[经典力学]牛顿自然哲学的数学原理论文解读

Posted 2023-02-17 IT鹅

[经典力学]牛顿自然哲学的数学原理论文中文解读与质点运动学基本理论

致敬：
自然与自然的定律，都隐藏在黑暗之中。上帝说‘让牛顿来吧’，于是，一切变为光明。
Nature and its laws are hidden in darkness. God said, “Let Newton come,” and everything became bright。

牛顿自然哲学的数学原理
ISAAC NEWTON’S MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY
(1686–1713)

定义

DEFINITION I
The quantity of matter is the measure of the same, arising from its density and bulk conjointly.

The quantity of matter is the measure of the same, arising from its density and bulk conjointly.
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 whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter.

物质的量是一个相同不变的量，它由密度density和体积bulk产生。不论是任何物体与任何介质中，物质都存在这个
量quantity

DEFINITION II
The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.
The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

运动的量是一个不变（恒定）的量由物质的速度(velocity)和量(quantity)组成，整体运动是所有分运动的总和（一个力具有多个分力），因此，在一个加速运动(equal velocity)过程中，速度相等，运动是加倍的，通过两倍速度，实际上是四倍速度。

定义三 DEFINITION III
The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.

The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.
This force is always proportional to the body whose force it is and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inert nature of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia (vis inertiae) or force of inactivity. But a body only exerts this force when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered as both resistance and impulse; it is resistance so far as the body for maintaining its present state, opposes the force impressed; it is impulse so far as the body, by not easily giving way to the impressed force of another endeavours to change the state of that other. Resistance is 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 are taken to be so.

任何物体都存在一个力量(power),这种力量是与生俱来的力量，通过这种力量，无论物体处在任何位置，都将保持其当前状态，不论是处于静止状态或者运动状态，物体总是保持匀速运动right line的状态。

这种力总是与它受到的力成正比，不论物体是否运动，总是保持这种力，于是我们可以定义这种力称为
惯性力inertiae。

但是只有外界有外力施加到该物体上，试图改变它的状态时，物体才施加一种力，这种力量可以被认为是阻力resistance或者冲力impluse，对于物体运动状态而言，它是一种抵抗力；对于静止的物体而言，这是一种冲动impluse，外力无法轻易地改变物体运动状态。
阻力resistance通常用于描述运动中的物体，即滑动摩擦力sliding friction；
冲力impluse通常用于描述静止不懂的物体，即“静摩擦力static friction；
当然物体不是完全静止的，只我们将它认为是完全静止的状态。

DEFINITION IV
An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.
This force consists in the action only, and remains no longer in the body when the action is over. For a body maintains every new state it acquires by its inertia only. But impressed forces are of different origins, as from percussion, from pressure, from centripetal force.

外力施加在物体上，改变其状态，不论是静止状态还是沿直线运动状态，外力只存在于动作中，当运动结束该力不会留在其内部，因为物体只能通过其惯性来维持它所获得的每一个新状态，但外力不同如
敲击力percussion,压力pressure,向心力centripetal force。

DEFINITION V
A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.
Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the planets are continually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling continually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go 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 drawn aside continually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, or the quantity of its matter, or the greater the velocity with which it is projected, the less will it from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder, with a given velocity, and in a direction parallel to the horizon, is carried in a curved line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, 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, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might never fall to the earth, but go forwards into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be continually drawn aside towards the earth, out of the rectilinear way which by its innate force it would pursue; and would be made to 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 it was too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary that the force be of a just quantity, and it belongs to the mathematicians to find the force that may serve exactly to retain a body in a given orbit with a given velocity; and vice versa, to determine the curvilinear way into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.
The quantity of any centripetal force may be considered as of three kinds: absolute, accelerative, and motive.

向心力 centripetal force是物体相互吸引或者排斥，或以任何方式趋向某一点的力。重力(gravity)属于向心力，物体通过它倾向于地球中心，铁iron会倾向磁铁loadstone，这种力迫使行星脱离匀速直线运动，并使行星在曲线轨道上旋转。一块用细线拉着的石头如果通过手迫使它绕着中心点旋转，那么手会感受到石头在不断地倒退，随着细线旋转速度越大这个力对手的压力越大，与此相反的一个力，努力将石头拉向圆心，这个力称为向心力 centripetal force，物体会沿着轨道做匀速运动。
如果没有重力，抛射体不会偏向地球，而是沿着其他right line进行运动，假设没有空气阻力，那么物体则会匀速运动。因为重力存在，抛射体会偏离它运动路线，并根据它的重力多少与运动速度多少偏离地球。它受到的引力（万有引力）越小，则物体质量越小，物体运动的越远。如果一个铅球在火药的作用下在山顶发射，让铅球保持一定速度，则该铅球能绕地球一圈，或者铅球不会掉落在地球上，将射向天空，保持它固有的运动状态，进行运动。如果没有这个力，或者这个力太小，月球无法保持绕地球旋转的状态，这个力是一个量，数学家有责任找到准确地将该物体以给定速度固定在轨道上的力。
任何向心力可以被定义为：绝对的Absolute,加速的Accelerative,动机的Motive.

