M E C H 
AE: BD :: RE : MD; therefore, AE : BD :: AR 
: B M ; and, fince A E and B D are fpaces defcribed in 
the fame time by the uniform motions of A and B, A R 
and B M, which are proportional to them, will be de¬ 
fcribed in the fame time ; when, therefore, the centre of 
the body A is in R, the centre of the body B is in M, 
and the diftance M R = HEr= the fum of the radii of the 
bodies 5 lienee, they will be in the contart when they ar¬ 
rive at thofe points. Alfo M R, which joins their cen¬ 
tres, will pafs through the point of contact; and LC will 
be a tangent to them both. 
Prop. XXXII. Having given the motions, the quantities 
of matter, and the radii, of two fperical bodies which impinge 
obliquely upon each other, to find their motions after impaB .—• 
Let LN (fig. 59.) be the plane which touches the bodies 
at the point of impart; produce AB, which joins the 
centres of the bodies, indefinitely both ways; through 
the centres A and B, draw E A F, G B H, parallel to LNj 
let C A, D B, represent the velocities of the bodies before 
impart ; refolve C A into the two Cl, I A, of which C I 
is parallel, and I A perpendicular, to LN; alfo refolve 
D B into two, D K parallel to L N, and K B perpendi¬ 
cular to it. Then C A and the angle C A I, which the 
direction of A’s motion makes with A I perpendicular to 
L N, being known, C I and I A are known ; in the fame 
manner, D K and K B are known. Now C I, D K, which 
are parallel to the plane L N, will not be afferted by the 
impart ; and IA, KB, which are perpendicular to it, are 
the velocities with which the bodies impinge dirertly 
upon each other; and their efferts may be calculated by 
Prop. XXVIII. when the bodies are perfertly hard ; and by 
Prop. XXIX. when they are elaftic. Let A R and B S be 
the velocities of the bodies after impart, thus determined ; 
take AF^CI, and BH=DK; complete the parallelograms 
R F, SH, and draw the diagonals A P, BQ; then the 
bodies will deferioe the lines AP, B Q,after impart, and in 
the fame time that they deferibe C A, D B, before impart. 
Cor. If A= B, and if the body which is (truck moves 
in a given dirertion and with a given velocity after im¬ 
part, the dirertion of the impinging body, and the velo¬ 
city of its motion, may be eafiiy found. Let the body 
D (fig. 60.) impinge againft the equal body C, and let 
C B be the direction in which C moves after impart; it 
is required to find the direction in which D will move. 
Draw D c, touching the ball C at c, the place where the 
ball D impinges ; produce B C to E, and through c draw 
A cF perpendicular to E B, and complete the rertangle 
F E. The force Dc may be refolved into the forces Ec, 
c F, of which Ec is employed to move the ball C in the 
dirertion C B and with the velocity Ec; but the force 
cFhas no fnare in the impulfe, and is wholly employed 
in making the body D move in the dirertion C A, and 
with the velocity C F. 
Of the CENTRE of GRAVITY. 
Definition. The centre of gravity of any body or fyf- 
tern cf bodies is that point about which the body or fyf- 
tem, aCted upon only by the force of gravity, will ba¬ 
lance itfelf in all poiitions; or, it is a point which when 
fupported, the body or fyltem will be fupported, however 
it may be fituated in other refperts. 
The centre of gravity of a body is not always within 
the body itfelf; thus, the centre of gravity of a ring is 
not in the fubftance of the ring, but in the axis of its cir- 
cumfcribing cylinder ; and the centre of gravity of a hol¬ 
low ftaff, or of a bone, is not in the matter of which it 
is conftituted, but fomewhere in its imaginary axis. 
Every body, however, has a centre of gravity, and fo has 
every fyltem of bodies, as will foon be made evident; but 
it will be proper to premife a few brief remarks with re- 
fpert to gravity itfelf, and its effert upon bodies fubjerted 
to its operation. 
It is a fact eltablifhed by general obfervation in all ages 
and all countries, that, whenever bodies are unfupported 
pr left to themfelves, they begin to move downwards in 
Vol.XIV. No. tool. 
