chapter 4 dynamics
Dynamics
THE WORLD had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seventeenth centuries of our era great Italians, in particular Leonardo da Vinci, the artist (bron 1452, died 1519), and Galileo (bron 1564,died 1642), rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena. Meanwhile the slow advance of mathematics and the accumulation of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. Also the very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves what happened; and the secret of the relation of mathematical theory and experiment in inductive reasoning was practically discovered. It was an act eminently characteristic of the age that Galileo, a philosopher , should have dropped the weights from the leaning tower of Pisa. There are always men of thought and men of action; mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action. 世界不得不等待一千八百年，直到希腊数学物理学家找到了继承者。在我们时代的十六和十七世纪，伟大的意大利人，特别是艺术家莱昂纳多·达·芬奇（生于1452年，卒于1519年）和伽利略（生于1564年，卒于1642年），重新发现了与自然现象的实验研究相结合的抽象数学思想的秘密，这一秘密早已为阿基米德所知。与此同时，数学的缓慢进步和准确天文知识的积累使自然哲学家们在研究方面处于更有利的位置。那个时代极度自我中心的自我主张，以及对个人体验的贪婪，促使其思想家们渴望亲自观察发生的一切；而数学理论与实验在归纳推理中的关系的奥秘也由此几乎被发现。伽利略作为一位哲学家，从比萨斜塔上投掷重物的举动是那个时代极具代表性的行为。总是有思想家和行动派；数学物理学是一个将思考冲动与行动冲动结合于同一群人的时代的产物。
This matter of the dropping of weights from the tower marks picturesquely an essential step in knowledge, no less a step than the first attainment of correct ideas on the science of dynamics, the basal science of the whole subject. The particular point in dispute was as to whether bodies of different weights would fall from the same height in the same time. According to a dictum of Aristotle, universally followed up to that epoch, the heavier weight would fall the quicker. Galileo affirmed that they would fall in the same time, and proved his point by dropping weights from the top of the leaning tower. The apparent exceptions to the rule all arise when , for some reason , such as extreme lightness or great speed , the air resistance is important. But neglecting the air the law is exact.
从塔上投掷物体的事件生动地标志着知识的一步飞跃，这不仅仅是一个简单的步骤，而是对整个学科基础科学——动力学科学的正确概念的首次确立。争论的核心问题是，不同重量的物体是否会在相同高度下同时落地。根据亚里士多德的断言，这一观点在当时被普遍接受，认为较重的物体会更快落下。而伽利略则主张它们会同时落地，并通过从比萨斜塔顶部投掷物体来证明了他的观点。看似与这一规则相违背的情况都出现在某些特殊条件下，比如物体极轻或者速度很快，从而空气阻力变得重要。然而，如果忽略空气阻力，这一法则是精确无误的。
Galileo's successful experiment was not the result of a mere lucky guess. It arose from his correct ideas in connection with inertia and mass . The first law of motion, as following Newton we now enunciate it , is — Every body continues in its state of rest or of uniform motion in a straight line, except so far as it is compelled by impressed force to change that state. This law is more than a dry formula: ‘ it is also a paean of triumph over defeated heretics. The point at issue can be understood by deleting from the law the phrase 'or of uniform motion in a straight line'. We there obtain what might be taken as the Aristotelian opposition formula: 'Every body continues in its state of rest except so far as it is compelled by impressed force to change that state.'
伽利略的成功实验并不是纯粹的幸运猜测的结果。它源于他对惯性和质量的正确理解。根据牛顿的描述，我们现在阐述的第一运动定律是——每个物体都保持静止状态或沿直线匀速运动状态，除非外力迫使它改变这种状态。这个定律不仅仅是一个枯燥的公式：它还是对被打败的异端分子的胜利颂歌。争论的焦点可以通过从定律中删除“或沿直线匀速运动状态”这一短语来理解。在这种情况下，我们得到的可以被视为亚里士多德的对立(与牛顿对立)公式：“每个物体都保持静止状态，除非外力迫使它改变这种状态。”
In this last false formula it is asserted that , apart from force , a body continues in a state of rest ; and accordingly that , if a body is moving , a force is required to sustain the motion; so that when the force ceases the motion ceases. The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause , or , if you like to put it so , as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion merely when the velocity is and remains zero. Thus, when a body is moving, we do not seek for any external influence except to explain changes in the rate of the velocity or changes in its direction. So long as the body in moving at the same rate and in the same direction there is no need to invoke the aid of any forces.
