The Essential Galileo (57 page)

S
AGR.
    This demonstration, Salviati, is rather long and difficult to keep in mind from a single hearing. Will you not, therefore, be good enough to repeat it?

S
ALV.
    As you like. But I would suggest instead a more direct and a shorter proof. This will, however, necessitate a different figure.

[168] SAGR. The favor will be that much greater. Nevertheless, I hope you will oblige me by putting into written form the proof just given, so that I may study it at my leisure.

S
ALV.
    I shall gladly do so. Now, let
A
denote a cylinder of diameter
DC
and the largest capable of sustaining its own weight; the problem is to find a larger cylinder that shall be at once the maximum and the unique one capable of sustaining its own weight. Let E be such a cylinder, similar to
A
, having an assigned length, and having the diameter
KL;
let
MN
be the third proportional to the two lengths
DC
and
KL;
let
MN
also be the diameter of another cylinder,
X
, having the same length as
E;
then, I say,
X
is the cylinder sought. For the resistance of the base
DC
is to the resistance of the base
KL
as the square of
DC
is to the square of
KL
, that is, as the square of
KL
is to the square of
MN
, or, as the cylinder
E
is to the cylinder
X
, that is, as the moment of
E
is to the moment of
X;
but the resistance of the base
KL
is to the resistance of the base
MN
as the cube of
KL
is to the cube of
MN
, that is, as the cube of
DC
is to the cube of
KL
, or, as the cylinder
A
is to the cylinder
E
, that is, as the moment of
A
is to the moment of
E;
hence it follows, by perturbed equidistance of ratios, that the moment of
A
is to the moment of
X
as the resistance of the base
DC
is to the resistance of the base
MN;
therefore, moment and resistance are related to each other in prism
X
precisely as they are in prism
A
.

Let us now generalize the problem. Then the proposition will read as follows:
Given a cylinder
AC
in which moment and resistance are related in any manner whatsoever, and given that
DE
is the length of another cylinder
,
then determine what its thickness must be in order that the relation between its moment and resistance shall be identical with that of the cylinder
AC
.

Using again the penultimate figure and almost in the same manner, we may say the following. Since the moment of the cylinder
FE
is to the moment of the portion
DG
as the square of
ED
is to the square of
FG
, that is, as the length
DE
is to
I;
and since the moment of the cylinder
FG
is to the moment of the cylinder
AC
as the square of
FD
is to the square of
AB
, or, as the square of
ED
is to the square of
I
, or, as the square of
I
is to the square of
M
, [169] that is, as the length
I
is to
O;
it follows, by equidistance of ratios, that the moment of the cylinder
FE
is to the moment of the cylinder
AC
as the length
DE
is to
O
, that is, as the cube of
DE
is to the cube of
I
, or, as the cube of
FD
is to the cube of
AB
, that is, as the resistance of the base
FD
is to the resistance of the base
AB
. This is what was to be proven.

From what has been demonstrated so far, you can plainly see the impossibility of increasing the size of structures to vast dimensions either in art or in nature. Thus, it would be impossible to build ships, palaces, or temples of enormous size in such a way that their oars, masts, beams, iron bolts, and, in short, all their other parts will hold together. Nor could nature produce trees of extraordinary size, because the branches would break down under their own weight. Likewise it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions; for these animals would have to be increased enormously in height and this increase could be accomplished only by employing a material that is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animals would be monstrous. This is perhaps what our wise poet had in mind, when he said, in describing a huge giant: “Impossible it is to reckon his height / So beyond measure is his size.”
¹⁶

To illustrate briefly, I have sketched a bone whose natural length has been increased three times and whose thickness has been multiplied until, for a correspondingly large animal, it would perform the same function which the small bone performs for its small animal. From the figures here shown you can see how out of proportion the enlarged bone appears. Clearly then if one wishes to maintain in a great giant the same proportion of limb as that found in an ordinary man, one must either find [170] a harder and stronger material for making the bones, or one must admit a diminution of strength in comparison with men of medium stature; for if his height be increased inordinately, he will fall and be crushed under his own weight. On the other hand, if the size of a body be diminished, the strength of that body is not diminished in the same proportion; indeed the smaller the body the greater its relative strength. Thus a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.

S
IMP.
    This may be so. But I am led to doubt it on account of the enormous size reached by certain fish, such as the whale which, I understand, is ten times as large as an elephant; yet they all support themselves.

