chapter 3 Methods of application

METHODS OF APPLICATION

THR WAY in which the idea of variables satisfying[ˈsætɪsfaɪɪŋ] a relation occurs in the applications of mathematics is worth thought, and by devoting[ dɪˈvoʊt] some time to it we shall clear up our thoughts on the whole subject.

变量满足关系的概念在数学的应用中出现的方式值得思考，通过花一些时间来研究，我们将能够澄清对整个主题的思路。 Let us start with the simplest of examples: Suppose[ səˈpoʊz] that <building costs 1s[先令]. per cubic[ˈkjuːbɪk] foot> and <that 20s. make $$1 [1英镑]. > Then in all the complex circumstances[ˈsɜːrkəmstænsɪz] [状况] which attend[əˈtend] the building of a new house, amid[əˈmɪd] all the various sensations and emotions of the owner, the architect[ˈɑːrkɪtekt], the builder, the workmen , and the onlookers[ ˈɑːnlʊkər] as the house has grown to completion[kəmˈpliːʃ(ə)n ], this fixed correlation[ˌkɔːrəˈleɪʃ(ə)n ] [关联] is by the law assumed[ əˈsuːmd] to hold[成立] between the cubic content and the cost to the owner, namely[ˈneɪmli ] [也就是,即] that if x be the number of cubic feet , and the cost, then . This correlation of x and y is assumed to be true for the building of any house by any owner. Also, the volume of the house and the cost are not supposed to have been perceived[pərˈsiːvd ] [感知] or apprehended[ˌæprɪˈhend] [理解] by any particular sensation or faculty[ˈfæk(ə)lti ] [才能], or by any particular man. They are stated[ˈsteɪtɪd ] [说明] in an abstract general way, with complete[kəmˈpliːt ] [完全的] indifference[ɪnˈdɪfrəns] [不感兴趣] to the owner's state of mind [心态] when he has to pay the bill[ bɪl].

让我们从最简单的例子开始：假设建筑成本为每立方英尺1先令，并且20先令等于1英镑。那么，在所有与建造新房子相关的复杂状况中，所有房主、建筑师、建造者、工人和旁观者的各种感受和情绪之中，当房子已接近完工时，根据假设的法则，这种固定的关联适用于房主的建筑体积和成本之间，也就是说，如果x是立方英尺数，而是成本，那么。这种x和y的关联被认为适用于任何房主建造的任何房屋。此外，房屋的体积和成本并不被认为是通过任何特定的感觉或能力，或由任何特定的人感知或理解的。它们以抽象而概括的方式说明，对房主支付账单时的心态完全不予考虑。

Now think a bit further as to what all this means. The building of a house is a complicated[ˈkɑːmplɪkeɪtɪd ] [复杂的] set of circumstances[ˈsɜːrkəmstænsɪz ]. It is impossible to begin to apply the law, or to test it, unless amid[ əˈmɪd ] [在……中] the general[一般的] course[ kɔːrs] [进程] of events it is possible to recognize[ˈrekəɡnaɪz] a definite set of occurrences[əˈkɜːrəns] [发生的事] as forming a particular instance of the building of a house.

现在进一步思考一下这一切意味着什么。盖房子是一系列复杂的情况。除非在事件的一般进程中能够识别出一组明确的事件，将其视为盖房子的特定实例，否则就不可能开始应用或测试规则。

In short, we must know a house when we see it , and must recognize the events which belong to its building. Then amidst[əˈmɪdst] [在……之中] these events, thus[] isolated[ˈaɪsəleɪtɪd] in idea from the rest of nature, the two elements of the cost and cubic comtent must be determinable;l and when they are both determined, if the law be true, they satisfy the general formula

简而言之，我们必须在看到房屋时认识它，并且必须认识与其建造相关的事件。然后在这些事件之中，因此在思想上与自然的其他部分隔离开来，成本和立方体内容这两个要素必须是可确定的；当它们都被确定后，如果法则成立，它们将满足通用公式。