在这个最后的错误公式中，断言除了力之外，一个物体会保持静止状态；因此，如果一个物体在运动，就需要有力来维持运动；所以当力消失时，运动也会停止。真正的牛顿定律持完全相反的观点。一个不受力作用的物体的状态是沿直线做匀速运动，而这种匀速直线运动的原因不应归结于任何外力或外部影响，或者，如果你愿意这样说，也不能将任何外力或影响视为这种匀速直线运动的必然伴随物。静止只是这种运动的一种特例，当速度为零并保持不变时就是静止。因此，当一个物体在运动时，我们不需要寻求任何外部影响，除非是为了解释速度的变化或方向的变化。只要物体以相同的速率和方向运动，就不需要引入任何力的帮助。
The difference between the two points of view is well seen by reference to the theory of the motion of the planets.
通过参考行星运动理论，可以清楚地看到这两种观点之间的区别。
Copernicus , a Pole, born at Thorn in West Prussia (born 1473, died 1543), showed how much simpler it was to conceive the planets, including the earth as revolving round the sun in orbits which are nearly circular; and later, Kepler, a German mathematician , in the year 1609 proved that , in fact , the orbits are practically ellipses, that is , a special sort of oval curves which we will consider later in more detail. Immediately the question arose as what are the forces which preserve the planets in this motion. According to the old false view , held by Kepler, the actual velocity itself required preservation by force. Thus he looked for tangential forces as in the accompanying figure (4). But according to Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun. Newton, therefore, had to search for a force which would bend the motion round into its elliptical orbit. This he showed must be a force directed towards the sun as in the next figure(5). In fact , the force is the gravitational attraction of the sun acting according to the law of the inverse square of the distance, which has been stated above.
“哥白尼，一位波兰人，生于普鲁士西部的托伦（生于1473年，卒于1543年），展示了将包括地球在内的行星设想为绕太阳运行在近乎圆形的轨道上是多么简单；而后来，德国数学家开普勒在1609年证明，实际上这些轨道几乎是椭圆形的，即一种特殊的椭圆曲线，我们将在后面更详细地讨论。随之而来的是一个问题：是什么力量使行星保持这种运动。根据旧的错误观点，开普勒认为实际的速度本身需要通过力量来保持。因此，他像在附图（4）中所示的那样寻找切向力。但根据牛顿定律，除了某种力量外，行星会以其现有速度沿直线永远运动，从而完全离开太阳。因此，牛顿不得不寻找一种力量，将运动弯曲成椭圆轨道。他表明这种力量必须是指向太阳的，如下一个图（5）所示。事实上，这个力是太阳的引力，根据上述提到的平方反比定律作用。”
The science of mechanics rose among the Greeks form a consideration of the theory of the mechanical advantage obtained by the use of l lever, and also from a consideration of various problems connected with the weights of bodies . It was finally put on its true basis at the end of the sixteenth and during the seventeenth centuries , as the preceding account shows, partly with the view of explaining the theory of falling bodies, but chiefly in order to give a scientific theory of planetary motions. But since those days dynamics has taken upon itself a more ambitious task , and now claims to be the ultimate science of which the others are but branches . The claim amounts to this: namely, that the various qualities of things perceptible to the senses are merely our peculiar mode of appreciating changes in position on the part of things existing in space. For example, suppose we look at Westminster Abbey【威斯敏斯特教堂】. It has been standing there, grey and immovable, for centuries past. But, according to modern scientific theory, that greyness, which so heightens our sense of the immobility of the building , is itself nothing but our way of appreciating the rapid motions of the ultimate molecules, which form the outer surface of the building and communicate vibrations to a substance called the ether. Again we lay our hands on its stones and note their cool, even temperature, so symbolic of the quiet repose of the building. But this feeling of temperature simply marks our sense of the transfer of heat from the hand to the stone, or from the stone to the hand ; and, according to modern science, heat is nothing but the agitation of the molecules of a body. Finally, the organ begins playing, and again sound is nothing but the result of motions of the air striking on the drum of the ear.
