Encyclopaedia Britannica > Volume 8, DIA-England

360
DYNAMICS.
Second
Law of
Motion.
y‘
C
It is evident that AGEH and AgFh are rhombuses;
because AO = OE, and Ao — oF. It is also plain, that
since bAd is half of BAD, the angle GAH is half of bAd.
It is therefore formed by a continual bisection of a right
angle. Therefore (G) the forces AG, AH, compose a
torce AE; and Ag, Ah, compose the force AF. There¬
fore the forces AG, AH, Ag, Ah, acting together, are
equivalent to the forces AE, AF, acting together. But
AG, Ag compose a force = 2 AI; and the forces AH, Ah
compose a force = 2 AL. Therefore the four forces act¬
ing together are equivalent to 2 AI + 2 AL, or to 4 AK.
But because AO is ^ AE, and the lines Gg, Oo, HA, are
evidently parallel, 4 AK is equal to 2 AQ, or to AC; and
the proposition is demonstrated.
(I.) Cor. Let us now suppose that by continual bisec¬
tion of a right angle we have obtained a very small angle
« of a rhombus; and let us name the rhombus by the mul¬
tiple of a, which forms its acute angle.
The proposition (G) is true of a, 2 a, 4 a, &c. The pro¬
position (H) is true of 3 a. In like manner, because (G)
is true of 4a and 8a, proposition (H) is true of 6 a; and
because it is true of 4 a, 6 a, and 8 a, it is true of 5 a and
7 a. And so on continually till we have demonstrated it
of every multiple of a that is less than a right angle.
(K.) Let HAS (fig. 6) be perpendicular to AC, and let
ABCD be a rhombus, whose acute angle BAD is some
multiple of 2 a that is less than a right angle. Let Ab, Cd
be another rhombus, whose sides Ab, Ad bisect the angles
11AB, SAD. Then the forces Ab, Ad compose a force AC.
Fig. 6.
R A >S
Draw Mi, rfS parallel to BA, DA. It is evident that
ARAB and ASe?D are rhombuses, whose acute angles are
multiples ot a, that are each less than a right angle.
Therefore (I) the forces AR and AB compose the force
Ab and AS, AD compose Ad; but AR and AS annihi¬
late each other’s effect, and there remains only the forces
AB, AD. Therefore Ab and Ad are equivalent to AB
and AD, which compose the force AC; and the proposi¬
tion is demonstrated.
(L.) Cor. Thus is the corollary of last proposition ex¬
tended to every rhombus, whose angle at A is some mul¬
tiple of a less than two right angles. And since a may be
Fig. 5.
A
taken less than any angle that can be named, the proposi-
tion may be considered as demonstrated of every rhombus •
and we may say,—
(M.) Two equal forces, inclined to each other in am
angle, compose a force which is measured by the diagonal
of the rhombus, whose sides are the measures of the consti¬
tuent forces.
(N.) Two forces AB, AC (fig. 7) having the direction
and proportion of the sides of a rectangle, compose a force
AD, having the direction and proportion of the diagonal.
Fig. 7.
Draw the other diagonal CB, and draw EAF parallel to
it: draw BE, CF parallel to DA.
AEBG is a rhombus ; and therefore the forces AE
and AG compose the force AB. AFCG is also a rhom¬
bus, and the force AC is equivalent to AF and AG.
Therefore the forces AB and AC, acting together, are
equivalent to the forces AE, AF, AG, and AG actino-
together, or to AE, AF, and AD acting together; but
AE and AF annihilate each other’s action, being opposite
and equal (for each is equal to the half of BC). There¬
fore AB and AC acting together are equivalent to AD,
or compose the force AD.
(0.) Two forces, which have the direction and propor¬
tions of AB, AC (fig. 8) the sides of any parallelogram,
compose a force having the direction and proportion of
the diagonal AD.
Fig. 8.
Se d
Draw AF perpendicular to BD, and BG and DE per¬
pendicular to AC.
Then AFBG is a rectangle, as is also AFDE ; and
AG is equal to CE. Therefore (N) AB is equivalent
to AF and AG. Therefore AB and AC acting together,
are equivalent to AF, AG, and AC acting together; that
is, to AF and AE acting together; that is (N) to AD;
or the forces AB and AC compose the force AD.
51. Hence arises the most general proposition.
Ifa material particle be urged at once by two pressures Com[
or incitements to motion, whose intensities are proportional tion o 1
to the sides of any parallelogram, and ivhich act in the di- 'nc‘t(e
rections of those sides, it is affected in the same manner as
it were acted on by a single force, whose intensity is measur¬
ed by the diagonal of the parallelogram, and which acts in
its direction; Or, two pressures, having the direction and
proportion of the sides of a parallelogram, generate a pres¬
sure having the direction and proportion of the diagonal.
52. Thus have we endeavoured to demonstrate from ab¬
stract principles the perfect similarity of the composition
of pressures, and the composition of forces measured by
the motions which they produce. We cannot help being
of the opinion that a separate demonstration is indispen-