自然哲学的数学原理-7

the third book the moon s theory and the profession of the equinoxes weremore fully deduced from their principles ; and the theory of the cometswas confirmed by more examples of the calculati >n of their orbits, donealso with greater accuracy.In this third edition the resistance of mediums is somewhat more largelyhandled than before; and new experiments of the resistance of heavybodies falling in air are added. In the third book, the argument to provethat the moon is retained in its orbit by the force of gravity is enlargedon ; and there are added new observations of Mr. Pound s of the proportionof the diameters of Jter to each other : there are, besides, added Mr.Kirk s observations of the comet in 16SO ; the orbit of that comet computed in an ellipsis by Dr. Halley ; and the ortit of the comet incomputed by Mr. Bradley,OOK I.THEMATHEMATICAL PRINCIPLESOFNATURAL PHILOSOPHYDEFINITIONS.DEFINITION I.77w? quantity of matter is the measure of the same, arising from itsdensity and hulk conjutictly.THUS air of a double density, in a double space, is quadruple in quantity ; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whateverdifferently condensed. I have no regard in this place to a medium, if anysuch there is, that freely pervades the interstices between the parts oibodies. It is this quantity that I mean hereafter everywhere under thename of body or mass. And the same is known by the weight of eachbody ; for it is proportional to the weight, as I have found by experimentson pendulums, very accurately made, which shall be shewn hereafter.DEFINITION II.The quantity of motion is the measure nf tlie same, arising from thevelocity and quantity of matter corjunctly.The motion of the whole i<! the sum of the motions of all the parts ; andtherefore in a body double in quantity, with equal velocity, the motion isiouble ; with twice the velocity, it is quadruple,DEFINITION III.The vis insita, or innate force of matter, is a power of resisting, hywhich every body, as much as in it lies, endeavours to persevere in itspresent stale, whether it be of rest, or of moving uniformly forwardin a right line.This force is ever proportional to the body whose force it is ; and differsnothing from the inactivity of the mass, but in our manner of conceivingT4 THE MATHEMATICAL PRINCIPLESit. A body, from the inactivity of matter, is not without difficulty put outof its state of rest or motion. Upon which account, this vis insita, may,by a most significant name, be called vis inertia, or force of inactivity.Hut a body exerts this force only, when another force, impressed upon it,endeavours to change its condition ; and the exercise of this force may boconsidered both as resistance and impulse ;it is resistance, in so far as thebody, for maintaining its present state, withstands the force impressed; itis impulse, in so far as the body, by not easily giving way to the impressedforce of another, endeavours to change the state of that other. Resistanceis usually ascribed to bodies at rest, and impulse to those in motion;but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly aretaken to be so.DKFLMTIOX IV.Ait impressed force is an action exerted upon a body, in order to changeits state, either of rest, or of moving uniformly forward in a rightline.This force consists in the action only; and remains no longer in thebody, when the action is over. For a body maintains every new state itacquires, by its vis inertice only. Impressed forces are of originsas from percussion, from pressure, from centripetal force.DEFINITION V.A centripetal force is that by which bodies are drawn or impelled, or anyway tend, towards a point as to a centre.Of this sort is gravity, by which bodies tend to the centre of the earthmagnetism, by which iron tends to the loadstone ; and that force, whatever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolvein curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends thesling, and that with so much the greater force, as it is revolved with thegreater velocity, and as soon as ever it is let go, flies away. That forcewhich opposes itself to this endeavour, and by which the sling perpetuallydraws back the stone towards the hand, and retains it in its orbit, becauseit is directed to the hand as the centre of the orbit, I call the centripetalforce. And the same thing is to be understood of all bodies, revolved inany orbits. They all endeavour to recede from the centres of their orbits ;and wore it not for the opposition of a contrary force which restrains themto, and detains them in their orbits, which I therefore call centripetal, wouldtiy off in right lines, with an uniform motion. A projectile, if it was notfor the force of gravity, would not deviate towards the earth, tut wouldOF NATUJIAL PHILOSOPHY. 7flgo off from it in a right line, and that with an uniform motion,, if the resistance of the air was taken away. It is by its gravity that it is drawnaside perpetually from its rectilinear course, and made to deviate towardsthe earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter,or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden balJ,projected from the top of a mountain by the force of gunpowder with agiven velocity, and in a direction parallel to the horizon, is carried in acurve line to the distance of two miles before it falls to the ground ; thesame, if the resistance of the air were taken away, with a double or decuplevelocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, tillat last it should fall at the distance of 10, 30, or 90 degrees, or even mightgo quite round the whole earth before it falls; or lastly, so that it mightnever fall to the earth, but go forward into the celestial spaces, and proceed in its motion in iiifiuitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and