On a Proposed Analytical Machine

In the first place I desire to record my indebtedness to Professor C. V. Boys, F.R.S., for the assistance which I owe to his kindness in entering into correspondence with me on the matter to which this paper is devoted.

首先，我要感谢 C. V. 博伊斯教授（皇家学会会员），承蒙他好意与我就本文所涉的问题通信，给予了帮助。

It would be difficult and very inadvisable to write on the present subject without referring to the remarkable work of Charles Babbage, who, having first invented two Difference Engines, subsequently (about eighty years ago) designed an Analytical Engine, which was shown to be at least a theoretical possibility; but unfortunately its construction had not proceeded far when its inventor died. Since Babbage’s time his Analytical Engine seems to have been almost forgotten; and it is probable that no living person understands the details of its projected mechanism. My own knowledge of Babbage’s Engines is slight, and for the most part limited to that of their mathematical principles.

论及这个题目，如果不提查尔斯·巴贝奇的卓越工作，那将是困难的，也是很不明智的。他先发明了两台差分机，后来（大约八十年前）又设计了一台分析机，并证明它至少在理论上是可能的；可惜它的建造尚未取得多少进展，发明者就去世了。自巴贝奇的时代以来，他的分析机似乎几乎被人遗忘了；很可能如今在世的人中，没有谁了解它设计中的机构细节。我本人对巴贝奇各台机器的了解很少，大体限于它们的数学原理。

It is unnecessary for me to prove the possibility of designing a machine capable of automatically solving all problems which can be solved by numbers. The principles on which an Analytical Machine may rest “have been examined, admitted, recorded, and demonstrated.” I would refer those who desire information thereon to the Countess of Lovelace’s translation of an article on Babbage’s Engine, which, together with copious notes by the translator, appears in R. Taylor’s “Scientific Memoirs,” vol. iii.; to Babbage’s own work, “Passages from the Life of a Philosopher”; and to the Report of the British Association for the year 1878, p. 92. These papers furnish a complete demonstration that the whole of the developments and operations of analysis are capable of being executed by machinery.

我无需证明设计一台能自动解决一切可用数字解决的问题的机器是可能的。分析机所能依据的原理，“已经被审视、承认、记录并证明过了。”想了解这方面情况的人，我建议去读洛夫莱斯伯爵夫人翻译的一篇关于巴贝奇分析机的文章——连同译者所加的详尽注释，载于 R. 泰勒编的《科学论文集》第三卷；巴贝奇本人的著作《一位哲学家的生平片段》；以及英国科学促进会 1878 年报告第 92 页。这些文献完整地证明了：分析学的全部展开和运算，都能由机器执行。

An Analytical Machine must have some means of storing the numerical data of the problem to be solved, and the figures produced at each successive step of the work (together with the proper algebraical signs); and, lastly, a means of recording the result or results. It must be capable of submitting any two of the numbers stored to the arithmetical operation of addition, subtraction, multiplication, or division. It must also be able to select from the numbers it contains the proper numbers to be operated on; to determine the nature of the operation to which they are to be submitted; and to dispose of the result of the operation, so that such result can be recalled by the machine and further operated on, should the terms of the problem require it. The sequence of operations, the numbers (considered as abstract quantities only) submitted to those operations, and the disposition of the result of each operation, depend upon the algebraical statement of the calculation on which the machine is engaged; while the magnitude of the numbers involved in the work varies with the numerical data of that particular case of the general formula which is in process of solution. The question therefore naturally arises as to how a machine can be made to follow a particular law of development as expressed by an algebraic formula. An eminently satisfactory answer to that question (and one utilized by both Babbage and myself) is suggested by the Jacquard loom, in which interesting invention a system of perforated cards is used to direct the movements of the warp and weft threads, so as to produce in the woven material the pattern intended by the designer. It is not difficult to imagine that a similar arrangement of cards could be used in a mathematical machine to direct the weaving of numbers, as it were, into algebraic patterns, in which case the cards in question would constitute a kind of mathematical notation. It must be distinctly understood that, if a set of such cards were once prepared in accordance with a specified formula, it would possess all the generality of algebra, and include an infinite number of particular cases.