DEFINITION VI
The absolute quantity of a centripetal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

The absolute quantity of a centripetal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.
Thus the magnetic force is greater in one loadstone and less in another, according to their sizes and strength of intensity.

向心力的绝对量是相同的量度，与将其从中心传播到周围空间的原因的功效成正比。
因此，根据它们的大小和强度强度，一个负载石的磁力较大，而另一个负载石的磁力较小。
DEFINITION VII
The accclerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.

The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.
Thus the force of the same loadstone is greater at a less distance, and less at a greater: also the force of gravity is greater in valleys, less on tops 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, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

向心加速度是一个相同的量，与给定的时间内产生的速度成正比。以地面为参考系不论在何处重力总是向球心。
DEFINITION VIII
The motive quantity of a centripetal force is the measure of the same, proportional to the motion which it generates in a given time.

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 the whole 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 sufficient to hinder the descent of the body.

These quantities of forces, we may, for the sake of brevity, call by the names of motive, accelerative, and absolute forces; and, for the sake of distinction, consider them with respect to the bodies that tend to the centre, to the places of those bodies, and to the centre of force towards which they tend; that is 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 the several parts taken together; the accelerative force to the place of the body, as a certain power diffused from the centre to all places around to move the bodies that are in them; and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause be some central body (such as is the magnet in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats.
Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity multiplied by the quantity of matter; and the motive force arises from the accelerative force multiplied by the same quantity of matter. For the sum of the actions of the accelerative force, upon the several particles of the body, is the motive force of the whole. Hence it is, that near the suffice of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body; but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one-half, the weight of 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 any sort towards a centre, promiscuously, and indifferently, one for another; considering those forces not physically, but mathematically: wherefore the reader is not to imagine that by those words I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

因此，体积越大，质量越小；体积越小，质量越大。存在一种量，靠近地球的地方受到的影响越大，离地球越远的地方受到的影响越小，这种量具有向心性tend to the centre。我们将这种力称为动机力motive，加速力accelerative ，绝对力Absolute forces。这些物体与位置与它们趋向中心有关。考虑到加速力与机动的关系，如速度与运动关系一样，因此物体运动的量是由速度乘以物体质量产生的，即动量momentum。因为加速力作用在粒子上作用力之和就是整体的动力。因此在地球周围所有物体受到的重力加速度相同，如果我们上升到更高的地区，那么受到的重力加速度减小，这里可以根据万有引力定律得出。

SCHOLIUM

I

I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.

绝对的，存粹的数学的时间，就其本性来说，均匀地流逝而与任何外在的情况无关。
《力学》中对于绝对时间观的描述，绝对时间观众首先认为时间的存在是绝对的，这种存在是独立于物体和物体运动形成的事物演化之外的，反之，物体却是必须在时间流逝中实现其运动并形成物体的变化。
II

Absolute space, in its own nature, without relation 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 commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative 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 which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed.
绝对空间，就起本性来说，与任何外在的情况下无关，始终保持着相似和不变。

经典物理学认为真实空间的存在是绝对的，也就是说没有物体和观察者的空间仍然存在，他们认为真是空间存在的几何性质是绝对的，也就是说没有物体的存在和物体的运动而发生变化，经典物理学认为绝对空间就是三维空间。

III

Place is a part of space which a body takes up, and is according to the 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 are always equal; but their suffices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the place of the whole is the same as the sum of the places of the parts, and for that reason, it is internal, and in the whole body.

物体占据空间位置的地点，称为物点Place整体移动的位置与部分移动的位置之和相同，整体位置与部分位置之和相同。

IV

Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of the cavity which 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 the ship, or of its cavity. But real, absolute rest, is the continuance of the body 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 is really at rest, the body, which relatively rests in the ship, will really and absolutely 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 will arise, partly from the true motion of the earth, in immovable space, partly from the relative motion of the ship on the earth; and if the body moves also relatively in the ship, its true motion will arise, partly from the true motion 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 from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved towards the east, with a velocity of 10,010 parts; while the ship itself, with a 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 the east, with I part of the said velocity; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10,001 parts, and relatively on the earth towards the west, with a velocity of g of those parts.

