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What I say is manifest because the medium subtracts from the whole gravity of a solid the weight of an equal volume of the medium, as Archimedes proves in his First Book on Floating Bodies; whenever the volume of the said solid increases, the more shall the medium subtract from its entire weight; and less, when by compression it shall be condensed and reduced to a lesser volume.
It was answered me that that proceeded not from the greater Levity, but from the shape, large and flat, which not being able to penetrate the resistance of the water, is the cause that it does not sink. I replied that any piece of ice, of whatsoever shape, floats upon water, a manifest sign that its being flat and broad has no part in its floating; and added that it was a clear proof of this to see a piece of ice of very broad shape being thrust to the bottom of the water, suddenly return to float on top, which, had it been heavier, and had its floating proceeded from its shape, unable to penetrate the resistance of the medium, would be altogether impossible. I concluded therefore that the shape was not a cause of the floating or sinking of bodies, but rather the cause was the greater or lesser gravity with respect to water; and therefore all bodies heavier than it, of whatever shape, indifferently go to the bottom, and the lighter, though of any shape, float indifferently on the top: and I suppose that those who believe otherwise were induced to that belief by seeing how the diversity of forms or shapes greatly changes the swiftness or slowness of the motion; so that bodies of broad flat shape descend far more leisurely in water than those of a more compact shape, even though they are made of the same substance: by which some might be induced to believe that the dilatation of the shape might expand it to such an ampleness that it should not only retard but wholly impede and remove the motion, which I hold to be false. . Upon this conclusion, in many days' discourse, much was spoken, and many experiments produced, of which your Highness heard and saw some, and in this discourse your Highness shall have all which has been produced against my assertion, and what has been suggested to my thoughts on this matter, and for confirmation of my conclusion: which if it shall suffice to remove that false opinion, I shall think I have spent my pains and time not unprofitably. And if that should not come to pass, I still say I will have the benefit of attaining the knowledge of the truth, by hearing my fallacies confuted, and true demonstrations produced by those of contrary opinion.
And to proceed with the greatest plainness and perspicuity, it is, I think, necessary first of all to declare what is the true, intrinsic, and complete cause of the ascending of some solid bodies in water and floating; or on the contrary, of their sinking; and especially as I cannot satisfy myself in that which Aristotle has left written on this subject.
I say then that the cause why some solid bodies descend to the bottom of water is the excess of their Gravity, above the Gravity of Water; and on the contrary, the excess of the water's Gravity over the Gravity of those others is the cause that those others do not descend, but rather that they rise from the bottom, and ascend to the surface. This was demonstrated by Archimedes in his book On Floating Bodies.
I, with a different method, and by other means, will endeavour to demonstrate the same, reducing the causes of such effects to more intrinsical and immediate Principles, in which also are discovered the causes of some admirable and almost incredible results, namely, that a very little quantity of Water, should be able, with its small weight, to raise and sustain a Solid Body, a hundred or a thousand times heavier than it. And because demonstrative order so requires, I shall define certain terms, and afterwards explain some propositions, which I may then use for my present purpose as things true and obvious.
I call "equal in specific weight" those substances of which equal volumes weigh equally. As if, for example, two balls, one of wax and the other of some wood of equal volume, were also equal in weight, we say that such wood and the wax have equal specific weight.
But we shall call two solids "equal in absolute weight" if they weigh equally, though they may be different in volume. As for example a mass of lead and another of wood that each weigh ten pounds I call equal in absolute weight, though the volume of the wood be much greater than the lead and consequently less in specific weight.
I say one substance has greater specific weight than another if a volume equal to a volume of the other shall weigh more. And so I say that lead has greater specific weight than tin, because if you take equal volumes of them, the lead weighs more. But I call one body heavier absolutely than another if it weighs more, without any reference to its volume. And thus a great piece of wood is said to weigh more than a little lump of lead, although the lead has greater specific weight than the wood. And the same is to be understood of lesser specific weight and lesser absolute weight.
These terms defined, I take from mechanics two principles: the first is, Weights absolutely equal, moved with equal velocity, are of equal force and moment in their operation.
Moment, amongst mechanicians means that virtue, that force, or that efficacy, with which the Mover moves, and the Moveable resists. Which virtue depends not only on the Weight alone, but also on the Velocity of the Motion, and on the different inclinations along which the motion occurs. For a descending weight makes a greater Impetus along a steep decline than along a less steep decline; and in sum, whatever is the occasion of such Virtue, it is always called Moment; nor in my judgment is the sense new in our idiom, for, if I mistake not, I think we often say `This is a weighty business, but the other is of little moment,' and `we consider lighter matters, and let pass those of moment;' a metaphor, I suppose, taken from mechanics.
