The Essential Galileo

by Galilei, Galileo, Finocchiaro, Maurice A.

  Let us begin, therefore, to examine one by one all the particulars that have been produced. I say that shapes, as simple shapes, not only do not operate in natural things, but neither are they ever separated from corporeal substance. Nor have I ever alleged them to be stripped of sensible matter. Likewise, I also freely admit that in our endeavoring [91] to examine the diversity of effects dependent upon the variety of shapes, it is necessary to apply them to materials that do not obstruct the various operations of those various shapes. And I admit and grant that I should be wrong if I would experiment about the influence of acuteness of edge with a knife of wax, applying it to cut an oak, because there is no acuteness in wax able to cut that very hard wood. But yet such an experiment with this knife would not be besides the purpose to cut curdled milk, or other very yielding matter; indeed, with such materials, wax is more appropriate than steel for finding the diversity depending upon more or less acute angles because that milk is indifferently cut with a razor and with a knife that has a blunt edge. It is necessary, therefore, that regard be had not only to the hardness, solidity, or gravity of the bodies which under diverse shapes are to divide and penetrate some materials, but also to the resistance of the materials to be divided and penetrated. But in making the experiment concerning our controversy, I have chosen a material that penetrates the resistance of the water and in all shapes descends to the bottom, and so my adversaries can charge me with no defect; indeed, I have proposed a more excellent method than they have, inasmuch as I have removed all other causes of descending or not descending to the bottom and retained the sole and pure variety of shapes, demonstrating that the same shapes all descend with the addition of only one grain in weight and return to the surface and float with its removal. It is not true, therefore (returning to the example introduced by them), that I have gone about experimenting on the efficacy of acuteness in cutting with materials unable to cut; rather, I have done so with materials proportioned to our occasion, since they are subjected to no other variation than that alone which depends on the shape being more or less acute.

  But let us proceed a little farther. Let us note how needlessly indeed they introduce the consideration that the material chosen ought to be proportionate for the making of our experiment; using the example of cutting, they declare that just as acuteness is insufficient to cut unless it exists in a material that is hard and apt to overcome the resistance of the wood or other material which we intend to cut, so the aptitude of descending or not descending in water can and should be recognized only in those [92] materials that are able to overcome the resistance and conquer the coarseness of the water. On this I say that it is indeed necessary to make a distinction and selection of this or that material on which to impress the shapes for cutting and penetrating this or that body based on whether the solidity or hardness of the said bodies shall be greater or less; but then I add that such distinction, selection, and caution would be superfluous and unprofitable if the body to be cut or penetrated should have no resistance or should not oppose at all the cutting or penetration; and if the knife were to be used in cutting mist or smoke, one of paper would be equally serviceable with one of Damascus steel. And so, because water does not have any resistance against penetration by any solid body, all choice of material is superfluous and needless; and the selection, which I said above to have been well made, of a material similar in gravity to water was made not because it was necessary for overcoming the coarseness of the water, but for overcoming its gravity with which only it resists the sinking of solid bodies; and for what concerns the resistance of the coarseness, if we carefully consider it, we shall find that all solid bodies (those that sink as well as those that float) are indifferently accommodated and apt to bring us to the knowledge of the truth in question. Nor will I be frightened off from believing these conclusions by the experiments that may be produced against me: that although many pieces of wood, cork, clay, and even thin plates of all sorts of stone and metal are ready by means of their natural gravity to move towards the center of the earth, nevertheless they are impotent (either because of their shape, as my adversaries think, or because of their lightness) to break and penetrate the continuity of the parts of the water and to disturb its union, and they continue to float without submerging in the least. Nor, on the other hand, shall I be moved by the authority of Aristotle, who in more than one place affirms the contrary of what experience shows me.