goround the whole earth, the moon also, either by the force of gravity, if itis endued with gravity, or by any other force, that impels it towards theearth, may be perpetually drawn aside towards the earth, out of the r&tilinearway, which by its innate force it would pursue; and would be madeto revolve in the orbit which it now describes ; nor could the moon without some such force, be retained in its orbit. If this force was too small,it would not sufficiently turn the moon out of a rectilinear course : if itwas too great, it would turn it too much, arid draw down the moon fromits orbit towards the earth. It is necessary, that the force be of a justquantity, and it belongs to the mathematicians to find the force, that mayserve exactly to retain a body in a given orbit, with a given velocity ; andvice versa, to determine the curvilinear way, into which a body projectedfrom a given place, with a given velocity, may be made to deviate fromits natural rectilinear way, by means of a given force.The quantity of any centripetal force may be considered as of threekinds; aboolu e, accelerative, and motive.DEFINITION VI.The absolute quantity of a centripetal force is the measure f>fthe sameproportional to the efficacy of the cause that propagates itfrom the centre, through the spaces round about.Thus the magnetic force is greater in one load-stone and less in anotheraccording to their sizes and strength of intensity.76 THE MATHEMATICAL PRINCIPLESDEFINITION VII.The accelerative quantity of a centripetal force is the measure, of thtsame, proportional to the velocity which it generates in a given time.Thus the force of the same load-stone is greater at a less distance, andless at a greater : also the force of gravity is greater in valleys, less ontops of exceeding high mountains ; and yet less (as shall hereafter be shown),at greater distances from the body of the earth ; but at equal distances, it is the same everywhere ; because (taking away, or allowing for, theresistance of the air), it equally accelerates all falling bodies, whether heavyor light, great or small.DEFINITION VIII.TJie motive quantity of a centripetal force, is the measure of the samt。proportional to the motion which it generates in a given twip.Thus the weight is greater in a greater body, less in a less body ; and.in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of thewhole body towards the centre, or, as I may say, its weight ; and it is always known by the quantity of an equal and contrary force just sufficientto Ifinder the descent of the body.These quantities of forces, we may, for brevity s sake, call by the namesof motive, accelerative, and absolute forces ; and, for distinction s sake, consider them, with respect to the bodies that tend to the centre ;to the placesof those bodies ; and to the centre of force towards which they tend ; thatis to say, I refer the motive force to the body as an endeavour and propensity of the whole towards a centre, arising from the propensities of theseveral parts taken together ; the accelerative force to the place of thebody, as a certain power or energy diffused from the centre to all placesaround to move the bodies that are in them : and the absolute force tothe centre, as endued with some cause, without which those motive forceswould not be propagated through the spaces round about ; whether thatcause be some central body (siuh as is the load-stone, in the centre of themagnetic force, or the earth in the centre of the gravitating force), oranything else that does not yet appear. For I here design only to give amathematical notion of those forces, without considering their physicalcauses and seats.Wherefore the accelerative force will stand in the same relation to themotive, as celerity does to motion. For the quantity of motion arises fromthe celerity drawn into the quantity of matter : and the motive force arisesfrom the accelerative force drawn into the same quantity of matter. Forthe sum of the actions of the accelerative force, upon the several ;articlesof the body, is the motive force of the whole. Hence it is, that near theOF NATURAL PHILOSOPHY. 77surface of the earth, where the accelerative gravity, or force productive ofgravity, in all bodies is the same, the motive gravity or the weight is asthe body : but if we should ascend to higher regions, where the accelerativegravity is less, the weight would be equally diminished, and would alwaysbe as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the weightof a body two or three times less, will be four or six times less.I likewise call attractions and impulses, in the same sense, accelerative,and motive ; and use the words attraction, impulse or propensity of anysort towards a centre, promiscuously, and indifferently, one for another ;considering those forces not physically, but mathematically : wherefore, thereader is not to imagine, that by those words, I anywhere take upon me todefine the kind, or the manner of any action, the causes or the physicalreason thereof, or that I attribute forces, in a true and physical sense, tocertain centres (which are only mathematical points) ; when at any time Ihappen to speak of centres as attracting, or as endued with attractivepowers.SCHOLIUM.Hitherto I have laid down the definitions of such words as are lessknown, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place andmotion, as being well known to all. Only I must observe, that the vulgarconceive those quantities under no other notions but from the relation theybear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absoluteand relative, true and apparent, mathematical and common.I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by anothername is called duration : relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration bythe means of motion, which is commonly used instead of true time ; suchas an hour, a day, a month, a year.II. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces ; which our senses determine by its position to bodies ; and which is vulgarly taken for immovable space ; such is the dimension of a subterraneous, an aereal, or celestialspace, determined by its position in respect of the earth. Absolute andrelative space, are the same in figure and magnitude ; but they do not remain always numerically the same. For if the earth, for instance, moves,a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into whichTS THE MATHEMATICAL PRINCIPLESthe air passes ;at another time it will be another part of the same, and so.absolutely understood, it will be perpetually mutable.III. Place is a part of space which a body takes up, and is according tothe space, either absolute or relative. I say, a part of space ; not the situation,nor the external surface of the body. For the places of equal solids arealways equal ; but their superfices, by reason of their dissimilar figures, areoften unequal. Positions properly have no quantity, nor are they so muchthe places themselves, as the properties of places. The motion of the wholeis the same thing with the sum of the motions of the parts ; that is, thetranslation of the whole, out of its place, is the same thing with the sumof the translations of the parts out of their places ; and therefore the placeof the whole is the same thing with the sum of the places of the parts, andfor that reason, it is internal, and in the whole body.IV. Absolute motion is the translation of a body from one absoluteplace into another ; and relative motion, the translation from one relativeplace into another. Thus in a ship under sail, the relative place of a bodyis that part of the ship which the body possesses ; or that part of its cavitywhich the body fills, and which therefore moves together with the ship :and relative rest is the continuance of the body in the same part of theship, or of its cavity. But real, absolute rest, is the continuance of thebody in the same part of that immovable space, in which the ship itself,its cavity, and all that it contains, is moved. Wherefore, if the earth isreally at rest, the body, which relatively rests in the ship, will really andabsolutely move with the same velocity which the ship has on the earth.But if the earth also moves, the true and absolute motion of the body willarise, partly from the true motion of the earth, in immovable space ; partlyfrom the relative motion of the ship on the earth ; and if the body movesalso relatively in the ship ;its true motion will arise, partly from the truemotion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship ; and fromthese relative motions will arise the relative motion of the body on theearth. As if that part of the earth, where the ship is, was truly movedtoward the east, with a velocity of 10010 parts; while the ship itself, witha fresh gale, and full sails, is carried towards the west, with a velocity expressed by 10 of those parts ; but a sailor walks in the ship towards theeast, with 1 part of the said velocity ; then the sailor will be moved trulyin immovable space towards the east, with a velocity of 10001 parts, andrelatively on the earth towards the west, with a velocity of 9 of those parts.Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are tr^y unequal, though they are commonly considered as equal, and used for a measure of time ; astronomers correct this inequality for their more accuratededucing of the celestial motions. It may be, that there is no such thingas an equable motion, whereby time may H accurately measured. All moOF NATURAL PHILOSOPHY. 79tions may be accelerated and retarded; but the true, or equable, progress ofabsolute time is liable to no change. The duration or perseverance of theexistence of things remains the same, whether the motions are swift or slow,or none at all : and therefore it ought to be distinguished from what areonly sensible measures thereof ; and out of which we collect it, by meansof the astronomical equation. The necessity of which equation, for determining the times of a phamomenon, is evinced as well from the experimentsof the pendulum clock, as by eclipses of the satellites of Jupiter.As the order of the parts of time is immutable, so also is the order ofthe parts of space. Suppose those parts to be moved out of their places, andthey will be moved (if the expression may be allowed) out of themselves.For times and spaces are, as it were, the places as well of themselves as ofall other things. All things are placed in time as to order of succession ;and in space as _to order of situation. It is from their essence or naturethat they are places ; and that the primary places of things should bemoveable, is absurd. These are therefore the absolute places ; and translations out of those places, are the only absolute motions.But because the parts of space cannot be seen, or distinguished from oneanother by our senses, therefore in their stead we use sensible measures ofthem. For from the positions and distances of things from any body considered as immovable, we define all places ; and then with respect to suchplaces, we estimate all motions, considering bodies as transferred from someof those places into others. And so, instead of absolute places and motions,we use relative ones; and that without any inconvenience in common affairs ; but in philosophical disquisitions, we ought to abstract from oursenses, and consider things themselves, distinct from what are only sensiblemeasures of them. For it may be that there is no body really at rest, towhich the places and motions of others may be referred.But