力学科学在希腊人中兴起，源于对杠杆作用下获得的机械优势理论的思考，同时也源于对与物体重量有关的各种问题的探讨。正如前述内容所示，力学最终在十六世纪末和十七世纪期间奠定了它的真正基础，部分是为了解释自由落体理论，但主要目的是提供一个关于行星运动的科学理论。然而，自那时起，动力学承担了一个更为宏大的任务，如今它声称自己是所有其他科学的终极学科，而其他科学仅仅是它的分支。这一主张的意义在于：我们感官所能察觉到的各种事物的性质，仅仅是我们以自己的方式来感知空间中物体位置变化的一种表现方式。例如，假设我们看着威斯敏斯特教堂。它已经静静地矗立在那里，灰色不动，经历了数个世纪。但根据现代科学理论，那灰色——极大增强了我们对建筑物不动性的感受——本身不过是我们感知建筑外表面形成的微小分子快速运动，并将振动传递给一种叫做以太的物质的方式。同样，当我们把手放在它的石头上，感受到它们的清凉与均匀的温度，这象征着建筑物的宁静安详。但这种温度的感觉，只是我们对从手到石头或从石头到手之间热传递的感知；而根据现代科学，热只是物体分子的运动。最后，风琴开始演奏，而声音不过是空气的运动撞击耳鼓所产生的结果。
This the endeavour to give a dynamical explanation of phenomena is the attempt to explain them by statements of the general form, that such and such a substance or body was in this place and is now in that place. Thus we arrive at the great basal idea of modern science, that all our sensations are the result of comparisons of the changed configurations of things in space at various times . It follows, therefore, that the laws of motion , that is , the laws of the changes of configurations of things , are the ultimate laws of physical science.
这是对现象进行动力学解释的努力，即试图通过一般形式的陈述来解释它们，例如某种物质或物体曾在某个地方，现在在另一个地方。因此，我们得到了现代科学的重要基础理念，即我们所有的感觉都是对空间中事物在不同时间内变化构型的比较的结果。因此，运动的法则，也就是事物构型变化的法则，是物理科学的最终法则。
In the application of mathematics to the investigation of natural philosophy, science does systematically what ordinary thought does casually. When we talk of a chair , we usually mean something which we have been seeing or feeling in some way; though most of our language will presuppose that there is something which exists independently of our sight of feeling. Now in mathematical physics the opposite course is taken , The chair is conceived without any reference to anyone in particular, or to any special modes of perception. The result is that the chair becomes in thought a set of molecules in space, or a group of electrons, a portion of the ether in motion, or however the current scientific ideas describe it. But the point is that science reduces the chair to things moving in space and influencing each other's motions. Then the various elements or factors which enter into a set of circumstances, as thus conceived , are merely the things, like lengths of lines, sizes of angles , areas, and volumes, by which the positions of bodies in space can be settled . Of course, in addition to these geometrical elements the fact of motion and change necessitates the introduction of the rates of changes of such elements , that is to say , velocities, angular velocities, accelerations , and suchlike things. Accordingly, mathematical physics deals with correlations between variable numbers which are supposed to represent the correlations which exist in nature between the measures of these geometrical elements and of their rates of change. But always the mathematical laws deal with variables, and it is only in the occasional testing of the laws by reference to experiments, or in the use of the laws for special predictions that definite numbers are substituted.
在数学应用于自然哲学的研究中，科学系统地做了普通思维随意进行的事情。当我们谈论一把椅子时，通常指的是我们以某种方式看到或感觉到的事物，尽管我们大多数的语言会假设这个事物独立于我们的视觉或感受而存在。现在，在数学物理中，采取了相反的做法。椅子被设想为不存在与任何特定的人或任何特定的感知方式有关。结果是，椅子在思维中变成了空间中的一组分子，或一群电子，或是运动中的一部分以太，或者以当前的科学观念描述它的任何方式（"however" 在这里用作副词，表示“以任何方式”。）。但重点是，科学将椅子简化为在空间中运动并相互影响的事物。然后，进入某一组被如此设想（前面的抽象）的情况中的各种元素或因素，都是那些事物，比如线段的长度、角度的大小、面积和体积，这些元素通过它们可以确定物体在空间中的位置。当然，除了这些几何元素之外，运动和变化的事实也要求引入这些元素变化率的概念，即速度、角速度、加速度等类似的东西。因此，数学物理处理的是可变数字之间的关联，这些数字被认为代表了自然中这些几何元素及其变化率之间存在的关联。但数学定律总是处理变量，只有在通过实验验证定律或用定律进行特定预测时，才会用具体的数字替代这些变量。
The interesting point about the world as thus conceived in this abstract way throughout the study of mathematical physics, where only the positions and shapes of things are considered together with their changes , is that the events of such an abstract world are sufficient to 'explain' our sensations. When we hear a sound , the molecules of the air have been agitated in a certain way: given the agitation, or air-waves as they are called, all normal people hear sound; and of there are no air-waves, there is no sound. And, similarly, a physical cause or origin, or parallel event (according as different people might like to phrase it) underlies our other sensations. Our very thoughts appear to correspond to the conformations and motions of the brain; injure the brain and you injure the thoughts. Meanwhile the events of this physical universe succeed each other according to the mathematical laws which ignore all special sensations and thoughts and emotions.
