On the Economy of Machinery and Manufactures, Chapter XX: On the Division of Mental Labour

(241.) We have already mentioned what may, perhaps, appear paradoxical to some of our readers,—that the division of labour can be applied with equal success to mental as to mechanical operations, and that it ensures in both the same economy of time. A short account of its practical application, in the most extensive series of calculations ever executed, will offer an interesting illustration of this fact, whilst at the same time it will afford an occasion for shewing that the arrangements which ought to regulate the interior economy of a manufactory, are founded on principles of deeper root than may have been supposed, and are capable of being usefully employed in preparing the road to some of the sublimest investigations of the human mind.

（241）我们前面已经提到一件在某些读者看来也许有些自相矛盾的事：劳动分工既可以用于机械操作，也同样可以成功地用于脑力操作，而且在两者中都能同样节省时间。简要介绍一下它在有史以来规模最大的一系列计算中的实际应用，就能有趣地说明这个事实；同时也借此表明，应当用来规范一家工厂内部经济的那些安排，所依据的原则比人们可能设想的要深刻，并且可以有效地用来为人类心灵某些最崇高的研究铺平道路。

(243.) The origin of the idea is related in the following extract:—

（243）这个想法的由来，在下面这段摘录中有所叙述：

"C'est à un chapitre d'un ouvrage Anglais, justement célèbre, qu'est probablement due l'existence de l'ouvrage dont le gouvernement Britannique veut faire jouir le monde savant:—

“那部英国政府打算让学术界享用的著作之所以存在，或许要归功于一部名副其实地享有盛誉的英国著作中的一章：

(244.) The ancient methods of computing tables were altogether inapplicable to such a proceeding. M. Prony, therefore, wishing to avail himself of all the talent of his country in devising new methods, formed the first section of those who were to take part in this enterprise out of five or six of the most eminent mathematicians in France.

（244）古代编制用表的方法完全不适用于这样的做法。因此，普罗尼先生希望利用全国的人才来设计新的方法，他从法国最杰出的五六位数学家中，组成了参与这项事业的第一组。

First Section.—The duty of this first section was to investigate, amongst the various analytical expressions which could be found for the same function, that which was most readily adapted to simple numerical calculation by many individuals employed at the same time. This section had little or nothing to do with the actual numerical work. When its labours were concluded, the formulæ on the use of which it had decided, were delivered to the second section.

第一组。——这一组的任务，是在能够表示同一函数的各种分析式中，研究哪一种最适合由许多人同时进行简单的数值计算。这一组与实际的数值工作几乎毫无关系。他们的工作完成后，他们决定采用的公式就交给第二组。

Second Section.—This section consisted of seven or eight persons of considerable acquaintance with mathematics: and their duty was to convert into numbers the formulæ put into their hands by the first section, an operation of great labour; and then to deliver out these formulæ to the members of the third section, and receive from them the finished calculations. The members of this second section had certain means of verifying the calculations without the necessity of repeating, or even of examining, the whole of the work done by the third section.

第二组。——这一组由七八位相当精通数学的人组成；他们的任务是把第一组交给他们的公式转化为数值，这是一项非常繁重的工作；然后把这些公式分发给第三组的成员，再从他们那里收回完成的计算。第二组的成员有一定的方法，可以验证计算结果，而不必重做、甚至不必检查第三组所做的全部工作。

Third Section.—The members of this section, whose number varied from sixty to eighty, received certain numbers from the second section, and, using nothing more than simple addition and subtraction, they returned to that section the tables in a finished state. It is remarkable that nine-tenths of this class had no knowledge of arithmetic beyond the two first rules which they were thus called upon to exercise, and that these persons were usually found more correct in their calculations, than those who possessed a more extensive knowledge of the subject.