But is the law true? Anyone who has had much to do with building will know that we have here put the cost rather[ˈræðər] high. It is only for an expensive type of house that it will work out[成立，得到解决] at this price. This brings out another point which must be made clear. While we are making mathematical calculations connected with the formula 20y=x, it is indifferent to us whether the law be true or false . In fact, the very meanings[确切的含义] assigned[əˈsaɪnd ] to x and y, as being a number of cubic feet and a number of pounds sterling, are indifferent. During the mathematical investigation[ ɪnˌvestɪˈɡeɪʃ(ə)n] [研究] we are , in fact, merely considering the properties of this correlation between a pair of variable numbers x and y. Our results will apply[əˈplaɪ ] equally[ˈiːkwəli] [同样地] well, if we interpret[ɪnˈtɜːrprət] [解释] y to mean a number of fishermen and x the number of fish caught , so that the assumed law is that on the average[ ˈævərɪdʒ] each fisherman catches twenty fish. The mathematical certainty[ˈsɜːrt(ə)nti ] of the investigation only attaches[əˈtætʃɪz ] to[连接到] the results considered as[看作] giving properties of the correlation 20y=x between the variable pair of numbers x and y. There is no mathematical certainty whatever[无论如何] [adv] about the cost of the actual[ˈæktʃuəl] building of any house. The law is not quite true and the result is gives will not be quite accurate[ ˈækjərət] [准确的]. In fact, it may well[adv] [相当地] be hopelessly[ˈhoʊpləsli ] [无可救药地] wrong.

但这条法则是否正确呢？任何与建筑有关的人都会知道，我们在这里将成本设定得相当高。只有对于昂贵类型的房屋，才可能以这个价格达成。这带出了另一个必须澄清的观点。在我们进行与公式 20y=x 相关的数学计算时，这个法则是真是假对我们而言都无关紧要。事实上，赋予 x 和 y 的确切含义——即立方英尺和英镑的数量——都是无关紧要的。在数学研究中，我们实际上只是在考虑这对变量 x 和 y 之间相关性质。如果我们解释 y 表示渔民的数量，x 表示捕获的鱼的数量，那么我们的结果同样适用良好，假设平均每位渔民捕捉二十条鱼。数学研究的确定性只与考虑到的结果相关联，这些结果被视为给出变量对 x 和 y 之间关系 20y=x 的性质。关于任何房屋实际建造成本，完全没有任何数学上的确定性。这条法则并非完全正确，我们得到的结果也不会完全准确。事实上，它很可能是彻底错误的。

Now all this no doubt seems very obvious[ˈɑːbviəs]. But in truth with more complicated instances there is no more common error than to assume that, because prolonged[ prəˈlɔːŋd] [长期的] and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely[ˌæbsəˈluːtli ] [完全地] certain [ ˈsɜːrt(ə)n] [必然的]. The conclusion[kənˈkluːʒ(ə)n] of no[全称否定] argument[ˈɑːrɡjumənt] can be more certain than the assumptions[əˈsʌmpʃən] from which it starts. All mathematical calculations about the course of nature[自然过程] must start from some assumed law of nature[自然法则], such, for instance, as the assumed law of the cost of building stated above. Accordingly[相应地], however accurately[ˈækjərətli ] [精确地] we have calculated that some event must occur, the doubt always remains[ rɪˈmeɪnz]—Is the law true? If the law states a precise[prɪˈsaɪs] [精确的] result, almost certainly[几乎肯定地] it is not precisely accurate; and thus even at the best[在最好的情况下] the result, precisely as calculated, is not likely to occur. But then we have no faculty[fæk(ə)lti] [n] capable[ˈkeɪpəb(ə)l ] [adj] [形容词短语作定语]of observation[ ˌɑːbzərˈveɪʃ(ə)n] [n] with ideal[aɪˈdiːəl] precision[prɪˈsɪʒ(ə)n], so, after all, our inaccurate laws may be good enough.

现在这一切无疑看起来非常显而易见。但实际上，在更复杂的情况下，最常见的错误莫过于去假设，因为进行了长时间和精确的数学计算，将结果应用于某个自然事实就是绝对确定的。任何论点的结论都不可能比它所基于的假设更为确定。所有关于自然过程的数学计算都必须从某些假定的自然法则出发，例如上述假定的建筑成本法则。因此，无论我们多么精确地计算某个事件必须发生，总会存在一个疑问——这个法则是否正确？如果这个法则给出了一个精确的结果，那么几乎可以肯定它并不完全准确；因此，即使在最好的情况下，精确计算的结果也不太可能发生。但是，我们没有具备理想精确度的观察能力，所以，总而言之，我们那些不够精确的法则可能已经足够好了。

