Chapter I: Part 1

[Transcriber's Notes

All apparent printer's errors retained. Variation in punctuation are
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page 24 in original text.

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introductions to paragraphs etc, some acting as footnotes with some
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* * * * *

A
DISCOURSE
_PRESENTED_
TO THE MOST SERENE
Don Cosimo II.
GREAT DUKE
_OF_
TUSCANY,

CONCERNING

The _NATATION_ of BODIES Vpon,
And _SUBMERSION_ In,
THE
WATER.

By GALILEUS GALILEI: Philosopher and
Mathematician unto His most Serene Highnesse.

Englished from the Second Edition of the ITALIAN,
compared with the Manuscript Copies, and reduced
into PROPOSITIONS:

By _THOMAS SALUSBURY_, Esq;

_LONDON_:

Printed by WILLIAM LEYBOURN:

_M DC LXIII._

* * * * *

[Decoration]

A DISCOVRSE

Presented to the Most Serene DON COSIMO II.

GREAT DUKE of _TUSCANY_:

CONCERNING

_The Natation of BODIES Upon, or Submersion_
_In, the WATER._

Considering (Most Serene Prince) that the publishing this present Treatise, of so different an Argument from that which many expect, and which according to the intentions I proposed in my [A] Astronomicall _Adviso_, I should before this time have put forth, might peradventure make some thinke, either that I had wholly relinquished my farther imployment about the new Celestiall Observations, or that, at least, I handled them very remissely; I have judged fit to render an account, aswell of my deferring that, as of my writing, and publishing this treatise.

[A] His Nuncio Siderio.

As to the first, the last discoveries of _Saturn_ to be tricorporeall, and of the mutations of Figure in _Venus_, like to those that are seen in the Moon, together with the Consequents depending thereupon, have not so much occasioned the demur, as the investigation of the times of the Conversions of each of the Four Medicean Planets about _Jupiter_, which I lighted upon in _April_ the year past, 1611, at my being in _Rome_; where, in the end, I assertained my selfe, that the first and neerest to _Jupiter_, moved about 8 _gr._ & 29 _m._ of its Sphere in an houre, makeing its whole revolution in one naturall day, and 18 hours, and almost an halfe. The second moves in its Orbe 14 _gr._ 13 _min._ or very neer, in an hour, and its compleat conversion is consummate in 3 dayes, 13 hours, and one third, or thereabouts. The third passeth in an hour, 2 _gr._ 6 _min._ little more or less of its Circle, and measures it all in 7 dayes, 4 hours, or very neer. The fourth, and more remote than the rest, goes in one houre, 0 _gr_ 54 _min._ and almost an halfe of its Sphere, and finisheth it all in 16 dayes, and very neer 18 hours. But because the excessive velocity of their returns or restitutions, requires a most scrupulous precisenesse to calculate their places, in times past and future, especially if the time be for many Moneths or Years; I am therefore forced, with other Observations, and more exact than the former, and in times more remote from one another, to correct the Tables of such Motions, and limit them even to the shortest moment: for such exactnesse my first Observations suffice not; not only in regard of the short intervals of Time, but because I had not as then found out a way to measure the distances between the said Planets by any Instrument: I Observed such Intervals with simple relation to the Diameter of the Body of _Jupiter_; taken, as we have said, by the eye, the which, though they admit not errors of above a Minute, yet they suffice not for the determination of the exact greatness of the Spheres of those Stars. But now that I have hit upon a way of taking such measures without failing, scarce in a very few Seconds, I will continue the observation to the very occultation of _JUPITER_, which shall serve to bring us to the perfect knowledge of the Motions, and Magnitudes of the Orbes of the said Planets, together also with some other consequences thence arising. I adde to these things the observation of some obscure Spots[1], which are discovered in the Solar Body, which changing, position in that, propounds to our consideration a great argument either that the Sun revolves in it selfe, or that perhaps other Starrs, in like manner as _Venus_ and _Mercury_, revolve about it, invisible in other times, by reason of their small digressions, lesse than that of _Mercury_, and only visible when they interpose between the Sun and our eye, or else hint the truth of both this and that; the certainty of which things ought not to be contemned, nor omitted.

[1] The Authors Observations of the Solar Spots

_Continuall observation hath at last assured me that these Spots
are matters contiguous to the Body of the Sun, there continually
produced in great number, and afterwards dissolved, some in a
shorter, some in a longer time, and to be by the Conversion or
Revolution of the Sun in it selfe, which in a Lunar Moneth, or
thereabouts, finisheth its Period, caried about in a Circle, an
accident great of it selfe, and greater for its Consequences._

As to the other particular in the next place [B] Many causes have moved me to write the present Tract, the subject whereof, is the Dispute which I held some dayes since, with some learned men of this City, about which, as your Highnesse knows, have followed many Discourses: The principall of which Causes hath been the Intimation of your Highnesse, having commended to me Writing, as a singular means to make true known from false, reall from apparent Reasons, farr better than by Disputing vocally, where the one or the other, or very often both the Disputants, through too greate heate, or exalting of the voyce, either are not understood, or else being transported by ostentation of not yeilding to one another, farr from the first Proposition, with the novelty, of the various Proposals, confound both themselves and their Auditors.

