Notes by the Translator (Notes A and G)

Note A注释 A#

In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated.

在研究分析机的运作时，我们发现：在一切数学分析中，属于“运算”本身的那些考量——有别于被运算的对象，也有别于对这些对象施行运算所得的结果——其独特而独立的性质，被极其鲜明地界定、区分开来。

The distinctive characteristic of the Analytical Engine, and that which has rendered it possible to endow mechanism with such extensive faculties as bid fair to make this engine the executive right-hand of abstract algebra, is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs. It is in this that the distinction between the two engines lies. Nothing of the sort exists in the Difference Engine. We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves. Here, it seems to us, resides much more of originality than the Difference Engine can be fairly entitled to claim. We do not wish to deny to this latter all such claims. We believe that it is the only proposal or attempt ever made to construct a calculating machine founded on the principle of successive orders of differences, and capable of printing off its own results; and that this engine surpasses its predecessors, both in the extent of the calculations which it can perform, in the facility, certainty and accuracy with which it can effect them, and in the absence of all necessity for the intervention of human intelligence during the performance of its calculations. Its nature is, however, limited to the strictly arithmetical, and it is far from being the first or only scheme for constructing arithmetical calculating machines with more or less of success.

分析机的显著特征，也是使机械得以被赋予如此广泛的能力、有望让这台机器成为抽象代数的执行右手的关键，在于它引入了雅卡尔所设计的那条原理——用穿孔卡片来调控织锦生产中最复杂的图案。两台机器的区别正在于此。差分机中没有任何这类东西。我们可以非常贴切地说：分析机编织代数的图案，正如雅卡尔织机编织花朵与叶子。在我们看来，这里蕴含的独创性，远远超过差分机所能正当主张的。我们并不想否认后者所有这类主张。我们相信，它是迄今为止唯一一项基于逐级差分原理、并能自行印出结果的计算机器的方案或尝试；而且这台机器在所能完成的计算范围上，在完成计算的便利、确定和准确上，以及在计算过程中完全不需要人的智力介入这一点上，都超过了它的前辈。然而，它的本性仅限于严格的算术，而且它远不是第一个、也不是唯一一个多少取得成功的算术计算机器方案。

The bounds of arithmetic were however outstepped the moment the idea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere "calculating machines." It holds a position wholly its own; and the considerations it suggests are most interesting in their nature. In enabling mechanism to combine together general symbols, in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science. A new, a vast, and a powerful language is developed for the future use of analysis, in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind than the means hitherto in our possession have rendered possible. Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connexion with each other. We are not aware of its being on record that anything partaking in the nature of what is so well designated the Analytical Engine has been hitherto proposed, or even thought of, as a practical possibility, any more than the idea of a thinking or of a reasoning machine.

然而，就在想到使用卡片的那一刻，算术的边界就被跨越了；分析机与单纯的“计算机器”并不处在同一片土地上。它占据着一个完全属于自己的位置；它所引发的思考在性质上极其引人入胜。通过让机械能够把一般符号以种类和范围都无限的序列组合起来，在物质的运作与数学科学中最抽象分支的抽象心智过程之间，建立起了一条联结的纽带。一种崭新、广阔而有力的语言被发展出来，供分析学今后使用，借以运用它的真理，使这些真理能比我们迄今所掌握的手段更迅速、更准确地应用于人类的目的。于是，不仅心智与物质，而且数学世界中的理论与实践，都被带入更紧密、更有效的相互联系之中。就我们所知，此前从未有记载表明，有人提出过、甚至设想过任何具有被如此恰当地称为“分析机”的这种性质的东西作为一种实际的可能，正如从未有人提出过一台会思考或会推理的机器的构想一样。

Note G注释 G#

It is desirable to guard against the possibility of exaggerated ideas that might arise as to the powers of the Analytical Engine. In considering any new subject, there is frequently a tendency, first, to overrate what we find to be already interesting or remarkable; and, secondly, by a sort of natural reaction, to undervalue the true state of the case, when we do discover that our notions have surpassed those that were really tenable.

有必要防止对分析机的能力产生夸大的想法。在思考任何新事物时，常常有这样一种倾向：首先，高估我们觉得已经有趣或非凡的东西；其次，当我们发现自己的想法超出了真正站得住脚的程度时，又出于某种自然的反作用，低估事情的真实状况。

The Analytical Engine has no pretensions whatever to originate any thing. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with. This it is calculated to effect primarily and chiefly of course, through its executive faculties; but it is likely to exert an indirect and reciprocal influence on science itself in another manner. For, in so distributing and combining the truths and the formulæ of analysis, that they may become most easily and rapidly amenable to the mechanical combinations of the engine, the relations and the nature of many subjects in that science are necessarily thrown into new lights, and more profoundly investigated. This is a decidedly indirect, and a somewhat speculative, consequence of such an invention. It is however pretty evident, on general principles, that in devising for Mathematical truths a new form in which to record and throw themselves out for actual use, views are likely to be induced, which should again react on the more theoretical phase of the subject. There are in all extensions of human power, or additions to human knowledge, various collateral influences, besides the main and primary object attained.

