MEC H 
larsfi, &c- but it is not of very great ufe in practical me¬ 
chanics, becaufe the wedge is fcarcely ever otlierwiffe urged 
than by percuflion. In cleaving of wood, the refiftance 
oppofing the force of the mallet (fuppofmg the fides of 
the wedge perfectly polifhed, and its edge a line without 
breadth) is the cohefion of the particles of the wood 
which are about to be feparated ; and. this being a kind 
of preffive force adting again ft the Tides of the wedge, it is 
by many philofophers thought abfurd to attempt to com¬ 
pare it with the percuftive force of the mallet; for the 
greateft finite prefiive force nnift, in their opinion, give 
way to the lealt percuftive one, and there cannot be an 
equilibrium between two fuch different forces. “Any 
percuftive force (fay they), acting on a moveable body, 
generates a finite quantity of motion in an indefinitely 
j'mall portion of time ; but the time will be finite in which 
any given prefiive force whatever, adting on the fame 
body, can generate or deftroy the fame quantity of mo¬ 
tion. Therefore, a body being urged in a certain direc¬ 
tion by any preffive force whatever, and in the contrary 
diredlion by any percuftive one, the prefiive force will be 
fome finite time in deftroying the quantity of motion 
which the percuftive one generated in an inftant. Con- 
fequently, how great foever the preffive force may be, and 
how fmall foever the percuftive one, the body vvijl be 
moved (at lealt for fome ftiort time) by this laft force.” 
Indeed, after the ltroke is given, the preffive force may 
quickly prevail, and force back the body which the im- 
pulfe of the other force had driven forward. And fo it 
would frequently be in the operation of cleaving wood, 
if the fides of the wedge were perfedtly fmooth. For, af¬ 
ter the ftroke of the mallet, the wedge, unlefs it's weight 
were equivalent to the attraction of the parts of the wood 
about to be feparated, would prefently be forced back 
from the place to which it had been driven by the mallet. 
And it is chiefly the roughnefs of the fides of the wedge, 
and of the parts of the wood in contadt with it, which, in 
that operation, keeps the wedge from receding. It is 
that roughnefs too, and the bluntnefs of the edge, which 
fometimes prevent the wedge from being moved by the 
ftroke of the mallet: for, were it not obltrudted by fuch 
roughnefs and bluntnefs, it would, according to what we 
juft now obferved, be always driven forward, «ven by the 
leaft percuftive force. 
Of thefe fix fimple machines, all the mod; compound en¬ 
gines in the world are made. As the fcrdw includes the 
inclined plane, and two equally-inclined planes make 
the wedge, we have all the mechanical powers combined 
together in a common jack, if it be turned by the fly ; for 
then we have alfo the lever, the wheel and axle, and the 
pulleys. 
Some authors have deduced the properties of the me¬ 
chanical powers immediately from the third law of motion, 
contending that, if the power and weight be fuch as would 
fuftain each other, and the machine be put into motion, 
the momenta of the power and weight are equal; and 
confequently, that the power X the velocity of the power 
“ the weight X the velocity of the weight; or, The 
power’s velocity : the weight’s velocity ;; the weight : 
the power. 
Though this conclufion be juft, the reafoning by which 
it is attempted to be proved is inadmiflible, becaufe the 
third law of motion relates to the adtion of one body im¬ 
mediately upon another. It may however be deduced 
from the foregoing Propofitions; and, as it is, in many 
cafes, the fimpleft method of eltimating the power of a 
machine, it may not be improper to eftablifh its truth. 
In the application of the rule, two things muft be at¬ 
tended to : ift, We muft eftimate the velocity of the power 
or weight in the direction in which it adb. zdly, We 
muft eftimate that part only of the power or weight which 
is effedtive. Thefe two confiderations are fuggelted by 
the feco.nd law of motion, according to which, motion is 
communicated in the diredlion of the force imprefjed , and 
is proportional to that force. 
