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But yet, for all this, any great mass floating in a standing lake may be moved by any miniscule force -- only it is true that a lesser force moves it more slowly. But if the water's resistance to division were in any way detectable, it would follow that the said mass should, in spite of the application of some noticeable force, remain at rest, which is not so. Indeed, I will say further, that if we should retire into the more internal contemplation of the nature of water and other fluids, perhaps we should discover the constitution of their parts to be such that they not only do not oppose division, but that they have nothing in them to be divided, so that the resistance that is observed in moving through the water is like that which we meet in passing through a great throng of people, where we are impeded not by any difficulty in the division, for none of those persons comprising the crowd is divided, but only by moving those persons sideways. And thus we find resistance in thrusting a stick into a heap of sand, not because any part of the sand is to be cut in pieces, but only to be moved and raised. Two kinds of penetration, therefore, offer themselves to us, one in bodies whose parts are continuous, and here division seems to be necessary; the other in aggregates of parts which are not continuous, but only contiguous, and here there is no necessity of dividing, but only of moving. Now I am not well resolved whether water and other fluids may be considered continuous or only contiguous; yet I find myself indeed inclined to think that they are rather contiguous (if in Nature there is no other manner of aggregating than by union or by the touching of boundaries): and I am induced to this by the great difference that I see between the conjunction of the parts of a hard or solid body, and the conjunction of the same parts when the same body shall be made liquid and fluid. If, for example, I take a mass of silver or other solid and hard metal, I shall find, in dividing it into two parts, not only the resistance that arises from simply displacing it, but an incomparably larger resistance, depending on that virtue, whatever it is, that holds the parts united. And so, if we would divide again those two parts each into two more, and so on successively into others and others, we should still find a similar resistance, but always less as the parts are smaller and smaller; but if, lastly, using the thinnest and sharpest instruments, namely the very tenuous particles of Fire, we shall resolve it (perhaps) into its last and least particles, there shall not be left in them any longer either resistance to division, or any capacity of being further divided, especially by instruments more crude than the acuities of fire: and what knife or razor put into well melted silver can we find that will divide a thing which even fire cannot divide? Certainly none. Because either the whole shall be reduced to the most minute and ultimate divisions, or if there remain parts still capable of other subdivisions, they cannot be divided unless we use knives sharper than fire: but a stick or rod of iron, moved in the melted metal, is no such thing. I consider the parts of water and other liquids to be of a similar constitution and consistency, namely incapable of division, or if not absolutely indivisible, yet at least not to be divided by a chip or other solid body, palpable to the hand, since the thing which cuts must always be sharper than the thing to be cut. Solid bodies, therefore, merely move the water and do not divide it, when they are put into it; for the parts of the water are already divided to the smallest extreme, and therefore they are capable of being moved, either many of them at once, or few, or very few, and they readily make way for every small corpuscle that sinks; for if a small, light body descends through the air, then arriving at the surface of the water it meets with particles of water even smaller and with less resistance against motion and displacement than its own force to move, whereupon it sinks, and moves as many of them as is proportionate to its power. There is not, therefore, any resistance to division in water, nay there are not even any divisible parts. I add, moreover, that in case there should be found some small resistance (which is absolutely false), perhaps in attempting to move an enormous floating machine with a hair, or in trying by the addition of one small grain of lead to sink, or by removal of it to raise a very broad plate of matter of specific weight equal to water (which likewise will not happen, if we proceed with dexterity), then we may observe that that resistance is a very different thing from that which my adversaries produce as the cause why a lead plate or an ebony chip floats, for one may make a chip of ebony which, being put upon the water, will float and cannot be submerged, no not by the addition of a hundred grains of lead placed upon it, and afterwards being wetted, not only sinks, even if the lead is taken away, but if a quantity of cork or some other light substance be fastened to it, it will still not be enough to keep it from sinking to the bottom. So you see that even if it might be granted that there is a certain small resistance of division found in the substance of the water, yet this has nothing to do with the cause that supports the chip above the water with a resistance a hundred times greater than anyone can find in the parts of the water. Nor let them tell me that only the surface of the water has such a resistance, and not the internal parts, or that such resistance is found to be greatest at the beginning of submersion, as it also seems that motion meets with greater opposition in the beginning than in the continuation of it, because, first, I will permit that the water may be stirred, so that the upper parts are mixed with the middle and lower parts, or that the top parts be completely removed, and only those in the middle made use of, and yet you shall see the same effect, for all that: Moreover, that hair which draws a beam through the water also has to divide the upper part, and to begin the motion, and yet it does begin it, and it does divide it. And finally, let the ebony chip be put in the middle, between the bottom and the top of the water, and let it be kept there for awhile and settled, and afterwards set free, and it will instantly begin its motion, and will continue it to the bottom. Nay, more, the chip, so soon as it is placed upon the water, has not only begun to move and divide it, but it submerges a good way into it.
Let us accept it, therefore, as a true and undoubted conclusion, that water has no resistance against simple division, and that it is not possible to find any solid body of any shape which, being put into the water, will have its motion upwards or downwards, according as it is lighter or heavier than water, taken away from it or prohibited by the materiality of the said water (even if the excess or defect of specific weight is insensibly small). When, therefore, we see the ebony chip, or some other substance heavier than water, stay in the confines of the water and the air, without sinking, we must have recourse to some other cause of that effect than the broad shape, unable to overcome the resistance with which the water opposes division, since there is no resistance; and we can expect no effect from something that does not exist. It remains most true, therefore, as we have said before, that what is put upon the water is not the same as what is put into the water. Because what is put into the water is the pure ebony chip, which sinks, being heavier than water, and what is put upon the water is a composition of ebony and enough air that both together are lighter in specific weight than the water, and therefore they do not descend.