第三组。——这一组的成员人数在六十到八十人之间，他们从第二组那里领取一些数，只用简单的加法和减法，就把完成的用表交还给第二组。值得注意的是，这一组十分之九的人，除了要他们运用的那头两条算术法则之外，对算术一无所知；而这些人的计算，通常比那些对这门学科懂得更多的人还要准确。

(246.) The proceeding of M. Prony, in this celebrated system of calculation, much resembles that of a skilful person about to construct a cotton or silk-mill, or any similar establishment. Having, by his own genius, or through the aid of his friends, found that some improved machinery may be successfully applied to his pursuit, he makes drawings of his plans of the machinery, and may himself be considered as constituting the first section. He next requires the assistance of operative engineers capable of executing the machinery he has designed, some of whom should understand the nature of the processes to be carried on; and these constitute his second section. When a sufficient number of machines have been made, a multitude of other persons, possessed of a lower degree of skill, must be employed in using them; these form the third section: but their work, and the just performance of the machines, must be still superintended by the second class.

（246）普罗尼先生在这套著名的计算体系中的做法，很像一位要建造棉纺厂、丝织厂或任何类似工厂的能干的人。他凭着自己的才智，或者借助朋友们的帮助，发现某种改进了的机器可以成功地用于他的事业，于是画出机器的设计图，他本人就可以被看作构成了第一组。接下来，他需要能够制造他所设计的机器的技术工程师来协助，其中有些人应当懂得要进行的各道工序的性质；这些人构成他的第二组。机器造好足够的数量之后，就必须雇用许许多多技能较低的人来操作它们；这些人构成第三组：但他们的工作，以及机器的正常运转，仍然必须由第二组来监督。

(247.) As the possibility of performing arithmetical calculations by machinery may appear to non-mathematical readers to be rather too large a postulate, and as it is connected with the subject of the division of labour, I shall here endeavour, in a few lines, to give some slight perception of the manner in which this can be done,—and thus to remove a small portion of the veil which covers that apparent mystery.

（247）用机器来进行算术计算的可能性，在不懂数学的读者看来，也许是一个过于大胆的假设；而且这与劳动分工的题目有关，所以我打算在这里用几行文字，让读者对这件事怎样能够做到略有感知，从而揭开遮盖着这一表面神秘之事的面纱的一小角。

(248.) That nearly all tables of numbers which follow any law, however complicated, may be formed, to a greater or less extent, solely by the proper arrangement of the successive addition and subtraction of numbers befitting each table, is a general principle which can be demonstrated to those only who are well acquainted with mathematics; but the mind, even of the reader who is but very slightly acquainted with that science, will readily conceive that it is not impossible, by attending to the following example.

（248）几乎所有遵循某种规律的数表，无论这规律多么复杂，都可以在或大或小的程度上，仅仅通过适合每个表的数的逐次加减的恰当安排编制出来，这是一条一般原理，只能向精通数学的人加以证明；但是，即使是对这门学科了解得很少的读者，只要留意下面这个例子，也会很容易想到这并非不可能。

The subjoined table is the beginning of one in very extensive use, which has been printed and reprinted very frequently in many countries, and is called a Table of Square Numbers.

下面这张表，是一种用途极广的表的开头，它曾在许多国家一再印行，叫作平方数表。

Terms of     A.          B.             C.
the Table.   Table.      First          Second
                         Difference.    Difference.

    1           1
                            3
    2           4                          2
                            5
    3           9                          2
                            7
    4          16                          2
                            9
    5          25                          2
                           11
    6          36                          2
                           13
    7          49

项数         A.          B.             C.
             表          一阶差         二阶差

    1           1
                            3
    2           4                          2
                            5
    3           9                          2
                            7
    4          16                          2
                            9
    5          25                          2
                           11
    6          36                          2
                           13
    7          49

Any number in the table, column A, may be obtained, by multiplying the number which expresses the distance of that term from the commencement of the table by itself; thus, 25 is the fifth term from the beginning of the table, and 5 multiplied by itself, or by 5, is equal to 25. Let us now subtract each term of this table from the next succeeding term, and place the results in another column (B), which may be called first-difference column. If we again subtract each term of this first difference from the succeeding term, we find the result is always the number 2, (column C;) and that the same number will always recur in that column, which may be called the second-difference, will appear to any person who takes the trouble to carry on the table a few terms further. Now when once this is admitted, it is quite clear that, provided the first term (1) of the Table, the first term (3) of the first differences, and the first term (2) of the second or constant difference, are originally given, we can continue the table of square numbers to any extent, merely by addition:—for the series of first differences may be formed by repeatedly adding the constant difference (2) to (3) the first number in column B, and we then have the series of numbers, 3, 5, 6, &c.: and again, by successively adding each of these to the first number (1) of the table, we produce the square numbers.

