6iS MEC H 
of motion, the quantity of abfolute motion is increafed, 
the fum of the motions A B artd AD, or B C, being 
greater than the motion AC. But the turn of the mo¬ 
tions, eflimated in a given direction, is no-way afl’efted 
by the ccmpofition or refolution of motion, or indeed by 
any a&ions or influences of bodies upon each other, that 
are equal and mutual, and have oppofite directions. 
For, fuppofe that the motions are to be eflimated in 
the direction A P, fig. 5. Let CP, B R, D Q, be per¬ 
pendicular to this direction in the points P, R, and Q ; 
then the motions AC, A B, AD, reduced to the direc¬ 
tion A P, are to be eflimated by A P, A R, and A Q, 
refpeCtively, the parts which are perpendicular to A P 
having no efFeCt in that direction. Let A P meet B C 
in S ; then, becaufe R P is to S P as B C (or A D) to C S, 
that is, as A Q to S P, it follows that A Q is equal to R P, 
and that AR-f-AQ is equal to A P; that is, that the 
fum of the motions AB and AD, reduced to any given 
direction AP, is equal to the compounded motion AC 
reduced to the fame direction. From which it is obvious, 
that, in general, when any number of motions are com¬ 
pounded together, or are refolved according to this gene¬ 
ral corollary, the fum of their motions continues invari¬ 
ably the fame, till fome foreign influence affeCts them. 
The ufefulnefs of the fame corollary has induced au¬ 
thors to invent other demonftrations for the farther illus¬ 
tration of it. We thall only add a proof of the fimpleft 
cafe, when the motions A B and A D are equal, and the 
angle BAD is a right one; as appears in fig. 6 and 7. 
In this cafe A B C D is a fquare, and the diagonal A C 
bifefts the angle BAD; and, becaufe the powers and 
motions of A D and A B are equal, and there can be no 
reafon why the direction of the compounded power or 
motion fhould incline to one of thefe more than to the 
other, it is evident that its direction muft be in the diago¬ 
nal AC ; and that the compounded power or motion is 
meafured by A C appears in the following manner : If 
it is not meafured by A C, firft let it be meafured by any 
right line A E lefs than A C ; join B D, interfeCting A C 
in K; upon A C take A M greater than A K, in the 
fame proportion that A C is greater than AE ; through 
the point M draw the right line F G parallel to B D, meet¬ 
ing A D in G, and A B in F; complete the parallelograms 
AMGH and A M F N ; then, becaufe thefe parallelo¬ 
grams are fquares as well asABCD, and A D is to A G 
as A K to A M ; that is, as A E to A C ; and A B to A F 
in the fame proportion ; and becaufe A E is fuppofed to 
be the power or motion compounded from AB and AD; 
it follows that the power or motion A D may be fuppofed 
to be compounded from the powers or motions A M and 
A H, and A B from A M and A N. But A H and A N, 
afting equally with oppofite directions, deftroy each other's 
effeCt ; lo that it would follow that the remaining powers 
or motions AM + AM(i.e. 2 AM), which are accu¬ 
mulated in the direction of the diagonal A C, ought to be 
equal to A B ; which is abfurd, for A M is greater than 
AK by the confirmation, and 2 AM greater than 2 AK 
or A C, which is fuppofed to be greater than A E. In like 
manner, itisfhown, fig. 7. that the compounded power or 
motion, in the diagonal A C, is not meafured by a right 
line greater than A C; and therefore it is meafured pre- 
cifely by the diagonal A C itfelf. 
Prop. I. If a body at ref, be acled upon at the fame time 
by three forces which are reprcfented in quantity and diretlion by 
the three fules of a triangle, taken in order, it will remain at ref. 
—Let A B, B C, and C A, fig. 8. reprefent the quantities 
and directions of three forces aCting at the fame time 
upon a body at A ; then, fince A B and BC are equiva¬ 
lent to A C ; A B, B C, and C A, are equivalent to A C 
and C A ; but A C and C A, which are equal and in op- 
pofite directions, keep the body at reft; therefore A B, 
B C, and C A, will alfo keep the body at reft. 