we may distinguish rest and motion, absolute and relative, one fromthe other by their properties, causes and effects. It is a property of rest,that bodies really at rest do rest in respect to one another. And thereforeas it is possible, that in the remote regions of the fixed stars, or perhapsfar beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regionswhether any of these do keep the same position to that remote body; itfollows that absolute rest cannot be determined from the position of bodiesin our regions.It is a property of motion, that the parts, which retain given positionsto their wholes, do partake of the motions of those wholes. For all theparts of revolving bodies endeavour to recede from the axis of motion ;and the impetus of bodies moving forward, arises from the joint impetusof all the parts. Therefore, if surrounding bodies are moved, those thatare relatively at rest within them, will partake of their motion. Uponwhich account, the true and absolute motion of a body cannot be Jeter8C THE MATHEMATICAL PRINCIPLESmined by the translation of it from those which only seem to rest; for theexternal bodies ought not only to appear at rest, but to be really at rest.For otherwise, all included bodies, beside their translation from near thesurrounding ones, partake likewise of their true motions ; and though thattranslation were not made they would not be really at rest, but only seemto be so. For the surrounding bodies stand in the like relation to thesurrounded as the exterior part of a whole does to the interior, or as theshell does to the kernel ; but, if the shell moves, the kernel will alsomove, as being part of the whole, without any removal from near the shell.A property, near akin to the preceding, is this, that if a place is moved,whatever is placed therein moves along with it; and therefore a body,which is moved from a place in motion, partakes also of the motion of itsplace. Upon which account, all motions, from places in motion, are noother than parts of entire and absolute motions ; and every entire motionis composed of the motion of the body out of its first place, and themotion of this place out of its place ; and so on, until we come to someimmovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than byimmovable places : and for that reason I did before refer those absolutemotions to immovable places, but relative ones to movable places. Nowno other places are immovable but those that, from infinity to infinity, doall retain the same given position one to another ; and upon this accountmust ever remain unmoved ; and do thereby constitute immovable space.The causes by which true and relative motions are distinguished, onefrom the other, are the forces impressed upon bodies to generate motion.True motion is neither generated nor altered, but by some force impressedupon the body moved : but relative motion may be generated or alteredwithout any force impressed upon the body. For it is sufficient only toimpress some force on other bodies with which the former is compared,that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motionsuffers always some change from any force impressed upon the movingbody ; but relative motion docs not necessarily undergo any change by suchforces. For if the same forces are likewise impressed on those other bodies,with which the comparison is made, that the relative position may be preserved, then that condition will be preserved in which the relative motionconsists. And therefore any relative motion may be changed when thetrue motion remains unaltered, and the relative may be preserved when thetrue suffers some change. Upon which accounts; true motion does by nomeans consist in such relations.The effects whicli distinguish absolute from relative motion arc, theforces of receding from the axis of circular motion. For there are no suchforces in a circular motion purely relative, but in a true and absolute circular motion., they are greater or less, according t the quantity of theOF NATURAL PHILOSOPHY. 1motion. If a vessel, hung: by & }ong cord, is so often turned ubout that thecord is strongly twisted, then filled with water, and held at rest togetherwith the water ; after, by the sudden action of another force, it is whirledabout the contrary way, and while the cord is untwisting itself, the vesselcontinues for some time in this motion ; the surface of the water will atfirst be plain, as before the vessel began to move : but the vessel; by gradually communicating its motion to the water, will make it begin sensibly^to revolve, and recede by little and little from the middle, and ascend to thesides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise,till at last, performing its revolutions in the same times with the vessel,it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion ; and the true and absolutecircular motion of the water, which is here directly contrary to the relativejdiscovers itself, and may be measured by this endeavour. At first,when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis ; the water showed no tendencyto the circumference, nor any ascent towards the sides of the vessel, butremained of a plain surface, and therefore its true circular motion had notyet begun. But afterwards, when the relative motion of the water haddecreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis ; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired itsgreatest quantity, when the water rested relatively in the vessel. Andtherefore this endeavour does not depend upon any translation of the waterin respect of the ambient bodies, nor can true circular motion be definedby such translation. There is only one real circular motion of any onerevolving body, corresponding to only one power of endeavouring to recede