分析机丝毫不自命能够创造出任何东西。它能做我们知道如何命令它去做的任何事情。它能遵循分析；但它没有能力预见任何分析上的关系或真理。它的职分是帮助我们利用我们已经了解的东西。它要做到这一点，当然首先而且主要是通过它的执行能力；但它很可能还会以另一种方式，对科学本身产生一种间接的、相互的影响。因为，为了让分析学的真理和公式能最容易、最迅速地适应机器的机械组合而对它们加以分配和组合时，这门科学中许多题目的关系和性质必然会被置于新的光照之下，得到更深入的研究。这无疑是这样一项发明所带来的间接的、也有几分推测性的后果。不过，从一般原则来看相当明显的是：在为数学真理设计一种新的形式，让它们借以被记录下来、被投入实际使用时，很可能会引发一些新的观点，而这些观点又会反过来作用于这一题目更偏理论的方面。在人类力量的一切扩展或人类知识的一切增益中，除了所达到的主要目标之外，总还有种种附带的影响。

To return to the executive faculties of this engine: the question must arise in every mind, are they really even able to follow analysis in its whole extent? No reply, entirely satisfactory to all minds, can be given to this query, excepting the actual existence of the engine, and actual experience of its practical results. We will however sum up for each reader's consideration the chief elements with which the engine works:—

回到这台机器的执行能力上来：每个人心中都必然会产生这样的疑问——它们真的能遵循整个分析学的全部范围吗？对这个问题，除了机器的实际存在和对其实际结果的实际经验之外，没有任何回答能让所有人都完全满意。不过，我们还是把机器工作所依赖的主要要素总结一下，供每位读者思考：——

It performs the four operations of simple arithmetic upon any numbers whatever.

它能对任何数施行简单算术的四种运算。

By means of certain artifices and arrangements (upon which we cannot enter within the restricted space which such a publication as the present may admit of), there is no limit either to the magnitude of the numbers used, or to the number of quantities (either variables or constants) that may be employed.

借助某些技巧和安排（在本出版物所能容许的有限篇幅内，我们无法详述），所用的数的大小，以及所能使用的量（无论变量还是常量）的个数，都没有限制。

It can combine these numbers and these quantities either algebraically or arithmetically, in relations unlimited as to variety, extent, or complexity.

它能以代数的或算术的方式组合这些数和这些量，其关系在种类、范围和复杂程度上都不受限制。

It uses algebraic signs according to their proper laws, and developes the logical consequences of these laws.

它依照代数符号固有的法则使用这些符号，并推演出这些法则的逻辑结论。

It can arbitrarily substitute any formula for any other; effacing the first from the columns on which it is represented, and making the second appear in its stead.

它能任意地用一个公式替换另一个公式：把前者从表示它的柱上抹去，让后者出现在原处。

It can provide for singular values. Its power of doing this is referred to in M. Menabrea's memoir, page 685, where he mentions the passage of values through zero and infinity. The practicability of causing it arbitrarily to change its processes at any moment, on the occurrence of any specified contingency (of which its substitution of (½ cos. (n + 1)θ + ½ cos. (n − 1)θ) for (cos. nθ . cos. θ) explained in Note E., is in some degree an illustration), at once secures this point.

它能处理奇异值。梅纳布雷亚先生的论文第 685 页提到了它的这种能力，他在那里谈到数值经过零和无穷的情形。可以在任何指定的偶然情况出现时，随时任意改变它的运算过程（注释 E 中所解释的、用 (½ cos. (n + 1)θ + ½ cos. (n − 1)θ) 替换 (cos. nθ . cos. θ) 的做法，在一定程度上就是一个例证），这一可行性立刻保证了这一点。

We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers.