We will now turn to an actual[ˈæktʃuəl] case, that[代词] of[所属] Newton and the Law of Gravity[ˈɡrævəti]. This law states that any two bodies attract[əˈtrækt] one another[表相互作用] with a force proportional[prəˈpɔːrʃən(ə)l ] [%adj1%] to the product of their masses[ˈmæsɪz], and inversely[ˌɪnˈvɜːrsli] proportional to the square[skwer] of the distance between them. Thus if m and M are the masses of the two bodies, reckoned[ˈrekən] in lbs [paʊndz] [pounds]. say[假设], and d miles is the distance between them, the force on either body, due to the attraction[əˈtrækʃ(ə)n] of the other and directed[ dəˈrektɪd] towards it, is proportional to ; thus this force can be written as equal to , where k is a definite[ˈdefɪnət] number depending on the absolute magnitude[ˈmæɡnɪtuːd] [数值] [n] of this attraction and also on the scale by which we choose to measure forces . It is easy to see that, if we wish to reckon[[ˈrekən]] in terms of [按照] forces such as the weight of mass of 1 lb., the number which k represents must be extremely[ɪkˈstriːmli ] small; for when m and M and d are each put [pp] equal to 1, becomes the gravitational[ ɡrævɪˈteɪʃənl] [adj] attraction of two equal masses of 1 lb. at the distance of one mile[maɪl] , and this is quite inappreciable[ ˌɪnəˈpriːʃəbəl] [微不足道的].

我们现在将转向一个实际案例，即牛顿和万有引力定律。这个定律指出，任何两个物体之间的相互吸引力与它们质量的乘积成正比，与它们之间距离的平方成反比。因此，如果 m 和 M 分别是两个物体的质量，以磅计算，并且 d 英里是它们之间的距离，则由于彼此之间的相互吸引力，作用在任一物体上的力与 成正比；因此，这个力可以写成 的形式，其中 k 是一个确定的数字，取决于这种引力的绝对大小，也取决于我们选择的力量衡量标准的尺度。很容易看出，如果我们希望按照诸如1磅质量的重量等力量来计算的话，常数 k 所代表的数字必须非常小；因为当 m、M 和 d 分别取 1 时， 就变成了在一英里的距离上两个相等的 1 磅质量之间的万有引力，而这是相当微不足道的。

However, we have now got our formula for the force of attraction. If we call this force F, it is , giving the correlation between the variables F,m,M,and d. We all know the story of how it was found out. Newton, it states[据说], was sitting in an orchards[ˈɔːrtʃərd] and watched the fall of an apple, and then the law of universal[ˌjuːnɪˈvɜːrs(ə)l] gravitation[ ˌɡrævɪˈteɪʃ(ə)n] burst[bɜːrst] [爆炸] upon his mind. It may be that the final formulation[ˌfɔːrmjuˈleɪʃ(ə)n] [构想] of the law occurred to him in an orchard, as well as elsewhere—and he must have been somewhere. But for our purposes it is more instructive[ ɪnˈstrʌktɪv] [有启发性的] to dwell [dwel] upon [仔细思考] the vast[væst] [巨大的] amount[ əˈmaʊnt] [数量] of preparatory[prɪˈpærətɔːri] [预备性的] thought, the product of many minds and many centuries[ˈsentʃəriz], which was necessary before this exact law could be formulated[ ˈfɔːrmjuleɪt] [用公式表示]. In the first place, the mathematical habit[ˈhæbɪt] [习惯] of mind and the mathematical procedure[ prəˈsiːdʒər ] [步骤] explained in the previous two chapters had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance. Again, what are the meanings of the terms[tɜːrmz] [术语] employed[ ɪmˈplɔɪd], Force, Mass,Distance? Take the easiest of these terms, Distance. It seems very obvious to us to conceive[kənˈsiːv] [设想] all material things as forming a definite geometrical whole, such that the distances of the various parts are measurable in terms of some unit length, such as a mile or a yard. This is almost the first aspect[ ˈæspekt ] [方面] of a material structure[构造] which occurs[ əˈkɜːrz] [呈现] to us. It is the gradual[ˈɡrædʒuəl ] [平缓的] outcome[结果] of the study of geometry and of the theory[ˈθiːəri ] of measurement[ˈmeʒərmənt]. Even now, in certain[某个] cases, other modes of thought are convenient[kənˈviːniənt]. In a mountainous[ˈmaʊntənəs] country distances are often reckoned[ˈrekənd] in hours. But leaving distance, the other terms, Force and Mass, are much more obscure[ əbˈskjʊr] [模糊的]. The exact comprehension[ˌkɑːmprɪˈhenʃ(ə)n] of the ideas which Newton meant[ment] [打算] to convey[ kənˈveɪ ] [传送] by these words was of [%bo1% ] slow growth, and , indeed, Newton himself was the first man who had thoroughly[ˈθɜːrəli ] mastered[ˈmæstərd] the true general[ˈdʒen(ə)rəl] [一般] principles[原理] of Dynamics.

