MECH 
valent to Afi| AB, $C, CD, DE, and EA, are equi¬ 
valent to A E and E A ; that is, they will keep the body 
at reft. 
Prop. VI. If any number of lines, taken in order, reprcfent 
the quantities and direElions of forces which keep a body at ref, 
thefe lines will form a polygon. —Let A B, B C, CD, and 
DE, fig. ix. reprefent forces which keep a body at reft; 
then the point E coincides with A. If not, join A E ; 
then A B, B C, C D, and D E, are equivalent to A E ; 
and the body will be put in motion by a Angle force A E, 
which is contrary to the fuppotition ; therefore the point 
E coincides with A, and the lines form a polygon. 
This and the laft propolition are true when the forces 
aft in different planes. 
Of the MECHANICAL POWERS. 
The mechanical powers are the mod timple inftruments 
nfed for the purpofe of fupporting weights, or communi¬ 
cating motion to bodies; and by the combination of which, 
all machines, however complicated, are conlirufted. 
Thefe powers are fix in number, viz. the lever ; the 
wheel and axle; the pulley; the inclined plane; the 
fcrew; and the wedge. 
When two forces aft upon each other by means of any 
machine, one of them is, for the fake of diftinftion, called 
the power, and the other the weight. The weight is the 
refinance to be overcome, or the effeft to be produced. 
The power is the force, whether animate or inanimate, 
which is employed to overcome that refiftance, or to pro¬ 
duce the required effeft. 
Of the Lever. 
A lever is an inflexible bar or'rod, moving freely 
round a point called its fulcrum, or centre of motion. 
Levers have been generally divided into three kinds. In 
levers of the firft kind the fulcrum is fituated between 
the power and the weight, as in fteelyards, fciffars, pin¬ 
cers, &c. Levers of the fecond kind have the weight be¬ 
tween the power and the fulcrum, as in cutting-knives 
fattened at the point of the blade, and in the oars of a 
boat where the water is regarded as the fulcrum. In le¬ 
vers of the third kind, the power is between the weight 
and the fulcrum, as in tongs, (beers for (heep, See. The 
bones of animals are generally confidered as levers of the 
third kind ; for the mufcles, by the contraction of which 
the power or moving force is generated, are fixed much 
nearer to the joints or centres of motion than the centre 
of gravity of the weight to be railed. 
The properties of the lever cannot be deduced imme¬ 
diately from the propofitions laid down in the laft fection, 
becaufe the forces afting upon the lever are not applied 
at a point, which is always fuppofed to be the cafe in the 
compofition and refolution of forces; they may however 
be derived from the following principles or axioms, the 
truth of which will readily be admitted. 
i. If two weights balance each other upon a ftraight 
lever, the preffure upon the fulcrum is equal to the (um 
of the weights, whatever be the length of the lever. 
z. If a weight be fupported upon a lever which reds 
on two fulcrums, the preffure upon the fulcrums is equal 
to the whole weight. 
3. Equal forces, afting perpendicularly at the extremi¬ 
ties of equal arms of a lever, exert the fame effort to turn 
the lever round. 
Prop. VII. If two equal weights aEl perpendicularly upon 
a ftraight lever, the effort to put it in motion, round any fulcrum, 
will be the fame as if they ailed together at the middle point 
between them.—'Let A and B, fig. 12, 13. be two equal 
weights, afting perpendicularly upon the lever FB, whole 
fulcrum is F. Bifeft A B in C; make CE = CF; and 
at E fuppofe another fulcrum to be placed. Then, fince 
the two weights A and B are fupported by E and F, and 
thefe fulcrums are fimilarly fituated with refpeft to the 
equal weights, each firft a ins an equal preffure ; and there¬ 
fore the weight fuftained by E is equal to half the fum of 
the weights. Now let the weights A and B be placed at 
VOL. XIV. No. 1000. 
