MECHANICS. C33 
survature, agreeably to tbe theory of central forces in the 
circle 3 which equally reduces to the fame principle the 
motion in the direction of the radius of curvature. Such 
are the means that were long employed to determine the 
motions of ifolated bodies, ailed upon by accelerating 
forces, of whatever magnitude or direction. Newton fol¬ 
lowed this method : he only clothed his folutions in a fyn- 
thefis, which frequently conceals the greatelf difficulties 
under the appearance of elegance and fimplicity. 
The Mechanics of Euler, publiffied in 1736, contain the 
whole theory of rectilinear or curvilinear motion in an 
ifolated body, ailed upon by any accelerating forces what¬ 
ever, either in vacuo, or in a refilling medium. The au¬ 
thor has every where followed the analytical method ; 
which, by reducing all the branches of this theory to uni¬ 
formity, greatly facilitates our underltanding it, as Euler 
manages this method with an elegance and fagacity, of 
which before him there was no example. He not only re- 
folves a number of difficult problems, fome of which- were 
then new, but he even improves the analyfis itfelfby new 
and delicate folutions, to which his fubjett gives occafion. 
As to the principles of mechanics for putting the pro¬ 
blems into equations, he employs thofe mentioned above. 
Though this manner of laying the foundation of the cal¬ 
culation was fufficiently commodious, the fame end might 
be attained by means ftill more fimple. This was, to re- 
folve at every inftant the forces and the motions intoother 
forces and other motions, parallel to fixed lines of a given 
pofition in fpace. Nothing more then is neceflary, but to 
apply the equations of the principle of the vis inertias to 
tliefe forces and motions ; in which cafe there is no need 
qf recurring to the theorem of Huygens. This fimple and 
happy idea, of which Maclaurin firlt made ufe in his Trea- 
tife on Fluxions, has thrown new light on mechanics, and 
Angularly facilitated the folution of various problems. 
When the body moves conftantly in one plane, two fixed 
axes only are to be taken, which are fuppofed to be per¬ 
pendicular to each other, for the fake of greater fimplicity ; 
but, when we are obliged by the nature of the forces to 
change the path continually in all direftions, and to de- 
fcribe a curve of double curvature, three fixed axes are to 
be employed, perpendicular to each other, or forming the 
edges of a right-angled parallelopiped. 
In the year 1743, d’Alembert publiffied his Traite de 
Dynamique, founded upon a new principle in mechanics. 
This principle was firlt employed by James Bernouilli in 
his folution of the problem of the centre of ofcillation ; but 
d'Alembert had the honourof generaliiing it, and giving it 
all that fimplicity and fertility of which it was fufceptible. 
He ffiowed, that, in whatever manner the bodies of one 
fyfteinait upon another, their motions may always be de- 
compofed into two others at every inllant; thofe of the 
one being deftroyed the inllant following, and thofe of 
the other retained 3 and that the motions retained are ne- 
cefiarily known from the conditions of equilibrium be¬ 
tween thofe which are dellroyed. This principle is evi¬ 
dently a confequence of the laws of motion and equili¬ 
brium, and has the advantage of reducing all the problems 
of dynamics to pure geometry and the principles of Itatics. 
By means of it, d’Alembert relolved a number of beauti¬ 
ful problems which had efcaped his predeceflbrs, and par¬ 
ticularly that of the preceffion of the equinoxes, which 
had occupied the attention of Newton. In this treatife 
d’Alembert has likewife reduced the whole of mechanics 
to three principles, the vis inertias, compound motion, 
and equilibrium ; and has illullrated his views on this 
fbbjeft by that profound and luminous reafoning which 
charadterifes all his writings. 