然而，我们现在已经得到了引力公式。如果我们将这个力量称为 F，那么它就是 ，这就给出了变量 F、m、M 和 d 之间的关系。我们都知道这是如何被发现的故事。据说，牛顿坐在果园里看着一颗苹果掉落，然后宇宙万有引力定律突然出现在他脑海中。也许这个定律的最终形式确实是在果园里想出来的，也可能是在别的地方——而他肯定是在某个地方。但就我们的目的而言，更值得关注的是需要在确立这个确切定律之前，必须进行的大量准备性思考，这是许多思想和许多世纪的产物。首先，必须培养数学思维习惯和在前两章中解释的数学程序；否则，牛顿永远无法想出一个代表任意两个质量之间的力的公式。再次，所使用的术语 Force、Mass、Distance 的含义是什么？从这些术语中，最简单的是 Distance。对我们来说，设想所有物质都构成一个确定的几何整体，因此各部分之间的距离可以用某种长度单位（如英里或码）来测量，这似乎是非常明显的。这几乎是我们所见过的物质结构的第一个方面。这是几何学研究和测量理论的渐进结果。即使在某些情况下，其他思维方式也是方便的。在多山的国家，距离经常以小时计算。但离开距离，其他术语，如力和质量，就显得更加模糊。牛顿打算通过这些话传达的概念的确切理解是缓慢发展的，事实上，牛顿本人是第一个彻底掌握动力学真正一般原理的人。

Throughout[θruːˈaʊt] the middle ages, under the influence of Aristotle, the science was entirely[ ɪnˈtaɪərli ] misconceived[ˌmɪskənˈsiːv]. Newton had the advantage[ ədˈvæntɪdʒ] of coming after a series of great men, notably[ˈnoʊtəbli] [adv] [尤其] Galileo, in Italy, who in the previous two centuries had reconstructed[ˌriːkənˈstrʌktɪd] [重建] the science and had invented[ ɪnˈventɪd ] [创造] the right way of thinking about it. He completed their work. Then, finally, having the ideas of force, mass, and distance, clear and distinct[dɪˈstɪŋkt] [清楚的] in his mind, and realizing their importance and their relevance [ˈreləvəns] [相关性] to the fall of an apple and the motions[ˈmoʊʃ(ə)n] [运动] of the planets[ˈplænɪt], he hit upon[想到] the law of gravitation and proved[pruːvd] [证明] it to be the formula always[adv] satisfied[ˈsætɪsfaɪd] in these various[ˈveriəs] motions.

在中世纪期间，受亚里士多德的影响，科学完全被误解了。牛顿有一个优势，他是在一系列伟大的人物之后出现的，尤其是在意大利的伽利略，他们在前两个世纪中重建了科学，并创造了正确的思考方式。他完成了他们的工作。最后，当他心中对力量、质量和距离的想法清晰而明确时，意识到它们对于苹果落下和行星运动的重要性和相关性，他想到了万有引力定律，并证明了它在这些不同运动中始终成立的公式。

The vital[ˈvaɪt(ə)l ] [至关重要的] point in the application of mathematical formulae is to have clear ideas and a correct estimate[ˈestɪmeɪt] [估计] of their relevance [ ˈreləvəns] to the phenomena [fəˈnɑːmɪnə] [现象] under observation[ ˌɑːbzərˈveɪʃ(ə)n]. No less than ourselves, our remote ancestors[ˈænsestərz] [祖先] were impressed[ɪmˈprest] [留下深刻印象] with the importance of natural[adj] phenomena and with the desirability[dɪˌzaɪərəˈbɪləti] [渴望，必要性] [n] of taking energetic[ˌenərˈdʒetɪk] [积极的] measures [措施] to regulate[ˈreɡjuleɪt] [控制] the sequence[ ˈsiːkwəns] of events. Under the influence of irrelevant[ɪˈreləvənt] [adj] ideas they executed elaborate[ɪˈlæbərət] [复杂的] religious[ rɪˈlɪdʒəs] [宗教的] ceremonies[ˈserəmənɪz] [仪式] to aid[促进] the birth of the new moon, and performed[执行] sacrifices[ˈsækrɪfaɪsɪz ] [供奉] to save the sun during the crisis[ ˈkraɪsɪs] [关键时刻] of an eclipse. There is no eason to believe that they were more stupid than we are. But at that epoch[ ˈepək] [时代] there had not been opportunity for the slow accumulation[ əˌkjuːmjəˈleɪʃn] [积累] [n] of clear and relevant ideas.