I will further confirm what I say. Gentlemen, my antagonists, we are agreed that the excess or defect of the gravity of the solid in comparison with the gravity of the water is the true and proper cause of floating or sinking. Now if you want to show that besides that cause there is another cause which is so powerful that it can hinder and remove the sinking of those same solids that by their gravity should sink, and if you want to say that this is breadth or ampleness of shape, then you are obliged, whenever you would show such an experiment, first to make certain that the solid which you put into the water is not lighter in specific weight than water, for if you do not do so, anyone might say, with reason, that it is not the shape but the levity which causes it to float. But I say that when you place an ebony chip on the water, you do not place there a solid heavier in specific weight than water, but one lighter, for besides the ebony, there is in the water a volume of air, united with the ebony, and such that the two together make a composition lighter in specific weight than water. See therefore that you remove the air and put the ebony alone into the water, for in this way you will immerse a solid heavier than water, and if this does not go to the bottom, then you have philosophized well, and I have done it badly.
Now since we have found the true cause of the floating of those bodies which otherwise, being heavier than water, would descend to the bottom, I think that for the perfect and distinct knowledge of this business, it would be good to proceed in a way of discovering demonstratively certain particulars that attend these effects, and to find, for several shapes, what their specific weight ought to be that they could float by virtue of the contiguous air.
 Let therefore, for better illustration, DFNE be a vessel containing water, and suppose we have a plate or chip whose thickness is contained between lines IC and OS, and let it be of a substance exceeding the specific weight of water, so that being put upon the water, it sinks and descends below the level of the said water, leaving the little ramparts AI and BC, which are at the greatest height they can be, so that if the plate IS should descend any further, the little banks or ramparts would no longer hold, but expelling the air AICB, they would spill over the surface IC and would sink the plate. The height AIBC is therefore the greatest height that the little banks of water allow. Now I say that from this, and from the ratio of the specific weight of the substance to the specific weight of water, we may easily find the maximum thickness that we may make the said plates, such that they still can float: for if the substance of the plate or chip IS were, for example, twice as heavy as water, a chip of that substance shall be, at the most, of a thickness equal to the greatest height of the banks, that is, as thick as AI is high: which we will thus demonstrate. Let the solid IS have twice the specific weight of water, and let it be a regular prism or cylinder, that is, it has its two flat surfaces above and below, alike and equal, and at right angles with the other lateral surfaces, and let its thickness IO be equal to the greatest altitude of the banks of water: I say that if it is put upon the water, it will not sink: for since the altitude AI is equal to the altitude IO, the volume of the air ABCI will be equal to the volume of the solid CIOS: and the whole volume AOSB double to the volume IS; and since the volume of the air AC neither increases nor diminishes the weight of the volume IS, and the solid IS was supposed to have double the specific weight of water, therefore a volume of water equal to AOSB, the submerged volume compounded of air AICB and solid IOSC, weighs just as much as the same sumberged volume AOSB. But when a volume equal to a submerged solid weighs as much as the said solid, it descends no farther, but rests, as has been demonstrated by Archimedes and by us. Therefore IS shall descend no farther, but shall rest. And if the solid IS is 1-1/2 times the specific weight of water, it will float as long as its thickness is no more than twice as much as the maximum height of the ramparts of water, and the altitude OI, being double AI, the submerged volume AOSB will be 1-1/2 times the volume of the solid IS. And because the air AC neither increases nor diminishes the weight, therefore water of volume AOSB weighs as much as the submerged substance, and therefore it rests.
And briefly, in general, whenever the excess of the specific weight of the solid over the specific weight of water has the same proportion to the specific weight of water that the altitude of the rampart has to the thickness of the solid, that solid shall not sink, but if it is ever so little thicker, it shall sink. Let the solid IS be heavier in specific weight than water, and of such thickness that the height of the rampart AI is to the thickness of the solid IO as the excess of the specific weight of the said solid IS over that of water is to the specific weight of water. I say that the solid IS shall not sink, but if it is never so little thicker, it shall go to the bottom. For since AI is to IO as the excess of the weight of the solid IS over the weight of a volume of water equal to the volume IS is to the weight of the said volume of water, therefore compounding as AO is to OI, so shall the weight of the solid IS be to the weight of a volume of water equal to the volume IS: and turning it around, as IO is to OA, so shall the weight of a a volume of water equal to the volume IS be to the weight of the solid IS: but as IO is to OA, so is a volume of water IS to a volume of water equal to the volume ABSO: and so is the weight of a mass of water IS to the weight of a volume of water AS: therfore as the weight of a volume of water equal to the volume IS is to the weight of the solid IS, so is the same weight of a volume of water IS to the weight of a mass of water AS: therfore the weight of the solid IS is equal to the weight of a volume of water equal to the volume AS: but the weight of the solid IS is the same with the weight of the solid AS, compounded of the solid IS and of the air ABCI. Therefore the whole compounded solid AOSB weighs as much as the water that would be comprised in the place of the said compound AOSB; and therefore it shall make an equilibrium and rest, and that same solid IOSC shall sink no farther. But if its thickness IO should be increased, it would be necessary also to increase the altitude of the rampart AI, to maintain the due proportion: but by what has been supposed, the altitude of the rampart AI is the greatest that the nature of the water and air allow, without the waters repulsing the air adherent to the surface of the solid IC and flooding the space AICB: therefore a solid of greater thickness than IO and of the same substance as the solid IS shall not rest without submerging, but shall descend to the bottom: which was to be demonstrated.
