MECHANICS. 
a body would acquire in falling down AB, (by Prop. LXX.) 
the random will be the greatelt pofiible, and will be equal 
to four times C F, or twice B A. But the body caft per¬ 
pendicularly upwards with the fame velocity would rife 
to the height B A ; therefore the greatelt random, with 
a given velocity, is double the height to which the body, 
thrown perpendicularly with the fame velocity, would rife. 
Prop. LXXIII. The randoms of projcElilcs, whofe elevations 
are given, are as the fqnares of their velocities. —If a body be 
thrown in a direflion B E, its random (Prop. LXIX.) will 
be equal to four times D E, or four times the fine of dou¬ 
ble the angle of elevation, in a circle whofe diameter A B 
is the height from which the body mult fall to acquire 
the velocity with which it is projected. But, the veloci¬ 
ty s being fuppofed variable, AB the diameter will be 
direffly as the velocity; fince, the greater velocity a body 
moves with, the greater fpace it will fall through in a 
given time. And becaufe, in the triangle E D C, the 
angle at D, being a right angle, is always invariable, and 
that the angle ECD, which is double of E A D, that is, 
of the given angle of elevation E B I, is given, the tri¬ 
angle ECD, in every variation of AB, is always equi¬ 
angular and fimilar to itfelf, and E D is always as E C : 
but EC, being a radius, is as AB : therefore ED, the 
fine of the given angle of elevation, is as AB the dia¬ 
meter. Confequently four times the fine E D, that is, the 
/ random, is as A B. But the height A B from which a 
body mud fall to acquire any velocity, is as the fquare 
of that velocity : therefore the random is as the fquare of 
the velocity. 
The theory of the motion of projefliles juft exhibited 
depends upon three fuppofitions, all of which are inac¬ 
curate. ift. That the force of gravity is the fame in 
every point of the curve, adly. That it a£ls in parallel 
lines. 3dly. That the body moves in a non-refifting me¬ 
dium. Of tbefe fuppofitions, however, the two fir ft pro¬ 
duce no error which deferves notice, but the third is a 
fource of confiderable difference between this theory and 
experiments, particularly when the initial velocity of the 
projtfliie is great. The refiftance of the air is variable, 
according to the different velocities and magnitudes of 
the projectiles; on this account the trajectory of the pro¬ 
jectile is not a parabola, nor any known and regular 
curve ; its vertex is not in the middle, but more remote 
from the point of projection than from the other extre¬ 
mity ; the time of def'cent alfo, though through a lefs 
portion of the curve, is longer than the time of afcent; 
and the part of the curve through which the body de- 
fcends is lefs curved than that through which it afcends. 
Thefe circumftances are very -perceptible to the fight in 
the motion of (tones, arrows, balls, and (hells ; and even 
in a jet of water or mercury we may trace the fame par¬ 
ticulars, tinlefs the velocity be (mall, when the path near¬ 
ly coincides with a parabola. Befides this, a body pro¬ 
jected with any confiderable velocity is not only deflected 
from a parabolic path in a vertical direction, but is made 
to deviate laterally, and change the plane of motion; in 
fome experiments indeed this deviation has been equal 
to -g or of the aftual range. Such material difcrepancies 
between the theory and the practice have induced feve- 
ral philofophers at different times to inftitute courfes of 
experiments, in order to improve the theory by a coinpa- 
rifon with their refults ; the molt extenlive and important 
of thefe are the experiments by M. Robins, fir Benj. 
Thompfon,- and Dr. Hutton; for accounts of which, lee 
the article Gunnery, vol. ix. p. m-n6. 
Of the STRENGTH and STRESS of MATERIALS. 
The refiftance cf ('olids, or that force with which the 
quiefcent parts of ioiid bodies oppofe the motion of 
others contiguous to them, is generally confidered as of 
two kinds: in one of which the refitting and refitted parts, 
though contiguous, conftitute fepsrate mattes; this will 
be confidered in another place, under the title of Frio 
Vol. XIV. No, 1002. 