[B] The occasion inducing the Author to write this Treatise.

Moreover, it seemed to me convenient to informe your Highnesse of all the sequell, concerning the Controversie of which I treat, as it hath been advertised often already by others: and because the Doctrine which I follow, in the discussion of the point in hand, is different from that of _Aristotle_; and interferes with his Principles, I have considered that against the Authority of that most famous Man, which amongst many makes all suspected that comes not from the Schooles of the Peripateticks, its farr better to give ones Reasons by the Pen than by word of mouth, and therfore I resolved to write the present discourse: in which yet I hope to demonstrate that it was not out of capritiousnesse, or for that I had not read or understood _Aristotle_, that I sometimes swerve from his opinion, but because severall Reasons perswade me to it, and the same _Aristotle_ hath tought me to fix my judgment on that which is grounded upon Reason, and not on the bare Authority of the Master[2]; and it is most certaine according to the sentence of _Alcinoos_, that philosophating should be free. Nor is the resolution of our Question in my judgment without some benefit to the Universall[3], forasmuch as treating whether the figure of Solids operates, or not, in their going, or not going to the bottome in Water, in occurrences of building Bridges or other Fabricks on the Water, which happen commonly in affairs of grand import, it may be of great availe to know the truth.

[2] _Aristotle_ prefers Reason to the Authority ofan Author.

[3] The benefit of this Argument.

I say therfore, that being the last Summer in company with certain Learned men, it was said in the argumentation; That Condensation was the propriety of Cold[4], and there was alledged for instance, the example of Ice: now I at that time said, that, in my judgment, the Ice should be rather Water rarified than condensed[5], and my reason was, because Condensation begets diminution of Mass, and augmentation of gravity, and Rarifaction causeth greater Lightness, and augmentarion of Masse: and Water in freezing, encreaseth in Masse, and the Ice made thereby is lighter than the Water on which it swimmeth.

[4] Condensation the Propriety of Cold, according to the
Peripateticks.

[5] Ice rather water rarified, than condensed, and why:

_What I say, is manifest, because, the medium subtracting from the
whole Gravity of Sollids the weight of such another Masse of the
said Medium; as_ Archimedes _proves in his_ [C] First Booke De
Insidentibus Humido; _when ever the Masse of the said Solid
encreaseth by Distraction, the more shall the_ Medium _detract from
its entire Gravity; and lesse, when by Compression it shall be
condensed and reduced to a lesse Masse._

[C] In lib: 1. of Natation of Bodies Prop. 7.

[Sidenote: Figure operates not in the Natation of Sollids.]

It was answered me, tha{t} that proceeded not from the greater Levity, but from the Figure, large and flat, which not being able to penetrate the Resistance of the Water, is the cause that it submergeth not. I replied, that any piece of Ice, of whatsoever Figure, swims upon the Water, a manifest signe, that its being never so flat and broad, hath not any part in its floating: and added, that it was a manifest proofe hereof to see a piece of Ice of very broad Figure being thrust to the botome of the Water, suddenly return to flote atoppe, which had it been more grave, and had its swimming proceeded from its Forme, unable to penetrate the Resistance of the _Medium_, that would be altogether impossible; I concluded therefore, that the Figure was in sort a Cause of the Natation or Submersion of Bodies, but the greater or lesse Gravity in respect of the Water: and therefore all Bodyes heavier than it of what Figure soever they be, indifferently go to the bottome, and the lighter, though of any figure, float indifferently on the top: and I suppose that those which hold otherwise, were induced to that beliefe, by seeing how that diversity of Formes or Figures, greatly altereth the Velosity, and Tardity of Motion; so that Bodies of Figure broad and thin, descend far more leasurely into the Water, than those of a more compacted Figure, though both made of the same Matter: by which some might be induced to believe that the Dilatation of the Figure might reduce it to such amplenesse that it should not only retard but wholly impede and take away the Motion, which I hold to be false. Upon this Conclusion, in many dayes discourse, was spoken much, and many things, and divers Experiments produced, of which your Highnesse heard, and saw some, and in this discourse shall have all that which hath been produced against my Assertion, and what hath been suggested to my thoughts on this matter, and for confirmation of my Conclusion: which if it shall suffice to remove that (as I esteem hitherto false) Opinion, I shall thinke I have not unprofitably spent my paynes and time. and although that come not to passe, yet ought I to promise another benefit to my selfe, namely, of attaining the knowledge of the truth, by hearing my Fallacyes confuted, and true demonstrations produced by those of the contrary opinion.

And to proceed with the greatest plainness and perspicuity that I can possible, it is, I conceive, necessary, first of all to declare what is the true, intrinsecall, and totall Cause, of the ascending of some Sollid Bodyes in the Water, and therein floating; or on the contrary, of their sinking and so much the rather in asmuch as I cannot satisfie myselfe in that which _Aristotle_ hath left written on this Subject.

[Sidenote: The cause of the Natation & submersion of Solids in
the Water.]