这个世界从这样抽象的角度出发进行思考——在数学物理学研究中，只考虑事物的位置、形状及其变化——有趣的一点是，这样一个抽象世界的事件足以“解释”我们的感知。当我们听到声音时，空气中的分子以某种方式被激发：给定这种激发，或称为空气波，所有正常人都会听到声音；如果没有空气波，就没有声音。同样地，其他感知背后也有物理原因、起源或类似事情（根据不同的人喜欢如何表达）。我们的思想似乎与大脑的构象和运动相对应；伤害大脑就会伤害思想。同时，这个物理宇宙中的事件依照数学定律依次发生，而这些定律忽略了所有特殊的感知、思想和情感。
Now, undoubtedly , this is the general aspect of the relation of the world of mathematical physics to our emotions,sensations, and thoughts; and a great deal of controversy has been occasioned by it and much ink spilled. The whole, situation has arisen, as we have seen, from the endeavour to describe an external world 'explanatory' of our various individual sensations and emotions, but a world, also, not essentially dependent upon any particular sensations or upon any particular individual. Is such a world merely but one huge fairy tale? But fairy tales are fantastic and arbitrary: if in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations. Now, to a large degree, this scientific world does submit itself to this test and allows its events to be explored and predicted by the apparatus of abstract mathematical ideas. It certainly seems that here we have an inductive verification of our initial assumption. It must be admitted that no inductive proof is conclusive; but if the whole idea of a world which has existence independently of our particular perceptions of it be erroneous, it requires careful explanation why the attempt to characterize it , in terms of that mathematical remnant of our ideas which would apply to it, should issue in such a remarkable success.
现在，无疑地，这是数学物理世界与我们的情感、感官和思想之间关系的总体概貌；由此引发了大量争议，许多笔墨因此耗费。正如我们所见，整个局面源于试图描述一个外部世界——它能够“解释”我们各自不同的感官体验和情感，但同时也是一个不依赖于任何特定感官或特定个体的世界。这样一个世界仅仅是一个巨大的童话故事吗？但童话故事是幻想的、随意的：如果这个世界确实存在，那么它应当允许被精确描述，准确确定其各个部分及其相互关系。现在，科学世界在很大程度上确实服从这一测试，并允许通过抽象数学思想的工具对其事件进行探索和预测。看来，我们在此确实获得了对我们最初假设的归纳验证。必须承认，任何归纳证明都不是决定性的；但是，如果这个独立于我们特定感知的世界的整体想法是错误的，那么就需要对为何我们试图通过适用于它的数学思想残余来刻画它的尝试能够取得如此显著的成功，给出细致的解释。
It would take us too far afield to enter into a detailed explanation of the other laws of motion. The remainder of this chapter must be devoted to the explanation of remarkable ideas which are fundamental, both to mathematical physics and to pure mathematics: there are the ideas of vector quantities and the parallelogram law for vector addition. We have seen that the essence of motion is that a body was at A and is now at C. This transference from A to C requires two distinct elements to be settled before it is completely determined, namely its magnitude (i.e. the length AC) and its direction. Now anything , like this transference, which is completely given by the determination of a magnitude and a direction is called a vector. For example, a velocity requires for its definition the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction. The existence and the independence of these two elements in the determination of a velocity are well illustrated by the action of the captain of a ship, who communicates with different subordinates respecting them: he tells the chief engineer the number of knots at which he is to steam, and the helmsman the compass bearing of the course which he is to keep. Again the rate of change of velocity , that is velocity added per unit time, is also a vector quantity: it is called the acceleration. Similarly a force in the dynamical sense is another vector quantity. Indeed, the vector nature of forces follows at once according to dynamical principles from that of velocities and accelerations; but this is a point which we need not go into. It is sufficient here to say that a force acts on a body with a certain magnitude in a certain direction.