In consequence of what has been demonstrated, many and various conclusions may be deduced, by which the truth of my principal proposition is more and more confirmed, and the imperfection of all former arguments concerning the present question come to be discovered.
And first we deduce from the things demonstrated that all substances, however heavy, even gold itself, the heaviest of all substances known to us, may float upon water. For since its specific weight is considered to be almost twenty times greater than that of water, and since moreover the greatest height that the rampart of water can be extended to, without breaking the contiguity of the air adherent to the surface of the solid put upon the water, is predetermined, if we should make a plate of gold so thin that it does not exceed the nineteenth part of the height of the said rampart, this put lightly upon the water shall rest, without going to the bottom: and if ebony shall chance to be in the proportion 8/7 heavier in specific weight than water, the greatest thickness that can be allowed to an ebony chip, so that it may be able to stay above water without sinking, would be seven times more than the height of the rampart. Tin, for example, eight times heavier than water, floats if the thickness of the plate is not more than the seventh part of the altitude of the rampart.
And here I will not omit to note, as a second corollary dependent on the things demonstrated, that breadth of shape not only is not the cause of floating of those bodies that otherwise would sink, but in fact determining which chips of ebony or plates of iron or gold will float does not depend on it. Rather, the determination depends only on the thickness of those shapes of ebony or gold, completely ignoring the consideration of length and breadth, as having no relevance to the effect. It has already been shown that the only cause of floating of the said plates is their being made lighter than water by means of the connection with the air which descends together with them, and makes a space in the water, which place so occupied, if it is enough to hold water that would weigh equally with the plate, and if the surrounding water does not run in and fill that space, the plate shall remain suspended and sink no farther. Now let us see on which of the three dimensions of the solid this resting state depends, what and how much the volume of it ought to be, so that the assistance of the contiguous air may make it lighter in specific weight than water, so that it may rest without submersion. It shall undoubtedly be found that the length and breadth have nothing to do with this determination, but only the height, or, if you will, the thickness: for if we take a plate or chip, as for example, of ebony, whose height has to the maximum height of the rampart the above mentioned proportion, which makes it float indeed, yet not if we ever so little increase its thickness; I say that keeping that thickness and increasing its area to twice, four times, or ten times its size, or diminishing it by dividing it into four, or six, or twenty, or a hundred parts, it shall still in the same manner continue to float: but increasing its thickness only a hair's breadth, it will always submerge, although we should multiply the surface a hundred times a hundred times. Now since that thing is a cause which, being added, we add the effect, and being removed, we remove the effect; and since by increasing or decreasing the length and breadth in any manner, the effect of going or not going to the bottom is not added or removed: I conclude that the greatness or smallness of the surface has no influence upon floating or sinking. And if the proportion of the height of the rampart of water to the thickness of the solid is as prescribed above, it is clear from what has been demonstrated above that the greatness or smallness of the surface makes no difference. And since prisms and cylinders which have the same base are in volume proportional to one another as their heights, these prisms and cylinders, or chips, whether broad or narrow, provided only that they have equal thickness, all have the same proportion to their adjoining air, which has for its base the area of the chip and for its height the maximum height of a rampart of water. Thus the air and the chip always form a compound solid with a weight equal to the weight of the same volume of water: thus all the said solids float in the same manner.
We conclude in the third place that all sorts of shapes of whatever substance, although heavier than water, not only float with the help of the said ramparts, but some shapes, even though of the heaviest substance, stay wholly above the water, wetting only the under surface that touches the water: And these shall be all figures which from the bottom upwards grow lesser and lesser; which we shall exemplify here as pyramids or cones. We will demonstrate therefore that: It is possible to form a pyramid of any proposed substance whatsoever which, being put with its base upon the water, not only rests without submerging, but without wetting more than its base.
For the explication of this we must first demonstrate the following lemma, namely that solids with volumes inversely proportional to their specific weights are equal in absolute weight.
 Let AC, and B be two solids, and let the volume AC be to the volume B as the specific weight of the solid B is the specific weight of the solid AC: I say the solids AC and B are equal in absolute weight, that is, equally heavy. For if the volume AC is equal to the volume B, then, by the assumption, the specific weight of B shall be equal to the specific weight of AC, and being equal in volume, and of the same specific weight, they shall absolutely weigh one as much as the other. But if their volumes are unequal, let the volume AC be greater, and in it take the part C equal to the volume B. And, because the volumes B and C are equal, the absolute weight of B has the same proportion to the absolute weight of C that the specific weight of B has to the specific weight of C, or of CA, which has the same specific weight. But consider what proportion the specific weight of B has to the specific weight of AC, just the same proportion, by the assumption, that the volume CA has to the volume B; that is, to the volume C. Therefore, the absolute weight of B to the absolute weight of C is as the volume AC to the volume C. But as the volume AC is to the volume C, so is the absolute weight of AC to the absolute weight of C. Therefore the absolute weight of B has the same proportion to the absolute weight of C that the absolute weight of AC has to the absolute weight of C. Therefore the two solids AC and B are equal in absolute weight, which was to be demonstrated.