I say then the Cause why some Sollid Bodyes descend to the Bottom of Water, is the excesse of their Gravity, above the Gravity of the Water; and on the contrary, the excess of the Waters Gravity above the Gravity of those, is the Cause that others do not descend, rather that they rise from the Bottom, and ascend to the Surface. This was subtilly demonstrated by _Archimedes_ in his Book Of the NATATION of BODIES: Conferred afterwards by a very grave Author, but, if I erre not invisibly, as below for defence of him, I shall endeavour to prove.

I, with a different Method, and by other meanes, will endeavour to demonstrate the same, reducing the Causes of such Effects to more intrinsecall and immediate Principles, in which also are discovered the Causes of some admirable and almost incredible Accidents, as that would be, that a very little quantity of Water, should be able, with its small weight, to raise and sustain a Solid Body, an hundred or a thousand times heavier than it.

And because demonstrative Order so requires, I shall define certain Termes, and afterwards explain some Propositions, of which, as of things true and obvious, I may make use of to my present purpose.

DEFINITION I.

_I then call equally Grave_ in specie, _those Matters of which equall
Masses weigh equally._

As if for example, two Balls, one of Wax, and the other of some Wood of equall Masse, were also equall in Weight, we say, that such Wood, and the Wax are _in specie_ equally grave.

DEFINITION II.

_But equally grave in Absolute Gravity, we call two Sollids,
weighing equally, though of Mass they be unequall._

As for example, a Mass of Lead, and another of Wood, that weigh each ten pounds, I call equall in Absolute Gravity, though the Mass of the Wood be much greater then that of the Lead.

_And, consequently, less Grave_ in specie.

DEFINITION III.

_I call a Matter more Grave_ in specie _than another, of which a
Mass, equall to a Mass of the other, shall weigh more._

And so I say, that Lead is more grave _in specie_ than Tinn, because if you take of them two equall Masses, that of the Lead weigheth more.

DEFINITION IV.

_But I call that Body more grave absolutely than this, if that
weigh more than this, without any respect had to the Masses._

And thus a great piece of Wood is said to weigh more than a little lump of Lead, though the Lead be _in specie_ more heavy than the Wood. And the same is to be understood of the less grave _in specie_, and the less grave absolutely.

These Termes defined, I take from the Mechanicks two Principles: the first is, that

AXIOME. I.

_Weights absolutely equall, moved with equall Velocity, are of
equall Force and Moment in their operations._

_DEFINITION V._

Moment, amongst Mechanicians, signifieth that Vertue, that Force,
or that Efficacy, with which the Mover moves, and the Moveable
resists.

_Which Vertue dependes not only on the simple Gravity, but on the
Velocity of the Motion, and on the diverse Inclinations of the
Spaces along which the Motion is made: For a descending Weight
makes a greater Impetus in a Space much declining, than in one less
declining; and in summe, what ever is the occasion of such Vertue,
it ever retaines the name of Moment; nor in my Judgement, is this
sence new in our Idiome, for, if I mistake not, I think we often
say; This is a weighty businesse, but the other is of small moment:
and we consider lighter matters and let pass those of Moment; a
Metaphor, I suppose, taken from the Mechanicks._

As for example, two weights equall in absolute Gravity, being put into a Ballance of equall Arms, they stand in _Equilibrium_, neither one going down, nor the other up: because the equality of the Distances of both, from the Centre on which the Ballance is supported, and about which it moves, causeth that those weights, the said Ballance moving, shall in the same Time move equall Spaces, that is, shall move with equall Velocity, so that there is no reason for which this Weight should descend more than that, or that more than this; and therefore they make an _Equilibrium_, and their Moments continue of semblable and equall Vertue.

The second Principle is; That

AXIOME II.

_The Moment and Force of the Gravity, is encreased by the Velocity
of the Motion._

So that Weights absolutely equall, but conjoyned with Velocity unequall, are of Force, Moment and Vertue unequall: 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 Stiliard of unequall Arms, at which Weights absolutely equall being suspended, they do not weigh down, and gravitate equally, but that which is at a greater distance from the Centre, about which the Beam moves, descends, raising the other, and the Motion of this which ascends is slow, and the other swift: and such is the Force and Vertue, 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 Beames moving about the Centre, the end of that would go ten times as far as the end of this, a Weight suspended at the greater distance, may sustain and poyse another ten times more grave absolutely than it: and that because the Stiliard moving, the lesser Weight shall move ten times faster than the bigger. It ought alwayes 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; that then the other is made along the same Horizontall plain: and in summe, alwayes both in like Inclinations. This proportion between the Gravity and Velocity is found in all Mechanicall Instruments: and is considered by _Aristotle_, as a Principle in his _Mechanicall Questions_; whereupon we also may take it for a true Assumption, That

AXIOME III.

_Weights absolutely unequall, do alternately counterpoyse and
become of equall Moments, as oft as their Gravities, with
contrary proportion, answer to the Velocity of their Motions._

That is to say, that by how much the one is less grave than the other, by so much is it in a constitution of moving more swiftly than that.