分析机必须有某种手段，来存储待解问题的数值数据，以及每一步工作产生的数字（连同适当的代数符号）；最后，还要有记录一个或多个结果的手段。它必须能对所存储的任意两个数进行加、减、乘、除的算术运算。它还必须能从所存的数中选出要参加运算的那些数；决定对它们施行何种运算；并处置运算的结果，使这个结果在问题的条件需要时，能被机器调出来作进一步运算。运算的顺序、参加这些运算的数（仅作为抽象的量来看），以及每次运算结果的处置，都取决于机器正在进行的计算的代数表述；而工作中涉及的数的大小，则随正在求解的一般公式的那个具体情形的数值数据而变化。于是自然产生一个问题：怎样才能使机器遵循一个由代数公式所表达的特定展开规律？对此，提花织机提供了一个极为令人满意的答案（巴贝奇和我都采用了它）：在这项有趣的发明中，一套穿孔卡片用来指挥经线和纬线的运动，从而在织物上织出设计者所要的图案。不难想象，在数学机器中也可以用类似的卡片装置，来指挥把数字仿佛编织成代数图案；这样，这些卡片就构成了一种数学记号。必须清楚地认识到：一套这样的卡片一旦按照某个指定的公式准备好，就具有代数的全部一般性，包含无穷多个特殊情形。

Babbage’s Jacquard-system and mine differ considerably; for, while Babbage designed two sets of cards—one set to govern the operations, and the other set to select the numbers to be operated on—I use one sheet or roll of perforated paper (which, in principle, exactly corresponds to a set of Jacquard-cards) to perform both these functions in the order and manner necessary to solve the formula to which the particular paper is assigned. To such a paper I apply the term formula-paper. Each row of perforations across the formula-paper directs the machine in some definite step in the process of calculation—such as, for instance, a complete multiplication, including the selection of the numbers to be multiplied together. Of course a single formula-paper can be used for an indefinite number of calculations, provided that they are all of one type or kind (i.e. algebraically identical).

巴贝奇的提花系统和我的大不相同：巴贝奇设计了两套卡片——一套掌管运算，另一套选择参加运算的数——而我只用一张或一卷穿孔纸（它在原理上与一套提花卡片完全对应），按求解该纸所对应的公式所必需的顺序和方式，来完成这两种功能。我把这样的纸称为公式纸。公式纸上每一行横向的孔，都指挥机器完成计算过程中的某个确定步骤——例如一次完整的乘法，包括选择要相乘的数。当然，一张公式纸可以用于无数次计算，只要它们都属于同一类型（即在代数上相同）。

In my machine each Variable is stored in a separate shuttle, the individual figures of the Variable being represented by the relative positions of protruding metal rods or “type,” which each shuttle carries. There is one of these rods for every figure of the Variable, and one to indicate the sign of the Variable. Each rod protrudes a distance of from 1 to 10 units, according to the figure or sign which it is at the time representing. The shuttles are stored in two co-axial cylindrical shuttle-boxes, which are divided for the purpose into compartments parallel to their axis. The present design of the machine provides for the storage of 192 Variables of twenty figures each; but both the number of Variables and the number of figures in each Variable may, if desired, be greatly increased. It may be observed, too, that the shuttles are quite independent of the machine, so that new shuttles, representing new Variables, can be introduced at any time.

在我的机器中，每个变量存放在一个单独的梭子里；变量的各位数字，由梭子所带的、伸出的金属杆即“字杆”的相对位置来表示。变量的每一位数字有一根这样的杆，另有一根表示变量的正负号。每根杆按它当时所表示的数字或符号，伸出 1 到 10 个单位的长度。梭子存放在两个同轴的圆柱形梭箱里，梭箱为此分隔成平行于轴线的格子。按目前的设计，机器可以存储 192 个各有二十位数字的变量；但如有需要，变量的个数和每个变量的位数都可以大大增加。还可以看到，梭子完全独立于机器，因此随时可以放入代表新变量的新梭子。

When two Variables are to be multiplied together, the corresponding shuttles are brought to a certain system of slides called the index, by means of which the machine computes the product. It is impossible precisely to describe the mechanism of the index without drawings; but it may be compared to a slide-rule on which the usual markings are replaced by movable blades. The index is arranged so as to give several readings simultaneously. The numerical values of the readings are indicated by periodic displacements of the blades mentioned, the duration of which displacements are recorded in units measured by the driving shaft on a train of wheels called the mill, which performs the carrying of tens, and indicates the final product. The product can be transferred from thence to any shuttle, or to two shuttles simultaneously, provided that they do not belong to the same shuttle-box. The act of inscribing a new value in a shuttle automatically cancels any previous value that the shuttle may have contained. The fundamental action of the machine may be said to be the multiplying together of the numbers contained in any two shuttles, and the inscribing of the product in one or two shuttles. It may be mentioned here that the fundamental process of Babbage’s Engine was not multiplication but addition.

