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Babbage's "Calculating Engine"; but giving much information on many collateral subjects. From this we have already had occasion to quote. The following passage deserves a place here:
There are nevertheless many persons who, admitting the great ingenuity of the contrivance, have, notwithstanding, been accustomed to regard it more in the light of a philosophical curiosity, than an instrument for purposes practically useful. This mistake-than which it is not possible to imagine a greater-has arisen mainly from the ignorance which prevails of the extensive utility of those numerical tables which it is the purpose of the engine in question to produce. There are also some persons, who, not considering the time requisite to bring any invention of this magnitude to perfection in all its details, incline to consider the delays which have taken place in its progress as presumptions against its practibility. These persons should, however, before they arrive at such a conclusion, reflect upon the time which was necessary to bring to perfection engines infinitely inferior in complexity and mechanical difficulty. Let them remember that-not to mention the invention of that machine-the improvements alone introduced into the steam-engine by the celebrated Watt occupied a period of not less than 20 years of the life of that distinguished person, and involved an expenditure of capital amounting to £50,000. The calculating machine is a contrivance even new in its details. inventor did not take it up already imperfectly formed, after having received the contributions of human ingenuity exercised upon it for a century or more. It has not, like almost all other great mechanical inventions, been gradually advanced to its present state through a series of failures, through difficulties encountered and overcome by a succession of projectors. It is not an object on which the light of various minds has thus been shed. It is, on the contrary, the production of solitary and individual thought-begun, advanced through each successive stage of improvement, and brought to perfection by one mind. Yet this creation of genius, from its first rude conception to its present state, has cost little more than half the time and not one-third of the expense consumed in bringing the steam-engine-previously far advanced in the course of improvement-to that state of comparative perfection in which it was left by Watt. Short as the period of time has been which the inventor has devoted to this enterprise, it has, nevertheless, been demonstrated, to the satisfaction of many scientific men of the first eminence, that the design in all its details, reduced, as it is, to a system of mechanical drawings, is complete; and requires only to be constructed in conformity with those plans to realize all that its inventor has promised.
This was high testimony in favour of the new and more comprehensive machine upon which the inventor was then engaged. But an equally high testimony to the ability of the writer of that art. has to be recorded, in the fact that two separate inventors actually constructed calculating machines from merely reading the descriptions given by Dr. Lardner therein of Mr. Babbage's appliances.
The first of these constructions was by MR. DEACON, of Beaufort House, Strand, a well-known mechanist, who, simply for his own satisfaction, constructed a small model of the calculating part of such a machine, which however was only shown to a few friends, and never made generally known. The other was the Swedish, of which we shall speak in due order.
The reader will have surmised, from the nature of the report by the Royal So., as also from Dr. Lardner's art., that some difficulties had originated regarding the progress of Mr. Babbage's Difference Engine No. 2. This was so. By 1834 the sum of £17,000 or more had been expended, and yet the machine was very far from complete. Mr. Babbage wavered regarding the completion. The Gov. began to waver on the subject of the expense-the limit of which appeared undefined. Several years' correspondence ensued. The reason for all this is now rendered clear. Mr. Babbage was doubtful "whether the discoveries which he was then advancing might not ultimately supersede the work already executed." [Passages, etc., p. 88.] Mr. Babbage had in fact conceived the idea of his Analytical Engine, of which we shall presently have to speak. In a letter to the Chancellor of the Exchequer, under date 20th Jan., 1836, he speaks of his new conception thus:
It is not only capable of accomplishing all those other complicated calculations which I had intended, but it also performs all calculations which were peculiar to the Difference Engine, both in less time, and to a greater extent; in fact it completely supersedes the Difference Engine.
We must now leave Mr. Babbage some few years to determine upon his future course of action in regard to the prosecution of his rival conceptions. [See 1848.]
In 1840 there was pub., in Paris, General Plan, No. 25, of Mr. Babbage's Great Calculating or Analytical Engine.
In 1842 there appeared in the Bibliothèque Universelle de Genève, General Menabrea's Sketch of the Analytical Engine invented by Charles Babbage. This was afterwards translated by the late Countess Lovelace [Lord Byron's daughter—“Ada, fair daughter of my home and heart"], with extensive notes by the translator.