 Having demonstrated this, I say that it is possible to make of any assigned substance a pyramid or cone upon any base which, being put upon the water, shall not submerge, nor wet any more than its base. Let the maximum height of the rampart be the line DB, and the diameter of the base of the cone to be made of some assigned substance BC, at right angles to DB: and as the specific weight of the substance of the pyramid or cone to be made is to the specific weight of water, so let the height of the rampart DB be to the third part of the pyramid or cone ABC, described upon the base whose diameter is BC. I say that the said cone ABC and any other cone lower than it shall rest upon the surface of the water BC without sinking. Draw DF parallel to BC, and consider the prism or cylinder EC, which has three times the volume of the cone ABC. And because the cylinder DC has the same proportion to the cylinder CE that the altitude DB has to the altitude BE, but the cylinder CE is to the cone ABC as the altitude EB is to the third part of the altitude of the cone, therefore, by equality of proportion, the cylinder DC is to the cone ABC as DB is to the third part of the altitude BE: but as DB is to the third part of BE, so is the specific weight of the cone ABC to the specific weight of water: therefore as the volume of the solid DC is to the volume of the cone ABC, so is the specific weight of the said cone to the specific weight of water, and therefore, by the preceding lemma, the cone ABC weighs in absolute weight as much as a volume of water equal to the volume DC: but the water which is driven out of its place by the imposition of the cone ABC is as much as would precisely lie in the place DC, and is equal in weight to the cone that displaces it: therefore there shall be an equilibrium, and the cone shall rest without further submerging. And it is clear that making upon the same base a cone of less altitude, it shall also be less heavy, and so much the more it shall rest without submersion.
It is clear also that one may make cones and pyramids of any substance whatsoever, heavier than water, which being put into the water with the apex or point downwards, rest without submersion. Because if we make use of what has been demonstrated above for prisms and cylinders, and we make cones of the same substance on bases equal to the said cylinders and three times as high as the cylinders, they shall rest afloat, because in volume and weight they shall be equal to those cylinders, and by having their bases equal to those of the cylinders, they shall leave equal volumes of air included within the ramparts.
This, which for the sake of example has been demonstrated for prisms, cylinders, cones, and pyramids, might be proved for all other solid figures, but it would require a whole volume (such is the multitude and variety of their details and particulars) to comprehend the individual demonstration of them all, and of their several segments: but in order to avoid prolixity in the present Discourse, I will content myself that from what I have said anyone of ordinary intelligence may comprehend that there is no substance so heavy, no not gold itself, of which one may not form all sorts of shapes which, by virtue of the air adherent to them, and not by the water's resistance to penetration, remain afloat, and do not sink: Nay, further, I will show, for the removal of that error, that a pyramid or cone may float with the point downward, and the same put with the base downward will sink, and it will be impossible to make it float. Now just the contrary would happen if the difficulty of penetrating the water were what had hindered the descent, since the said cone is far more apt to pierce and penetrate with its sharp point than with its broad and spacious base.
 And to demonstrate this, let the cone be ABC, twice as heavy as water, and let its height be triple the height of the rampart DAEC: I say, first, that being put lightly into the water with the point downwards, it shall not descend to the bottom: for the cylinder of air contained between the ramparts DACE is equal in volume to the cone ABC, so that the whole volume of the solid compounded of the air DACE and of the cone ABC is double the cone ACB: and because the cone ABC is supposed to be of a substance double in specific weight to water, as much water as the whole volume DABCE weighs as much as the cone ABC: and therefore there shall be an equilibrium, and the cone ABC shall descend no lower.
 Let therefore the cone be ABD, double in specific weight to water, and let its height be triple the height of the rampart of water LB: it is already clear that it shall not say wholly out of the water since the cylinder is contained between the ramparts LBDP, equal to the cone ABD, and since the substance of the cone is double in specific weight to water, it is evident that the weight of the said cone shall be double to the weight of the volume of water equal to the cylinder LBDP: therefore it shall not rest in this state, but shall descend. And much less, I say further, shall the said cone stay afloat if a part be submerged, as you may see by comparing the water with both the part immersed and the part above the water. Let us therefore submerge the part NTOS of the cone ABD, and keep the point NSF above water. The altitude of the cone FNS shall either be more than half the whole altitude of the cone FTO, or it shall not be more. If it be shall be more than half, the cone FNS shall be more than half of the cylinder ENSC in volume: for the altitude of the cone FNS shall be more than 3/2 the altitude of the cylinder ENSC: and because the material of the cone is supposed to be double in specific weight to water, the water which would be contained within the rampart ENSC would be less heavy absolutely than the cone FNS; so that the whole cone FNS cannot be sustained by the rampart: but the part immersed NTOS, by being double in specific weight to water, shall tend to the bottom: therefore the whole cone FTO, as well in respect of the part submerged as the part above water shall descend to the bottom. But if the altitude of the point FNS shall be half the altitude of the whole cone FTO, the same altitude of the said cone FNS shall be 3/2 the altitude EN, and therefore ENSC shall be double to the cone FNS; and as much water in volume as the cylinder ENSC would weigh as much as the part of the cone FNS. But because the other immersed part NTOS is double in specific weight to the water, a volume of water equal to that compounded of the cylinder ENSC and of the solid NTOS shall weigh less than the cone FTO, by as much as the weight of a volume of water equal to the solid NTOS: therefore the cone shall also descend.
