then the last or greatest weight, is the proper measure of the absolute strength of the bar. And the same is the case with a rope, or cord, &c.-So much then for the longitudinal strength and stress of bodies. Proceed we now to consider those of their transverse actions.

PROPOSITION XLVIII.

243. The Strength of a Beam or Bar, of Wood or Metal, &c, in a Lateral or Transverse Direction, to resist a Force acting Laterally, is Proportional to the Area or Section of the Beam in that Place, drawn into the Distance of its Centre of Gravity from the Place where the Force acts, or where the Frac ture will end.

LET AB represent the beam or bar, supported at its two ends, and on which is laid a weight w, to cause a transverse fracture abee. The force w acting downwards there, the fracture will commence or open across the fibres, in the opposite or

B

lowest line ab; from thence, as the weight presses down the upper line ee, the fracture will open more and more below, and extend gradually upwards, successively, to the parallel lines of fibres cc, dd, &c, till it arrive at, and finally open in the last line of fibres ee, where it ends; when the whole fracture is in the form of a wedge, widest at the bottom, and ending in an edge or line ee at top. Now the area ae contains and denotes the sum of all the fibres to be broken, or torn asunder; and as they are supposed to be all equal to one another, in absolute strength, that area will denote the aggregate or whole strength of all the fibres in the longitudinal direction, as in the foregoing proposition. But, with regard to lateral strength, each fibre must be considered as acting at the extremity of a lever whose centre of motion is in the line ee: thus, each fibre in the line ab, will resist the fracture, by a force proportional to the product of its individual strength into its distance ae from the centre of motion; consequently the resistance of all the fibres in ab, will be expressed by ab x ae. In like manner, the aggregate rcsistance of another course of fibres, parallel to ab, as cc, will be denoted by cc × ce; and a third, as dd, by dd x de; and so on throughout the whole fracture. So that the sum of all these products will express the total strength or resist

ance

ance of all the fibres, or of the beam in that part. But, by art. 222, the sum of all these products, is equal to the product of the area aeeb, into the distance of its centre of gravity from ee. Hence the proposition is manifest.

244. Corol. 1. Hence it is evident that the lateral strength of a bar, must be considerably less than the absolute longitudinal strength, considered in the former proposition, and will be broken by a much less force, than was there necessary to draw the bar asunder lengthways. Because, in the one case the fibres must be all separated at once, in an instant; but in the other, they are overcome and broken successively, one after another, and in some portion of time. For instance, take a walking stick, and stretching it lengthways, it will bear a very great force before it can be drawn asunder; but again taking such a stick, apply the middle of it to the bended knee, and with the two hands drawing the ends towards you, the stick is broken across by a small force.

245. Corol. 2. In square beams, the lateral strengths are as the cubes of the breadths or depths.

246. Corol. 3. And in general, the lateral strengths of any bars, whose sections are similar figures, are as the cubes of the similar sides of the sections.

247. Corol. 4. In cylindrical beams, the lateral strengths are as the cubes of the diameters.

248. Corol. 5. In rectangular beams, the lateral strengths are to each other, as the breadths and square of the depths.

249. Corol. 6. Therefore a joist laid on its narrow edge, is stronger than when laid on its flat side horizontal, in proportion as the breadth exceeds the thickness. Thus, if a joist be 10 inches broad, by 24 thick, then it will bear 4 times more when laid on edge, than when laid flat. Which shows the propriety of the modern method of flooring, with very thin, but deep joists.

250. Corol. 7. If a beam be fixed firmly by one end into a wall, in a horizontal position, and the fracture be caused by a weight suspended at the other end, the process would be the same, only that the fracture would commence above, and terminate at the lower side; and the prop. and all the corollaries would still hold good.

251. Carol. 8. When a cylinder or prism is made hollow, it is stronger than when solid, with an equal quantity of mate

rials and length, in the same proportion as its outer diameter is greater. Which shows the wisdom of Providence in making the stalks of corn, and the feathers and bones of animals, &c, to be hollow. Also, if the hollow beam have the hollow or pipe not in the middle, but nearest to that side where the fracture is to end, it will be so much the stronger.

252. Corol. 9. If the beam be a triangular prism, it will be strongest when laid with the edge upwards, if the fracture commence or open first on the under side; otherwise with the flat side upwards; because in either case the centre of gravity is the farther from the ending of the fracture. And the same thing is true, and for the same reason, for any other shape of the prism. On the same account also, a square beam is stronger when laid, or when acting anglewise, than when on a flat side.

PROPOSITION XLIX.

253. The Lateral Strengths of Prismatic Beams, of the same materials, are Directly as the Areas of the Sections and the Distances of their Centres of Gravity; and Inversely as their Lengths and Weights.