We have next to speak of the second machine brought into existence by the practical descriptions of the Edinburgh Reviewer. It was constructed by HERR GEORGE SCHEUTZ, at that time the editor of a technological journal at Stockholm, and a practical printer. After reading the article in question, and satisfying himself of the practicability of construction, he laid the matter on one side to wait for an opportunity. Three years later, or in 1837, his son, HERR EDWARD SCHEUTZ, then a student at the Royal Technological Institute at Stockholm, undertook the construction of the machine, his father giving him the use of work-room, lathe and tools, with such other appliances as severe economy enabled him to procure. An application was made to the Gov. for assistance, and refused. By 1840, after many trials and modifications, a model was so far completed as to be able to calculate correctly "series with terms of 5 figures and one difference also of 5 figures." By April, 1842, its power was extended to "calculate similar series, with 2 or 3
orders of differences." In 1843, the machine, with the printing apparatus, was ready for inspection by the Royal Swedish Academy of Sciences, and after various trials a certificate was signed by several of the leading members of that body, under date 18th September, 1843, and containing the following:
The apparatus in question is composed of 3 parts. 1. The Calculating Machine. It cannot compute series of a higher degree than the third, nor does it give complete terms exceeding 5 figures; but in the nature of the mechanism there is nothing to prevent its extension to the working of series of any degree whatever, and to terms of as many figures as the purpose may require. 2. The Printing Machine. Every term given by the calculating apparatus is expressed by printed figures closely arranged in lines, as in a printed table, the lines being impressed on some softer material adapted to receive galvanoplastic or stereotyped copies. All the lines succeed each other very correctly in the same vertical column. 3. The Numbering Machine. With the printing machine another apparatus is combined, which prints the arguments before every term. The machine is put in motion by turning the handle of a winch, by means of which, and without further manipulations, the calculations as well as the printing and arranging of figures and lines are effected.
The inventors, furnished with this certificate, sought orders in various countries, but without success; and the machine remained shut up in its case for the ensuing 7 years.[See 1853 and 1854.]
In 1843 was pub. by Mr. Babbage-Statement of the Circumstances respecting Mr. Babbage's Calculating Engines.
By 1848 MR. BABBAGE had completely mastered the details of his Analytical Engine— that is, had reduced them to diagrams, or working drawings, capable of being understood and executed by skilled workmen. The scope and capabilities of this new machine we propose briefly to notice, and in doing this we shall not depart from the language of its inventor:
The circular arrangement of the axes of the Difference Engine round large central wheels led to the most extended prospects. The whole of arithmetic now appeared within the grasp of mechanism. The most important part of the Analytical Engine was undoubtedly the mechanical method of carrying the tens. On this I laboured incessantly, each succeeding improvement advancing me a step or two. The difficulty did not consist so much in the more or less complexity of the contrivance as in the reduction of the time required to effect the carriage. Twenty or thirty different plans or modifications had been drawn. At last I came to the conclusion that I had exhausted the principle of successive carriage. I concluded that nothing but teaching the Engine to foresee, and then to act upon this foresight, could ever lead me to the object I desired, namely, to make the whole of any unlimited number of carriages in one unit of time. . . .
Yet he did accomplish even this.
This new and rapid system of carrying the tens when two numbers are added together reduces the actual time of the addition of any number of digits, however large, to nine units of time for the add., and one unit for the carriage. Thus, in ten units of time, any two numbers, however large, might be added together. A few more units of time, perhaps 5 or 6, were required for making the requisite previous arrangements. Having thus advanced as nearly as seemed possible to the minimum of time requisite for arithmetical operations, I felt renewed power and increased energy to pursue the far higher object I had in view.
To describe the successive improvements of the Analytical Engine would require many vols... To those who are acquainted with the principles of the Jacquard loom, and who are also familiar with analytical formulæ, a general idea of the means by which the Engine executes its operations may be obtained without much difficulty.... It is known as a fact that the Jacquard loom is capable of weaving any design which the imagination of man may conceive. It is also the constant practice for skilled artists to be employed by manufacturers in designing patterns. These patterns are then sent to a peculiar artist, who, by means of a certain machine, punches holes in a set of pasteboard cards in such a manner that when those cards are placed in a Jacquard loom, it will then weave upon its produce the pattern designed by the artist.