Another general principle in dynamics was about this 
•time difcovered feparately by Euler, Daniel Bernouilli, 
and the chevalier d’Arcy, and received the name of “ the 
confervation of the momentum of rotatory motion.” Ac¬ 
cording to the two firlt philofophers, the principle may 
be thus defined : In the motion of feveral bodies round a 
fixed centre, the Cum of the products of the mafs of each 
body multiplied by the velocity of its motion round the 
centre, and by its diftance from that centre, is always in¬ 
dependent of the mutual aition which the bodies may 
exert upon each other, and always preferves itfelf the 
fame, provided the bodies are not influenced by any ex¬ 
ternal caufe. This principle was given by Daniel Ber¬ 
nouilli in the Memoirs of the Academy of Berlin for 
1746 ; and in the fame year by Euler in the firlf volume 
of his works. They were both led to the difeovery, while 
inveftigating the motion of feveral bodies in a.tube of a 
given form, and which can only turn round a fixed point. 
The principle difcovered by d’Arcy was given in a me¬ 
moir dated 1746, and publiffied in the Memoirs of the 
Academy for 1747. He fhowed that the fum of the pro¬ 
ducts of the mafs of each body by the area which its 
radius veilor deferibes round a fixed point, is always pro¬ 
portional to the times. The identity of this principle, 
which is a generalifation of Newton’s theorem about the 
areas deferibed by the planetary bodies, with that of Euler 
and Bernouilli, will be eafily perceived, if we confider 
that the element of the circular arc, divided by the ele¬ 
ment of the time, expreffes the velocity of circulation 3 
and that the element of the circular arc, multiplied by 
the diftance from the centre, gives the element of the area 
deferibed round that centre ; fo that the principle of Euler 
is only a differential expreffion of the principle of d’Arcy, 
which he afterwards exprefied in this form, that the fum 
of the products of the maffes of each body by their velo¬ 
cities, and by the perpendiculars drawn from the centre 
to their lines of direction, is a conllant quantity. 
The principle of leaft adtion, which was firft propofed 
by Maupertuis in 1744, confifts in this, that when feveral 
bodies, ailing upon one another, experience any change 
in their motion, this change is always fuel), that the quan¬ 
tity of aition (or the produit of the mafs by the fpace 
and the velocity) employed by nature to produce it, is 
the leaft polfible. From this principle Maupertuis de¬ 
duced the laws of the refledlion and refradlion of light, 
and thofe of the collifion of bodies. (Mein. Acad. Paris, 
1744. and Mem. Acad. Berlin, 1746.) He afterwards ex¬ 
tended its application to the laws of motion ; and made 
the principle fo general as to comprehend the laws of 
equilibrium, the uniform motion of the centre of gravity 
in the percuffion of bodies, and the confervation of aitive 
forces. This celebrated principle was attacked by Koenig, 
profeffor of mathematics at the Hague, in the Leipfic Adts 
for 1751, who not only attempted to (how its falfity, but 
afferted that Leibnitz had firft deferibed it in 1707, in a 
letter to Hermann. The paper of Koenig gave rile to a 
long and violent difpute about the accuracy of the prin¬ 
ciple, and the authenticity of the letter of Leibnitz. The 
academy of Berlin interfered in behalf of their prefident, 
and gave importance to a controverfy which was too per- 
fonal to merit the attention which it received. In his 
Traite dcs Ifoperimctries, printed at Laufanne in 1744, Euler 
extended the principle of lead adtion, and (bowed, “that 
in the trajectories deferibed by means of centra! force*, 
the integral of the velocity multiplied by the element of 
the curve, is either a maximum or a minimum." This re¬ 
markable property, which Euler recognifed only in the 
cafe of infulated bodies, was generalifed by Lagrange into 
this new principle, “ that the fum of the products of the 
maffes by the integrals of the velocities, multiplied by the 
elements of the fpaces deferibed, is always a maximum or 
a minimum.” In the Memoirs of Turin, Lagrange has 
employed this principle to refolve feveral difficult pro¬ 
blems in dynamics 3 and he has ffiown, that when it is 
combined with the confervation of active forces, and de¬ 
veloped according to the rules of his method of variations, 
it furnilhes diredtly all the equations neceflary for the 
folution of each problem, and gives rife to a fimple and 
general method of treating the various problems concern¬ 
ing the motion of bodies. 
An important difeovery in rotatory motion, was'at this 
time made by profeffor Segner. In a paper, entitled Spe¬ 
cimen 
