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Gravity, with the Centres of PercufTion, afford a fine view; and (et the whole Doctrine in the moft agreeable Light. See Centre.
In each Syftem, the Bafe whereby the Body breaks, moves on the Axis of Equilibrium* which is an immoveable Line in the fame Bafe; but in the fecond, the Fibres of this Bafe are conti- nually ftretching more and more; and that in the fame Ratio as they recede farther and farther from the Axis of Equilibrium, and of Confequence are ftill exerting a greater and greater Part of their whole Force.
Thefe unequal Extcntions, like all other Forces, muft have fome common Centre where they all meet; and with regard to which they make equal Efforts on each Side ; And as they are precifely in the fame Proportion as the Velocities which the fe- veral Points of a Rod moved circularly would have to one ano- ther ; the Centre of Extention of the Bafe whereby the Body breaks, or tends to break, muft be the fame with its Centre of Percuffion — Galileo's Hypothecs, where Fibres ftretch equally, and break all at once, correfponds to the Cafe of a Rod mo- ving parallel to it felf; where the Centre of Extenfion or Per- cuifion does not appear, as being confounded with the Centre of Gravity.
The Bafe of Fraction being a Surface whole particular Nature determines its Centre of Percuffion ; 'tis necelTary it be firft known to find on what Point of the vertical Axis of that Bafe it is placed; and how far it is from the Axis of Equilibrium. — In- deed, we know, in the general, that it always acts with fo much more Advantage as it is further from it, in regard it aits by a longer Arm of a Lever; and of Confequence 'tis the unequal Re- fiflence of the Fibres in M. Mariotte's Hypothecs, which produces that Centre of Percuffion ; but this unequal Refiflence is grea- ter or lefs, according as the Centre of Percuffion is plac'd more or lefs high on the vertical Axis of the Bafe, in the different Sur- faces of the Bafe of the Fracture.
To exprefs this unequal Refiflence, accompanied with all the Variations it is capable of, regard muft be had to the Ratio be- tween the Diftance of the Centre of Percuffion from the Axis of Equilibrium, and the Length of the vertical Axis of the Bafe. —In which Ratio, the firft Term, or the Numerator, is always lefs than the fecond or the Denominator: So that the Ratio is always a Fraction lefs than Unity; and the unequal Refiflence of the Fibres in M. Mariotte's Hypothefis is fo much the greater, or which amounts to the fame, approaches fo much nearer to the equal Refiflence in Galileo's Hypothefis, as the two Terms of the Ratio are nearer to an Equality.
Hence it follows, that the Refiflence of Bodies, in M. Mariotte's Syftem is to that in Galileo's, as the leaft of the Terms in the Ratio is to the greateft. — Hence, alfo, the Refiflence being lefs than what Galileo imagined, the relative Weight muft alfo be lefs; fo that the Proportion already mentioned between the ab- folute and relative Weight, caniaot fubfift in the new Syftem, without an Augmentation of the relative Weight, or a Diminu- tion of the abfolute Weight : Which Diminution is had by mul- tiplying the Weight by the Ratio, which is always lefs than Uni- ty. This done, we find that the abfolute Weight multiplied by the Ratio, is to the relative Weight, as the Diftance of the Cen- tre of Gravity of the Body from the Axis of Equilibrium, is to the Diftance of the Centre of Gravity of the Bale of Fracture, from the fame Axis. Which is precifely the fame thing with the general Formula given by M. Varignon, for the Syftem of M. Mariotte. In effect, after conceiving the relative Weight of a Body, and its Refiflence equal to its abfolute Weight, as the two contrary Forces applied to the two Arms of a Lever, in the Hy- pothefis of Galileo ; there needs nothing to convert it into that of M. Mariotte-) but to imagine that the Refiflence, or the abfolute Weight is become lefs ; every thing clfe remaining the fame.
We have here only confidered Bodies as to be broke by their own Weight.— It will amount to the fame, if we fuppofe them void of Weight themfelves, and to be broken by a Weight ap- plied to their Extremities : Only it is to be obferved, that a fo- reign Weight acts by an Arm of a Lever equal to the whole Length of the Body ; whereas their own Weight being all united in their Centre of Gravity, is only the Diftance of that Centre from the Axis of Equilibrium,
One of the moft curious, and perhaps the moft ufeful Quefti- ons in this Refearch, is to find what Figure a Body muft have, that its Refiflence may be equal in all its Parts ; whether it be conceived as charg'd with a foreign Weight ; or as only fuftain- ing its own Weight.— We fliall, here, only confidet the latter Cafe, from which the former will be eatily determined,
For a Body, then, fufpended horizontally, to refift equally in all its Parts; 'tis necefTary fome Part of it being conceived as cut off in a Plane parallel to the Bafe of Fracture of the Body; the Weight of the Part retrench'dbe to its Refiflence, in the fame Ratio, as the Weight of the whole to its Refiflence; thefe four
Powers acting by Arms of Levers proper to themfelves. Now
the Weight of any Body thus conceived, is its whole Weight multiplied by the Diftance of the Centre of Gravity of the Bo- dy, from the Axis of Equilibrium ; and the Refiflence is the Plane of the Bafe of Fracture multiplied by the Diftance of the Centre of Gravity of the Bafe from the fame Axis ; Confequently thefe
four Quantities are to be proportional in the whole, and in each Part of a Solid of equal Refiflence.