进入其他运动定律的详细解释将使我们偏离主题。本文的其余部分必须专注于一些基础性的重要思想，这些思想对数学物理和纯数学都至关重要：即向量量和向量加法的平行四边形法则。我们已经看到，运动的本质在于一个物体从A点移动到C点。这种从A到C的转移需要确定两个不同的要素，才能完全确定，即其大小（即AC的长度）和方向。现在，任何像这种转移的事物，如果完全由大小和方向的确定给出，就被称为向量。例如，速度的定义需要指定一个大小和一个方向。它必须以某个特定的英里每小时的速度朝着某个方向行驶。这两个要素在速度确定中的存在和独立性通过船长的行动得到了很好的说明：他与不同的下属进行沟通，他告诉首席工程师应以多少节的速度行驶，并告诉舵手应保持的航向角度。同样，速度的变化率，即单位时间内的速度增加，也是一个向量量：称为加速度。类似地，动力学意义上的力也是另一种向量量。事实上，力的矢量性质根据动力学原理立即从速度和加速度的矢量性质中推导出来；但这一点我们不需要深入探讨。这里足够说的是，力以一定的大小和方向作用于物体。
Now all vectors can be graphically represented by straight lines. All that has to be done is to arrange: (1) a scale according to which units of length correspond to units of magnitude of the vector — for example , one inch to a velocity of 10 miles per hour in the case of velocities, and one inch to a force of 10 tons weight in the case of forces — and (2) a direction of the line on the diagram corresponding to the direction of the vector. Then a line drawn with the proper number of inches of length in the proper direction represents the required vector on the arbitrarily assigned scale of magnitude. This diagrammatic representation of vectors is of the first importance. By its aid we can enunciate the famous ' parallelogram law' for the addition of vectors of the same kind but in different direction.
现在所有向量都可以用直线在图形上表示。需要做的就是安排： (1) 一个比例尺，其中长度单位对应于向量的大小单位——例如，在速度的情况下，一英寸对应于每小时10英里的速度，而在力的情况下，一英寸对应于10吨重的力；(2) 图中线条的方向对应于向量的方向。然后，用适当数量的英寸长度和适当方向绘制的线条表示在任意指定的大小比例尺上所需的向量。这种向量的图示表示非常重要。借助它，我们可以表述著名的“平行四边形法则”，用于同类但方向不同的向量相加。
Consider the vector AC in Figure 6 as representative of the changed position of a body from A to C : we will call this the vector of transportation. It will be noted that, if the reduction of physical phenomena to mere changes in positions , as explained above, is correct , all other types of physical vectors are really reducible in some way or other to this single type. Now the final transportation from A to C is equally well effected by a transportation from A to B and a transportation from B to C , or , completing the parallelogram ABCD , by a transportation from A to D and a transportation from D to C . These transportations as thus successively applied are said to be added together. This is simply a definition of what we mean by the addition of transportations. Note further that , considering parallel lines as being lines drawn in the same direction, the transportations B to C and A to D may be conceived as the same transportation applied to bodies in the two initial positions B and A . With this conception we may talk of the transportation A to D as applied to a body in any position, for example at B. Thus we may say that the transportation A to C can be conceived as the sum of the two transportations A to B and A to D applied in any order. Here we have the parallelogram law for the addition of transportations : namely, if the transportations are A to B and A to D , complete the parallelogram ABCD, and then the sum of the two is the diagonal AC.
