6.54 MECHANIC S. 
the aggregate of the ftrength of the fibres in the longitu¬ 
dinal direction. But, with refpeft to lateral ftrength, we 
muft confider each fibre as acting at the extremity of a 
lever vvhofe centre of motion is on the lineal; thus, 
each fibre in the line cd will rend’the breach by a force 
proportional to the product of its individual ftrength into 
its diftance db from the centre of motion, and confe- 
quentiy the refiftance of all the fibres in cd will be re- 
prefented by c cl x bd. In like manner, the aggregate re¬ 
finance of another courfe of fibres parallel to ab, as oo, 
will be reprefented by ooy.bo ; and of a third, as ii, by 
the reftangle iixbi ; and fo throughout. Therefore the 
fium of all thefe prodndls will expref's the total ftrength or 
refiftance of the beam in that part. But (Prop. XXXIII.) 
the fum of all thefe produdls is equal to the product of 
the area abed into the diftance of its centre of gravity 
from ab ; whence the propofition is manifeft. 
Cor. i. In fquare heaths, the lateral ftrengths are at 
the cubes of tire breadths or depths. 
Cor. 2. In cylindrical beams, the lateral ftrengths are as 
the cubes of the diameters. 
Cor. 3. The lateral ftrengths of any beams wbofe fec- 
tions are fimilar figures, are as the cubes of correfponding 
dimenlions of tire fedions. 
Cor. 4. In rectangular beams, the lateral ftrengths are 
conjointly as the breadths and fquares of the depths. For 
the areas are CC breadth x depth, and the diltances of the 
centre of gravity are OC depth ; confequently, ftrength CC 
breadth X depth 2 . 
Cor. 5. The lateral ftrength of a beam with its nar¬ 
rower face upwards, is to its ftrength with the broader 
face upwards, as the breadth of the broader face, to the 
breadth of the narrow’er. For, bd 2 : db 2 :: d : b. 
Cor. 6 . If a beam were fixed firmly to one end into a 
wall, arid the fraCture were caufed by a weight fufpended 
at the other end, the procefs of nature would be fimilar, 
only that the breach would terminate at the lower part of 
the beam ; and the propofition and five firlt corollaries 
would ftill obtain. 
Prop. LXXV. The lateral frengths of prifmatic beams of 
the fame materials arc as the areas of the feblions and the diftances 
of their centres of gravity, direftly, and as their lengths and 
weights, inverfely. —Let B C, G H, (fig. 85 ) be two beams 
of like materials fixed in horizontal pontions to the up¬ 
right wall AB, by their ends B, G. Let Abe the area 
of the end G of the beam G H, G tlie diftance of the cen¬ 
tre of gravity of that end from its loweft point, L its length, 
W its weight, and S its ftrength ; and let a, g, l, w, and s, 
be correfponding particulars in the beam C B. Then 
S : s ; ; ■ a -~ : A . G . / . w : a . g . L . W. 
For, the direCt ftrength, or effort tending to preferve the 
adhefion of the fibres, varies as the product of A . G, 
« • g, by the laft Prop, while the efforts tending to de- 
Itroy their adhefion, and which are therefore in the inverfe 
ratio of the ftrengths, vary both in proportion to the 
weights of the beams and the diftances at which thofe 
weights a6t; but the weights of the beam may evidently 
be confidered as aCting at their centres of gravity, the 
diftances of which from the end fupported vary as the 
length of the beams ; and confequently the efforts tending 
to deftroy the adhefion of the beams are as L . W, / . w. 
Whence, by incorporating the direft and inverfe ratios, 
we obtain that ftated in the propofition. 
Cor. 1. Had the beams been confidered as fixed at both 
ends, the fame thing would follow, with this difference 
only, that a beam when fixed at both ends is as Jlrong as one of 
equal breadth and depth, and but half the length, which is fixed 
only at one end. For, it the longer beam were bilefted, 
each of its halves would be fitualed with refpeCt to its 
fixed end in the fame manner as the fliorter beam with re- 
/peet to its fixed end. 