The analogy of the Analytical Engine with this well-known process is nearly perfect. The A. E. consists of two parts: 1. The store in which all the variables to be operated upon, as well as all those quantities which have arisen from the result of other operations, are placed. 2. The mill into which the quantities about to be operated upon are always brought. Every formula which the Analytical Engine can be required to compute consists of certain algebraical operations to be performed upon given letters, and of certain other modifications depending on the numerical value assigned to those letters. There are therefore two sets of cards: the first, to direct the nature of the operations to be performed-these are called operation cards; the other, to direct the particular variables on which those cards are required to operate-these latter are called variable cards. Now the symbol of each variable or constant is placed at the top of a column capable of containing any required number of digits.
Under this arrangement, when any formula is required to be computed, a set of operation cards must be strung together, which contain the series of operations in the order in which they occur. Another set of cards must then be strung together, to call the variables into the mill, in the order in which they are required to be acted upon. Each operation card will require 3 other cards, two to represent the variables and constants and their numerical values upon which the previous operation card is to act, and one to indicate the variable on which the arithmetical result of this operation is to be placed. But each variable has below it, on the same axis, a certain number of figure-wheels marked on their edges with the ten digits; upon these any number the machine is capable of holding can be placed. Whenever variables are ordered into the mill, these figures will be brought in, and the operation indicated by the preceding card will be performed upon them. The result of this operation will then be replaced in the store.
The Analytical Engine is therefore a machine of the most general nature. Whatever formula it is required to develope, the law of its development must be communicated to it by two sets of cards. When these have been placed, the Engine is special for that particular formula. The numerical value of constants must then be put on the columns of wheels below them, and on setting the Engine in motion, it will calculate and print the numerical results of that formula.
Every set of cards made for any formula will at any future time recalculate that formula with whatever constants may be required. Thus the A. E. will possess a library of its own. Every set of cards once made will at any future time reproduce the calculations for which it was first arranged. The numerical value of its constants may then be inserted.
Besides the sets of cards which direct the nature of the operations to be performed, and the variables
or constants which are to be operated upon, there is another class of cards, called number cards. These are much less general in their uses than the others, although they are necessarily of much larger size.
The A. E. will contain: 1. Apparatus for printing on paper, one, or if required, two copies of its results. 2. Means for producing a stereotype mould of the T. or results it computes. 3. Mechanism for punching on blank pasteboard cards or metal plates the numerical results of any of its computations. Of course the Engine will compute all the T. which it may itself be required to use. . . .
So much for the mechanism of this almost Human Engine. We have yet to glance at its powers of operation; and here again we shall follow the learned Professor. We are now introduced to a conversation between the inventor and his friend Prof. MacCullagh, late of Dublin, on this very subject:
After a long conversation on the subject, he inquired what the machine could do, if, in the midst of algebraic operations, it was required to perform logarithmic or trigonometrical operations. My answer was, that whenever the A. E. should exist, all the developments of formulæ would be directed by this condition-that the machine should be able to compute their numerical value in the shortest possible time. I then added that if this answer were not satisfactory, I had provided means by which, with equal accuracy, it might compute by logarithmic or other T. I explained that the T. to be used must, of course, be computed and punched on cards by the machine, in which case they would undoubtedly be correct. I then added that when the machine wanted a tabular number, say the logarithm of a given number, that it would ring a bell and then stop itself. On this the attendant would look at a certain part of the machine and find that it wanted the logarithm of a given number, say of 2303. The attendant would then go to the drawer containing the pasteboard cards representing its T. of logarithms. From amongst these he would take the required logarithmic card, and place it in the machine. Upon this the Engine would first ascertain whether the assistant had or had not given him the correct logarithm of number; if so, it would use it and continue its work. But if the Engine found the attendant had given him a wrong logarithm, it would then ring a louder bell and stop itself. On the attendant again examining the Engine, he would observe the words "wrong tabular number," and then discover that he really had given the wrong logarithm, and he would have to replace it by a right one.
Tables are used for saving the time of continually computing individual numbers. But the computations to be made by the Engine are so rapid that it seems most prob. that it will make shorter work by computing directly from proper formula than by having recourse even to its own T.