考虑图6中的向量AC，作为物体从A移动到C的变化位置的代表：我们称之为运输向量。如果上述将物理现象简化为位置变化的观点是正确的，那么所有其他类型的物理向量实际上都可以以某种方式简化为这种单一类型。现在，从A到C的最终运输也可以通过从A到B和从B到C的运输来实现，或者通过完成平行四边形ABCD，先从A到D再从D到C进行运输。这些运输方式在如此连续应用时被称为相加。这只是我们所说的运输相加的定义。此外，考虑平行线为朝同一方向绘制的线段，从B到C和从A到D的运输可以被视为相同的运输应用于两个初始位置B和A。基于这种理解，我们可以讨论将运输从A到D应用于任何位置的物体，例如在B处。因此，我们可以说，从A到C的运输可以被视为以任意顺序应用的两个运输A到B和A到D的和。在这里，我们得到了运输相加的平行四边形法则：即如果运输是从A到B和从A到D，则完成平行四边形ABCD，两个运输的和就是对角线AC。
All this at first sight may seem to be vert artificial. But it must be observed that nature itself presents us with the idea. For example, a steamer is moving in the direction AD (cf.Fig.6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at B; but during that minute his starting point A on the deck has moved to D, and his path on the deck has moved from AB to DC. So that , in fact , his transportation has been from A to C over the surface of the sea. It is , however, presented to us analysed into the sum of two transportations, namely, one from A to B relatively to the steamer, and one from A to D which is the transportation of the steamer.
乍一看，这一切可能显得非常人造。但是必须注意到，自然本身向我们呈现了这个概念。例如，一艘轮船正朝着AD方向移动（参见图6），而一个人则在甲板上走。如果轮船静止，他在一分钟内会到达B；但在这一分钟内，他在甲板上的起点A已经移动到D，而他在甲板上的路径从AB变为DC。因此，实际上，他的运输是从A到C，穿越海面。然而，这个过程被我们分析为两个运输的总和，即一个相对于轮船从A到B的运输，以及一个从A到D的轮船运输。
By talking into account the element of time , namely one minute, this diagram of the man's transportation AC represents his velocity. For if AC represented so many feet of transportation, it now represents a transportation of so many feet per minute, that is to say, it represents the velocity of the man. Then AB and AD represent two velocities , namely, his velocity relatively to the steamer, and the velocity of the steamer, whose 'sum' makes up his complete velocity. It is evident that diagrams and definitions concerning transportations are turned into diagrams and definitions concerning velocities by conceiving the diagrams as representing transportations per unit time. Again, diagrams and definitions concerning velocities are turned into diagrams and definitions concerning accelerations by conceiving the diagrams as representing velocities added per unit time.
通过考虑时间因素，即一分钟，这个男人的运输图AC表示了他的速度。因为如果AC代表如此多英尺的运输，现在它代表每分钟如此多英尺的运输，也就是说，它表示了这个人的速度。然后，AB和AD表示两个速度，即他相对于轮船的速度和轮船的速度，它们的“和”构成了他的总速度。显然，通过将图示视为单位时间内的运输，关于运输的图示和定义被转化为关于速度的图示和定义。同样，关于速度的图示和定义通过将图示视为单位时间内的速度的叠加而转化为关于加速度的图示和定义。
Thus by the addition of vector velocities and of vector accelerations, we mean the addition according to the parallelogram law.
因此，通过向量速度和向量加速度的相加，我们表示了向量加法根据平行四边形法则。
Also , according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law.
此外，根据运动定律，一个力在特定质量的物体中产生的向量加速度完全代表了该力。因此，当它们的共同作用按照平行四边形法则被计算时，力被认为是相加的。
Hence for the fundamental vectors of science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a 'resultant' vector according to the rule of the parallelogram law.
因此，对于科学的基本向量，即运输、速度和力，相同类型的任何两个向量的相加都是根据平行四边形法则产生一个“合成”向量。
By far the simplest type of parallelogram is a rectangle, and in pure mathematics it is the relation of the single vector AC to the two component vectors , AB and AD , at right angles (cf.Fig.7), which is continually recurring. Let x,y, and r units represent the lengths of AB, AD, and AC, and let m units of angle represent the magnitude of the angle BAC. Then the relations between x, y, r, and m, in all their many aspects are the continually recurring topic of pure mathematics; and the results are of the type required for application to the fundamental vectors of mathematical physics. This diagram is the chief bridge over which the results of pure mathematics pass in order to obtain application to the facts of nature.
迄今为止，最简单的平行四边形类型是矩形，在纯数学中，单一向量AC与两个成分向量AB和AD（相互垂直，参见图7）之间的关系是不断重复出现的。设x、y和r单位表示AB、AD和AC的长度，设m单位表示角BAC的大小。则x、y、r和m之间的关系在纯数学的各个方面都是不断重复出现的话题；而这些结果是适用于数学物理基本向量所需的类型。这个图示是纯数学结果通向自然事实的主要桥梁。