Again, because the solid NTOS is seven times the cone FNS, and the cylinder EN is double it, the proportion of the solid NTOS to the cylinder ENSC is seven to two. Therefore the whole solid compounded of the cylinder ENSC and of the solid NTOS is much less than double the solid NTOS: therefore the single solid NTOS is much heavier than a volume of water equal to the volume compounded of the cylinder ENSC and of NTOS. From this it follows that even if one should remove and take away the part of the cone FNS, the remaining part NTOS would go to the bottom. And if we should push the cone FTO down farther, it would be so much the more impossible that it would sustain itself afloat, the submerged part NTOS always increasing, and the volume of air contained within the ramparts diminishing, and growing ever less, the more the cone sinks.
Such a cone, therefore, that with its base upward and its point downward floats, being put into the water with its base downward must of necessity sink. They have argued far from the truth, therefore, those who have taken the water's resistance to division as the cause of floating, as a passive principle, and the breadth of the shape which does the dividing as the efficient principle.
I come in the fourth place to summarize and conclude the argument I have proposed to my adversaries, namely that it is possible to form solid bodies of any shape and size whatsoever that of their own nature would go to the bottom, but by the help of air contained in the rampart float without submerging.
The truth of this proposition is sufficiently clear in all those solid figures whose uppermost surface is plane: for if we make such figures of a material with the same specific weight as water and put them into the water so that the whole volume is covered, it is clear that they shall rest in all places, provided that such a material equal in weight to water may be exactly adjusted: and they shall as a consequence rest or lie even with the level of the water, without making any rampart. If, therefore, for this material such shapes are apt to rest without submerging, though deprived of the help of the rampart, it is clear that they may admit so much increase of their weight (without increasing their volume) as is the weight of as much water as would be contained within the rampart that is made about their upper plane surface: by the help of which being sustained, they shall rest afloat, but being wetted, they shall descend, having been made heavier than water. In shapes, therefore, that have a plane upper surface we may clearly comprehend that the rampart added or removed may prohibit or permit the descent: but in those shapes that go lessening upwards towards the top, some persons may, and very plausibly too, doubt whether the same may be done, and especially by those which terminate in a very sharp point, such as are your cones and small pyramids. Touching these, therefore, as more dubious than the rest, I will endeavour to demonstrate the same result of going or not going to the bottom, no matter what size they are.
 Let therefore the cone be ABD, made of a material specifically as heavy as water. It is clear that being put all under water, it shall rest in all places (always provided that it shall weigh exactly as much as the water, which is almost impossible to effect) and that any small weight being added to it, it shall sink to the bottom: but if it shall descend downwards gently, I say that it shall make the rampart ESTO, and that there shall stay out of the water the poinit AST, triple in height to the rampart ES: which is clear, for the material of the cone having the same weight as the water, the part submerged SBDT becomes indifferent to move downwards or upwards; and the cone AST, being equal in volume to the water that would be contained in the concave of the rampart ESTO, is also equal to it in weight: and therefore there shall be a perfect equilibrium, and consequently a rest.
Now here a doubt arises, whether the cone ABD may be made heavier, in such a way that when it is put wholly under water it goes to the bottom, but yet not so much as to take from the rampart the virtue of sustaining it so that it doesn't sink, and the reason of the doubt is this: that although at such time as the cone ABD is specifically as heavy as water, the rampart ESTO sustains it, not only when the point AST is triple in height to the altitude of the rampart ES but also when a lesser part is above water; (for although in the descent of the cone the point AST diminishes little by little, and likewise the rampart ESTO, yet the point diminishes in greater proportion than the rampart, in that it diminishes according to all the three dimensions, but the rampart according to two only, the altitude still remaining the same; or, if you will, because the cone ST diminishes according to the cubes of the lines that successively become the diameters of the bases of emergent cones, and the ramparts diminish according to the proportion of the squares of the same lines; implying the proportions of the conical points are the 3/2 power of the proportions of the cylinders contained within the ramparts; so that if, for example, the height of the emergent point were double, or equal to the height of the rampart, in these cases, the cylinder contained within the rampart would be much greater than the said point, because it would be either 3/2 or triple, by reason of which it would perhaps serve over and above to sustain the whole cone, since the part submerged would no longer weigh anything); yet, nevertheleess, when any weight is added to the whle volume of the cone, so that also the part submerged is not without some excess of gravity above the gravity of the water, it is not clear whether the cylinder contained within the rampart, in the descent that the cone shall make, can be reduced to such a proportion to the emergent point, and to such an excess of volume above the volume of it, as to compensate the excess of the cone's specific weight above the specific weight of the water. and the scruple arises because although in the descent made by the cone, the emergent point AST diminishes, whereby there is also a diminution of the excess of the cone's gravity above the gravity of the water, yet the case stands so, that the rampart also contracts itself, and the cylinder contained in it diminishes. Nevertheless it shall be demonstrated that the cone ABD being of any supposed size, and made first of a material exactly equal in gravity to water, if there may be affixed to it some weight, by means of which it may descend to the bottom when submerged under water, it may also by virtue of the rampart stay above without sinking.