Prop. II. If a body be kept at ref by three forces, and two 
$f them be reprefented in quantity and diredion by two fides, A B, 
Si C, of a triangle, fig. 8. the third fde, taken in order, will 
A N I C S. 
reprefent the quantity and diredion of the other force.— Since 
A B, BC, reprefent the quantities and directions of two 
of the forces, and AB, B C, are equivalent to A C, the 
third force muft be fuitained by A C ; therefore C A mutt 
reprefent the quantity and direction of the third force. 
Cor. If three forces keep a body at reft, they aCt in the 
fame plane ; becaufe the three fides of a triangLe are in 
the fame plane. (Euc. 2. %i.) 
Prop. III. If a body be kept at ref by three forces, ading 
upon it at the fame time, any three lines, which are in the direc¬ 
tions of thefe forces, and form a triangle, will reprefent. than .—- 
Let three forces, acting in the directions A B, A C, AD, 
fig. 9. keep the'body A at reft ; then A B, AC, AD, 
are in the fame plane. In A B take any point, B, and 
through B draw B I parallel to A C, meeting D A pro¬ 
duced in I; then will A B, B I, and I A, reprefent the 
three forces. 
For, A B being taken to reprefent the force in that di¬ 
rection, if B I do not reprefent the force in the direction 
A C or B I, let B F be taken to reprefent it; join A F ; 
then, fince three forces keep the body at reft, and AB, B F, 
reprefent the quantities and directions of two of them, 
FA will reprefent the third, that is, F A is in the direc¬ 
tion AD, which is impoffible ; (Euc. 11. i. Cor.) there¬ 
fore B I reprefents the force in the direction AC; 
and confequently I A reprefents the third force, as in 
Prop. II. 
Any three lines, refpeCtively parallel to A B, B I, I A, 
and forming a triangle, will be proportional to the fides 
of the triangle A B I, and therefore proportional to the 
three forces. 
Cor. 1. If a body be kept at reft by three forces, any 
two of them are to each other inverfely as the fines of the 
angles which the lines of their directions make with the 
direction of the third force.—Let A B I be a triangle 
whofe fides are in the directions of the forces; then thefe 
fides reprefent the forces ; and A B : B I :: fin. B I A : 
fin. BAI :: fin. I AC : fin. BAI :: fin. CAD ; 
fin. BAD. 
Cor. 2. If a body, at reft, be aCted upon at the fame 
time by three forces, in the directions of the fides of a 
triangle taken in order, and any two of them be to each 
other inverfely as the fines of the angles which their di¬ 
rections make with the direction of the third, the body will 
remain at reft. For, in this cafe, the forces will be pro¬ 
portional to the three fides of the triangle, and confe¬ 
quently they will fuftain each other, as in Prop. I. 
Prop. IV. If a body be kept at ref by three forces, and 
lines be drawn at right angles to the diredions in which they 
ad, forming a triangle, the fides of this triangle will reprefent 
the quantities of the forces. —Let AB, B C, C A, fig. 10. be 
the directions in which the forces aft; and let them form 
the triangle ABC; then the lines AB, B C, C A, will 
reprefent the forces. Draw the perpendiculars D H, E I, 
FG, forming a triangle GHI; then, fince the four an¬ 
gles of the quadrilateral figure AD H F are equal to four 
right angles, and the angles at D and F are right angles, 
the remaining angles DHF, D A F, are equal to two 
right angles, or to the two angles DHF, D H G ; confe¬ 
quently, the angle D A F is equal to the angle IHG. In 
the fame manner it may be fliown,-that the angles ABC, 
B C A, are refpeCtively equal to G I H, H G I; therefore 
the triangles ABC and GHI are equiangular; hence, 
the fides about their equal angles being proportiona 1 , the 
forces, which are proportional to the lines A B, B C, C A, 
are-proportional to the lines HI, I G, G H. 
Cor. If the lines D H, El, F G, be equally inclined 
to the lines D B, EC, FA, and form a triangle G H I, 
the fides of this triangle will reprefent the quantities of 
the forces. 
Prop. V. If any number of forces, reprefented in quantity 
and diredion by the fides of a polygon, taken in order, ad at 
the fame time upon a body at ref, they will keep it at ref. —Let 
AB, B C, CD, DE, and E A, fig. n. reprefent the 
forces; then, fince AB, BC, CD, and DE, are equi- 
4 valent 
