Section 9 of Passages from the Life of a Philosopher This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org Passages from the Life of a Philosopher by Charles Babbage Chapter 8 of the Analytical Engine Man wrongs and time avenges Byron, the prophecy of Dante 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 A vague glimpse even of an analytical engine at length opened out and I pursued with enthusiasm the shadowy vision The drawings and the experiments were of the most costly kind. Draftsmen of the highest order were necessary to economize the labour of my own head whilst skilled workmen were required to execute the experimental machinery to which I was obliged constantly to have recourse In order to carry out my pursuits successfully I had purchased a house with above a quarter of an acre of ground in a very quiet locality. My coach house was now converted into a forge and a foundry whilst my stables were transformed into a workshop I built other extensive workshops myself and had a fireproof building for my drawings and draftsmen Having myself worked with a variety of tools and having studied the art of constructing each of them, I at length laid it down as a principle that, except in rare cases I would never do anything myself if I could afford to hire another person who could do it for me. The mechanical notation The complicated relations which then arose amongst the various parts of the machinery would have baffled the most tenacious memory I overcame that difficulty by improving and extending a language of signs, the mechanical notation, which in 1826 I had explained in a paper printed in the fill trans. By such means I succeeded in mastering trains of investigation so vast in extent that no length of years ever allotted to one individual could otherwise have enabled me to control. By the aid of the mechanical notation, the analytical engine became a reality, for it became susceptible of demonstration. Such works could not be carried on without great expenditure. The fluctuations in the demand and supply of skilled labour were considerable. The railroad mania withdrew from other pursuits the most intellectual and skillful draftsmen. One who had for some years been my chief assistant was tempted by an offer so advantageous that in justice to his own family he could scarcely have declined it. Under these circumstances I took into consideration the plan of advancing his salary to one guinea per day. Whilst this was in compliance, I consulted my venerable surviving parent. When I had fully explained the circumstances, my excellent mother replied, My dear son, you have advanced far in the accomplishment of a great object which is worthy of your ambition. You are capable of completing it. My advice is pursue it even if it should oblige you to live on bread and cheese. This advice entirely accorded with my own feelings. I therefore retained my chief assistant at his advanced salary, carrying the tens by anticipation. 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 affect the carriage. Twenty or thirty different plans and modifications had been drawn. At last I came to the conclusion that I had exhausted the principle of successive carriage. I concluded also that nothing but teaching the engine to foresee and then to act upon that 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. One morning where I had spent many hours in the drawing office in endeavouring to improve the system of successive carriages, I mentioned these views to my chief assistant and added that I should retire to my library and endeavour to work out the new principle. He gently expressed a doubt whether the plan was possible, to which I replied that not being able to approve its impossibility, I should follow out a slight glimmering of light which I thought I perceived. After about three hours examination I returned to the drawing office with much more definite ideas upon the subject. I had discovered a principle that proved the possibility and I had contrived mechanism which I thought would accomplish my object. I now commenced the explanation of my views which I soon found were but little understood by my assistant. Nor was this surprising since in the course of my own attempt at explanation I found several defects in my plan and was also led by his questions to perceive others. All these I removed one after another and ultimately terminated as a late hour my morning's work with the conviction that anticipating carriage was not only within my power but that I had devised one mechanism at least by which it might be accomplished. Many years after my assistant on his return from a long residence abroad called upon me and we talked over the progress of the analytical engine. I referred back to the day on which I had made that most important step and asked him if he recollected it. His reply was that he perfectly remembered the circumstance for that on retiring to my library he seriously thought that my intellect was beginning to become deranged. The reader may perhaps be curious to know how I spent the rest of that remarkable day. After working as I constantly did for ten or eleven hours a day I had arrived at this satisfactory conclusion and was revising the rough sketches of the new contrivance when my servant entered the drawing office and announced that it was seven o'clock that I dined in Park Lane and that it was time to dress. I usually arrived at the house of my friend about a quarter of an hour before the appointed time in order that we might have a short conversation on subjects on which we were both much interested. Having mentioned my recent success in which my host thoroughly sympathised I remarked that it had produced an exhilaration of the spirits which not even his excellent champagne could rival. Having enjoyed the society of Hallam, of Rogers and of