数学公式应用中的关键是对观察对象现象的相关性有清晰的想法和正确的估计。我们的遥远祖先和我们自己一样，都深刻认识到自然现象的重要性，以及采取积极措施调节事件顺序的必要性。在无关想法的影响下，他们举行了复杂的宗教仪式以促进新月的诞生，并进行供奉以在日食的关键时刻拯救太阳。没有理由相信他们比我们更愚蠢。但是在那个时代，还没有机会慢慢积累清晰且相关的想法。

The sort of [这种] way in which physical[ˈfɪzɪk(ə)l ]sciences grow into a form[形式] capable[ˈkeɪpəb(ə)l ] [%adj1%] of treatment[ˈtriːtmənt] by mathematical methods is illustrated[ ˈɪləstreɪtɪd ] [被阐明] by the history of the gradual[ ˈɡrædʒuəl] [逐渐的] growth of the science of electromagnetism[ ɪˌlektroʊˈmæɡnətɪzəm] [电磁学]. Thunderstorms[ˈθəndərˌstɔːrmz] [雷暴] are events on a grand[ɡrænd] [宏大的] scale [ skeɪl] [在宏大的规模上] , arousing[əˈraʊzɪŋ] [激发] terror[ˈterər] [恐惧] in men and even animals. From the earliest times they [电] must have been objects of wild and fantastic[fænˈtæstɪk ] hypotheses[ haɪˈpɑːθəsiːz ], though it may be doubted[ˈdaʊtɪd] whether our modern scientific discoveries in connexion[kəˈnekʃən ] with electricity[ɪˌlekˈtrɪsəti ] are not more astonishing[ əˈstɑːnɪʃɪŋ] [惊人的] than any of the magical [ ˈmædʒɪk(ə)l] explanations of savages[ ˈsævɪdʒɪz]. The Greeks knew that amber[琥珀] (Greek, electron[ɪˈlektrɑːn]) when rubbed[rʌbd] [擦] would attract light and dry bodies. In A.D.1600, Dr.Gilbert, of Colchester [科尔切斯特], published the first work on the subject on which any scientific method is followed. He made a list of substances possessing[pəˈzesɪŋ] [拥有] properties similar to those of amber; he must also have the credit[ˈkredɪt] [赞扬] connecting, however vaguely[ˈveɪɡli] [模糊地] , electric and magnetic phenomena. At the end of the seventeenth and throughout the eighteenth century knowledge advanced. Electrical [ ɪˈlektrɪkl] machines were made, sparks [spɑːrks] were obtained [əbˈteɪnd] from them; and the Leyden [ˈlaɪdən] Jar [dʒɑːr] [莱顿瓶] was invented, by which these effects [ɪˈfekts] could be intensified [ ɪnˈtensɪfaɪd] [增强]. Some organized[系统的] [adj] knowledge was being obtained; but still no relevant mathematical ideas had been found out. Franklin, in the year 1752, sent a kite into the clouds and proved[pruːvd] that thunderstorms were electrical.

物理科学如何逐步发展成为可以用数学方法处理的形式，这一过程可以通过电磁学科学的历史来阐明。雷暴是在宏大的规模上发生的事件，引发人类甚至动物的恐惧。自古以来，电一定是野心勃勃和奇幻假设的对象，尽管我们现代关于电的科学发现是否比野蛮人的任何神奇解释更为惊人，这可能是值得怀疑的。希腊人知道琥珀（希腊语，electron）在摩擦后会吸引轻的和干燥的物体。公元1600年，科尔切斯特的吉尔伯特博士出版了关于电学科学第一部采用科学方法的著作。他列出了一些具有类似琥珀性质的物质；他还应当因为即使模糊地联系电和磁现象而受到赞扬。在十七世纪末和整个十八世纪期间，知识得到了进步。电机被制造，从中获得了火花；莱顿瓶也被发明，这些东西增强了关于电的影响。一些系统的知识正在被获取；但是仍然没有发现相关的数学理念。1752年，富兰克林将一只风筝送入云层，并证明雷暴是电气现象。

Meanwhile[ˈmiːnwaɪl] [与此同时] from the earliest epoch[ˈepək ] [时代] (2634 B.C.) the Chinese had utilized [ ˈjuːtəlaɪzd] [利用] the characteristic[ˌker.ək.təˈrɪs.tɪk] [adj] [独特的] property of the compass [ˈkʌmpəs] needle [ˈniːd(ə)l], but do not seem to have connected it with any theoretical [ˌθiːəˈretɪk(ə)l ] ideas. The really profound[prəˈfaʊnd] [深刻的] changes in human life all have [都有] their ultimate[ˈʌltɪmət] [根本的] origin [起源] in knowledge pursued[pərˈsuːd] for its own sake[seɪk] [目的]. The use of the compass was not introduced into Europe till the end of the twelfth[twelfθ] century A.D., more than 3,000 years after its first use in China. The importance which the science of electromagnetism[ɪˌlektroʊˈmæɡnətɪzəm] has since[从此] assumed[呈现] in every department of human life is not due to the superior[suːˈpɪriər] [更高的] practical[ˈpræktɪk(ə)l ] [实际的] bias[ˈbaɪəs] [偏好] of Europeans, but to the fact that in the West electrical and magnetic[mæɡˈnetɪk] phenomena were studied by men who were dominated[ˈdɑːmɪneɪtɪd] by abstract theoretic interests.