Let therefore the cone ABD be of any supposed size and of the same specific weight as water. It is clear that if it is put lightly into the water it shall rest without descending; and the point AST, triple in height to the height of the rampart ES, emerges out of the water: Now suppose the cone ABD pushed down, so that only the point AIR emerges, halfway up the point AST, with the rampart about it CIRN. And because the cone ABD is to the cone AIR as the cube of the line ST is to the cube of the line IR, but the cylinder ESTO is to the cylinder CIRN as the square of ST to the square of IR, the cone AST shall be eight times the cone AIR and the cylinder ESTO four times the cylinder CIRN. But the cone AST is equal to the cylinder ESTO: therefore the cylinder CIRN is double to the cone AIR: and the water which might be contained in the rampart CIRN would be double in volume and in weight to the cone AIR, and therefore would be able to sustain double the weight of the cone AIR: therefore, if to the whole cone ABD there be added as much weight as the weight of the cone AIR, that is to say, the eight part of the weight of the cone AST, it also shall be sustained by the rampart CIRN, but without that, it shall go to the bottom: the cone ABD being, by the addition of the eighth part of the weight of the cone AST made specifically heavier than water. But if the altitude of cone AIR were two thirds of the altitude of the cone AST, the cone AST would be to the cone AIR as 27 to 8; and the cylinder ESTO to the cylinder CIRN as 9 to 4, that is as 27 to 12; and therefore, the cylinder CIRN to the cone AIR as 12 to 8; and the excess of the cylinder CIRN above the cone AIR, to the cone AST as 4 to 27: therefore if to the cone ABD be added so much weight as is the 4/27 part of the weight of the cone AST, which is a little more than 1/7, it also shall continue to float, and the height of the emergent point shall be double the height of the rampart. What has been demonstrated here for cones holds exactly for pyramids, although the one or the other should be very sharp in their point or cusp. And from this we conclude that the same result shall happen all the more readily in other shapes, as their tops are less sharp, since they will be assisted by more spacious ramparts.
All shapes, therefore, of whatever size, may go, or not go, to the bottom, according as their tops shall be wetted or not wetted. And this result holds for all shapes, without a single exception. Shape has, therefore, no part in the production of this effect of sometimes sinking and sometimes again not sinking, but only the air above the shape, which is sometimes conjoined and sometimes separated: which cause, finally, whoever shall rightly, and, as we say, with both his eyes, consider this business, will find that it is reduced to, indeed, that it really is the same with the true, natural and primary cause of floating or sinking, to wit, the excess or deficiency of the specific weight of water in relation to the specific weight of that solid magnitude that is placed into the water. For just as a lead plate the thickness of the back of a knife, when it is put into the water by itself, goes to the bottom, but if you fasten to it a piece of cork four fingers thick, it stays floating, since now the solid that is put into the water is not, as before, heavier than water but lighter, just so the ebony chip, of its own nature heavier than water, and therefore descending to the bottom when it is put into the water by itself alone, if it shall be put upon the water conjoined with a quantity of air that descends together with the ebony, and such that it makes with it a compound less heavy than as much water in volume, as equals the volume submerged and depressed beneath the level of the water's surface, it shall not descend any farther, but shall rest, for no other than the universal and most common cause, which is that solid magnitudes less heavy in specific weight than water do not go to the bottom. So that if one should take a lead plate, as for example a finger thick, and as broad as a hand in every direction, and should attempt to make it float, putting it lightly on the water, he would be wasting his time, because as soon as it sinks a hair's breadth beyond the possible height of the ramparts of water, it dives down and sinks; but if, while it is going downwards, one should make a certain bank or rampart about it that would prevent the water from covering it, which banks should rise so high that they would be able to contain as much water as should weigh equally with the said plate, it would without all question descend no lower, but would rest, as being sustained by virtue of the air contained within the aforesaid ramparts: and, in short, there would be a vessel formed by this means with a bottom of lead. But if the thinness of the lead shall be such that a very small height of rampart would suffice to contain as much air as might keep it afloat, it shall also rest without the artificial banks or ramparts, but yet not without the air, because the air by itself makes banks sufficient for a small height, to resist the flooding of the water: so that what floats in this case is, as it were, a vessel filled with air, by virtue of which it continues afloat.
I will, finally, attempt to remove all difficulties with another experiment, if there should still be any doubt left in anyone, concerning this continuity of the air with the thin floating plate, and afterwards put an end to this part of my discourse.