  I return, therefore, to assert that there is no solid of such lightness, or of such shape, that being put upon the water does not divide and penetrate its coarseness. Indeed, if anyone with a more perspicacious eye shall return to observe more exactly the thin boards of wood, he shall see part of their thickness to be under water; their lower surface is not the only part that kisses the upper surface of the water, as those of necessity must have believed who have said that such [93] boards are not submerged, not being able to divide the tenacity of the parts of the water. Moreover, he shall see that when the very thin slivers of ebony, stone, or metal float, they not only have broken the continuity of the water, but also are under its surface with all their thickness, and more and more according as the materials are heavier; thus, a thin plate of lead shall be lower than the surface of the surrounding water by at least twelve times the thickness of the plate, and gold shall dive below the level of the water almost twenty times the thickness of the plate, as I shall show anon.

  But let us proceed to evince that the water yields and allows itself to be penetrated by the lightest solid; and thereby demonstrate how, even from materials that are not submerged, we may come to know that shape accomplishes nothing about the going or not going to the bottom, given that the water allows itself to be penetrated equally by every shape.

  Make a cone or pyramid of cypress, fir, or other wood of similar weight, or of pure wax, and let its height be very great, namely a palm or more; and put it into the water with the base downwards. First, you shall see that it will penetrate the water and will not be at all impeded by the width of the base; nor yet shall it sink all under water, but the part near the vertex shall lie above it. From this it is manifest that such a solid does not refrain from sinking out of an inability to divide the continuity of the water, having already divided it with its broad part, which in the opinion of my adversaries is less apt to make the division. The pyramid being thus positioned, note what part of it is submerged. Then, turn it with the vertex downwards. You shall see that it shall not penetrate the water more than before. Instead, if you observe how far it shall sink, every person expert in geometry may measure that those parts that remain out of the water are equal to a hair in the one as well as in the other experiment. Thus, one may manifestly conclude that the acute shape, which seemed most apt to part and penetrate the water, does not part or penetrate it any more than the large and spacious.

  Now, whoever wants to make an easier experiment can take two cylinders of the same material, one long and thin, the other short but very broad; let him put them in water, not sideways but erect and endways. If he diligently measures the parts of the one and the other, he shall see that in each of them the part submerged retains exactly the same ratio to the part out of the water, and that no [94] greater part is submerged of the long and thin one than of the other more spacious and broad, although the latter rests upon a very large surface of water and the former upon a very small one. Therefore, the diversity of shape produces neither ease nor difficulty in parting and penetrating the continuity of water; consequently, it cannot be the cause of sinking or not sinking. One may likewise discover that the variation of shapes does not cause the rising from the bottom of the water towards the surface: take some wax and mix it with a large quantity of lead filings, so that it becomes considerably heavier than water; then make it into a ball and place it at the bottom of the water; then fasten to it as much cork, or other light material, as just suffices to raise it and draw it towards the surface; finally, changing the same wax into a thin plate, or into any other figure, that same cork s
hall raise it in the same manner to a hair.

  This does not silence my antagonists. But they say that the whole argument hitherto made by me matters little to them; and that it serves their purpose to have demonstrated in only one particular case and for a material and a shape of their choice, namely, for a board and ball of ebony, that when placed in water the latter descends to the bottom and the former stays atop floating. The material being the same, and the two bodies differing in nothing but in shape, they affirm that they have with all perspicacity demonstrated and sensibly manifested what they undertook, and lastly, that they have attained their goal. Nevertheless, I believe and think I can demonstrate that the same experiment proves nothing against my conclusion.

  First, it is false that the ball descends and the plate does not. For the plate shall also descend if you do to both shapes what the words of our controversy require: that is, if you place them both into the water.

  The words were these: “My antagonists are of the opinion that shape would alter solid bodies in regard to the descending or not descending and the ascending or not ascending in the same medium; for example, in the same water, if a solid of spherical shape shall descend to the bottom, being reduced to some other shape it shall not descend. I hold the contrary and affirm that if a solid corporeal body shall go to the bottom when reduced into a spherical shape, or any other, it shall do the same under whatsoever other shape, etc.”