S
ALV.
    Your question, Simplicio, suggests another principle, one that had hitherto escaped my attention and that enables giants and other animals of vast size to support themselves and to move about as well as smaller animals do. This result may be secured by increasing the strength of the bones and other parts intended to carry not only their weight but also the superincumbent load. But there is another way: keeping the proportions of the bony structure constant, the skeleton will hold together in the same manner or even more easily provided one diminishes, in the proper proportion, the weight of the bony material, of the flesh, and of anything else which the skeleton has to carry. It is this second principle that is employed by nature in the structure of fish, making their bones and muscles not merely light but entirely devoid of weight.

S
IMP.
    The trend of your argument, Salviati, is evident. Since fish live in water, which on account of its density or (as others would say) heaviness diminishes the weight of bodies immersed in it, you mean to say that, for this reason, the bodies of fish will be devoid of weight and will be supported without injury to their bones. But this is not all; for although the remainder of the body of the fish may be without weight, there can be no question but that their bones have weight. Take the case of a whale's rib, having the dimensions of a beam; who can deny its great weight or [171] its tendency to go to the bottom when placed in water? One would, therefore, hardly expect these great masses to sustain themselves.

S
ALV.
    A very shrewd objection! And now, in reply, tell me whether you have ever seen fish stand motionless at will under water, neither descending to the bottom nor rising to the top, without the exertion of force by swimming?

S
IMP.
    This is a well-known phenomenon.

S
ALV.
    The fact then that fish are able to remain motionless under water is a conclusive reason for thinking that the material of their bodies has the same specific gravity as that of water; accordingly, if in their make-up there are certain parts that are heavier than water, there must be others that are lighter, for otherwise they would not produce equilibrium. Hence, if the bones are heavier, it is necessary that the muscles or other constituents of the body should be lighter, in order that their buoyancy may counterbalance the weight of the bones. In aquatic animals, therefore, circumstances are just reversed from what they are with land animals, inasmuch as in the latter the bones sustain not only their own weight but also that of the flesh, while in the former it is the flesh that supports not only its own weight but also that of the bones. We must therefore cease to wonder why these enormously large animals inhabit the water rather than the land, that is to say the air.

S
IMP.
    I am convinced. I only wish to add that what we call land animals ought really to be called air animals, seeing that they live in the air, are surrounded by air, and breathe air.

S
AGR.
    I have enjoyed Simplicio's discussion, including both the question raised and its answer. Moreover, I can easily understand that one of these giant fishes, if pulled ashore, would perhaps not sustain itself for any great length of time, but would be crushed under its own mass as soon as the connections between the bones gave way.

[§10.5 Day III: A New Science of Motion]
¹⁷

[190] My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small. Nevertheless, I have discovered
¹⁸
some properties of it that are worth knowing and that have not hitherto been either observed or demonstrated. Some superficial properties have indeed been noted, such as, for instance, that the natural motion of a heavy falling body is continuously accelerated. But in just what proportion this acceleration occurs has not yet been shown. For, as far as I know, no one has yet demonstrated that the distances traversed during equal intervals of time by a body falling from rest stand to one another in the same ratio as the odd numbers beginning with unity. It has been observed that missiles and projectiles describe a curved path of some sort. However, no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in demonstrating. And, what I consider more important, this will open the doors to a vast and most excellent science, of which my work is merely the beginning; then other minds more acute than mine will explore its remote corners.

This discussion is divided into three parts. The first part deals with motion that is steady or uniform. The second treats of motion as we find it accelerated in nature. The third deals with violent motions, or projectiles.

[§10.6 Day III: Definition of Uniform Acceleration]
¹⁹

[196] S
ALV.
    The preceding is what our Author has written concerning uniform motion. We turn now to a newer and more discriminating discussion, dealing with naturally accelerated motion, such as that generally experienced by heavy falling bodies. The title is “On Naturally Accelerated Motion,” and here is the introduction:

[197] The properties belonging to uniform motion have been discussed in the preceding section; but accelerated motion remains to be considered.

And first of all, it seems desirable to investigate and explain the definition that best corresponds to the accelerated motion which nature uses. For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions that are not met with in nature, and they have very commendably established the properties which these curves possess in virtue of their definitions. But we have decided to consider the properties of bodies falling with an acceleration such as actually occurs in nature and to make our definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties that have been, one after another, demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself in all her various other processes, to employ only those means that are most common, simple, and easy. For I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds.