M E C H 
■conjointly as the areas and the diflances of the centres of 
gravity of the feftions from A or from B, according as the 
fractures terminate at the one or the other point ; but the 
areas of the annulus in fig. 86, and of the circle in fig. 87, 
are here equal, and the centres of gravity of both are at 
their centres of magnitude ; wherefore, tince the radii vary 
as the diameters, the Strengths in this cafe vary in the fame 
ratio. 
Cor. 1. Since, when the area of a circular feftion is given, 
its diameter is greater when the feftion is an annulus titan 
when it is a circle without any cavity 5 and fince the power 
with which the parts of the cylinder refill extraneous 
force is greater in the fame proportion ; it follows, accord¬ 
ing to the theory thus Stated, that the ftrength may be in- 
creafed indefinitely without augmenting the quantity of 
matter. This conclusion is, however, manifeltiy errone¬ 
ous ; becaufe, after the diameter of the tube exceeds a 
certain magnitude, (which can only be afeertained by 
experiment^) it will become flaccid, and bend under the 
fmallett additional weight. The fa ft is, the reafoning in 
the proportion is founded upon the preemption that the 
figure of the Section will be conftantly circular ; and will 
therefore only hold true under thofe limits in which the 
preffure or ftroke upon the tube will not caufe its feftion 
to degenerate from the circle to an ellipfis or any other 
form. 
Cor. 2. When the two diameters of the end of a tube are 
given, the diameter of a folid cylinder of equal weight 
may be eatily found (the lengths being fuppofed the fame), 
by taking the fquare-root of the difference of the fquares 
of the diameters of the tube; or the fquare-root of the 
prod lift of their fum and difference. Or, the fame may be 
effected geometrically by this Ample procefs: From one 
end of the exterior diameter AB (fig. 86.) fet of] AE equal to 
the interior diameter CD, and join E B, which will be the 
diameter of the feftion fought. For E B 2 mult be equal 
to A E 2 —CD 2 , which it is by this conftruftion, and by 
Euc. i. 47. iii. 31. Or, if B E be drawn a tangent to the 
inner circle till it cuts the exterior one in two points E 
and B, it will be the diameter fought. For the triangles 
B E A, BFG, are then fimilar, the angles at E and F 
being right angles ; confequently, E A= 2 F G = C D ; as 
in the preceding conftruftion. 
Cor. 3. The lateral ftrengths of tubes and folid cylinders 
of equal length and fimilar materials, are as the areas of 
their ends and their diameters conjointly. 
Scholium. From this propofition Galileo juftly concludes, 
that Nature in a thoufand operations greatly augments the 
ftrength of fubftances without increasing their weight ; as 
is manifelted in the bones of animals and the feathers of 
birds, as well as in molt tubes, or hollow trunks, which, 
though light, greatly refift any effort to bend or break 
them. “ Thus (lays he), if a wheat-ftraw which fupports 
an ear that is heavier than the whole Italk were made of the 
fame quantity of matter, but folid, it would bend or break 
with tar greater eafe than it now does. And with the 
fame reafon Art has obferved, and Experience confirmed, 
that a hollow cane, or tube of wood or metal, is much 
Stronger and more firm than if, while it continued of the 
fame weight and length, it were folid, as it would then, 
of confequence, be not fo thick ; and therefore Art has 
contrived a method to make lances hollow within, when 
they are required to be both light and Strong;” in this 
inftance, as in many others, imitating the wifdom of Nature. 
In ail f'uch instances, however, there is an obvious dis¬ 
tinction between the works of Nature and thofe of Art. 
“ In the former (as M. Girard remarks, when treating 
the fame fubjeft), the caufe and the effeft effentially agree 3 
the one cannot undergo any modification without the 
other experiencing a correfpondent change ; or, to fpeak 
more precifely, a new effeft always refults from a new 
caufe. In the produftions of human induftry, on the 
contrary, there is no neceffary proportion between the 
effeft and caufe; if, for example, a determinate weight 
is to be raifed, it is indifferent whether we ufe the 
\ N I C S. 6.55 
thread which has precifely the adequate force, or the, 
cable which has a Superabundant one; while, if the fame 
weight had refted naturally fufpended, it would have 
done fo by means of fibres peculiarly appropriated in 
their organization to the objeft, and whofe difpofitioa 
would have prefented the moli advantageous form. Per- 
feftion refides in a Single point, at which Nature arrives 
without effort; while man is obliged, by repeated trials, 
to pafs over an immenfe fpace which Separates him from 
it.” 
