M E C H A N I C Sj 
*>:o 
Call the (rue weight x * and the apparent weights, when 
it is fufpended at A and B, a and b refpeftively ; then 
a : x :: A C : B C, and x : b :: A C : B C ; therefore 
a : x :: x : b. 
Cor. 6. If a weight C, fig. 16. be placed upon a lever 
which is fupported upon two props A and B in an hori- 
aontal polition, The preflure upon A : the prell'ure upon 
B :: BC : AC. For, if B be conceived to be the ful¬ 
crum, we have this proportion : The weight fuftained 
by A : the weight C :: B C : A B; in the fame manner, 
if A be confidered as the fulcrum, then, The weight C : 
the weight fuftained by B :: AB : C A. Therefore, ex 
esquo, The weight fuftained by A : the weight fuftained 
by B :: B C : A C. 
Cor. 7. If a given weight P, fig. 17. be moved along the 
graduated arm of a fcraight lever, the weight W, which it 
will balance at A, is proportion-id to CD, the diltance at 
which the given weight acts. 
When there is an equilibrium, WxAC = PxDC; 
and AC and P are Invariable; therefore W Ct DC. 
Prop. IX. If two forces, acling upon the arms of any lever, 
keep it at ref, they are to each other inverfely as the perpendi¬ 
culars drawn from the centre of motion to the directions in which 
the forces ad. 
Cafe 1. Let two forces, A and B, fig. 18. a 61 perpendicu¬ 
larly upon the arms C A, C B, of the lever A C B, whofe 
fulcrum is C, and keep each other at reft. Produce B C 
to D, and make C D == C A ; then the effort of A to 
move the lever round C, will be the fame, whether it be 
fuppofed to aft perpendicularly at the extremity of the 
arm C A, or CD; and on the latter fuppofition, fince 
there is an equilibrium, A : B :: CB : CD; therefore 
A : B :: C B : C A. 
Cafe 2. When the directions AD, BH, fig. 19. in which 
the forces aft, are not perpendicular to the arms.—Take 
A D and B H to reprefent the forces ; draw C M and C N 
at right angles to tliofe directions; alfo draw AF perpen¬ 
dicular, and DF parallel, to AC, and complete the pa¬ 
rallelogram G F; then the force A D is equivalent to the 
two A F, AG, of which, AG aits in the direction of 
the arm, and therefore can have no effefl in cauling or 
preventing any angular motion in the lever about C. Let 
B H be refolved, in the fame manner, into the two B I, B K, 
of. which B I is perpendicular to, and B K in the direc¬ 
tion of, the arm CB; then BK will have no eff'edt in 
caufing or preventing any angular motion in the lever 
about C ; and fince the lever is kept at reft, A F and B I, 
which produce this effeft, and aft perpendicularly upon 
the arms, are to each other, by the firft cafe, inverfely as 
the arms ; that is, A F : B I :: C B : C A, or A F X 
C A = B I X C B. Alfo in the fimilar triangles A D F, 
ACM, As AF : AD :: CM : C A, and A F X 
C A = A D X C M ; in the fame manner, B 1 X CE = 
B H X C N 5 therefore, A D X CM = BH X CN, and 
AD : BH :: CN :■ CM. 
Cor. 1. Let a body I K, fig. 20. be moveable about the 
centre C, and two forces aft upon it at A and B, in the di- 
ndtions AD, BH, which coincide with the plane ACB ; 
join AC, CB ; then this body may be confidered as a lever 
ACB, and, drawing the perpendiculars CM, C N, there 
will be an equilibrium, when The force afting at A : the 
force acting at B : CN : CM. 
Cor. 2. The effort of the force A, to turn the lever 
round, is the fame, at whatever point in the direftion 
JV 1 D it is applied: becaufe the perpendicular CM re¬ 
mains the fame. 
Cor. 3. Since CA : CM :: rad. r fin. CAM, CM = 
C A X fin- CAM , , _ ., 
-j- ; and in the fame manner, CN = 
CB X fin! CBN . . 
-“j-; therefore, when there is an equi¬ 
librium, The power at A : the weight at B :: 
CBxfin. CBN C A X fin. CAM 
- - » -p—-— ;; CB X nn. 
rad. rad. 
CBN C A X fin. CAM. 
for. 4. If the lever A C B, fig. ax. be ftraight, and the 
direftions AD, B H, parallel, A : B :: BC : AC j 
becaufe in this cafe, fin. CAM = fin. C B H. Hence 
alfo, Ax ACrBxBC. 
Cor. 5. If two weights balance each other upon a ftraight 
lever in any one polition, they will balance each other in 
any other polition of the lever ; for the weights aft in pa¬ 
rallel direftions, and the arms of the lever are invariable. 
Cor. 6. If a man, balanced in a common pair of feales, 
prefs upwards by means of a rod, againft any point in the 
beam, except that from which the fcale is fufpended, he 
will preponderate.—Let the aftion upwards take place at 
D, fig. 22. then the fcale, by the re-aftion downwards, 
v.iil be brought into the fituation E ; and the effeft will 
he the fame as if D A, A E, D E, conllituted one mafs ; 
that is, drawing EF perpendicular to C A produced, as 
if the fcale were applied at F ; confequently the weight, 
neceffary to maintain the equilibrium, is greater than If 
the fcale were fuffered to hang freely from A, in the pro¬ 
portion of C F : C A. 
Cor. 7. Let AD, fig. 23. reprefent a wheel, bearing a 
weight at its centre C ; A B an obftacle over which it is- 
to be moved by a force afting in the direftion C E ; join 
C A, draw C D perpendicular to the horizon, and from A 
draw A G, A F, at right angles to C E, C D. Then C A 
may be confidered as a lever whofe centre of motion is A, 
C D the direftion in which the weight afts, and C E the 
direftion in which the power is applied ; and there is an 
equilibrium on this lever when The power : the weight 
:: AF : AG. Suppofing the wheel, the weight, and 
the obftacle, given; the power is the leaft when A G is the 
greateft; that is, when C E is perpendicular to C A, or 
parallel to the tangent at A. 
Cor. 8. Let two forces afting in the direftions AD, BH, 
upon the arms of the lever ACB, fig. 24. keep each other 
in equilibrio; produce D A and H B till they meet in P'j 
join CP, and draw CL parallel to PB ; then will P L, 
L C, reprefent the two forces, and P C the preffure upon 
the fulcrum. For, if PC be made the radius, CM and 
C N are the fines of the angles C P M, C P N, or CP L, 
PCL; and PL : LC :: fin. PCL : fin. LPC : CN 
: CM; therefore PL, LC, reprefent the quantities and 
direftions of the two forces, which may be fuppofed to 
be applied at P, and which are fuftained by the re-adtion 
of the fulcrum ; confequently, CP represents the quan¬ 
tity and direftion of that re-adtion, or P C reprefents the 
preffure upon the fulcrum. 
Prop. X. In a combination of fraight levers, A B, CD, 
fig. 25, 26. Plate II. uhofe centres of motion are E and F, 
if they ad perpendicularly upon each ether, and the diredions in 
which the power and weight are applied be alfo perpendicular to 
the arms, there is an equilibrium when P : W :: E B X F D 
: E A X F C.—For, The power at A : the weight at B 
or C :: EB : F. A ; and, The weight at C : the weight 
at D :: FD : FC; therefore, P : W :: EB XFD ; 
E A X FC. By the fame method we may find the pro¬ 
portion between the power and the weight, when there is 
an equilibrium, in any other combination of levers. 
Cor. If E and F be confidered as the power and weight, 
A and D the centre of motion, we have, as before, E 1 
F :: F D X B A : AE X C D. Hence, The preffure 
upon E : the preffure upon F :: FD X BA : AE x CD. 
Prop. XI. Any weights will keep each other in equilibrio on 
the arms of a fraight lever, when the produds, which arifefroin 
multiplying each weight by its diflance from the fulcrum, are 
equal on each fide of the fulcrum .—The weights A, B, D, 
and E, F, fig. 27. will balance each other upon the lever, 
A F, whofe fulcrum is C, if Ax AC + B XBC-j* 
D X D C = E X E C -f F X F C. In C F take any point 
X, and let the weights r, s, t, placed at X, balance 
refpeftively, A, B s D; then Ax AC = rx XC; B 
XBC = j X XC; D X D C — t X X C ; or, A X 
A C —B X B C D X D C — r -p s -j- / X X C. In the 
fame manner, let p and q, placed at Y, balance refpec- 
tively, E and F 5 Jhem p-pjxYC = ExEC-|-FX 
F C j 