Cor. 2. When the ftrength of a beam is very cojifidcra- 
ble in relation to its weight, we may, inftead of the pro* 
r ■ . _ A . G a . g 
pofition, take S : s :: —-— ; — ~ . 
Cor. 3. Cylinders and fquare prifms have their lateral 
ftrengths proportional to the cubes of the diameters, or 
depths, directly, and their lengths and weights, inverfely. 
Cor. 4. Similar prifms and cylinders have their ftrengths 
inverfely proportional to their like linear dimenfions. 
For, the cubes of the diameters or depths vary as the cubes 
of the lengths, and the weights and lengths are as the 
cubes of the lengths and the lengths conjointly, or in the 
quadruplicate ratios of the lengths; therefore, the ftrengths 
are as L 3 to l 3 dire&iy, and L 4 to / 4 inverfely, or in¬ 
verfely as the lengths L and l. 
From the preceding deductions it follows, that greater 
beams and bars muft be in greater danger of breaking than 
the lefs fimilar ones: and that, thougn a lefs beam may 
be firm and l'ecure, yet a greater fimilar one may be made 
fo long as neceffarily to break by its own weight. Hence 
Galileo juftly concludes, that what appears very firm, and 
fucceeds well, in models, may be very weak and unftable, 
or even fall to pieces by its weight, when it comes to be 
executed in large dimenfions, according to the model. 
From the fame principles he argues that there are neceffi- 
rily limits in the works of nature and art, which they can¬ 
not furpafs in magnitude 5 that immenfely-great fhips, 
palaces, temples, &c. cannot be erected, their yards, 
beams, bolts, &c. falling afunder by reafon of their weight. 
Were trees of a very enormous magnitude, their branches 
would, in like manner, fall off. Large animals have not 
ftrength in proportion to their fize ; and, if there were 
any land-animals much larger than thofe we know', they 
could hardly move, and would be perpetually lubjeffed to 
moft dangerous accidents. As to the animals of the fea, 
indeed, the cafe is different, as the gravity of the water 
fultains thofe animals in great meafure; and in fa£t thefe 
are known to be fometimes vaftly larger than the greateft 
land-animals. It is,,fays Galileo, impoftible for Nature 
to give bones for men, horfes, or other animals, fo formed 
as to fubfift, and proportionally to perform their offices, 
when fuch animals fhould be enlarged to immenfe heights, 
unlefs (be ufes matter much' firmer and more refilling than 
ftie commonly does ; or fhould make bones of a thicknefs 
out of all proportion ; whence the figure and appearance 
of the animal muft be monftrous. This he fuppofes the 
Italian poet hinted at, when he Laid, 
Non fi puo compartir quanto fid lungo, 
Si fmifuratamente e tutto groJJ'o. 
Whatever Height we to the giant give, 
He cannot without equal thicknefs live. 
And, this fentiment being fuggefted to us by perpetual 
experience, we naturally join the idea of greater ftrength 
and force with the groffer proportions, and that of agility 
with the more delicate ones. The fame admirable philo- 
fopher likewife remarks, iii connexion with this fuhjett, 
that a greater column is in much more danger of being 
broken by a fall than a fimilar fmall one ■, that a man is in 
greater danger from accidents than a child 5 that an infedt 
can fuftain a weight many times greater than itfelfj whereas a 
much larger animal, as a horfe, could fcarcely carry another 
horfe of his own fize. The ingenious Itudent may eafily 
extend thefe pradical remarks to any cafes which may 
come before him. 
Prop. LXXVI, The lateral Jlrengths of two cylinders (of the 
fame matter) of equal weight and length, one oj which is hollow 
and ike other/olid, are to each other as the diameters of their ends. 
—Let ABE, III K, (figs. 86 and 87.) be the ends of two 
cylinders of equal length, and containing equal quantities 
of matter, the former being the feftion of a tube confti- 
tuted of cylinders having a common axis ; then is the 
ftrength of the tube to that of the folid cylinder as All 
to III. For the lateral ftrengths (Prop. LXXIV.) are 
3 conjointly 