Next we have some insight into the scope of its operations:
The A. E. I propose will have the power of expressing every number it uses to fifty places of figures. It will multiply any two such numbers together, and then, if required, will divide the product of one hundred figures by number of fifty places of figures. Supposing the velocity of the moving parts of the Engine to be not greater than 40 feet p. minute, I have no doubt that 60 additions or subtractions may be completed and printed in 1 minute. One multiplication of two numbers, each of 50 figures in 1 minute. One division of a number having 100 places of figures by another of 50 in 1 minute.
Again we are told that "two great principles were embodied to an unlimited extent:" 1. The entire control over arithmetical operations, however large, and whatever might be the number of their digits. 2. The entire control over the combinations of algebraic symbols, however lengthened those processes may be required. The inventor fairly states: "The possibility of fulfilling these two conditions might reasonably be doubted by the most accomplished mathematician as well as by the most ingenious mechanician." He then proceeds:
The difficulties which naturally occur to those capable of examining the questions, as far as they relate to arithmetic, are these: (a). The number of digits in each constant inserted in the Engine must be without limit. (b). The number of constants to be inserted in the Engine must also be without limit. (c). The number of operations necessary for arithmetic is only 4; but these 4 may be repeated an unlimited number of times. (d). These operations may occur in any order, or follow an unlimited number of laws.
Next we learn that the following conditions relate to the algebraic portion of the Analytical Engine:
(e). The number of litteral constants must be unlimited. (f). The number of variables must be without limit. (g). The combinations of the algebraical signs must be unlimited. (h). The number of functions to be employed must be without limit.
This enumeration included 8 conditions, each of which is absolutely unlimited as to the number of its combinations. Now it is obvious that no finite machine can include infinity. It is also certain that no question necessarily involving infinity can ever be converted into any other in which the idea of infinity under some shape or other does not enter. It is impossible to construct machinery occupying unlimited space; but it is possible to construct finite machinery, and to use it through unlimited time. It is this substitution of the infinity of time, for the infinity of space, which I have made use of, to limit the size of the Engine, and yet to retain its unlimited power.
The inventor then proceeds briefly to point out the means by which he had effected this change:
Since every calculating machine must be constructed for the calculation of a definite number of figures, the first datum must be to fix upon that number. In order to be somewhat in advance of the greatest number that may ever be required, I chose 50 places of figures as the standard for the Analytical Engine. The intention being that in such a machine 2 numbers, each of 50 places of figures, might be multiplied together, and the resultant product of 100 places might then be divided by another number of 50 places. It seems to me probable that a long period must elapse before the demands of science will exceed this limit.
He then enters upon a number of scientific details, for the purpose of elucidating the 8 conditions above indicated. We cannot follow these; but we may with advantage cite an occasional passage :
The same reasoning will show that if the numbers of digits of each factor are between 100 and 150, then the time required for the operation will be nearly 9 times that of a pair of factors having only 50 digits.
At another part of the machine a series of number cards, resembling those of Jacquard, but delivered to and computed by the machine itself, can be placed. These can be called for by the Engine itself in any order in which they may be placed, or according to any law the Engine may be directed to use. Hence the condition (b) is fulfilled, viz, an unlimited number of constants can be inserted in the machine in an unlimited time.
I propose in the Engine I am constructing to have places only for a 1000 constants, because I think it will be more than sufficient. But if it were required to have 10, or even a 100 times that number, it would be quite possible to make it, such is the simplicity of its structure of that portion of the Engine.
Thus it appears that the whole of the conditions which enable a finite machine to make calculations of unlimited extent are fulfilled in the A. E. The means I have adopted are uniform. I have converted the infinity of space, which was required by the conditions of the problem, into the infinity of time. The means I have employed are in daily use in the art of weaving patterns.
As soon as an A. E. exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise-By what course of calculation can these results be arrived at by the machine in the shortest time?