与此同时，自公元前2634年的最早时代起，中国人就利用了指南针的特性，但似乎没有将其与任何理论联系起来。人类生活中真正深刻的变化都根源于纯粹为了知识本身而追求的知识。指南针的使用直到公元十二世纪末才引入欧洲，距离它在中国首次使用已经过去了三千多年。电磁学科自那时在人类生活的各个领域中的重要性的提升，并非源自欧洲人对实际的偏好，而是因为在西方，电和磁现象是被抽象理论兴趣所驱使的人们所研究的事实。

The discovery[dɪˈskʌvəri ] of the electric current [电流] is due to two Italians, Galvani[伽尔伐尼] in 1780, and Volta[伏打] in 1792. This great invention opened a new series of phenomena for investigation [以供研究]. The scientific world had now three separate[ˈseprət], though allied[ˈælaɪd ] [有关联的], groups of occurrences[ əˈkɜːrəns] [现象] on hand— the effects of 'statical[ˈstætɪkəl ] [静态的]' electricity [ ɪˌlekˈtrɪsəti ] [n] arising [əˈraɪzɪŋ] from [由..引起] frictional[ˈfrɪkʃənəl] [摩擦的] electrical machines , the magnetic phenomena, and the effects due to electric currents. From the end of the eighteenth century onwards[向前看], these three lines of investigation were quickly interconnected and the modern science of electromagnetism was constructed which now threatens to transform human life.

电流的发现归功于两位意大利人，分别是1780年的伽尔伐尼和1792年的伏打。这一伟大的发明开启了一系列新的现象供研究。科学界现在有了三个独立但相关的现象领域——静电机摩擦产生的静电效应、磁现象，以及由电流产生的效应。从十八世纪末开始，这三条研究线路迅速相互联系，构建起现代电磁学科，如今这一学科正威胁着改变人类生活。

The experimental investigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped full-grown and full armed, from the brain of the 'Newton of Electricity'. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

安培通过实验研究确立了电流间机械作用的规律，这是科学上最辉煌的成就之一。整个理论和实验，仿佛是从“电学的牛顿”的大脑中完全成熟和全副武装地跳出来的。它在形式上是完美的，在准确性上是无懈可击的，并且它总结在一个公式中，所有现象都可以从这个公式中推导出来，而且这个公式将永远是电动力学的基本公式。

The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831-2. Faraday was asked: 'What is the use of this discovery?' He answered:'What is the use of a child—it grows to be a men.' Faraday's child has grown to be a man and is now the basis of all the modern applications of electricity. Faraday also reorganized the whole theoretical conception of the science. His ideas, which had not been fully understood by the scientific world, were extended and put into a directly mathematical form by Clerk Maxwell in 1873. As a result of his mathematical investigations, Maxwell recognized that, under certain conditions, electrical vibrations ought to be propagated. He at once suggested that the vibrations which form light are electrical. This suggestions has since been verified, so that now the whole theory of light is nothing but a branch of the great science of electricity. Also Herz, a German, in 1888, following on Maxwell's ideas, succeeded in producing electric vibrations by direct electrical methods. His experiments are the basis of our wireless telegraphy.

1831-1832年，迈克尔·法拉第发现了电流之间以及电流与磁体之间的感应定律。当被问及“这个发现有什么用？”时，法拉第回答道：“一个孩子有什么用——他会长大成人。”法拉第的孩子已经长大成人，并成为现代所有电力应用的基础。法拉第还重新组织了整个科学的理论概念。他的思想当时并未被科学界完全理解，直到1873年克拉克·麦克斯韦将这些思想扩展并直接用数学形式表达出来。通过数学研究，麦克斯韦认识到，在特定条件下，电振动应该能够传播。他立即提出，形成光的振动是电振动。这个建议后来得到了验证，所以现在整个光的理论实际上只是伟大电学科学的一个分支。1888年，德国人赫兹在麦克斯韦的思想基础上，通过直接的电学方法成功地产生了电振动。他的实验成为我们无线电报的基础。

In more recent years even more fundamental discoveries have been made, and the science continues to grow in theoretic importance and in practical interest. This rapid sketch of its progress illustrates how, by the gradual introduction of the relevant theoretic ideas, suggested by experiment and themselves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple assumed laws, supply the explanation to the complex tangle of the course of events.