  But to be in the water means to be placed in the water; and by [95] Aristotle’s own definition of place, to be placed implies to be surrounded by the surface of the ambient body; therefore, the two shapes shall be in the water when the surface of the water shall embrace and surround them. But when my adversaries show the board of ebony not descending to the bottom, they put it not into the water but upon the water; there, being held by a certain impediment (as by and by we will show), it is surrounded part by water and part by air.

  This is contrary to our agreement, which was that the bodies should be in the water, and not part in water and part in air.

  This is again made manifest by the fact that the question being debated was about the things that go to the bottom as well as about those that rise from the bottom to float. And who does not see that things placed at the bottom must have water around them?

  It is now to be noted that the plate of ebony and the ball, put into the water, both sink, but the ball more swiftly and the plate more slowly, and slower and slower according as it is broader and thinner; and the true cause of this slowness is the breadth of the shape. But these plates that descend slowly are the same that float when put lightly upon the water. Therefore, if what my adversaries affirm were true, the same identical shape in the same identical water would cause sometimes rest and other times slowness of motion. This is impossible, because every particular shape that descends to the bottom has of necessity its own determinate slowness, proper and natural unto it, according to which it moves, so that every other slowness (greater or lesser) is improper to its nature; for example, if a plate of one square palm descends naturally with six degrees of slowness, it is impossible that it should descend with ten or twenty unless some new impediment hinders it. Much less can it, by reason of the same shape, rest and wholly cease to move; but is it necessary that whenever it rests there be some greater impediment than the breadth of the shape. Therefore, it must be something else, and not the shape, that keeps the plate of ebony above water; the only effect of the shape is the retardation of the motion, according to which it descends more slowly than the ball. Let it be said, therefore, in accordance with the best reasoning, that the true and sole cause of the ebony’s going to the bottom is the excess of its gravity over the gravity of [96] the water; and the cause of the greater or lesser slowness is the breadth of this shape or the smallness of that. But it can by no means be allowed that the quality of the shape is the cause of its rest; for by making the slowness greater according as the shape expands, there cannot be an expansion so immense that there may not be found a corresponding immense slowness not yet reduced to nullity of motion; besides, the shapes produced by my antagonists as causes of rest are the same that also go to the bottom.

  I will not omit another reason also founded upon experience and, if I am not mistaken, manifestly showing that the introduction of the breadth of shape and the resistance of the water against penetration have nothing to do with the effect of descending, or ascending, or resting in the water. Take a piece of wood or other material a ball of which ascends from the bottom of the water to the surface more slowly than a ball of ebony of the same size descends to the bottom,4 so that it is manifest that the ball of ebony more readily divides the water in descending than the other in ascending; for example, let the wood be walnut-tree. Then make a board of walnut-tree, similar and equal to the ebony board of my antagonists, that floats; and if it be true that this floats above water by reason of the shape being unable through its breadth to pierce the coarseness of the same, then unquestionably the other of walnut-tree when placed unto the bottom should stay there, being less apt through the same impediment of shape to divide the said resistance of the water. But if we should find and by experience see that not only the thin plate but every other shape of the same walnut-tree will go up to float (as undoubtedly we do find and see), then I would ask my opponents to forbear to attribute the floating of the ebony to the shape of the board; for the resistance of the water is the same to the ascent as well as to the descent, and the force of the walnut’s ascending is less than the force of the ebony’s going to the bottom.

  Indeed, I will say more. If we shall consider gold in comparison to water, we shall find that gold exceeds water almost twenty times in gravity; thus the force and impetus with which a ball of gold goes to the bottom is very great. On the contrary, there is no lack of materials, such as virgin wax and some woods, which are only about two percent lighter than water; thus, their ascent in water is very slow and a thousand times weaker in impetus than the descent of gold. Nevertheless, a thin [97] leaf of gold floats without descending to the bottom; and on the contrary, we cannot make a cake of wax or of the said wood which, when placed at the bottom of the water, shall rest there without ascending. Now, if the shape can obstruct the penetration and impede the descent of gold, which has such a great impetus, how can it not suffice to resist the same penetration of the other material in ascending, when it has scarcely a thousandth part of the impetus that the gold has in descending? It is necessary, therefore, that whatever suspends the gold leaf or thin board of ebony upon the water be something that is lacking to the other leaves and boards of materials less heavy than water, which rise up to the surface without any obstruction when placed at the bottom and left at liberty. But they do not lack flatness and breadth of shape. Therefore, the spaciousness of the shape is not what makes the gold and ebony float.