As for example, two weights equal absolutely, being put into a balance of equal arms, they stand in equilibrium, neither one going down, nor the other up: because the equality of the distances of both, from the center on which the balance is supported, and about which it moves, causes that those weights, the said balance moving, shall in the same time move equal spaces, that is, shall move with equal velocity, so that there is no reason for which this weight should descend more than that, or that more than this. And therefore they are in equilibrium, and their moments are similar and of equal virtue.
The second principle is, The Moment and Force of Weight is increased by the Velocity of the Motion; So that weights absolutely equal, but with unequal velocity, are unequal in force, moment, and virtue: and the more potent, the more swift, according to the proportion of the velocity of the one to the velocity of the other. Of this we have a very pertinent example in the Balance or Steelyard of unequal arms, at which weights absolutely equal being suspended, they do not weigh down and gravitate equally, but that which is at a greater distance from the center about which the beam moves, descends, raising the other, and the motion of this which ascends is slow, and that other swift: and such is the Force and Virtue, which from the Velocity of the Mover is conferred on the Moveable, which receives it, that it can exquisitely compensate as much more weight added to the other slower Moveable: so that if of the arms of the balance one were ten times as long as the other, whereupon in the beam's moving about the center, the end of that would go ten times as fast as the end of this, a weight suspended at the greater distance may sustain and balance another ten times more heavy absolutely than it: and that because if the steelyard moves, the lesser weight shall move ten times faster than the greater. It ought always therefore to be understood, that motions are according to the same inclinations, namely that if one of the moveables move perpendicularly to the horizon, then the other makes its motion by the like perpendicular, and if the motion of one were to be made horizontally, then the other is made along the same horizontal: and in sum, always both in like inclinations. This proportion between the weight and velocity is found in all Mechanical Instruments: and is considered by Aristotle, as a Principle in his Mechanical Questions; whereupon we also may take it for a true assumption that Weights absolutely unequal are of equal moment if their velocity of motion be inversely proportional to their weights. That is to say, that by however much the one is less heavy than the other, by so much must it move more swiftly.
Having established these things, we may begin to inquire which bodies completely sink in water and go to the bottom, and which float on the top, so that being thrust by violence under water, they return to float, with part of their volume visible above the surface: and this we will do by considering the respective operations of the said solids, and of the water. The operation which follows submersion and sinking is this: that in sinking, the solid, being weighed downward by its proper gravity, drives away the water from the place where it successively enters, and the water so driven away rises and ascends above its former level; and being itself a heavy body, it resists doing this. And because the descending solid immerses itself more and more, a greater and greater quantity of water ascends, until the whole solid is submerged. And therefore it is necessary to compare the Moment of the resistance of the water to ascension with the Moment of the weight of the solid pressing down. And if the Moment of the resistance of the water shall equal the Moment of the solid before it is totally immersed, they shall doubtless be in equilibrium, and the body shall sink no farther. But if the Moment of the solid shall always exceed the Moment with which the ascending water successively resists, the solid will not only submerge wholly under the water, but shall descend to the bottom. But if, lastly, in the instant of total sumbersion equality shall obtain between the Moments of the solid and the resisting water, then the said solid shall be able to rest indifferently in whatsoever part of the water.
It is clear by now that one must compare the gravity of the water and of the solid; and this comparison might at first sight seem sufficient to conclude and determine which are the solids that float and which are those that sink to the bottom in water, asserting that those shall float which are less heavy in specific weight, and those sink which are heavier in specific weight. For it seems apparent that if the solid sinks continually, it raises as much water in volume as its own bulk submerged; whereupon it is impossible that a solid which is less heavy in specific weight than water should wholly sink, as being unable to raise a weight greater than its own, as a volume of water equal to its own volume would be; and likewise it seems necessary that a heavier solid must go to the bottom, having a force more than sufficient for raising water of a volume equal to its own, but of lesser absolute weight. Nevertheless the business proceeds otherwise; and although the conclusions are true, yet the causes assigned are deficient. Nor is it true that the solid, in sinking, raises and drives away a volume of water equal to the part of itself which is submerged, and the more so as the vessel which holds the water is narrower. And, in fact, a solid may completely submerge under the water without raising as much water as one tenth or one twentieth of its volume. And taking it the other way, a very small quantity of water may raise a very large solid body, though this solid should weigh absolutely one hundred times as much or more than the said water, provided that the substance of that solid be less heavy in specific weight than water. And thus a great beam, of weight say 1000, may be raised and born afloat by water which weighs not even 50: and this happens when the Moment of the water is compensated by the Velocity of its Motion.
But because such things, propounded thus in abstract, are somewhat difficult to comprehend, it would be good to demonstrate them by particular examples; and for facility of demonstration, we will suppose that the vessels in which we are to put the water and place the solids should have sides perpendicular to the plane of the horizon, and that the solid that is to be put into such a vessel should be either a straight cylinder or else an upright prism.
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