves in the diagonal A C. 
The converfe of this corollary is, that the motion in the 
diagonal A C, may be refolved into the motions in the 
fides of the parallelogram A B and AD. For it is mani¬ 
feft that if A K, fig. 3. be taken equal to A D with an 
oppofite direction, and the parallelogram A K B C be 
completed, the right line A B will be the diagonal of this 
parallelogram ; confequently, by the two laft articles, the 
motion A C compounded with the motion A K, equal and 
oppofite to the motion A D, produces the motion A B ; 
that is, if from the motion A C; in the diagonal, you 
fubduft the motion A D in one of the fides, there will re¬ 
main the motion A B in the other fide of the parallelo¬ 
gram ABCD. 
This doftrine will receive farther illuftration by re- 
folving each of the morions A B and A D into two mo¬ 
tions, one in the direction of the diagonal A C, and the 
other in the direftion perpendicular to it; that is, by re- 
folving the motion A B, fig. 4,. into the motions A M and 
A N, and the motion A D into the motions A K and A L. 
For, the triangles A D K and BCM being equal and fimi- 
lar, D K is equal to B M, or A L to A N ; fo that, the 
motions A L and A N being equal and oppofite, they de- 
flroy each other’s effeft ; and, it being an obvious and 
general principle, that the motion of a body in a right 
line is no-way affefted by any two equal powers or mo¬ 
tions that aft in direftions perpendicular to that line, and 
oppofite to each other, it thus appears how the body A is 
determined to move in the diagonal A C ; and, becaufe 
A K is equal to M C, it appears how the remaining mo¬ 
tions A M and A K are accumulated in the direftion A C, 
fo as to produce a motion meafured by A C. It appears 
likewife how abfolute motion is loft in the compofuion of 
motion; for the parts of the motions A B and A D that 
are reprefented by A N and A L, being equal and oppo¬ 
fite, deftroy each other’s effeft; and the other parts A N[ 
and A K, only, remain in the direftion of the compound¬ 
ed motion A C j while, on the contrary, in the refolution 