A N I C S. fix 
Prop. XXII. The velocity of a body in any one dir eel ion 
A B being 1 given , to cjlimatc its velocity in any other dircEl on 
BP. (fig. 48.)—Suppofe the motion of A to be produced 
by a force adding in the direction BP, by means of a 
firing which paffes over a pulley at P; produce P B to O, 
making PO—PA; join AO; then O B is the fpace 
which meafures the approach of A to P. Now let the 
pulley be removed to fuch a diftance that the angle at P 
may be conlidered as evanefeent, and the power will al¬ 
ways adt in the fame direction B P ; alio, the angles at A 
and O are equal ; and they are right angles, becaufe the 
three angles of the triangle APO are equal to two right 
angles, and the angle at P vanifhes ; therefore, the fpace 
deferibed in the diredlion OP, or BP, is determined by 
draw ing A O perpendicular to O P. If the fpaee deferibed 
in the diredlion xy, which is parallel to O P, be required, 
produce A O to x, and from B draw By at right angles 
to xy-, then the figure OByx is a parallelogram, and 
O B = xy the fpace required. Alfo, if the motion in the 
diredlion A B be uniform, the motion in the diredlion 
BP, or xy, is uniform ; fince AB : OB :: rad. : cof. 
ABO. Hence, The velocity in the direction AB : the 
velocity in the direction BP :: AB : OB. 
Prop. XXIII. If a power and weight fujlain each other on 
any machine, and the whole be put in motion , The velocity of the 
■power : the velocity of the weight :: the weight : the power. 
Cafe i. In the lever A C B, fig. 49. let a power and 
weight, adting in the directions A D, B II, fuftain each 
other, and let the machine be moved uniformly round the 
centre C, through a very fmall angle AC a. Join A a, 
B£; draw CM ,ax, at right angles to MD; and CN, 
by, at right angles to N B; then, A’s velocity : B’s velo¬ 
city :: A x : By. Now the triangles A * a, MCA, are 
fimilar; becaufe /.tACA/AMC-f/MCA; and 
/_a AC = /AMC; therefore, ,/.t'A«=/MCA; and 
the angles at M and x are right angles; conlequently, the 
remaining angles are equal; and 
Axi A aw C M : C A ; 
alfo, in the fim. As AC a, B C b, A a : B b :: C A : C B ; 
and in the fim. As B by, BCN, B b : By : : C B : CN; 
by compofition, Ax : By : CM : CN the weight : 
the power, (Prop. IX. Cor. 1.) or, The power’s velocity 
: the weight’s velocity :: the weight : the pow : er. 
Cafe 2. In the wheel and axle, if the power be made to 
deferibe a fpace equal to the circumference of the wheel 
with an uniform motion, the weight will be uniformly 
raifed through a fpace equal to the circumference of the 
axle; hence, The power’s velocity : the weight’s velocity 
:: the circumference of the wheel : the circumference of 
the axle :: the radius of the wheel : the radius of the axle 
:: the weight : the power. 
Cafe 3. In the fmgle fixed pulley, if the weight be uni¬ 
formly raifed 1 inch, the power will uniformly delcribe 1 
inch in the diredlion of its adiion ; therefore, The power’s 
velocity : the weight’s velocity :: the weight : the power. 
Cafe 4. In the Jingle moveable pulley where the firings are 
parallel, if the weight be raifed 1 inch, each of the firings 
is fhortened 1 inch, and the power deferibes s inches; 
therefore, P’s velocity : W’s velocity :: W : P. 
Cafe 5. In the fyfiem of pulleys deferibed at Prop. XIV. 
if the weight be railed 1 inch, each of the firings at the 
lower block is fhortened 1 inch, and the power deferibes 
winches; therefore, P’s velocity : W’s velocity a W : P. 
In this fyftem of pulleys, whilll 1 inch of the firing paffes 
over the pulley A, z inches pafs over the pulley B, 3 over 
C, 4 over D, &c. 
Hence it follows, that if, in the folid block A, (fig. 30.) 
the grooves A, C, E, &c. be cut, whofe radii are 1, 3, 5, 
&c. and in the block B, the grooves B, D, F, &c. whoie 
radii are z, 4, 6, &c. and a firing be paffed round thefe 
grooves as in the annexed figure; the grooves will anlvver 
the purpofe of fo many diftindt pulleys, and^ every point 
in each moving with the velocity of the firing in contadt 
with it, the whole fridlion will be removed to the two 
centres.of motion in the blocks A and B, 
4 * 