LET AB and CD represent the two beams fixed horizontally, by their ends, into an upright wall AC. Now, by the last prop. the strength of either beam, considered as without or

W

B

D

independent of weight, is as its section drawn into the distance of its centre of gravity from the fixed point, viz. as sc, where s denotes the transverse section at A or c, and c the distance of its centre of gravity above the lowest point A or

C.

But the effort of their weight, w or w, tending to separate the fibres and break the beam, are, by the principle of the lever, as the weight drawn into the distance of the place where it may be supposed to be collected and applied, which is in the middle of the length of the beam; that is, the effort of the weight upon the beam is as w x AB. Hence the prop. is manifest.

254. Corol. 1. Any extraneous weight or force also, any-. where applied to the beam, will have a similar effect to break the beam as its own weight; that is, its effect will be as wx d, as the weight drawn into the length of lever or distance from a where it is applied.

255. Corol.

255. Corol. 2. When the beam is fixed at both ends, the same property will hold good, with this difference only, that in this case the beam is of the same strength, as another of an equal section, and only half the length, when fixed only at one end. For, if the longer beam were bisected, or cut in halves, each half would be in the same circumstances with respect to its fixed end, as the shorter beam of equal length.

256. Corol. 3. Square prisms and cylinders have their lateral strengths proportional to the cubes of the depths, or diameters, directly, and to their lengths and weights inversely.

Corol. 4. Similar prisms and cylinders have their strengths inversely proportional to their like linear dimensions, the smaller, being comparatively larger in that proportion. For their strength increases as the cube of the diameter or of their length; but their stress, from their weight and length of lever, as the 4th power of the length.

257. Scholium. From the foregoing deductions it follows that, in similar bodies of the same texture, the force which tends to break them, or to make them liable to injury by accidents, in the larger bodies, increases in a higher proportion than the force which tends to preserve them entire, or to secure them against such accidents; their disadvantage, or tendency to break by their own weight, increasing in the same proportion as their length increases: so that, though a smaller beam may be firm and secure, yet a large and similar one may be so long as to break by its own weight. Hence it is justly concluded, that what may appear very firm and successful in a model or small machine, may be weak and infirm, or even fall in pieces by its own weight, when it is executed on large dimensions according to the model.

A

For, in similar bodies, or engines, or in animals, the greater must be always more liable to accidents than the smaller, and have a less relative strength, that is, the greater have not a strength in so great a proportion as their magnitude. A greater column, for instance, is in much more danger of breaking by a fall, than a similar smaller one. man is in more danger from accidents of this kind than a child. An insect can bear and carry a load many times heavier than itself; whereas a large animal, as a horse, for instance, can hardly support a burden equal to his own weight.

From the same principle it is also justly inferred, that there are e necessarily limits in all the works of nature and

art,

art, which they cannot surpass in magnitude. Thus, for instance, were trees to be of a very enormous size, their branches would break and fall off by their own weight. Large animals have not strength in proportion to their size: and if there were any land animals much larger than those we know, they would hardly be able to move, and would be perpetually subjected to most dangerous accidents.

As to the sea animals indeed, the case is different, as the pressure of the water in a great measure sustains them; and accordingly we find they are vastly larger than land animals.

From what has been said it clearly follows, that to make bodies, or engines, or animals, of equal relative strength, the larger ones must have grosser proportions, or a higher degree of thickness, than they have of length. And this sentiment being suggested to us by continual experience, we naturally join the idea of greater strength and force with the grosser proportions, and of agility with the more delicate ones. In architecture, where the appearance of solidity is no less regarded than real firmness and strength, in order to satisfy a judicious eye and taste, the various orders of the columns serve to suggest different ideas of strength. But, by the same principle, if we should suppose animals vastly large, from the gross proportions a heaviness and unwieldiness would arise, which would make them useless to themselves, and disagreeable to the eye. In this, as in all other cases, whatever generally pleases taste, not vitiated by prejudice of education, or by fabulous and marvellous relations, may be traced till it appears to have a just foundation in

nature.

PROPOSITION L.

258. If a Weight be placed, or a Force act, on any part of a Horizontal beam, supported at both ends, the Stress upon that part, will be as the Rectangle or Product of its two Distances from the supported ends.

THAT is, the stress upon the beam AB, at c, by the weight w, ́ is as AC X BC. For, by the nature of the lever, the effect of the weight w, on the lever AC, is AC. w; and the effect of this force acting at c, on the lever BC, is AC. W. BC AC. BC W.

And, the weight w being given, the effect or stress is as AC.

BC.

259. Coral.