I imagine myself to be debating with some of my opponents, Whether shape has any influence upon the increase or decrease of the resistance that any weight feels against its being raised in the air? and I suppose that I am to maintain the affirmative, asserting that a volume of lead, reduced to the shape of a ball, shall be raised with less force than if the same had been made into a thin and broad plate, because this broad shape has a very great quantity of air to penetrate, and that other, more compact and contracted shape, has very little. And to demonstrate the truth of my opinion, I will hang on a small thread first the ball, or bullet, and put that into the water, tying the thread that holds it to one end of a balance that I hold in the air, and to the other end I gradually add weight until at last it brings up the ball of lead out of the water: to do which, suppose a weight of thirty ounces suffices. I afterwards reduce the said lead into a flat and thin plate, which I likewise put into the water, suspended by three threads, which hold it parallel to the surface of the water, and putting weights to the other end in the same manner, until the plate is raised and drawn out of the water, I find that thirty six ounces will not suffice to separate it from the water and raise it through the air: and arguing from this experiment, I affirm that I have fully demonstrated the truth of my propositionHere my opponent desires me to look down, showing me a thing which I had not noticed before, to wit, that in the ascent of the plate out of the water, it draws after it another plate (if I may so call it) of water, which before it divides and parts from the bottom surface of the lead plate is raised above the level of the other water more than the thickness of the back of a knife: then he goes to repeat the experiment with the ball, and makes me see that it is but a very small quantity of water which cleaves to its compacted and contracted shape: and then he adds that it is no wonder if in separating the thin and broad plate from the water, we meet with much greater resistance than in separating the ball, since together with the plate we are to raise a great quantity of water, which does not happen with the ball. He tells me moreover that our question is whether the resistance of elevation is greater in a dilated lead plate than in a ball, and not whether a lead plate with a great quantity of water resists more than a ball with a very little water. He shows me, in closing, that putting the plate and the ball first into the water in order to make proof of their resistance in air is beside the point, which was about elevating in air, and of things placed in the air, and not of the resistance that is made between air and water, and by things which are partly in air and partly in water: and lastly, they make me feel with my hand that when the thin plat is in the air, and free from the weight of the water, it is raised with the very same force that raises the ball . Seeing and understanding thes things, I know not what to do, except to admit I am convinced, and to thank such a friend, for having made me see that which I never till then had observed. And being prompted by this result to tell my adversaries that our question is, whether a chip and a ball of ebony go equally to the bottom in water, and not a ball of ebony and a chip of ebony joined with another flat body of air; and furthermore, that we speak of sinking and not sinking to the bottom in water, and not of that which happens at the surface to bodies that are partly in the air and partly in the water; nor much less do we treat of the greater or lesser force necessarey to separate this or that body from the air; not omitting to tell them, finally, that air resists and gravitates upwards in water, just as much as water (if I may so speak) gravitates and resists downwards in the air, and that the same force is required to sink a bladder under water that is full of air as to raise it in the air if it is full of water, removing the consideration of the weight of the film or skin, and considering the water and the air only. And it is likewise true that the same force is required to sink a cup or similar vessel under water while it is full of air, as to raise it above the surface of the water, keeping it with the mouth downward while it is full of water, which is constrained in the same manner to follow the cup which contains it and to rise above the other water into the region of the air, as the air is forced to follow the same vessel under the surface of the water, until in this case the water, spilling over the brim of the cup, breaks in, driving out the air, and in that case the said brim coming out of the water, and arriving to the confines of the air, the water falls down and the air rushes in to fill the cavity of the cup: and from this it follows that he no less transgresses the articles of our question who produces a chip conjoined with much air, to see if it will descend to the bottom in water, than he who makes proof of the resistance against elevation in air with a plate of lead joined with a similar quantity of water.
I have said all that I could at present think of to maintain the assertion I undertook. It remains that I examine what Aristotle has written on this matter, towards the end of his book De Caelo, wherein I shall note two things. The one, that since it is true, as has been demonstrated, that shape has nothing to do with moving or not moving itself upwards or downwards, it seems that Aristotle, when he first considered this speculation, was of the same opinion, as in my opinion may be deduced from the examination of his words. It is true, indeed, that in attempting afterwards to propose a reason for this effect, and not in my view gotten it quite right (which I will examine in the second place), it seems that he is brought to admit the breadth of shape to play a role in this operation.
As to the first particular, hear the precise words of Aristotle: "Shapes are not the causes of moving simply upwards or downwards, but of moving more slowly or swiftly, and by what means this comes to pass it is not difficult to see".
Here I first note that there are four terms under consideration, namely, Motion, Rest, Slowly, and Swiftly. And since Aristotle names shapes as causes of Slowness and Swiftness, excluding them from being the cause of absolute and simple Motion, it seems necessary that he exclude them on the other side from being the cause of Rest, so that his meaning is this: Shapes are not the causes of moving or not moving absolutely, but of moving quickly or slowly: and if anyone should say here that the opinion of Aristotle is to exclude shapes from being causes of motion but not from being causes of rest, I would demand whether we ought with Aristotle to understand that all shapes universally are, in some manner, the causes of rest in those bodies which otherwise would move, or else some particular shapes only, as for example broad and thin shapes. If all indifferently, then because every Body has some shape, every Body would be at rest, which is false. But if only some particular shapes may be in some way a cause of rest, as, for example, the broad, then the others would be in some way the cause of motion: for if one can infer that shape has a role as a cause of rest from seeing bodies of a contracted shape move, which after being broadened into plates are seen to rest, then with the same reason it may be affirmed that the compact and contracted shape has a role in causing motion, as the remover of that which impeded it: which again is directly opposite to what Aristotle says, namely that shapes are not the causes of Motion. Besides, if Aristotle had admitted, and not excluded, shapes as causes of rest in some bodies, which if they were molded into some other shape would move, he would have argued in an irrelevant and dubious manner in the words immediately following, on why large, thin plates of lead or iron rest upon water, since the cause would be obvious, namely the breadth of the shape. Let us conclude, therefore that the meaning of Aristotle in this place is to affirm that shapes are not the causes of moving absolutely or not moving, but only of moving swiftly or slowly: which we ought all the more to believe, since it is indeed an idea and opinion most true. Now the opinion of Aristotle being such, and consequently appearing at first sight rather contrary than favorable to my opponents, their interpretation must necessarily be not exactly the same as that, but it was set down in part as it was understood by some of them, and in part by others: and it may easily be indeed so, being an interpretation agreeable to the more famous interpreters, that the adverb simply or absolutely, put in the text, ought not to modify the verb move but the noun causes. So that the purport of Aristotle's words is to affirm, That shapes are not the causes absolutely of moving or not moving, but yet are causes secundum quid, viz. in some way; for which they are called auxiliary and comcomitant causes. And this proposition is taken and asserted as true by Signor Buonamico, Book 5 Chapter 28, where he writes "There are other concomitant causes by which some things float and others sink, among which the shapes of bodies is the most important, etc.".