some few others of that delightful circle I retired and joined one or perhaps two much more extensive reunions. Having thus forgotten science and enjoyed society for four or five hours I returned home. About one o'clock I was asleep in my bed and thus continued for the next five hours. This new and rapid system of carrying the tens when two numbers are added together reduced the actual time of the addition of any number of digits however large to nine units of time for the addition 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 five or six were required for making the requisite previous arrangements. Having thus advanced as nearly as seemed possible to the minimum of time requisite for our 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 volumes. I only propose here to indicate a few of its more important functions and to give to those whose minds are duly prepared for it some information which will remove those vague notions of wonder and even of its impossibility with which it is surrounded in the minds of some of the most enlightened. Jacquard Loom. To those who are acquainted with the principles of the Jacquard Loom and who are also familiar with analytical formulae a general idea of the means by which the engine executes its operations may be obtained without much difficulty. In the exhibition of 1862 there were many splendid examples of such looms. It is known as a fact that the Jacquard Loom is capable of weaving any design which the imagination of man can 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 these cards are placed in a Jacquard Loom it will then weave upon its produce the exact pattern designed by the artist. Weaving formulae. Now the manufacturer may use, for the warp and weft of his work, threads which are all of the same colour. Let us suppose them to be unbleached or white threads. In this case the cloth will be woven all of one colour but there will be a damask pattern on it such as the artist designed. But the manufacturer might use the same cards and put into the warp threads of any other colour. Every thread might even be of a different colour. Or of a different shade of colour. But in all these cases the form of the pattern will be precisely the same. The colours only will differ. The analogy of the analytical engine with this well known process is nearly perfect. The analytical engine consists of two parts. First 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. Second 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 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 in the variables into the mill the order in which they are required to be acted upon. Each operation card will require three 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 10 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. Law of development the analytical engine is therefore a machine of the most general nature whatever formula it is required to develop 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 its 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 analytical engine 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 which is perhaps difficult to apprehend these descriptions without a familiarity both with analytical forms and mechanical structures. I will now therefore confine myself to the mathematical view of the analytical engine and illustrate by example some of its supposed difficulties. An excellent friend of mine the late Professor McCulloch of Dublin was discussing with me at breakfast the various powers of the analytical engine. 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 trigonometric operations. Its use of tables my answer was that whenever the analytical engine should exist all the developments of formula 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 tables discovers a mistake. I explained that the tables 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 table 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 the 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 of course he would have to replace it by the right one. Upon this Professor McCulloch naturally asked why if the machine could tell whether the logarithm was the right one it should have asked the attendant at all. I told him that the means employed were so ridiculously simple that I would not at that moment explain them. But that if he would come again in the course of a few days I should be ready to explain it. Three or four days after Bessel and Jacobi who had just arrived in England were sitting with me inquiring about the analytical engine when fortunately my friend McCulloch was announced. The meeting was equally agreeable to us all and we continued our conversation. After some time Bessel put to me the very same question which McCulloch had previously asked. On this Jacobi remarked that he too was about to make the same inquiry when Bessel had asked the question. I then explained to them the following very simple means by which that verification was accomplished. Knows what it wants. Besides the set 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. Any number which the analytical engine is capable of using or of producing can if required be expressed by a card with certain holes in it. The card illustrated contains 11 vertical rows for holes each row having 9 or any less number of holes. In this example the tabular number is 3622939 whilst its number in the order of the table is 2303. In fact the former number is the logarithm of the letter. The analytical engine will contain first apparatus for printing on paper one or if required two copies of its results. Second means for producing a stereotype mold of the tables or results it computes. Third mechanism for punching on blank pasteboard cards or metal plates the numerical results of any of its computations stops and rings a bell. Of course