A N I C S. 6ii 
vertical lines, and continue thus to move until they meet 
with fomething which interrupts their motion or prevents 
their further defeent. This is obferved to take place not 
only with refpfrt to large and weighty bodies, but to 
fmaller ones, and even to tlie molt minute particles into 
which they can be feparated, provided they are not fo 
fmall as to elude the obfervation of our fenfes. And, if 
certain fubftances, fuch as fmoke, and vapours, See. feem 
to contradirt this univerfal fart ; it is becaufe they are 
only in appearance left to themfelves, while in reality they 
are fupported, and put into ail al’cending motion, by the 
aCtion of the fluids. See. that compofe the atmofphere 
which furrounds the earth. All bodies, and their molt 
intimate particles, tend towards a point w'hich is either 
accurately or very nearly the centre of the terraqueous 
globe; yet this tendency is certainly not effential to mat¬ 
ter; it is an effort which matter of itfelf is not able to 
make, being indifferent to either motion or reft. We are 
authorifed, then, to conclude, that this tendency to mo¬ 
tion is caufed by a power not exifting in the matter on 
which our obfervations are made, but in fomething ex¬ 
terior; and this force, without attempting to explain its 
nature and effence. We delignate by the term Gravity. 
The general fart or event of bodies falling is denoted by 
the verbal noun Gravitation ; and it is a part or con- 
fequence of a more univerfal property, not here entered 
upon, that of the mutual Attraction of the different 
bodies in the univerfe towards each other. 
Since gravity impreffes, or has a tendency to imprefs, 
on every particle of bodies, in an inftant, a certain velo¬ 
city with which they would begin to fall if they were not 
fupported; and fince, abftrarting the influence of the air, 
this velocity would be the fame for each of the molecules 
of bodies, whatever be their fubftance, it will not be dif¬ 
ficult to attach a juft and fcientific meaning to that which 
is commonly called weight-, it is the effort necelfary to 
prevent a body from falling. But bodies fall in confe- 
quence of the artion of the force of gravity upon each of 
their particles ; and they can be prevented from falling by 
a force equal and oppoiite to the refultant or equivalent 
of all thefe aftions. Hence, we may readily diftinguilh 
between the effect of gravity and that of weight, by adopt¬ 
ing the language of Condorcet, when belays, “ the former 
is the power of tranfmitting, or a tendency to tranfmit, 
into every particle of matter a certain velocity which is 
abfolutely independent on the number of material parti¬ 
cles; and the fecond is the effort which muft be exercifed 
to prevent a given mafs from obeying the law of gravity. 
Weight , accordingly, depends on the mafs ; but gravity 
has no dependence at all upon it.” 
Every particle of which bodies are compofed receiving 
from gravity equal folicitations towards the centre of the 
earth, it follows, that, if the fupports of bodies, whether 
large or minute, were taken away, and they were permit¬ 
ted to fall from equal altitudes, they would arrive at the 
furface of the earth after equal portions of time ; and this 
is confirmed by experience ; for under the exhaulted re¬ 
ceiver of the air-pump (where the refiftance of the air is 
removed) the heavieft metals and the lighted: feathers, or 
down, fall in the fame time. If, therefore, a body is di¬ 
vided into ever fo many parts, each of them left to itfelf 
would arrive at the furface of the earth after the fame 
time as would have been employed by the whole body in 
defeending. All bodies being more or lefs porous, and 
poffefling different degrees of denfity, they will contain a 
greater or lefs number of equal moleculae in the fame vo¬ 
lume or bulk; hence all bodies of equal bulk are not 
equal in weight. But, fince the weight is equal to the! 
fum of all the efforts exercifed by gravity upon the con- 
ftituent moleculae of a body, it is proportional to its den¬ 
fity 01 to its mafs. If p, p', p", See. be the feveral particles 
of which a body is compofed, and M iis mafs, then will 
Mz=p+p'+p"+ 8 cc. and, if g reprefent the force of gra¬ 
vity loliciting each particle, we (hall have the weight = 
g^l~gpfgp'-\-gp"-\- See. 
8 A 
When 