V N I C 3. 629 
Cj the middle point between A and B, and confequently 
the middle point between E and F ; then, fince E and F 
fupport the whole weight C, and are fimilarly fituated 
with refpeft to it, the fulcrum E fupports half the weight; 
that is, the preffure upon E is the fame, whether the 
weights are placed at A and B, or collefted in C, the 
middle point between them; and therefore, the effort to 
put the lever in motion round F is the fame on either 
fuppofition. 
Cor. If a weight be formed into a cylinder A B, fig. 14. 
which is every-where of the fame denftty, and placed pa¬ 
rallel to the horizon, the effort of any part A D, to put 
the whole in motion round C, is the fame as if this part 
were collefted at E, the middle point of AD. For the 
weight AD may be fuppofed to con lift of pairs of equal 
weights, equally diftant from the middle point. And 
what is here affirmed of weights, is true of any forces 
which are proportional to the weights, and aft in the 
fame direftions. 
Prop. VIII. Tiov weights, or two forces, aiding perpendi¬ 
cularly upon a ftraight lever, will balance each other, when tin$ 
are reciprocally proportional to their dftances from the fulcrum. 
Cafe 1. When the weights aft on contrary fides of the 
fulcrum.-—Let x and y be the two weights, and let them 
be formed into the cylinder A B, fig. 14. which is every¬ 
where of the fame denfity. Bifeft AB in C ; then this 
cylinder will balance itfelf upon the fulcrum C. Divide 
A B into two parts in D, fo that A D : D B :: x : y, 
and the weights of AD and DB will be refpeftively x and 
y\ bifeft A D in E, and D B in F ; then, fince A D and 
D B keep the lever at reft, they will keep it at reft when 
they are collefted at E and F; that is, x, when placed at 
E, will balance y, when placed at F ; and x : y :: AD 
: BD :: AB — BD : AB —AD :: 2 C B — 2 B F : 
2 A C — 2 A E :: 2 C F : 2 C E :: CF : C E. 
Cafe 2. When the two forces aft on the fame fide of 
the centre of motion.—Let A B, fig. 13. be a lever whofe 
fulcrum is C 5 A and B two weights afting perpendicu¬ 
larly upon it; and let A : B :: B C : A C. Then thefe 
weights will balance each other, as appears by the former 
Cafe. Now, fuppofe a power fufficient to fuftain a weight 
equal to the fum of the weights A and B, to be applied 
at C, in a direction oppofite to that in which the weights 
aft ; then will this power fupply the place of the fulcrum ; 
(Ax. 1.) alfo, a fulcrum placed at A, or B, and fuftain- 
ing a weight A, or B, will fupply the place of the body 
there, and the equilibrium will remain. Let B be the 
centre of motion; then we have a ftraight lever whofe 
centre of motion is B, and the two forces A and A -f- B, 
afting perpendicularly upon it at the points A and C, 
fuftain each other; alfo, A : B :: BC : AC; therefore 
A : A + B :: BC : BA. 
Cor. 1. If two weights, or two forces, afting perpendi¬ 
cularly on the arms of a ftraight lever, keep each other 
in equilibrio, they are inverfely as their diftances from 
the centre of motion. For the weights will balance when 
they are in that proportion, and, if the proportion be al¬ 
tered by increafing or diminifhing one of the weights, its 
effort to turn the lever round will be altered, or the equi¬ 
librium will be deflroyed. 
Cor. 2. Since A : B :: BC : AC, when there is an 
equilibrium upon the lever A B, whofe fulcrum is C, by 
multiplying extremes and means, AX AC = HxBC, 
Cor. 3. When the power and weight aft on the fame 
ffde of the fulcrum, and keep each other in equilibrio, 
the weight fuftained by the fulcrum is equal to the differ¬ 
ence between the power and the weight. 
Cor. 4. In the common balance, the arms of the lever 
are equal; confequently, the power and weight, or two 
weights, which fuftain each other, are equal. In the falfe 
balance, one arm is longer than the other; therefore the 
weight, which is fufpended at this arm, is proportionally 
lefs than the weight which it luftains at the other. 
Cor. 5. If the fame body be weighed at the two ends 
of a falfe balance, its true weight is a mean proportional 
between the apparent weights. 
7 ^ 
Cali 