the engine will compute all the tables which it may if be required to use. These cards will therefore be entirely free from error. Now when the engine requires a tabular number it will stop ring a bell and ask for such number. In the case we have assumed it asks for the logarithm of 2303. When the attendant has placed a tabular card in the engine the first step taken by it will be to verify the number of the card given it by subtracting its number from 2303 the number whose logarithm it asked for. If the remainder is zero then the engine is certain that the logarithm must be the right one since it was computed and punched by itself. Thus the analytical engine first computes and punches on cards its own tabular numbers. These are brought to it by its attendant when demanded but the engine itself takes care that the right card is brought to it by verifying the number of that card by the number of the card which it demanded. The engine will always reject a wrong card by continually ringing a loud bell and stopping itself until supplied with the precise intellectual food it demands. It will be an interesting question which time only can solve to know whether such tables of cards will ever be required for the engine. 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 probable that it will make shorter work by computing directly from proper formula than by having recourse even to its own tables. The analytical engine I propose will have the power of expressing every number it uses to 50 places of figures. It will multiply any two such numbers together and then if required divide the product of 100 figures by number of 50 places of figures. Arithmetical difficulties supposing the velocity of the moving parts of the engine to be not greater than 40 feet per minute. I have no doubt that 60 additions or subtractions may be completed and printed in one minute. One multiplication of two numbers each of 50 figures in one minute. One division of a number having 100 places of figures by another of 50 in one minute. In the various sets of drawings of the modifications of the mechanical structure of the analytical engines already numbering upwards of 30 two great principles were embodied to an unlimited extent. First the entire control over arithmetical operations however large and whatever might be the number of their digits. Second the entire control over the combinations of algebraic symbols however lengthened those processes may be required. The possibility of fulfilling these two conditions might reasonably be doubted by the most accomplished mathematician as well as by the most ingenious mecanition. The difficulties which naturally occur to those capable of examining the question 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 four but these four may be repeated an unlimited number of times. D. These operations may occur in any order or follow an unlimited number of laws. Algebraical difficulties the following conditions relate to the algebraic portion of the analytical engine. E. The number of literal constants must be unlimited. F. The number of variables must be without limit. G. The combinations of the algebraic signs must be unlimited. H. The number of functions to be employed must be without limit. This enumeration includes eight 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. A. I shall now proceed briefly to point out the means by which I have affected this change. Larger numbers treated. Since every calculating machine must be constructed for the calculation of a definite number of figures first data 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 two 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. To this it may be added that the addition and subtraction of numbers in an engine constructed for n places of figures would be equally rapid whether n were equal to 5 or 5,000 digits. With respect to multiplication and division the time required is greater. Thus if A times 10 to the 50 plus B and A prime times 10 to the 50 plus B prime are two numbers each of less than 100 places of figures then each can be expressed upon two columns of 50 figures and A, B, A prime and B prime are each less than 50 places of figures. They can therefore be added and subtracted upon any column holding 50 places of figures. The product of two such numbers is A A prime 10 to the 100 plus A B prime 10 to the 50 plus B B prime. This expression contains four pairs of factors A A prime A B prime A prime B B prime each factor of which has less than 50 places of figures each multiplication can therefore be executed in the engine the time however of multiplying two numbers each consisting of any number of digits between 50 and 100 will be nearly four times as long as that of two such numbers of less than 50 places of figures. The same reasoning will show that if the number of digits of each factor are between 100 and 150 then the time required for the operation will be nearly nine times that of a pair of factors having only 50 digits. Thus it appears that whatever may be the number of digits the analytical engine is capable of holding. If it is required to make all the computations with k times that number of digits then it can be executed by the same engine but in an amount of time equal to k squared times the former. Hence the condition A or the unlimited number of digits contained in each constant employed is fulfilled. It must however be admitted that this advantage is gained at the expense of diminishing the number of the constants the engine can hold. An engine of 50 digits when used as one of 100 digits can only contain half the number of variables. An engine containing m columns each holding n digits if used for computations requiring k n digits can only hold m over k constants or variables of punching cards b. The next step is therefore to prove