Having prefatically explicated these things, we may begin to enquire, what Bodyes those are which totally submerge in Water, and go to the Bottom, and which those that by constraint float on the top, so that being thrust by violence under Water, they return to swim, with one part of their Mass visible above the Surface of the Water: and this we will do by considering the respective operation of the said Solids, and of Water: Which operation followes the Submersion and sinking; and this it is[6], That in the Submersion that the Solid maketh, being depressed downwards by its proper Gravity, it comes to drive away the water from the place where it successively subenters, and the water repulsed riseth and ascends above its first levell, to which Ascent on the other side it, as being a grave Body of its own nature, resists: And because the descending Solid more and more immerging, greater and greater quantity of Water ascends, till the whole Sollid be submerged; its necessary to compare the Moments of the Resistance of the water to Ascension, with the Moments of the pressive Gravity of the Solid: And if the Moments of the Resistance of the water, shall equalize the Moments of the Solid, before its totall Immersion[7]; in this case doubtless there shall be made an _Equilibrium_, nor shall the Body sink any farther. But if the Moment of the Solid, shall alwayes exceed the Moments wherewith the repulsed water successively makes Resistance[8], that Solid shall not only wholly submerge under water, but shall descend to the Bottom. But if, lastly, in the instant of totall Submersion, the equality shall be made between the Moments of the prement Solid, and the resisting Water[9]; then shall rest, ensue, and the said Solid shall be able to rest indifferently, in whatsoever part of the water. By this time is manifest the necessity of comparing the Gravity of the water, and of the Solid[10]; and this comparison might at first sight seem sufficient to conclude and determine which are the Solids that float a-top, and which those that sink to the Bottom in the water, asserting that those shall float which are lesse grave _in specie_ than the water, and those submerge, which are _in specie_ more grave. For it seems in appearance, that the Sollid in sinking continually, raiseth so much Water in Mass, as answers to the parts of its own Bulk submerged: whereupon it is impossible, that a Solid less grave _in specie_, than water, should wholly sink, as being unable to raise a weight greater than its own, and such would a Mass of water equall to its own Mass be. And likewise it seems necessary, that the graver Solids do go to the Bottom, as being of a Force more than sufficient for the raising a Masse of water, equall to its own, though inferiour in weight. Nevertheless the business succeeds otherwise: and though the Conclusions are true, yet are the Causes thus assigned deficient, nor is it true, that the Solid in submerging, raiseth and repulseth Masses of Water, equall to the parts of it self submerged; but the Water repulsed, is alwayes less than the parts of the Solid submerged[11]: and so much the more by how much the Vessell in which the Water is contained is narrower: in such manner that it hinders not, but that a Solid may submerge all under Water, without raising so much Water in Mass, as would equall the tenth or twentieth part of its own Bulk: like as on the contrary, a very small quantity of Water, may raise a very great Solid Mass[12], though such Solid should weigh absolutely a hundred times as much, or more, than the said Water, if so be that the Matter of that same Solid be _in specie_ less grave than the Water. And thus a great Beam, as suppose of a 1000 weight, may be raised and born afloat by Water, which weighs not 50: and this happens when the Moment of the Water is compensated by the Velocity of its Motion.

[6] How the submersion of Solids in the Water, is effected.

[7] What Solids shall float on the Water.

[8] What Solids shall sinke to the botome.

[9] What Solids shall rest in all places of the Water.

[10] The Gravitie of the Water and Solid must be compared in all
Problems, of Natation of Bodies.

[11] The water repelled is ever less than the parts of the Sollid
submerged.

[12] _A_ small quantity of water, may float a very great Solid
Mass.

But because such things, propounded thus in abstract, are somewhat difficult to be comprehended, it would be good to demonstrate them by particular examples; and for facility of demonstration, we will suppose the Vessels in which we are to put the Water, and place the Solids, to be inviron'd and included with sides erected perpendicular to the Plane of the Horizon, and the Solid that is to be put into such vessell to be either a streight Cylinder, or else an upright Prisme.

_The which proposed and declared, I proceed to demonstrate the truth of what hath been hinted, forming the ensuing Theoreme._

_THEOREME I._

[Sidenote: The Proportion of the water raised to the Solid
submerged.]

The Mass of the Water which ascends in the submerging of a Solid,
Prisme or Cylinder, or that abaseth in taking it out, is less
than the Mass of the said Solid, so depressed or advanced: and
hath to it the same proportion, that the Surface of the Water
circumfusing the Solid, hath to the same circumfused Surface,
together with the Base of the Solid.