Prop. LXXVIT. Of all hollow cylinders, whofe lengths and 
the diameters of the exterior and interior circles continue the fame , 
thofe have the greateft lateral ftrength, in which the interior touches 
the exterior circle in the highefi part, provided the cylinders are 
fixed at both ends, and in a horizontal pofiiion ; or, zvhen they 
touch in the lowefi part, if the cylinders arc fixed only at. one 
end: the cylinders in both cafes being conceived to exert their 
ftrength againft weights adling vertically. —Here, fince the 
diameters of the exterior and interior circles are fuppofed 
invariable, the area of the fpace they include will be like- 
wife invariable, fo that the ftrengths of the cylinders will 
be proportional to the diftances of the centres of gravity 
of their feftions from the point where the fracture would 
end on the fuppofition the cylinders were broken. Now, 
when a beam is fixed at both ends, and broken by a weight 
laid between thofe ends, the breach would manifeltiy 
commence at the lower part, and terminate at the upper; 
and, when it is fixed at one end only, and the weight acts’ 
at the other, the breach would commence on the upper 
fide, and terminate at the lower ; confequently, in the firSt 
inftance, the centre of gravity of the feftion muff be at 
the greateft distance from its higheft point, and in the Se¬ 
cond inftance at the greateft distance from the lowed: point, 
to enfure a maximum of ftrength. 
Let figs. 88 and 89 reprefent the feftions of the tubes, BC 
and A D being the diameters of the hollow part of both ; 
let the area of the circle whofe diameter is A B be M ; the 
diltance of its centre of gravity from A being called G ; 
the area of the Smaller circle, m ; the distance of its centre 
of gravity from A, g ; the area of the fpace between the two 
circles (=M— m ) p ; the diltance of its centre of gravity 
from A, y. Then, by the nature of the centre of gravity, 
will M G— mg-fjxy. In this equation M, G, m, and p., 
are invariable ; therefore y muff vary inverfely as g ; and, 
confequently, when y is a maximum g is a minimum, and 
vice vtrfa. But g is a minimum when the inner circle 
touches the outer one in A, as in fig. 89 ; and, it is a 
maximum when the interior and exterior circles touch in 
B, as in fig. 88. Therefore the maximum and the mi¬ 
nimum values of y obtain when the circles touch in A 
and B refpeftively ; and the comparative ftrengths of the 
tubes are as expreffed in the propofition. 
In cafes aftually arifing in praftice, this Prop, as well 
as Prop. LXXVI. will require Some modification, and for 
the fame reafon as is there ftated. The ftrengths of the 
tubes, however, will increafe as the circles approach nearer 
to each other, until they reach a certain limit, which can 
only be determined by experiment for each different kind 
of refitting body, and for various'proportions of the ex¬ 
terior and interior circles. 
General Scholia. We have already adverted to a general 
maxim, which, on account of its great importance, we 
beg to State again ; it is this : “ When feveral pieces of 
timber, iron, or any ether material, are introduced into a 
machine or ftrufture of any kind, the parts not only of 
the fame piece, but of the different pieces in the fabric, 
ought to be fo adjlifted with refpeft to magnitude, that 
the ftrength may be in every part as near as polfible in a 
conftant proportion to. the Strain to, which they will be 
fubjefted.” Thus, in the conftruftion of any engine, the 
weight and preffure upon every part Should be inveltigated, 
and the Strength Should be apportioned accordingly. All 
levers, for inftance, Should be made Strongeft where they 
are molt Strained ; as levers of the firlt kind, at the ful¬ 
crum 3 leyers of the fecond kind, where the weight afts ; 
and 