表中 A 栏的任何一个数，都可以用表示这一项与表头距离的那个数自乘而得到；例如，25 是从表头数起的第五项，而 5 自乘，也就是乘以 5，等于 25。现在我们用这个表的每一项去减它后面的一项，把结果放在另一栏（B）中，这一栏可以叫作一阶差栏。如果我们再用这个一阶差的每一项去减后面的一项，就会发现结果总是数 2（C 栏）；而在这一栏——可以叫作二阶差——里，总会重复出现同一个数，任何人只要费点力气把这个表再往下多算几项，就会看出来。一旦承认了这一点，就很清楚了：只要事先给出表的第一项（1）、一阶差的第一项（3）和二阶差即常差的第一项（2），我们就可以只用加法，把平方数表一直延续下去：因为一阶差的数列，可以把常差（2）反复加到 B 栏的第一个数（3）上得到，于是我们就有了 3、5、6 等一系列数；再把这些数逐次加到表的第一个数（1）上，就得出了平方数。

(249.) Having thus, I hope, thrown some light upon the theoretical part of the question, I shall endeavour to shew that the mechanical execution of such an engine, as would produce this series of numbers, is not so far removed from that of ordinary machinery as might be conceived. Let the reader imagine three clocks, placed on a table side by side, each having only one hand, and each having a thousand divisions instead of twelve hours marked on the face; and every time a string is pulled, let them strike on a bell the numbers of the divisions to which their hands point. Let him further suppose that two of the clocks, for the sake of distinction called B and C, have some mechanism by which the clock C advances the hand of the clock B one division, for each stroke it makes upon its own bell: and let the clock B by a similar contrivance advance the hand of the clock A one division, for each stroke it makes on its own bell. With such an arrangement, having set the hand of the clock A to the division I., that of B to III., and that of C to II., let the reader imagine the repeating parts of the clocks to be set in motion continually in the following order: viz.—pull the string of clock A; pull the string of clock B; pull the string of clock C.

（249）我希望已经就这个问题的理论部分作了一些说明，现在我要努力表明，制造这样一台能产生这一系列数的机器，在机械执行上，与普通机器的距离并不像人们可能设想的那样遥远。请读者想象三座钟，并排放在一张桌上，每座钟只有一根指针，钟面上刻的不是十二个小时，而是一千个分度；每拉一次绳子，钟就在铃上敲出指针所指的分度数。再请读者假定，其中两座钟（为了区别，称为 B 和 C）装有某种机构，使钟 C 每在自己的铃上敲一下，就把钟 B 的指针推进一个分度；钟 B 也用类似的装置，每在自己的铃上敲一下，就把钟 A 的指针推进一个分度。这样安排好以后，把钟 A 的指针拨到第 I 分度，钟 B 的拨到第 III 分度，钟 C 的拨到第 II 分度，然后请读者想象三座钟的报时部件按下面的顺序连续运转：拉钟 A 的绳子；拉钟 B 的绳子；拉钟 C 的绳子。

(250.) We have seen, then, that the effect of the division of labour, both in mechanical and in mental operations, is, that it enables us to purchase and apply to each process precisely that quantity of skill and knowledge which is required for it: we avoid employing any part of the time of a man who can get eight or ten shillings a day by his skill in tempering needles, in turning a wheel, which can be done for sixpence a day; and we equally avoid the loss arising from the employment of an accomplished mathematician in performing the lowest processes of arithmetic.

（250）由此可见，劳动分工无论在机械操作中还是在脑力操作中，其效果都在于：它使我们能够为每一道工序购买并运用恰好所需的那么多技能和知识；我们避免了让一个凭淬炼针的手艺每天能挣八到十先令的人，花任何时间去转动一个一天六便士就能雇人转动的轮子；同样，我们也避免了让一位造诣深厚的数学家去做最低级的算术运算所造成的损失。