_Let the Vessell be A B C D, and in it the Water raised up to the Levell E F G, before the Solid Prisme H I K be therein immerged; but after that it is depressed under Water, let the Water be raised as high as the Levell L M, the Solid H I K shall then be all under Water, and the Mass of the elevated Water shall be L G, which is less than the Masse of the Solid depressed, namely of H I K, being equall to the only part E I K, which is contained under the first Levell E F G. Which is manifest, because if the Solid H I K be taken out, the Water I G shall return into the place occupied by the Mass E I K, where it was continuate before the submersion of the Prisme. And the Mass L G being equall to the Mass E K: adde thereto the Mass E N, and it shall be the whole Mass E M, composed of the parts of the Prisme E N, and of the Water N F, equall to the whole Solid H I K: And, therefore, the Mass L G shall have the same proportion to E M, as to the Mass H I K: But the Mass L G hath the same proportion to the Mass E M, as the Surface L M hath to the Surface M H: Therefore it is manifest, that the Mass of Water repulsed L G, is in proportion to the Mass of the Solid submerged H I K; as the Surface L M, namely, that of the Water ambient about the Sollid, to the whole Surface H M, compounded of the said ambient water, and the Base of the Prisme H N. But if we suppose the first Levell of the Water the according to the Surface H M, and the Prisme allready submerged H I K; and after to be taken out and raised to E A O, and the Water to be faln from the first Levell H L M as low as E F G; It is manifest, that the Prisme E A O being the same with H I K, its superiour part H O, shall be equall to the inferiour E I K: and remove the common part E N, and, consequently, the Mass of the Water L G is equall to the Mass H O; and, therefore, less than the Solid, which is without the Water, namely, the whole Prisme E A O, to which likewise, the said Mass of Water abated L G, hath the same proportion, that the Surface of the Waters circumfused L M hath to the same circumfused Surface, together with the Base of the Prisme A O: which hath the same demonstration with the former case above._

_And from hence is inferred, that the Mass of the Water, that riseth in the immersion of the Solid, or that ebbeth in elevating it, is not equall to all the Mass of the Solid, which is submerged or elevated, but to that part only, which in the immersion is under the first Levell of the Water, and in the elevation remaines above the first Levell: Which is that which was to be demonstrated. We will now pursue the things that remain._

And first we will demonstrate that,

THEOREME II.

[Sidenote: The proportion of the water abated, to the Solid
raised.]

_When in one of the above said Vessels, of what ever breadth,
whether wide or narrow, there is placed such a Prisme or
Cylinder, inviron'd with Water, if we elevate that Solid
perpendicularly, the Water circumfused shall abate, and the
Abatement of the Water, shall have the same proportion to the
Elevation of the Prisme, as one of the Bases of the Prisme, hath
to the Surface of the Water Circumfused._

Imagine in the Vessell, as is aforesaid, the Prisme A C D B to be placed, and in the rest of the Space the Water to be diffused as far as the Levell E A: and raising the Solid, let it be transferred to G M, and let the Water be abased from E A to N O: I say, that the descent of the Water, measured by the Line A O, hath the same proportion to the rise of the Prisme, measured by the Line G A, as the Base of the Solid G H hath to the Surface of the Water N O. The which is manifest: because the Mass of the Solid G A B H, raised above the first Levell E A B, is equall to the Mass of Water that is abased E N O A. Therefore, E N O A and G A B H are two equall Prismes; for of equall Prismes, the Bases answer contrarily to their heights: Therefore, as the Altitude A O is to the Altitude A G, so is the Superficies or Base G H to the Surface of the Water N O. If therefore, for example, a Pillar were erected in a waste Pond full of Water, or else in a Well, capable of little more then the Mass of the said Pillar, in elevating the said Pillar, and taking it out of the Water, according as it riseth, the Water that invirons it will gradually abate, and the abasement of the Water at the instant of lifting out the Pillar, shall have the same proportion, that the thickness of the Pillar hath to the excess of the breadth of the said Pond or Well, above the thickness of the said Pillar: so that if the breadth of the Well were an eighth part larger than the thickness of the Pillar, and the breadth of the Pond twenty five times as great as the said thickness, in the Pillars ascending one foot, the water in the Well shall descend seven foot, and that in the Pond only 1/25 of a foot.

[Sidenote: Why a Solid less grave _in specie_ than water, stayeth
not under water, in very small depths:]

This Demonstrated, it will not be difficult to show the true cause, how it comes to pass, that,

THEOREME III.

_A Prisme or regular Cylinder, of a substance specifically less
grave than Water, if it should be totally submerged in Water,
stayes not underneath, but riseth, though the Water circumfused
be very little, and in absolute Gravity, never so much inferiour
to the Gravity of the said Prisme._

Let then the Prisme A E F B, be put into the Vessell C D F B, the same being less grave _in specie_ than the Water: and let the Water infused rise to the height of the Prisme: I say, that the Prisme left at liberty, it shall rise, being born up by the Water circumfused C D E A. For the Water C E being specifically more grave than the Solid A F, the absolute weight of the water C E, shall have greater proportion to the absolute weight of the Prisme A F, than the Mass C E hath to the Mass A F (in regard the Mass hath the same proportion to the Mass, that the weight absolute hath to the weight absolute, in case the Masses are of the same Gravity _in specie_.) But the Mass C E is to the Mass A F, as the Surface of the water A C, is to the Superficies, or Base of the Prisme A B; which is the same proportion as the ascent of the Prisme when it riseth, hath to the descent of the Water circumfused C E.

Therefore, the absolute Gravity of the water C E, hath greater proportion to the absolute Gravity of the Prisme A F; than the Ascent of the Prisme A F, hath to the descent of the said water C E. The Moment, therefore, compounded of the absolute Gravity of the water C E, and of the Velocity of its descent, whilst it forceably repulseth and raiseth the Solid A F, is greater than the Moment compounded of the absolute Gravity of the Prisme A F, and of the Tardity of its ascent, with which Moment it contrasts and resists the repulse and violence done it by the Moment of the water: Therefore, the Prisme shall be raised.