From this Proportion M. Varignon eafily deduces two Solids, which fliall refift equally in all their Parts. Galileo had found one before : That difcovered by M. Varignon, is in form of a Trum- pet, and is to be fix'd into the Wall at its greater End; fb that its Bignefs and Weight is always diminiih 'd in Proportion as its Length, or the Arm of the Lever whereby its Weight ads, increafes. 'Lis added, (which feems very remarkable) that howfoever different the two Syfterns may be, the Solids of equal Refiflence are the fame in both.
For the Resistence of a Solid fupported at each Extreme; at, of a Beam between tivo Walls. See Beam.
Resistence of Fluids, in Hydroftaticks, is the Force where- with Bodies moving in fluid Mediums, are impeded and retarded in their Motions. See Fluid and Medium.
Latus of the Resistence of fluid Mediums.
A Body moving in a Fluid, is refifled from two Caufes ; the firft, the Cohefion of the Parts of the Fluid.— For a Body in its Motion feparating the 1 Parts of a Liquid, muft overcome the Force with which thofe Parts cohere. See Cohesion.
The fecond is the Inertia, or inactivity of Matter, whereby a certain Force is required to move the Particles from their Places, in order to let the Body pais. See Vis Inertle.
The Retardation from the firft Caufe, is always the fame in the fame Space, the fame Body remaining ; be the Velocity what it will.— Hence, the Refiflence increafes as the Space run through; in which Ratio, the Ve.ocity alio increafes; therefore the Re- fiflence is as the Velocity it felf. See Velocity.
The Refiflence from the fecond Caufe, when the fame Body moves through different Liquids, with the fame Velocity, fol- lows the Proportion of Matter to be removed in the fame timej which is as the Dcnfity of the Liquid. See Density.
When the fame Body moves through the fame Liquid with different Velocities, this Refiflence increafes in Proportion to the Number of Particles itruck in an equal Time; which Number is as the Space run through in that Line, that is, as the Veloci- ty. But farther it increafes in Proportion to the Force with which the Body ftrikes againft every Part j which Force is alfo as the Velocity of the Body. And therefore if the Velocity be Triple, the Refiflence is Triple, from a triple Number of Parts to be re- moved.— It is alfo triple from a ftroke three times ftronger againft every Particle ; therefore the whole Refiflence is ninefold, that is, as the Square of the Velocity. Hence a Body moved in a Liquid, is refifled partly in a Ratio of the Velocity, and partly, in a duplicate Ratio of it.
The Refiflence from the Cohefion of Parts in Liquids, except glutinous ones, is not very fcnfible in refpect of the other Re- fiflence; which, increafes in aRatio of the Square of 'the Velo- cities, but the firft in a Ratio of the Velocity itfelf. By how much the Velocity increafes, by fo much more do thefe Re- fiflences differ ; wherefore in fwifter Motions the Refiflence alone is to be confidered, which is as the Square of the Velocity.
If a Liquid be included in a Veflel of a prifmatical Figure, and there be moved along in it with equal Velocity, and in a Direction parallel to the Sides of the Prifm, two Bodies, the one fpherical, the other cylindric; fo that the Diameter of the Bafe of the lat- ter be equal to the Diameter of the Sphere; and the Cylinder be moved in the Direction of its Axis; thefe Bodies will fuffer the fame Refiflence.
To demonftrate this, fuppofe the Bodies at reft, and that the Liquid moves in the Veflel, with the fame Velocity that the Bo- dies had; by this the relative Motion of the Bodies, and the Li- quid is not changed : Confequently the Actions of the Bodies on the
Liquid, and of the Liquid on the Bodies, are not changed
The Retardation which the Liquor fuffers in paffing by the Bo- dy, arifes only from this, that in that Place it is reduced to a nar- rower Space; but the Capacity of the Veilel is equally diminifli- ed by each Body; therefore each Body produces an equal Retar- dation. And becaufe Action and Re-a6tion are equal to one ano- ther, the Liquid acts equally upon each Body ; wherefore alfo each Body will be equally retarded, when the Bodies arc moved, and the Liquid is at reft.
Refiflence and Retardation are ufed indifferently for each other, as being both in the fame Proportion ; and the fame Refiflence always generating the fame Rerardation But with regard to dif- ferent Bodies, the fame Refiflence frequently generates different Retardarions; the Refiflence being the Quantity of Motion, and the Retardation the Celerity.— Fo? the Difference and Meafure of the two, fee Retardation.
The Retardations from this Refiflence may be compared toge- ther, by comparing the Refiflence with the Gravity. — It isdemon- ftrated that the Refiflence of a Cylinder, which moves in the Di- rection of its Axis, (to which the Refiflence of a Sphere i of the fame Diameter is equal) is equal to the Weight of a Cylinderi of that Liquid through which the Body moves, having its Bafe e- qual to the Body's Bafe, and its Height equal to half the Height, from which a Body falling in vacuo, may acquire the Velocity with which the Cylinder is moved through the Liquid.
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