b. Fizz to show that a finite engine can be used as if it contained an unlimited number of constants. The method of punching cards for tabular numbers has already been alluded to. Each analytical engine will contain one or more apparatus for printing any numbers put into it and also an apparatus for punching on base board cards the holes corresponding to those numbers. At another part of the machine a series of number cards resembling those of jackard but delivered to and computed by the machine itself can be placed or according to any law the engine may be directed to use. Hence the condition b is fulfilled. Namely 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 for only a thousand constants because I think it will be more than sufficient. But if it were required to have ten or even a hundred times that number it would be quite possible to make it such as the simplicity of its structure of that portion of the engine. A thousand variables. C. The next stage in the arithmetic is the number of times the four processes of addition, subtraction, multiplication and division can be repeated. It is obvious that four different cards would give the orders for the four rules of arithmetic. Now there is no limit to the number of such cards which may be strung together according to the nature of the operations required. Consequently the condition c is fulfilled d. The fourth arithmetical condition d that the order of succession in which these operations can be varied is itself unlimited follows as a matter of course. The four remaining conditions which must be fulfilled in order to render the analytical engine as general as the science of which it is the powerful executive relate to algebraic quantities with which it operates. The thousand columns each capable of holding any number of less than 51 places of figures may each represent a constant or a variable quantity. These quantities I have called by the comprehensive title of variables and have denoted them by v subscript n with an index below. In the machine I have designed n may vary from 0 to 999 but after any one or more columns have been used for variables if those variables are not required afterwards they may be printed upon paper and the columns themselves again used for other variables. In such cases the variables must have a new index thus m v n are proposed to make n vary from 0 to 99. If more variables are required these may be supplied by variable cards which may follow each other in unlimited succession. Each card will cause its symbol to be printed with its proper indices for the sake of uniformity I have used v with as many indices as may be required throughout the engine. This however does not prevent the printed result of a development from being represented by any letters which may be thought to be more convenient. In that part in which the results are printed type of any form may be used according to the taste of the proposer of the question. It thus appears that the two conditions e and f which require that the number of constants and variables should be unlimited are both fulfilled. The condition g requiring that the number of combinations of the four algebraic signs shall be unlimited is easily fulfilled by placing them on cards in any order of succession the problem may require. The last condition h namely that the number of functions to be employed must be without limit might seem at first sight to be difficult to fulfill. But when it is considered that any function of any number of operations performed upon any variables is but a combination of the four simple signs of operation with various quantities it becomes apparent that any function whatever may be represented by two groups of cards the first being signs of operation placed in the order in which they succeed each other and the second group of cards representing the variables and constants placed in the order of succession in which they are acted upon by the former. End of section 9. Section 10 of passages from the life of a philosopher. This is a LibriVox recording all LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org recording by Avayee in June 2019 passages from the life of a philosopher by Charles Babbage section 10 of the analytical engine part 2. A finite machine may make unlimited calculation thus it appears that the whole of the conditions which enable a finite machine to make calculations of unlimited extent are fulfilled in the analytical engine. 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. It is accomplished by systems of cards punched with various holes strung together to any extent which may be demanded. Two large boxes, the one empty and the other filled with perforated cards are placed before and behind a polygonal prism which revolves at intervals upon its axis and advances through a short space after which it immediately returns. A card passes over the prism just before each stroke of the shuttle the cards that have passed hang down until they reach the empty box placed to receive them into which they arrange themselves one over the other. When the box is full another empty box is placed to receive the coming cards and a new full box on the opposite side replaces the one just emptied. As the suspended cards on the entering side are exactly equal to those on the side at which the others are delivered they are perfectly balanced so that whether the formulae to be computed be excessively complicated or very simple the force to be exerted always remains nearly the same. Discussions at Turin In 1840 I received from my friend Monsieur Planat a letter pressing me strongly to visit Turin at the then approaching meeting of Italian philosophers in that letter Monsieur Planat stated that he had inquired anxiously of many of my countrymen about the power and mechanism of the analytical engine. He remarked