Having thus completely mastered the details of the proposed Analytical Engine, Mr. Babbage then completed the drawings of the Difference Engine No. 2. In the mean time certain portions of the machine had been completed, and were in working order. These latter remained in Mr. Babbage's possession for some time previous to their removal to King's College. During this period many persons of great scientific eminence saw these parts in actual operation. On one occasion there were present Dr. Lloyd, then Provost of Trinity College, and Dr. Robinson, of Armagh. Mr. Babbage proceeds to tell us what took place:
I then proceeded to explain the mechanism of the Engine, and to cause it to calculate T. One of the party remarked two axes in front of the machine which had not hitherto been performing any work, and inquired for what purpose they were so placed. I informed him that these axes had been so placed in order to illustrate a series of calculations of the most complicated kind, to which they contributed. I observed that the T. thus formed were of so artificial and abstract a nature that I could not foresee the time when they would be of any use. This remark additionally excited their curiosity, and they requested me to set the machine at work to compute such a T. Having taken a simple case of this kind, I set the Engine to do its work, and then told them, That it was now prepared to count the natural numbers; but that it would only obey this law as far as the millionth term. That after that term it would commence a series, following a different, but known law, for a very long period. That after this new law had been fulfilled for another long period, it would then suddenly abandon it, and calculate the term of a series following another new law, and so on throughout all time. Of course it was impossible to verify these assertions by making the machine actually go through the calculations; but after having made the Engine count the natural numbers for some time, I proceeded to point out the fact that it was impossible, by its very structure, that the machine could record any but the natural numbers before it reached the number 999,990. This I made evident to my friends by showing them the actual structure of the Engine. Having demonstrated this to their entire satisfaction, I put the machine on to the number 999,990, and continued to work the Engine, when the result I had predicted soon arrived. After the millionth term a new law was taken up, and my friends were convinced that it must, from the very structure of the machine, continue for a very long time, and then inevitably give place to another new law, and so on throughout all time.
He then reverts to the Analytical Engine; but we had better let Mr. Babbage pursue his own narrative:
When they were quite satisfied about this fact, I observed that in a new engine, which I was then contemplating, it would be possible to set it so that: 1. It should calculate a T. for any given length of time according to any given law. 2. That at the termination of that time it should cease to compute a T. according to that law; but that it should commence a new T. according to any other given law that might be desired, and should then continue this computation for any other given period. 3. That this succession of a new law, coming in and continuing during any desired time, and then giving place to other new laws, in endless but unknown succession, might be continued indefinitely. I remarked that I did not conceive the time could ever arrive when the results of such calculations would be of any utility. I added, however, that they offered a striking parallel with, although at an immeasurable distance from, the successive creations of animal life, as developed by the vast epochs of geological time.
I continued the subject, and pointed out the application of the same reasoning to the nature of miracles. The same machine could be set in such a manner that these laws might exist for any assigned number of times, whether large or small; also that it was not necessary that these laws should be different, but the same law might, when the machine was set, be ordered to re-appear, after any desired interval. Thus we might suppose an observer watching the machine, to see a known law continually fulfilled, until after a lengthened period, when a new law has been appointed to come in. This new law might after a single instance cease, and the first law might again be restored, and continue for another interval, when the second law might again govern the machine as before for a single instance, and then give place to the original law. This property of a mere piece of mechanism may have a parallel to the laws of human life.
But the workings of machinery run parallel to those of the intellect. The Analytical Engine might be so set, that at definite periods, known only to its maker, a certain lever might become movable during the calculations then making. The consequence of moving it might be to cause the then existing law to be violated for one or more times, after which the orig. law would resume its reign. . . . It does not clearly appear how much of the Analytical Engine was ever actually constructed. Certain portions of Difference Engine No. 2 appeared also available for the A. E. In Passages from the Life of a Philosopher, pub. 1864, Mr. Babbage says [p. 449], "If I survive some few years longer, the Analytical Engine will exist, and its works will afterwards be spread over the world." He died in Oct. 1871, never having completed the A. E.; but he had prepared for the press a work thereon, which we notice under date 1864.