[Sidenote: The Proportion according to which the Submersion &
Natation of Solids is made.]

It followes, now, that we proceed forward to demonstrate more particularly, how much such Solids shall be inferiour in Gravity to the water elevated; namely, what part of them shall rest submerged, and what shall be visible above the Surface of the water: but first it is necessary to demonstrate the subsequent Lemma.

LEMMA I.

[Sidenote: The absolute Gravity of Solids, are in a proportion
compounded of their Specifick Gravities, and of their Masses.]

_The absolute Gravities of Solids, have a proportion compounded of
the proportions of their specificall Gravities, and of their
Masses._

Let A and B be two Solids. I say, that the Absolute Gravity of A, hath to the Absolute Gravity of B, a proportion compounded of the proportions of the specificall Gravity of A, to the Specificall Gravity of B, and of the Mass A to the Mass B. Let the Line D have the same proportion to E, that the specifick Gravity of A, hath to the specifick Gravity of B; and let E be to F, as the Mass A to the Mass B: It is manifest, that the proportion of D to F, is compounded of the proportions D and E; and E and F. It is requisite, therefore, to demonstrate, that as D is to F, so the absolute Gravity of A, is to the absolute Gravity of B. Take the Solid C, equall in Mass to the Solid A, and of the same Gravity _in specie_ with the Solid B. Because, therefore, A and C are equall in Mass, the absolute Gravity of A, shall have to the absolute Gravity of C, the same proportion, as the specificall Gravity of A, hath to the specificall Gravity of C, or of B, which is the same _in specie_; that is, as D is to E. And, because, C and B are of the same Gravity _in specie_, it shall be, that as the absolute weight of C, is to the absolute weight of B, so the Mass C, or the Mass A, is to the Mass B; that is, as the Line E to the Line F. As therefore, the absolute Gravity of A, is to the absolute Gravity of C, so is the Line D to the Line E: and, as the absolute Gravity of C, is to the absolute Gravity of B, so is the Line E to the Line F: Therefore, by Equality of proportion, the absolute Gravity of A, is to the absolute Gravity of B, as the Line D to the Line F: which was to be demonstrated. I proceed now to demonstrate, how that,

THEOREME IV.

[Sidenote: The proportion of water requisite to make a Solid
swim:]

_If a Solid, Cylinder, or Prisme, lesse grave specifically than the
Water, being put into a Vessel, as above, of whatsoever
greatnesse, and the Water, be afterwards infused, the Solid shall
rest in the bottom, unraised, till the Water arrive to that part
of the Altitude, of the said Prisme, to which its whole Altitude
hath the same proportion, that the Specificall Gravity of the
Water, hath to the Specificall Gravity of the said Solid: but
infusing more Water, the Solid shall ascend._

Let the Vessell be M L G N of any bigness, and let there be placed in it the Solid Prisme D F G E, less grave _in specie_ than the water; and look what proportion the Specificall Gravity of the water, hath to that of the Prisme, such let the Altitude D F, have to the Altitude F B. I say, that infusing water to the Altitude F B, the Solid D G shall not float, but shall stand in _Equilibrium_, so, that that every little quantity of water, that is infused, shall raise it. Let the water, therefore, be infused to the Levell A B C; and; because the Specifick Gravity of the Solid D G, is to the Specifick Gravity of the water, as the altitude B F is to the altitude F D; that is, as the Mass B G to the Mass G D; as the proportion of the Mass B G is to the Mass G D, as the proportion of the Mass G D is to the Mass A F, they compose the Proportion of the Mass B G to the Mass A F. Therefore, the Mass B G is to the Mass A F, in a proportion compounded of the proportions of the Specifick Gravity of the Solid G D, to the Specifick Gravity of the water, and of the Mass G D to the Mass A F: But the same proportions of the Specifick Gravity of G D, to the Specifick Gravity of the water, and of the Mass G D to the Mass A F, do also by the precedent _Lemma_, compound the proportion of the absolute Gravity of the Solid D G, to the absolute Gravity of the Mass of the water A F: Therefore, as the Mass B G is to the Mass A F, so is the Absolute Gravity of the Solid D G, to the Absolute Gravity of the Mass of the water A F. But as the Mass B G is to the Mass A F; so is the Base of the Prisme D E, to the Surface of the water A B; and so is the descent of the water A B, to the Elevation of the Prisme D G; Therefore, the descent of the water is to the elevation of the Prisme, as the absolute Gravity of the Prisme, is to the absolute Gravity of the water: Therefore, the Moment resulting from the absolute Gravity of the water A F, and the Velocity of the Motion of declination, with which Moment it forceth the Prisme D G, to rise and ascend, is equall to the Moment that results from the absolute Gravity of the Prisme D G, and from the Velocity of the Motion, wherewith being raised, it would ascend: with which Moment it resists its being raised: because, therefore, such Moments are equall, there shall be an _Equilibrium_ between the water and the Solid. And, it is manifest, that putting a little more water unto the other A F, it will increase the Gravity and Moment, whereupon the Prisme D G, shall be overcome, and elevated till that the only part B F remaines submerged. Which is that that was to be demonstrated.