近年来，出现了更加基础的发现，这门科学在理论重要性和实际应用方面继续增长。这一进展的简要概述说明了通过逐渐引入相关的理论概念，这些概念由实验提出，并且本身也促使新的实验产生，如何将大量孤立甚至琐碎的现象融合成一个连贯的科学体系。在这个体系中，从几个简单的假设定律出发，通过抽象的数学推导得出的结果，提供了对复杂事件过程的解释。

Finally, passing beyond the particular sciences of electromagnetism and light, we can generalize our point of view still further, and direct our attention to the growth of mathematical physics considered as one great chapter of scientific thought . In the first place, what in the barest outlines is the story of its growth?

最终，超越电磁学和光学等特定科学领域，我们可以进一步概括我们的观点，并将注意力集中在被视为科学思想重要篇章之一的数学物理学的发展上。首先，它的发展轮廓中最基本的故事是什么？

It did not begin as one science, or as the product of one band of men. The Chaldean shepherds watched the skies, the agents of Government in Mesopotamia and Egypt measured the land, priests and philosophers brooded on the general nature of all things. The vast mass of the operations of nature of all things. The vast mass of the operations of nature appeared due to mysterious unfathomable forces. "The wind bloweth where it listeth" expresses accurately the blank ignorance then existing of any stable rules followed in detail by the succession of phenomena. In broad outline, then as now, a regularity of events was patent. But no minute tracing of their interconnexion was possible, and there was no knowledge how even to set about to construct such a science.

它并非始于一门科学，或是某一群人的产物。迦勒底的牧羊人观察星空，美索不达米亚和埃及的政府代理测量土地，祭司和哲学家沉思万物的普遍本质。自然界一切事物的广泛运作似乎源于神秘莫测的力量。“风随意吹”准确地表达了当时对现象顺序中任何稳定规则的空白无知。大体上说，那时如今一样，事件的规律性是显而易见的。但是，无法进行它们相互联系的详细追踪，也不知如何着手构建这样一门科学。

Detached speculations, a few happy of unhappy or shots at the nature of things, formed the utmost which could be produced.

一些孤立的猜想和少数成功或不成功的尝试，是当时能够产生的最大成果。

Meanwhile land-surveys had produced geometry , and the observations of the heavens disclosed the exact regularity of the solar system. Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of hydrostatics and optics. Indeed, Archimedes, who combined a genius for mathematics with a physical insight must rank with Newton, who lived nearly two thousand year later, as one of the founders of mathematical physics. He lived at Syracuse, the great Greek city of Sicily. When the Romans besieged the town (in 212 to 210 B.C.), he is said to have burned their ships by concentrating on them, by means of mirrors, the sun's rays . The story is highly improbable, but is good evidence of reputation which he had gained among his contemporaries for his knowledge of optics. At the end of this siege he was killed. According to one account given by Plutarch, in his life of Marcellus[马尔库斯·克劳狄乌斯·马塞卢斯（罗马政治家）], he was found by a Roman soldier absorbed in the study of a geometrical diagram which he had traced in the sandy floor of his room .He did not immediately obey the orders of this captor, and so was killed. For the credit of the Roman generals it must be said that the soldiers had orders to spare him. The internal evidence for the other famous story of him is very strong; for the discovery attributed to him is one eminently worthy of his genius for mathematical and physical research. Luckily, it is simple enough to be explained here in detail. It is one of the best easy examples of the method of application of mathematical ideas to physics.

与此同时，土地测量产生了几何学，天文观测揭示了太阳系的精确规律。一些后来的希腊人，如阿基米德，对流体静力学和光学的基本现象有了深刻的见解。事实上，阿基米德将数学天赋与物理洞察力结合在一起，几乎可以与近两千年后的牛顿并列为数学物理学的奠基者之一。他生活在西西里岛的伟大希腊城市锡拉库萨。罗马人围困该城（公元前212至210年）时，据说他通过使用镜子将太阳光线聚焦在罗马舰队上，从而烧毁了他们的船只。这个故事可能性极小，但证明了他在同代人中因其光学知识而享有的声誉。围城结束时，他被杀害。根据普鲁塔克在《马塞卢斯传》中的一则记载，一名罗马士兵发现他沉浸在他在房间沙地上勾画的几何图形的研究中。他没有立即听从士兵的命令，因此被杀害。为了罗马将军的荣誉，必须指出士兵们其实是被命令保护他的。关于他的另一个著名故事的内部证据非常强大；他被归功于的发现无疑展示了他在数学和物理研究方面的天才。幸运的是，这个发现足够简单，可以在这里详细解释。它是将数学思想应用于物理学的最佳易懂示例之一。