The next machine we have to notice is one of Russian invention, by M. STAFFEL. The precise date of its invention we cannot ascertain. The mechanism is 18in. in length, 9in. in breadth, and 4in. in height. It consists of 3 rows of vertical cylinders: the first row contains 13; the 2nd 7; and the 3rd 7. Upon each of the cylinders in the first
row are 10 notches, corresponding with the units 1 to 10. Within each of these cylinders is a small pulley, in connexion with a lever, set in motion by a slider, which, when the cylinder has been turned from either 9 to 0, or o to 9, sets in motion the lever, and communicates its action to wheels which carry over the figures. The pulley connected with the cylinder the furthest from the handle, is in connexion with the hammer of a bell. The purpose of this bell is to give warning to the operator on committing an error, and constitutes a most important add. to the machine, particularly in the operation of division. Upon each of the cylinders in the 2nd row 10 units are placed. These 7 cylinders are so fixed upon their axes, that they can bodily be moved right and left, and fixed at any part, so that the cyphers on the two cylinders can be made to correspond. This cylinder is furnished with a spike, which lays hold of and works the third row of cylinders. The internal communication of each of the parts is brought about by means of a connecting wheel, furnished with 9 movable pegs, which are set in motion by means of an excentric incision in the dial.
The machine is capable of performing addition, subtraction, multiplication, and division; and of extracting the square root. The operation of addition is performed as follows: By simply placing one line of numbers upon the second row of cylinders (the index pointing to addition), and turning the handle till it stops, these numbers are trans. almost instantly to the first row of cylinders, and so on successively till all the numbers to be added are trans., and their sum is shown on the top row. In performing subtraction, the first part of the operation is the same as in addition, but on placing the 2nd line of of figures on the 2nd row of cylinders, the pointer being placed to subtraction, the handle is turned the opposite way, or against the motion of the sun, and the difference of the two numbers is shown on the upper line. The operation of multiplication is performed by placing the multiplier and the multiplicand on the 2nd and 3rd rows of cylinders, and then, the index pointing to multiplication, the product will be found on the first cylinder. The operation of division is very similar, excepting that the handle is turned as in
In the performance of the square root, the following add. mechanism is brought into play. Between every division of the cylinder, in row 2, a small wheel is placed, and near it a projecting piece which acts upon a lever. When the projecting piece is near the word "rod" engraved on the cylinder, on turning the handle the figures increase by I. This by other mechanism is connected with the other two rows of cylinders. The operation of the square root is performed directly, without any guessing at numbers; but is comparatively rather a long process.
M. Staffel has also invented a small machine for the performance of the add. and subtraction of fractions, whose denominators are 10, 12, and 15. By enlarging the machine, this number could be increased, and the power of the instrument extended. [See 1851.]
About 1850 M. THOMAS (de Colmar) invented a small machine adapted for the performance of the first four rules of arithmetic, and indirectly capable of being made to extract the square root. The instrument consists of two rows of cylinders, the first row containing 16, the 2nd 8-the former are movable, the operation at each step being changed tenfold. The principle of the instrument is that multiplication is in reality the continual addition of itself as many times as there are units in the multiplier; and division that of continued subtraction of the divisor. The instrument is adapted for the multiplication of numbers whose product is expressed by less than 16 figures. To multiply 5321 by 3256 the following is the process: The first number is placed on one series of cylinders; the number 6 is placed on one of the cylinders of the 2nd row. On the handle being turned (in this machine always in the same direction), the number 31,926 appears; the upper row is moved through one division; the handle turned again, and so on till in a very short time the number 17,325, 176 appears. [See 1851.]
This instrument, now called the "Arithmometer," and which is about 18in. long, by 6 in. wide, and 11⁄2 in. deep, has been greatly improved since it was first invented; and it is now in very general use in England and in France. [See 1871.}
HERR WERTHEIMER has invented several calculating machines, adapted for the performance of addition and subtraction of numbers and moneys of this and other countries. Each machine consists of a box, with a metal plate divided into 9 indexes, with semicircular notches, under which are placed a succession of holes. Round the indexes numbers are engraved, and the semicircular notches are furnished with teeth, and a pointer to insert between the notches, for the purpose of bringing the notch opposite any particular figure, from right to left. This latter operation is attended with some hazard, and may cause inaccuracy. For this and other reasons the machines have not come into general They are however very ingenious. [See 1851.]
M. SCHILT, a Swiss, has invented a simple calculating machine, or instrument, which can perform the first operation of arithmetic only.
M. LALANNE, a Frenchman, has invented a calculating rule, upon new principles. It consists of a graphic table formed entirely of right lines, with which all calculations usually performed by the sliding rule can be performed to within 1-2001 of the true result. In 1851 came about the great gathering of the mechanical and other arts of the world.