  What, then, shall we say that it is?5 For my part, I would say that it is the contrary of what causes the sinking; for sinking and floating are contrary effects, and the causes of contrary effects must be contrary. Now, when the flat plate of ebony and the thin leaf of gold go to the bottom, the cause of the sinking is unquestionably the excess of their gravity over the gravity of the water; therefore, of necessity, when they float, the cause of their staying above the water proceeds from their lightness. In this case, some circumstance perhaps not hitherto observed combines with the said plate, making it less heavy than water rather than heavier, as it was earlier when it did sink. But such a new lightness cannot derive from the shape, both because shapes do not increase or decrease the weight, and because the plate undergoes no change of shape when it sinks as compared to when it floats.

  Now, let us return to the thin leaf of gold or of silver, or the thin board of ebony; let us lay it lightly upon the water so that it stays there without sinking, and let us diligently observe the effect it produces. First, see how false the assertion of Aristotle and of our opponents is, to wit, that it stays above water through its inability to pierce and penetrate th
e resistance of the water’s coarseness. For it will manifestly appear, not only that the said leaves have penetrated the water, but also that they are considerably lower than the surface of the same; this surface is elevated around the leaves and forms, as it were, an embankment at the bottom of which they remain floating. Now, [98] according as the said leaves shall be heavier than water two, four, ten, or twenty times, it is necessary that their surface stay below the general surface of the surrounding water an equal number of times more than the thickness of those leaves, as we shall more distinctly show anon. In the meantime, for the easier understanding of what I say, let us examine the following figure. Let us suppose the surface of the water to extend along the lines FL and DB. Now, if one shall put upon it a board of a material whose specific gravity is greater than that of water, and one does this so lightly that the board does not submerge, it shall not rest above but enter with its whole thickness into the water. Moreover, it shall go down a little, as we see in the board AIOI; its thickness is wholly inside the water and is surrounded by the embankments LA and DO of the water, whose surface is notably higher than the surface of the board. See now whether it is true that the said board does not go to the bottom for having a shape that is inapt to penetrate the coarseness of the water.

  But if it has already penetrated and overcome the continuity of the water and is of its own nature heavier than the said water, why does it not proceed in its sinking but stop and suspend itself within that little cavity which with its weight it has made in the water? I answer: because in going down until its surface is level with that of the water, it loses part of its weight; and it loses the rest as it descends beneath the surface of the water, which makes ramparts and embankments around it. It sustains this loss by drawing and carrying along with it the air that is above and adheres to it by contact; this air manages to fill the cavity that is surrounded by the embankments in the water. Thus, in this case what descends and is placed in the water is not only the slice or plate of ebony (or iron), but a mixture of ebony and air, whose result is a solid that is no longer heavier than water, as was the simple ebony or simple gold. Now, if we consider exactly what, and how large, is the solid that enters into the water in this experiment, it will be found to be everything that lies beneath the surface of the water; this is an aggregate and mixture of an [99] ebony plate and an almost equal quantity of air, or a bulk compounded of a lead plate and ten or twelve times as much air. But, Gentlemen, you who are my antagonists, in our controversy we require that the material be the same and only its shape be changed. Therefore, you must remove the air, which being conjoined with the plate makes it become another body lighter than water, and you must put only the ebony into the water; then you shall certainly see the plate descend to the bottom; and if that does not happen, you have won the day. Now, to separate the air from the ebony, you need do no more than wet the surface of the said plate with the same water; for water being thus interposed between the plate and the air, the other surrounding water shall run together without any impediment and shall receive into itself the sole and bare ebony, as required.