COROLLARY I.

[Sidenote: _H_ow far Solids less grave _in specie_ than water, do
submerge.]

_By what hath been demonstrated, it is manifest, that Solids less
grave_ in specie _than the water, submerge only so far, that as
much water in Mass, as is the part of the Solid submerged, doth
weigh absolutely as much as the whole Solid._

For, it being supposed, that the Specificall Gravity of the water, is to the Specificall Gravity of the Prisme D G, as the Altitude D F, is to the Altitude F B; that is, as the Solid D G is to the Solid B G; we might easily demonstrate, that as much water in Mass as is equall to the Solid B G, doth weigh absolutely as much as the whole Solid D G; For, by the _Lemma_ foregoing, the Absolute Gravity of a Mass of water, equall to the Mass B G, hath to the Absolute Gravity of the Prisme D G, a proportion compounded of the proportions, of the Mass B G to the Mass G D, and of the Specifick Gravit{y} of the water, to the Specifick Gravity of the Prisme: But the Gravity _in specie_ of the water, to the Gravity _in specie_ of the Prisme, is supposed to be as the Mass G D to the Mass G B. Therefore, the Absolute Gravity of a Mass of water, equall to the Mass B G, is to the Absolute Gravity of the Solid D G, in a proportion compounded of the proportions, of the Mass B G to the Mass G D, and of the Mass D G to the Mass G B; which is a proportion of equalitie. The Absolute Gravity, therefore, of a Mass of Water equall to the part of the Mass of the Prisme B G, is equall to the Absolute Gravity of the whole Solid D G.

COROLLARY II.

[Sidenote: _A_ Rule to equilibrate Solids in the water.]

_It followes, moreover, that a Solid less grave than the water,
being put into a Vessell of any imaginable greatness, and water
being circumfused about it to such a height, that as much water
in Mass, as is the part of the Solid submerged, do weigh
absolutely as much as the whole Solid; it shall by that water be
justly sustained, be the circumfused Water in quantity greater or
lesser._

For, if the Cylinder or Prisme M, less grave than the water, _v. gra._ in Subsequiteriall proportion, shall be put into the capacious Vessell A B C D, and the water raised about it, to three quarters of its height, namely, to its Levell A D: it shall be sustained and exactly poysed in _Equilibrium_. The same will happen; if the Vessell E N S F were very small, so, that between the Vessell and the Solid M, there were but a very narrow space, and only capable of so much water, as the hundredth part of the Mass M, by which it should be likewise raised and erected, as before it had been elevated to three fourths of the height of the Solid: which to many at the first sight, may seem a notable Paradox, and beget a conceit, that the Demonstration of these effects, were sophisticall and fallacious: but, for those who so repute it, the Experiment is a means that may fully satisfie them. But he that shall but comprehend of what Importance Velocity of Motion is, and how it exactly compensates the defect and want of Gravity, will cease to wonder, in considering that at the elevation of the Solid M, the great Mass of water A B C D abateth very little, but the little Mass of water E N S F decreaseth very much, and in an instant, as the Solid M before did rise, howbeit for a very short space: Whereupon the Moment, compounded of the small Absolute Gravity of the water E N S F, and of its great Velocity in ebbing, equalizeth the Force and and Moment, that results from the composition of the immense Gravity of the water A B C D, with its great slownesse of ebbing; since that in the Elevation of the Sollid M, the abasement of the lesser water E S, is performed just so much more swiftly than the great Mass of water A C, as this is more in Mass than that which we thus demonstrate.

[Sidenote: _T_he proportion according to which water riseth and
falls in different Vessels at the Immersion and Elevation of
Solids.]

In the rising of the Solid M, its elevation hath the same proportion to the circumfused water E N S F, that the Surface of the said water, hath to the Superficies or Base of the said Solid M; which Base hath the same proportion to the Surface of the water A D, that the abasement or ebbing of the water A C, hath to the rise or elevation of the said Solid M. Therefore, by Perturbation of proportion, in the ascent of the said Solid M, the abasement of the water A B C D, to the abasement of the water E N S F, hath the same proportion, that the Surface of the water E F, hath to the Surface of the water A D; that is, that the whole Mass of the water E N S F, hath to the whole Mass A B C D, being equally high: It is manifest, therefore, that in the expulsion and elevation of the Solid M, the water E N S F shall exceed in Velocity of _M_otion the water A B C D, asmuch as it on the other side is exceeded by that in quantity: whereupon their Moments in such operations, are mutually equall.

