ones 
are in the fame ratio as the lengths of ct planes; he c 
cludes, that the weights will be in equilibrio on the in 
clined planes when they are to ea er as the lengths 
equal to ] 
the power is to the aries as ae height of the plane to 
its length. 
The virtual Abeta is that which a body in of nee 
ifpofed to receive, in cafe the equilibrium is difturbed 
or that which the body will really receive in the firft i 
h ufed by writers on the continent, in its moft 
If'a fyftem, compofed of any number 
of bodies or oe which are drawn ia es direétion be 
ing as pofitive the fmall {paces defcribed in the 
of the forces,:and negative thofe defcribed in a contrary 
dire€tion 
Galileo feems to have been the firft writer on mechanic 
who was acquainted with this principle, in his cone a 
*¢ Della Scienza Mecanica,’? and in his dialogues he pro- 
a it as a general property in the equilibrium of ma- 
ehir 
author of another principle, eh is a to the fame 
eff-&, or is rather a neceflary confequence of the principle 
of Galileo. It is, that when two weights are fo conneéted, 
that, being placed ir y ner, Lheir centre gravity 
neither rifes nor falls; then ia ail thefe fituations they will 
e in equilibrio. a celli applies this principle to the 
inclined plane, but it can be demonftrated to hold good in 
mac 
From this arifes Sneie: principle, 
‘which fome a s have recurred to, to tue 
roblems relating ie the equilibrium of force 
fyitem of ponderable bodies 
gravity is the loweft ae 
theory “ De aximis & Minimis,’? that the centre of 
gravity of a hie is ae pee when wd ap of 
varlous 
When a 
s is in equilibrio, the centre of 
its ent that is, when the centre of gravity 
neither ates nor -deteen by a an in cay {mall change in 
the pofition of the f 
John Be nonilli is the ‘firft author who perceive ed the 
” which is entirely employed in fhewing 
s principle, and its great uti ies when 
applied. to, the folate of different cafes in flatic 
From the fame fource originated another Ape ii pro- 
pofed by Maupertuis in the Memoirs of t c for 
1740, under the e of the ** Law of 
a Re of” and 
which was ee ds extended by E 
the Memoirs of the Academy of Berlin for 1751 e 
fame nature is the principle afflamed by M. Courtivron in 
the Memoir: Academy 1748 493 an 
which confiits in this, that 
eae of bodies can fucceffively take, a it in which the 
a isamaximum or minimum is wife that in 
Ww rn the fyftem muft be placed, to remain in equi ean : 
the wis viva of a fyftem being defined the fom of the 
pective maffes of which the fy{tem is compofed, ete 
each into the {quare of its ire 
that of virtual velocity a 
nen are derived) feems to b 
the moft generally ufeful. Praétical examples of the 
analy tic piocefes, by which general form 
tions for the equilibrium of any fyftem 
determined, are given by La Grange; and L ace 
demonflrates the principle on which the calculus is 
ounded. 
It fhould be obferved that force is here fuppofed to be 
the product of the mafs of a material point, by the velecity 
it oe receive if sage free. we confine thefe confi- 
derations to the cafe «¢ fingle material point, the con- 
con of  laeial orl will be a ind analogous to thole 
above-mentioned, but much fimplitied. 
The moft Sines ae to exprefs the ftate of 
equilibrium of a material point acted on by any number of 
forces is, that ie ae multiplied by aa element of its 
direction is equal ze hat is, if we fuppofe the point to 
Te its pofition an t jodinitely. {mall quantity in any direc- 
tion, then in the cafe of equilibrium, i every force be 
asl by the elementary {pace which the point has ap- 
proa to, or receded Hea a force eftimated in its 
iredtio: n, the produ will be z 
ie {uppofes the poiat or particle free ; but if it is con- 
will be perpendicular to it, or in the dire€tion of the radins 
of the curve. This re-action, therefore, may be confidered as 
anew force, and the re-aCtion multiplied by the elements of 
its dire€tion muft be added to the former equstion ; but if 
the variation of pofition, inftead of being taken arbitrarily, 
be taken upon the curve, fo as not to alter the conditions of 
the problem, fince the elementary variation of the radius is 
evidently equal to zero, the preceding equation ftill holds 
good. 
the forces is equal to zero. 
When the forces which act on a point, or a body, o 
of bodies, are not fo proportioned as to maintain the 
: this is the eee employed by La 
Grange, and fubfequently by La Place. La Grange, with 
the principle of virtual eo sane: the principle of 
d’Alembert, which is extremely fimple ; indeed it pre be 
I, onfidered. 