当两个变量要相乘时，相应的梭子被送到一组称为指标器的滑尺那里，机器借助它算出乘积。没有图纸，就无法准确描述指标器的机构；但可以把它比作一把计算尺，只是上面通常的刻度换成了可移动的刀片。指标器的安排使它能同时给出好几个读数。读数的数值由上述刀片的周期性位移来表示，位移的持续时间由驱动轴以单位计量，记录在一组称为运算器的齿轮上；运算器负责进位，并显示最后的乘积。乘积可以从那里转送到任何一个梭子，或同时转送到两个梭子，只要它们不属于同一个梭箱。往一个梭子里写入新值，会自动抹去它原先所存的任何值。可以说，这台机器的基本动作，就是把任意两个梭子中的数相乘，并把乘积写入一个或两个梭子。这里不妨一提：巴贝奇分析机的基本过程不是乘法，而是加法。

The machine prints all results, and, if required, the data, and any noteworthy values which may transpire during the calculation. It may be mentioned, too, that the machine may be caused to calculate and print, quite automatically, a table of values—such, for instance, as a table of logs, sines, squares, &c. It has also the power of recording its results by a system of perforations on a sheet of paper, so that when such a number-paper (as it may be called) is replaced in the machine, the latter can “read” the numbers indicated thereon, and inscribe them in the shuttles reserved for the purpose.

机器会打印出所有结果，如有需要，还可以打印数据以及计算过程中出现的任何值得注意的数值。还可以一提的是，机器可以完全自动地计算并打印一张数值表——例如对数表、正弦表、平方表等等。它还能用在一张纸上打孔的方式记录结果，这样，当这样一张数字纸（不妨这样称呼它）重新放进机器时，机器就能“读出”上面所表示的数，并把它们写入为此保留的梭子里。

Among other powers with which the machine is endowed is that of changing from one formula to another as desired, or in accordance with a given mathematical law. It follows that the machine need never be idle; for it can be set to tabulate successive values of any function, while the work of the tabulation can be suspended at any time to allow of the determination by it of one or more results of greater importance or urgency. It can also “feel” for particular events in the progress of its work—such, for instance, as a change of sign in the value of a function, or its approach to zero or infinity; and it can make any pre-arranged change in its procedure, when any such event occurs. Babbage dwells on these and similar points, and explains their bearing on the automatic solution (by approximation) of an equation of the *n*th degree; but I have not been able to ascertain whether his way of attaining these results has or has not any resemblance to my method of so doing.

机器所具有的其他能力之一，是能够按需要、或按某个给定的数学规律，从一个公式转到另一个公式。由此可知，机器永远不必闲着；因为它可以被设定去列出任何函数的逐次值，而列表工作又可以随时暂停，以便让它去求出一个或多个更重要、更紧迫的结果。它还能在工作过程中“感知”某些特定的事件——例如函数值的变号，或函数趋于零或无穷大；一旦出现这样的事件，它就能按预先的安排改变自己的程序。巴贝奇详细讨论过这些以及类似的问题，并说明了它们对于（用逼近法）自动求解 n 次方程的意义；但我无法确定，他达到这些结果的方法与我的方法是否有相似之处。

While the machine is in use its central shaft must be maintained at an approximately uniform rate of rotation—a small motor might be used for this purpose. It is calculated that a velocity of three revolutions per second would be safe; and such a velocity would ensure the multiplication of any two Variables of twenty figures each in about 10 seconds, and their addition or subtraction in about 3 seconds. The time taken to divide one Variable by another depends on the degree of convergency of the series derived from the divisor, but 1½ minutes may be taken as the probable maximum. When constructing a formula-paper, due regard should therefore be had to the relatively long time required to accomplish the routine of division; and it will, no doubt, be found advisable to use this process as sparingly as possible. The determination of the logarithm of any number would take 2 minutes, while the evaluation of aⁿ (for any value of n) by the exponential theorem, should not require more than 1½ minutes longer—all results being of twenty figures.

机器使用时，它的中央轴必须保持大致均匀的转速——可以用一台小电动机来做到这一点。据计算，每秒三转的速度是安全的；这样的速度可以保证在大约 10 秒内完成任意两个二十位变量的乘法，在大约 3 秒内完成它们的加法或减法。一个变量除以另一个变量所需的时间，取决于由除数导出的级数的收敛程度，但可以认为最多大约 1½ 分钟。因此，在编制公式纸时，应当考虑到完成除法程序所需的时间相对较长；无疑，应尽可能少用除法。求任何一个数的对数需要 2 分钟，而用指数定理计算 aⁿ（n 为任意值）所需的时间，最多再多 1½ 分钟——所有结果都是二十位的。