G53 
tion ; in the other kind, the refifling and refitted parts 
are not only contiguous, but cohere, being parts of the 
fame body or mafs ; and it is this which we now propofc 
to confider. 
This kind of refittance has exercifed the lagacity of 
fome of the moft eminent philofophers from the time of 
Galileo down to the prefent period ; and different theo¬ 
ries have been propofed by Mariotte", Leibnitz, Varignon, 
Euler, Lagrange, Girard, and perhaps others which have 
not come to our knowledge; but none of them are fo 
free from objection and' from error as might be wiftied. 
Indeed, the figure and conftitution of bodies are fo vari¬ 
able and irregular, that we cannot with the defirabie pre- 
cifioo determine thefe elements which fnould precede and 
regulate this difeuffion. Of the theories above adverted 
to, fome are certainly very ingenious; but at the fame 
time they are very complex and intricate, and cannot by 
any means be relied upon independent of experiment ; 
we therefore prefer the comparatively-fimplc theory of Ga¬ 
lileo, originally laid down in his dialogue “ On the Caufe 
of the Coherence of Solids,” with which the other hypo- 
thefes agree in the moft effential particulars, and which, 
when aided by proper experiments, may ferve as a fare 
approximation to the Itrength and ttrefs of the different 
parts of machines. 
That the refiftance of folids might be fubjeifled to cal¬ 
culation, Galileo fuppofed, fault, that bodies were com- 
pofed of folid fibres, parallel to one another; he then en¬ 
quired what was the force with which they refill the ac¬ 
tion of a power ftretching them in a direction parallel to 
their length, and found that it was proportional to the num¬ 
ber of integral fibres ; next, confidering the fibres as fob- 
jeded to an effort perpendicular to their length, he found 
that the refiftance of the integral fibres was proportional 
to their Aim multiplied by an arm of a lever which is al¬ 
ways at a certain part of the vertical dimenfions of a fo¬ 
lid in the plane of its rupture. The length of this arm 
of lever was regulated, according to Galileo, by the pofi- 
tion of the centre of gravity of the plane of rupture ; 
according to others, by the centre of perculfion, Sec. 
But the diftinclive charader of Galileo’s hypothefis con- 
fitts in this, “ that the refiftance of each of the fibres is 
independent of their quantity of extension at the inftant 
of their rupture.” 
Def. Strength, and Jlrefs or flrain, are terms, the former 
of which is ufed to denote the force or power with which 
any mafs or body refills a breach or change in its ftate, 
which a preffure or itroke upon it has a tendency to pro¬ 
duce ; and the latter are ufed indiferiminately to exprefs 
the force which is excited in any fuch mafs, and tending 
to break it. Thus, every part of a pillar is equally Jlrained 
by the load which it fupports. Hence, it is evident that 
we cannot make any ftnufture fit for its purpofe, unlefs 
the frength in every part be at lead equal to the frefs laid 
on, or the itrain excited, in that part ; and hence the ne- 
celfity of an acquaintance with the nature of the refift¬ 
ance of bodies, (o that there (hall be neither a furplus nor 
a deficiency of materials in any machine. 
Prop. LXX IV. The frength of a beam or bar to refjl a 
fradure by a force aiding laterally, is as the folid, made by a. 
fejdion of the beam in the place where the force is applied, into 
the difiance of its centre of gravity from the point or line-where 
the breach will end. 
Suppofe AB (fig. 84.) to be the beam (of uniform 
matter throughout), fixed firmly al its two ends A, B, at 
the middle of which is laid the w'eignt W. In the cafe 
of a rupture, we conceive the beam will be feparated firil 
in the line c d oppofiteto W and fartheft from it, and the 
feparation be gradually continued till it arrives at a b, 
which may, therefore, be confidered as a fixed line till tIre 
termination of the fraiffure. Now the area abed repre- 
fents the fum of all the fibres to be broken or torn ; and, 
as they are equal to each other both in Magnitude and 
flrength (by hypothefis), this area will likewife exprefs 
8 D the 