Concerning this proposition I meet with many doubts and difficulties, because it seems to me the words of Aristotle are not capable of such a construction and sense, and the difficulties are these.
First, in the order and disposition of the words of Aristotle, the particle Simpliciter, or, if you will, absolute, is joined with the verb to move, and separated from the noun causes, which greatly favors my argument, seeing that the writing and the text say, Shapes are not the cause of moving simply upwards or downwards. And if the words of a text, when transposed, take on a meaning different from that which they had in the order the Author wrote them, then it is not appropriate to invert them. And who will affirm that Aristotle, desiring to write a Proposition, would dispose the words in such a way that they would convey a different, nay, a contrary sense? Contrary, I say, because understood as they are written, they say that Shapes are not the causes of Motion, but inverted they say that Shapes are the causes of Motion, etc.
Moreover, if the intent of Aristotle had been to say that shapes are not the absolute causes of moving upwards or downwards, but only causes secundum quid, he would not have adjoined those words "but they are causes of the more swift or slow motion"; indeed, subjoining this would have been not only superfluous but false, because the whole tenor of the Proposition would import this much: Shapes are not the absolute causes of moving upwards or downwards, but they are the absolute cause of the swift or slow motion; which is not true, because the primary causes of greater or lesser velocity are, attributed by Aristotle in Book 4 of his Physics, Text 71, to the greater or lesser gravity of moveables, compared among themselves, and to the greater or lesser resistance of the medium, depending on their greater or lesser materiality; and these are given by Aristotle as the primary causes; and only these two are named in that place: and shape is considered afterwards, in Text 74, rather as an instrumental cause of the force of the gravity, which divides [a medium] either with shape or with impetus; and, indeed, shape by itself without the force of gravity or levity would have no effect.
I add that if Aristotle had the opinion that shape were in some way the cause of moving or not moving, the question he raises immediately after, doubting why a lead plate floats, would not be pertinent. For if he had just said that shape was in a certain way the cause of moving or not moving, he did not need to raise the question why a lead plate floats, and then ascribe the cause to its shape; and framing the argument in this manner: Shape is a cause secundum quid of not sinking: but now, if it be doubted why a thin lead plate does not sink, it shall be answered that it is because of its shape - a discourse that would be indecent in a child, not to say Aristotle. For where is the occasion of doubting? And who does not see that if Aristotle had held that shape was in some way a cause of floating, he would without the least hesitation have written, "Shape is in a certain way the cause of floating, and therefore the lead plate floats, in respect of its large, broad shape"; but if we take the proposition of Aristotle as I say, and as it is written, and indeed as it is true, the ensuing words come in very aptly, as much in the introduction of the words swift and slow as in the question, which offers itself very pertinently, and would say this: "Shapes are not the cause of moving or not moving absolutely upwards or downwards, but of moving more quickly or slowly: But if this is so, the cause is doubtful why a plate of lead or of iron, broad and thin, should float, etc." And the occasion of the doubt is obvious, because it seems at first glance that the shape is the cause of this floating, since the same lead, or a lesser quantity, but in another shape, goes to the bottom, and we have already affirmed that shape has no role in this effect.
Lastly, if the intent of Aristotle in this place had been to say that shapes, although not absolutely, are at least in some measure the cause of moving or not moving, I would have it considered that he names both motion upwards and motion downwards: and because in exemplifying it afterwards, he produces no other experiments than a lead plate and an ebony chip, substances that of their own nature go to the bottom, but by virtue of their shape (as our adversaries say) rest afloat; it is fit that they should produce some other experiment of substances which by their nature float, but retained by their shape rest at the bottom. But since it is impossible to do this, we conclude that in this place Aristotle attributes no role to shape on absolutely moving or not moving.
But although he has philosophized exquisitely in his investigations, still I cannot undertake to defend his solution to the doubts he raises. Rather, various difficulties that present themselves to me give me the occasion to suspect that he has not entirely displayed to us the true cause of the present conclusion; which difficulties I will propound one by one, ready to change my opinion whenever anyone shows me that the truth is different from what I say; and I am far more inclined to admit a truth than to maintain a contradiction.
Aristotle having raised the question why it is that broad plates of iron or lead float, he adds (as it were strengthening the occasion of doubting) "because other things, less, and less heavy, be they round or long, as for instance a needle, go to the bottom." Now here I doubt, or rather am certain, that a needle put lightly upon the water, rests afloat no less than thin plates of iron or lead.
I cannot believe, although it has been told to me, that some, to defend Aristotle, should say that he intends a needle not put in lengthwise, but endwise, point first; nevertheless, not to leave them even this very weak refuge, which in my judgment Aristotle himself would refuse, I say it ought to be understood that the needle must be put into the water according to the dimension named by Aristotle, which is the length: because if any other dimension than that which is named might or ought to be taken, I would say that even the plates of iron and lead sink to the bottom, if they be put into the water edgeways and not flat. But because Aristotle says "broad shapes do not go to the bottom," it is to be understood "put in flat": and therefore, when he says "long shapes, like a needle, although light, do not float," it ought to be understood "put in lengthwise."
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