_And, for ampler confirmation, and clearer explication of this, let
us consider the present Figure, (which if I be not deceived, may
serve to detect the errors of some Practick Mechanitians who upon a
false foundation some times attempt impossible enterprizes,) in
which, unto the large Vessell E I D F, the narrow Funnell or Pipe I
C A B is continued, and suppose water infused into them, unto the
Levell L G H, which water shall rest in this position, not without
admiration in some, who cannot conceive how it can be, that the
heavie charge of the great Mass of water G D, pressing downwards,
should not elevate and repulse the little quantity of the other,
contained in the Funnell or Pipe C L, by which the descent of it is
resisted and hindered: But such wonder shall cease, if we begin to
suppose the water G D to be abased only to Q D, and shall
afterwards consider, what the water C L hath done, which to give
place to the other, which is descended from the Levell G H, to the
Levell Q O, shall of necessity have ascended in the same time, from
the Levell L unto A B. And the ascent L B, shall be so much greater
than the descent G Q, by how much the breadth of the Vessell G D,
is greater than that of the Funnell I C; which, in summe, is as
much as the water G D, is more than the water L C: but in regard
that the Moment of the Velocity of the Motion, in one Moveable,
compensates that of the Gravity of another what wonder is it, if
the swift ascent of the lesser Water C L, shall resist the slow
descent of the greater G D?_

The same, therefore, happens in this operation, as in rhe Stilliard, in which a weight of two pounds counterpoyseth an other of 200, asoften as that shall move in the same time, a space 100 times greater than this: which falleth out when one Arme of the Beam is an hundred times as long as the other. Let the erroneous opinion of those therefore cease, who hold that a Ship is better, and easier born up in a great abundance of water, then in a lesser quantity[13], (_this was believed by_ Aristotle _in his Problems, Sect. 23, Probl. 2._) it being on the contrary true, that its possible, that a Ship may as well float in ten Tun of water, as in an Ocean.

[13] A ship flotes as well in ten Tun of Water as in an Ocean.

[Sidenote: A Solid specifiaclly graver than the water, cannot be
born up by any quantity of it.]

But following our matter, I say, that by what hath been hitherto demonstrated, we may understand how, that

COROLLARY III.

_One of the above named Solids, when more grave_ in specie _than the
water, can never be sustained, by any whatever quantity of it._

For having seen how that the Moment wherewith such a Solid, as grave _in specie_ as the water, contrasts with the Moment of any Mass of water whatsoever, is able to retain it, even to its totall Submersion, without its ever ascending; it remaineth, manifest, that the water is far less able to raise it up, when it exceeds the same _in specie_: so, that though you infuse water till its totall Submersion, it shall still stay at the Bottome, and with such Gravity, and Resistance to Elevation, as is the excess of its Absolute Gravity, above the Absolute Gravity of a Mass equall to it, made of water, or of a Matter _in specie_ equally grave with the water: and, though you should moreover adde never so much water above the Levell of that which equalizeth the Altitude of the Solid, it shall not, for all that, encrease the Pression, or Gravitation, of the parts circumfused about the said Solid, by which greater pression, it might come to be repulsed; because, the Resistance is not made, but only by those parts of the water, which at the Motion of the said Solid do also move, and these are those only, which are comprehended by the two Superficies equidistant to the Horizon, and their parallels, that comprehend the Altitude of the Solid immerged in the water.

I conceive, I have by this time sufficiently declared and opened the way to the contemplation of the true, intrinsecall and proper Causes of diverse Motions, and of the Rest of many Solid Bodies in diverse _Mediums_, and particularly in the water, shewing how all in effect, depend on the mutuall excesses of the Gravity of the Moveables and of the _Mediums_: and, that which did highly import, removing the Objection, which peradventure would have begotten much doubting, and scruple in some, about the verity of my Conclusion, namely, how that notwithstanding, that the excess of the Gravity of the water, above the Gravity of the Solid, demitted into it, be the cause of its floating and rising from the Bottom to the Surface, yet a quantity of water, that weighs not ten pounds, can raise a Solid that weighs above 100 pounds: in that we have demonstrated, That it sufficeth, that such difference be found between the Specificall Gravities of the _Mediums_ and Moveables, let the particular and absolute Gravities be what they will: insomuch, that a Solid, provided that it be Specifically less grave than the water, although its absolute weight were 1000 pounds, yet may it be born up and elevated by ten pounds of water, and less: and on the contrary, another Solid, so that it be Specifically more grave than the water, though in absolute Gravity it were not above a pound, yet all the water in the Sea, cannot raise it from the Bottom, or float it. This sufficeth me, for my present occasion, to have, by the above declared Examples, discovered and demonstrated, without extending such matters farther, and, as I might have done, into a long Treatise: yea, but that there was a necessity of resolving the above proposed doubt, I should have contented my self with that only, which is demonstrated by _Archimedes_, in his first _Book De Insidentibus humido_[14]: where in generall termes he infers and confirms the same Conclusions, namely, that Solids (_a_) less grave than water, swim or float upon it, the (_b_) more grave go to the Bottom, and the (_c_) equally grave rest indifferently in all places, yea, though they should be wholly under water.

[14] _Of Natation_ (a) _Lib. 1, Prop. 4._ (b) _Id. Lib. 1. Prop.
3._ (c) _Id. Lib. 1. Prop. 3._

[Sidenote: The Authors defence of _Archimedes_ his Doctrine,
against the oppositions of _Buonamico_.]