Small Scale Versus Large Scale

You Venetians display constant activity in your famous arsenal. It is a great way to learn about mechanics.

All types of instruments and machines are being constructed by many artisans who learned them partly by:

• inherited experience and
• their own observations

We asked him why they employed larger stocks, scaffolding and bracing for launching a big vessel than for a small one.

He answered that this was to prevent the ship parting under its own heavy weight, something that does not affect small boats.

The common opinion is here absolutely wrong.

I think the opposite is true – many machines can be constructed even more perfectly on a large scale than on a small.

A clock can be made more accurate on a large scale than on a small.

Some intelligent people maintain this same opinion. But they cut loose from geometry and argue that the better performance of the large machine is due to the imperfections and variations of the material.

Imperfections in the material cannot explain the deviations between machines in the concrete and in the abstract.

Even if imperfections did not exist, the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment;

The larger the machine, the greater its weakness.

I assume matter to be unchangeable and always the same.

So we are no less able to treat this constant and invariable property in a rigid manner than if it belonged to simple and pure mathematics.

You and the students of mechanics have a wrong opinion on the ability of machines and structures to resist external disturbances.

They think that when they are built of the same material and maintain the same ratio between parts, they are able equally, or rather proportionally, to resist or yield to such external disturbances and blows.

We can demonstrate by geometry that the large machine is not proportionately stronger than the small.

Every machine and structure, whether artificial or natural, has a limit beyond which neither art nor nature can pass.

If we take a wooden rod of a certain length and size, fitted into a wall at right angles, i. e. , (4) parallel to the horizon, it may be reduced to such a length that it will just support itself; so that if a hair’s breadth be added to its length it will break under its own weight and will be the only rod of the kind in the world.

Thus if its length is 100 times its width, you will not be able to find another rod whose length is also a hundred times its breadth and which, like the former, is just able to sustain its own weight and no more.

All the larger ones will break while all the shorter ones will be strong enough to support something more than their own weight.

And this which I have said about the ability to support itself must be understood to apply also to other tests; so that if a piece of scantling [corrente] will carry the weight of ten similar to itself, a beam [trave] having the same proportions will not be able to support ten similar beams.

Seemingly improbable facts will, even on scant explanation, drop the cloak which has hidden them and stand forth in naked and simple beauty.

A horse falling 3-4 cubits will break his bones. But a dog or cat falling from the same height or 8-10 cubits will suffer no injury.

Equally harmless would be the fall of a grasshopper from a tower or the fall of an ant from the distance of the moon.

Do not children fall with impunity from heights which would cost their elders a broken leg or perhaps a fractured skull?

And just as smaller animals are proportionately stronger and more robust than the larger, so also smaller plants are able to stand up better than larger.

An oak 200 cubits high would not be able to sustain its own branches if they were distributed as in a tree of ordinary size.

Nature cannot produce a horse as large as 20 ordinary horses or a giant ten times taller than an [53] ordinary man unless by miracle or by greatly altering the proportions of his limbs and especially of his bones, which would have to be considerably enlarged over the ordinary.

Likewise the current belief that, in the case of artificial machines the very (5) large and the very small are equally feasible and lasting is a manifest error.

Thus, for example, a small obelisk or column or other solid figure can certainly be laid down or set up without danger of breaking, while the large ones will go to pieces under the slightest provocation, and that purely on account of their own weight.

A large marble column was laid out so that its two ends rested each upon a piece of beam; a little later it occurred to a mechanic that, in order to be doubly sure of its not breaking in the middle by its own weight, it would be wise to lay a third support midway; this seemed to all an excellent idea; but the sequel showed that it was quite the opposite, for not many months passed before the column was found cracked and broken exactly above the new middle support.

Surely this is the explanation, and the moment the cause is known our surprise vanishes; for when the two pieces of the column were placed on level ground it was observed that one of the end beams had, after a long while, become decayed and sunken, but that the middle one remained hard and strong, thus causing one half of the column to project in the air without any support.

Under these circumstances the body therefore behaved differently from what it would have done if suppported only upon the first beams; because no matter how much they might have sunken the column would have gone with them.

This is an accident which could not possibly have happened to a small column, even though made of the same stone and having a length corresponding to its thickness, i.e.;, preserving the ratio between thickness and length found in the large pillar.

You will not be far wrong if you say eight times as much; nor does this phenomenon contradict the other even though in appearance they seem so different.

I am at your service if only I can call to mind what I learned from our Academician * who had thought much upon this subject and according to his custom had demonstrated everything by geometrical methods so that one might fairly call this a new science.

For, although some of his conclusions had been reached by others, first of all by Aristotle, these are not the most beautiful and, what is more important, they had not been proven in a rigid manner from fundamental principles.

Since I wish to convince you by demonstrative reasoning rather than to persuade you by mere probabilities, I shall suppose that you are familiar with present-day mechanics so far as it is needed in our discussion.

First of all it is necessary to consider what happens when a piece of wood or any other solid which coheres firmly is broken; for this is the fundamental fact, involving the first and simple principle which we must take for granted as well known.

To grasp this more clearly, imagine a cylinder or prism, AB, made of wood or other solid coherent material. Fasten the upper end, A, so that the cylinder hangs vertically. To the lower end, B, attach the weight C.

However great they may be, the tenacity and coherence between the parts of this solid, so long as they are not infinite, can be overcome by the pull of the weight C, a weight which can be increased indefinitely until finally the solid breaks like a rope.

And as in the case of the rope whose strength we know to be derived from a multitude of hemp threads which compose it, so in the case of the wood, we observe its fibres and filaments run lengthwise and render it much stronger than a hemp rope of the same thickness.

But in the case of a stone or metallic cylinder where the coherence seems to be still greater the cement which holds the parts together must be something other than filaments and fibres; and yet even this can be broken by a strong pull.

To solve the problems which you raise it will be necessary to make a digression into subjects which have little bearing upon our present purpose.