绝对运动Absolutemotion是物体从一个绝对位置到另一个位置的平移；和相对运动，从一个相对位置到另一个相对位置的平移。因此，在一艘航行的船中，一个身体的相对位置是船的一部分，它占据物点，相对静止是指在绝对空间内而言，我们的观察点在绝对空间内容，这艘船可以假定为一个绝对空间。如果一个物体在船内容中运动，其中地球同时在运动，它真正的运动会产生，部分来自地球运动。

Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the flowing of absolute time is not liable to any change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all: and therefore this duration ought to be distinguished from what are only sensible measures thereof; and from which we deduce it, by means of the astronomical equation. The necessity of this equation, for determining the times of a phenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.

匀速运动可能不存在，天文学家通过匀速运动测量时间。所有的运动都可以是加速的或减速的，但绝对时间的流动不会发生变化。事物存在的持续时间或不变性，不论物体运动快慢，时间总是保持持续且不改变。因此我们需要将绝对时间与感知时间分开

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they 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 of all 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 nature that they are places; and that the primary places of things should be movable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute motions.

由于时间部分顺序不可变，空间部分顺序也不可变，绝对地方的平移是唯一的绝对运动。

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of 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 our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

由于我们的器官无法空间的各个部分，我们应当将自己抽离出来，且考虑事物本身，我们通过其他参考系得知无绝对静止的物体

But we may distinguish rest and motion, absolute and relative, one from the 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 therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body, it follows that absolute rest cannot be determined from the position of bodies in our regions.

但是我们可以通过物体的属性，绝对性与真实性来判断物体的静止与运动。绝对与相对，是静止的属性，真正静止的物体确实相对于其他物体彼此静止，我们不能通过自己的位置来判断物体是否绝对静止。

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forwards arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, besides their translation from near the surrounding ones, partake likewise of their true motions; and though that translation were not made, they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel; but if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

如果周围的物体移动，它们内部相对静止的物体将参与它们的运动。一个物体的真实和绝对运动不能通过它从那些似乎静止的物体的平移来确定。因为外在的物体不仅应该看起来是静止的，而且应该是真正的静止的。

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 its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than by immovable places; and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute immovable space.

完全的和绝对的运动只能由不动的地方决定，之前我们用绝对运动代指不动的地方，而将相对运动代指可动的地方。现在没有地方是不可移动的，除了无限空间具有绝对不动的位置。只有保持不可移动才能构成不可移动的空间。

The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress 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 motion suffers always some change from any force impressed upon the moving body; but relative motion does not necessarily undergo any change by such forces. 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 motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Thus, true motion by no means consists in such relations.

区分真实运动和相对运动的原因是施加在物体上以产生运动的力。真正的运动既不会产生也不会改变，而是通过施加在运动物体上的某种力而产生的；但是可以在没有任何力施加在身体上的情况下产生或改变相对运动。真实的运动总是会因施加在运动物体上的任何力而发生一些变化。但相对运动不一定会因这种力而发生任何变化。

The effects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; thereupon, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion; the surface of the water will at first be plain, as before the vessel began to move; but after that, 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 the sides 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 absolute circular motion of the water, which is here directly contrary to the relative, becomes known, 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 tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, 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 continually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and, like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions.
Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place, and motion, their [sensible] measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. On this account, those violate the accuracy of language, which ought to be kept precise, who interpret these words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities with their relations and sensible measures.
It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindmost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion -did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to obtain the true motions from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise. For to this end it was that I composed it.

AXIOMS, OR LAWS OF MOTION

LAW I

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
PROJECTILES continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.