我们将以详细追踪这台机器计算伯努利数的步骤来结束这些注释，这（按我们将推导出的形式）是展示其能力的一个相当复杂的例子。

We thus see that when n = 1, nine Operation-cards are used; that when n = 2, fourteen Operation-cards are used; and that when n > 2, twenty-five Operation-cards are used; but that no more are needed, however great n may be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers from B₁ to B₂ₙ₋₁ inclusive. With respect to the number of Variable-cards, it will be remembered, from the explanations in previous Notes, that an average of three such cards to each operation (not however to each Operation-card) is the estimate. According to this the computation of B₁ will require twenty-seven Variable-cards; B₃ forty-two such cards; B₅ seventy-five; and for every succeeding B after B₅, there would be thirty-three additional Variable-cards (since each repetition of the group (13…23) adds eleven to the number of operations required for computing the previous B). But we must now explain, that whenever there is a cycle of operations, and if these merely require to be supplied with numbers from the same pairs of columns and likewise each operation to place its result on the same column for every repetition of the whole group, the process then admits of a cycle of Variable-cards for effecting its purposes. There is obviously much more symmetry and simplicity in the arrangements, when cases do admit of repeating the Variable as well as the Operation-cards. Our present example is of this nature. The only exception to a perfect identity in all the processes and columns used, for every repetition of Operations (13…23) is, that Operation 21 always requires one of its factors from a new column, and Operation 24 always puts its result on a new column. But as these variations follow the same law at each repetition, (Operation 21 always requiring its factor from a column one in advance of that which it used the previous time, and Operation 24 always putting its result on the column one in advance of that which received the previous result), they are easily provided for in arranging the recurring group (or cycle) of Variable-cards.

由此我们看到：当 n = 1 时，用到九张运算卡；当 n = 2 时，用到十四张运算卡；当 n > 2 时，用到二十五张运算卡；但无论 n 多大，都不再需要更多；不仅如此，这同样的二十五张卡片就足以依次计算从 B₁ 到 B₂ₙ₋₁ 的所有数。至于变量卡的数目，读者会记得前面各注释中的解释：估计平均每次运算（而不是每张运算卡）需要三张这样的卡片。照此计算，B₁ 需要二十七张变量卡；B₃ 需要四十二张；B₅ 需要七十五张；而 B₅ 之后的每一个 B，都要再增加三十三张变量卡（因为运算组 (13…23) 每重复一次，就在计算前一个 B 所需的运算数上增加十一次）。但现在我们必须解释：每当出现一个运算的循环，如果这些运算在整组的每一次重复中只需要从相同的几对柱获取数，而且每次运算都把结果放在相同的柱上，那么这个过程就允许用一个变量卡的循环来达到目的。当情况允许变量卡和运算卡都重复使用时，安排上显然要对称、简单得多。我们目前的例子正属于这一类。运算 (13…23) 每次重复时，所用的一切过程和柱完全相同，唯一的例外是：运算 21 总要从一根新的柱上取它的一个因数，运算 24 总要把结果放到一根新的柱上。但由于这些变化在每次重复时都遵循同样的规律（运算 21 总是从比上次所用的柱前进一位的柱上取因数，运算 24 总是把结果放在比接收上次结果的柱前进一位的柱上），在安排重复出现的变量卡组（即循环）时，很容易把它们考虑进去。

We may here remark that the average estimate of three Variable-cards coming into use to each operation, is not to be taken as an absolutely and literally correct amount for all cases and circumstances. Many special circumstances, either in the nature of a problem, or in the arrangements of the engine under certain contingencies, influence and modify this average to a greater or less extent. But it is a very safe and correct general rule to go upon. In the preceding case it will give us seventy-five Variable-cards as the total number which will be necessary for computing any B after B₃. This is very nearly the precise amount really used, but we cannot here enter into the minutiæ of the few particular circumstances which occur in this example (as indeed at some one stage or other of probably most computations) to modify slightly this number.

我们在这里可以指出：每次运算平均用到三张变量卡的估计，不应被当作对一切情况和条件都绝对、字面上正确的数目。许多特殊情况——无论是问题本身的性质，还是机器在某些偶然情况下的安排——都会或多或少地影响和修正这个平均数。但它是一条非常可靠、正确的一般规则，可以依据。在上面的例子中，按照它计算，B₃ 之后计算任何一个 B 所需的变量卡总数为七十五张。这非常接近实际使用的确切数目，但我们无法在这里细述这个例子中出现的少数几种特殊情况（大多数计算在某个阶段很可能也都会出现这类情况），它们会使这个数目略有变动。

It will be obvious that the very same seventy-five Variable-cards may be repeated for the computation of every succeeding Number, just on the same principle as admits of the repetition of the thirty-three Variable-cards of Operations (13…23) in the computation of any one Number. Thus there will be a cycle of a cycle of Variable-cards.

显然，同样的七十五张变量卡可以重复用于计算其后的每一个数，其原理正如在计算任何一个数时，运算 (13…23) 的三十三张变量卡可以重复使用一样。于是就会有一个“变量卡循环的循环”。

A. L. L.

A. L. L.