Hiero, King of Syracuse, had sent a quantity of gold to some goldsmith to form the material of a crown. He suspected that the craftsman had abstracted some of the gold and had supplied its place by alloying the remainder with some baser metal. Hiero sent the crown to Archimedes and asked him to test it. In these days an indefinite number of chemical tests would be available(表达对过去的假设，我们需要使用虚拟语气，would be). But then Archimedes had to think out the matter afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it, I have found it). This day, if we knew which it was , ought to be celebrated as the birthday of mathematical physics;

锡拉库萨的国王希罗曾经送了一批黄金给一位金匠，用来制作一顶王冠。他怀疑工匠偷取了一部分黄金，并用一些劣质金属掺杂剩余的黄金。希罗将王冠送给阿基米德，请他进行测试。在当时，无数的化学测试方法还没有出现（表达对过去的假设时，我们需要使用虚拟语气 would be）。但当时阿基米德必须自己重新思考这个问题。当他躺在浴缸里时，灵感突然闪现。他跳了起来，奔跑着穿过街道，直奔宫殿，高喊“Eureka! Eureka!”（我找到了，我找到了）。如果我们知道是哪一天，那一天应该被庆祝为数理物理学的诞生日。

the science came of age when Newton sat in his orchard. Archimedes had in truth made a great discovery. He saw that a boy when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. This law can be proved theoretically from the mathematical principles of hydrostatics and can also be verified experimentally. Hence, if W lb be the weight of the crown, as weighed in air, and w lb be the weight of the water which it displaces when completely immersed, W-w would be the extra upward force necessary to sustain the crown as hung in water.

当牛顿坐在他的果园里时，科学达到了成熟的阶段。实际上，阿基米德取得了一项伟大的发现。他发现，当一个物体浸入水中时，周围的水会对其产生一个向上的力，这个力的大小等于物体排开的水的重量。这个定律可以从流体静力学的数学原理上得到理论证明，也可以通过实验加以验证。因此，如果在空气中称得的王冠重量为W磅，而当王冠完全浸入水中时，它所排开的水的重量为w磅，那么W-w就是在水中悬挂王冠所需的额外向上的支撑力。

Now, this upward force can easily be ascertained by weighing the body as it hands in water,as shown in the annexed figure. If the weights in the right-hand scale come to F lb., then the apparent weight of the crown in water is F lb.; and we thus have and thus and(A) where W and F are determined by the easy, and fairly precise, operation of weighing. Hence , by equation(A)， is known. But is the ratio of the weight of the crown to the weight of an equal volume of water. This ratio is the same for any lump of metal of the same material: it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity. Thus to test if the crown were of gold , Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. If the two specific gravities agreed, the crown was pure; if they disagreed , it was debased.

其中 W 和 F 是通过简单且相当精确的称重操作确定的。因此，通过方程 (A)， 是已知的。但 是皇冠重量与等体积水重量的比值。对于相同材料的任何金属块，这个比值是相同的：现在称为材料的比重，并且仅取决于物质的内在性质，而不取决于其形状或数量。因此，为了测试皇冠是否为金，阿基米德只需取一块无可争议的纯金，并通过相同的过程找到其比重。如果两个比重一致，皇冠就是纯的；如果不一致，它就是掺假的。

This argument has been given at length, because not only is it the first precise example of the application of mathematical ideas to physics, but also because it is a perfect and simple example of what must be the method and spirit of the science for all time.

这个论点被详细阐述了，因为它不仅是数学思想应用于物理学的第一个精确例子，而且它也是科学永恒的方法和精神的一个完美而简单的例子。

The death of Archimedes by the hands of Roman soldier is symbolical of a world-change of the first magnitude: the theoretical Greeks, with their love of abstract science, were superseded in the leadership of the European world by the practical Romans. The Romans were a great race, but they were cursed with the sterility which waits upon practicality. They did not improve upon the knowledge of their forefathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.

阿基米德被罗马士兵杀害象征着一次重大的世界变革：热爱抽象科学的理论派希腊人被实用派的罗马人取代，成为欧洲世界的领导者。罗马人是一个伟大的种族，但他们因过于实用而陷入了不育的困境。他们没有改进祖先的知识，所有的进步都局限于工程学的次要技术细节。他们不够富有梦想，无法提出新的观点，从而更根本地控制自然力量。没有一个罗马人因为沉迷于数学图形的思考而丧命。