牛顿第一定律：

任何物体都保持静止的或沿一条直线作匀速运动的状态，除非作用于它的力迫使它改变这种状态。

LAW II

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
If any force generates n motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

牛顿第二定律：

运动的改变与所加的力成正比，并且发生在此力所沿的方向线上。

a ∝ ∑F

a ∝ 1/m

∑F ∝ m

∑F = ma

Newton’s Second Law:
F = m * dv/dt

LAW III

To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may say so) will be equally drawn back towards the stone; fro the distended rope, by the saame endeavour to relax or unbend itself, will draw the horse as muchas it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are inversely proportional to the bodies. This law takes place also in attractions, as will be proved in the next Scholium…

牛顿第三定律：

每一个作用总是有一个相等的反作用于它相对抗，或者说，两物体之间彼此的相互作用永远相等，并且各自指向其对方。

BOOK THREE

SYSTEM OF THE WORLD
(IN MATHEMATICAL TREATMENT)
IN THE PRECEDING BOOKS I have laid down the principles of philosophy; principles not philosophical but mathematical: such, namely, as we may 1 build our reasonings upon in philosophical inquiries. These principles are the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy; but, lest they should have appeared of themselves dry and barren, I have illustrated them here and there with some philosophical scholiums, giving an account of such things as are of more general nature, and which philosophy seems chiefly to be founded on; such as the density and the resistance of bodies, spaces void of all bodies, and the motion of light and sounds. It remains that, from the same principles, I now demonstrate the frame of the System of the World. Upon this subject I had, indeed, composed the third Book in a popular method, that it might be read by many; but afterwards, considering that such as had not sufficiently entered into the principles could not easily discern the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent the disputes which might be raised upon such accounts, I chose to reduce the substance of this Book into the form of Propositions (in the mathematical way), which should be read by those only who had first made themselves masters of the principles established in 0 the preceding Books: not that I would advise anyone to the previous study of every Proposition of those Books; for they abound with such as might cost too much time, even to readers of good mathematical learning. It is enough if one carefully reads the Definitions, the Laws of Motion, and the first three sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first two Books, as the references in this, and his occasions, shall require.

ULES OF REASONING IN PHILOSOPHY 哲学中的推理规则

RULE I

We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

RULE II

Therefore to the same natural effects we must, as far as possible, assign the same causes.
As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets.

RULE III

The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
For since the qualities of bodies are only known to us by experiments, we are to hold for universal all such as universally agree with experiments; and such as are not liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede from the analogy of Nature, which is wont to be simple, and always consonant to itself. We no other way know the extension of bodies than by our senses, nor do these reach it in all bodies; but because we perceive extension in all tht are sensible, therefore we ascribe it universally to all others also. That abundance of bodies are hard, we learn by experience; and because the hardness of the whole arises from the hardness of the parts, we therefore justly infer the hardness of the undivided particles not only of the bodies we feel but of all others. That all bodies are impenetrable, we gather not from reason, but from sensation. The bodies which we handle we find impenetrable, and thence conclude impenetrability to be an universal property of all bodies whatsoever. That all bodies are movable, and endowed with certain powers (which we call the inertia) of persevering in their motion, or in their rest, we only infer from the like properties observed in the bodies which we have seen. The extension, hardness, impenetrability, mobility, and inertia of the whole, result from the extension, hardness, impenetrability, mobility, and inertia of the parts; and hence we conclude the least particles of all bodies to be also all extended, and hard and impenetrable, and movable, and endowed with their proper inertia. And this is the foundation of all philosophy. Moreover, that the divided but contiguous particles of bodies may be separated from one another, is matter of observation; and, in the particles that remain undivided, our minds are able to distinguish yet lesser parts, as is mathematically demonstrated. But whether the parts so distinguished, and not yet divided, may, by the powers of Nature, be actually divided and separated from one another, we cannot certainly determine. Yet, had we the proof of but one experiment that any undivided particle, in breaking a hard and solid body, suffered a division, we might by virtue of this rule conclude that the undivided as well as the divided particles may be divided and actually separated to infinity.
Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain; that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and, all the planets one towards another; and the comets in like manner towards- the sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation.
For the argument from the appearances concludes with more force for the universal gravitation of all bodies than for their impenetrability; of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to be essential to bodies: by their vis insita I mean nothing but their inertia. This is immutable. Their gravity is diminished as they recede from the earth.

RULE IV

In experimental philosophy we are to look, upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may cither be made more accurate, or liable to exceptions.
This rule we must follow, that the argument of induction may not be evaded by hypotheses.

概念纠正：

right line我的意思是物体总是保持原来的运动状态，这里解释为匀速运动。我将LAW III中的Place解释成“物点”，实际是指物体占据的地点。其中我在LAW中引用了北大物理系教材的内容，在本文用现代理论解释了绝对时间，绝对空间等概念。

参考：
ISAAC NEWTON’S MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY
https://www.marxists.org/reference/subject/philosophy /works/en/newton.htm
https://physics.info/newton-second/
力学（物理类）舒幼生 编著 2005.09 北京大学出版社