that from all the information he could collect the case seemed to stand thus. He the two the legislative department of our analysis has been all powerful the executive all feeble. Your engine seems to give us the same control over the executive which we have hitherto only possessed over the legislative department. Considering the exceedingly limited information which could have reached my friend respecting the analytical engine I was equally surprised and delighted at his exact provisions of its powers. Even at the present moment I could not express more clearly and in fewer terms its real object. I collected together such of my models drawings and notations as I conceived to be best adapted to give an insight into the principles and mode of operating of the analytical engine. On mentioning my intention to my excellent friend the late Professor McCullough he resolved to give up a trip to the Tyrol and join me at Turin. We met at Turin at the appointed time and as soon as the first bustle of the meeting had a little abated I had the great pleasure of receiving at my own apartments for several mornings Messers Planar, Menabrea, Mossotti, McCullough, Plantamore and others of the most eminent geometers and engineers of Italy. Around the room were hung the formula, the drawings, notations and other illustrations which I had brought with me. I began on the first day to give a short outline of the idea. My friends asked from time to time further explanations of parts I had not made sufficiently clear. Messers Planar had at first proposed to make notes in order to write an outline of the principles of the engine. But his own laborious pursuits induced him to give up this plan and to transfer the task to a younger friend of his, Messers Menabrea, who had already established reputation as a profound analyst. These discussions were of great value to me in several ways. I was thus obliged to put into language the various views I had taken and I observed the effect of my explanations on different minds. My own ideas became clearer and I profited by many of the remarks made by my highly gifted friends. Mosotti's Difficulty One day Mosotti, who had been unavoidably absent from the previous meeting when a question of great importance had been discussed, again joined the party. Well aware of the acuteness and rapidity of my friends' intellect, I asked my other friends to allow me five minutes to convey to Professor Mosotti the substance of the preceding sitting. After putting a few questions to Mosotti himself, he placed before me distinctly his greatest difficulty. He remarked that he was now quite ready to admit the power of mechanism over numerical and even over algebraical relations, to any extent. But he added that he had no conception how the machine could perform the act of judgment sometimes required during an analytical inquiry when two or more different courses presented themselves, especially as the proper course to be adopted could not be known in many cases until all the previous portion had been gone through. Solution of Equations I then inquired whether the solution of a numerical equation of any degree by the usual but very tedious proceeding of approximation would be a type of the difficulty to be explained. He had once admitted that it would be a very eminent one. For the sake of perspicuity and brevity, I shall confine my present explanation to possible roots. I then mentioned the successive stages. Number of Operation Cards Used 1 a. Assertain the number of possible roots by applying Sturm's Theorem through the coefficients 2 b. Find a number greater than the greatest root 3 c. Substitute the powers of 10 commencing with that next greater than the greatest root and diminishing the powers by unity at each step for the value of x in the given equation. Continue this until the sign of the resulting number changes from positive to negative. The index of the last power of 10, call it n, which is positive, expresses the number of digits in that part of the root which consists of whole numbers. Call this index n plus 1. 4 d. Substitute successively for x in the original equation 0 times 10 to the power of n, 1 times 10 to the power of n, 2 times 10 to the power of n, 3 times 10 to the power of n, etc. until 9 times 10 to the power of n, until a change of sign occurs in the result. The digit previously substituted will be the first figure of the root sort. 5 e. Transform the original equation into another whose roots are less by the number thus found. The transformed equation will have a real root, the digit less than 10 to the power of n. 5 substitute 1 times 10 to the power of n minus 1, 2 times 10 to the power of n minus 1, 3 times 10 to the power of n minus 1, etc. successively for the root of this equation until a change of sign occurs in the result as in process 4. This will give the second figure of the root. This process of alternately finding a new figure in the root and then transforming the equation into another as in process 4 and 5 must be carried on until as many figures as are required where the whole numbers or decimals are arrived at. 7 g. The root thus found must now be used to reduce the original equation to one dimension lower. 8 h. This new equation of one dimension lower must now be treated by sections 3, 4, 5, 6 and 7 until the new root is found. 9 i. The repetition of sections 7 and 8 must go on until all the roots have been found. Now it will be observed that Professor Mosotti was quite ready to admit at once that each of these different processes could be performed by the analytical machine through the medium of properly arranged sets of jacquard cards. His real difficulty consisted in teaching the engine to know when to change from one set of cards to another and back again repeatedly at intervals not known to the person who gave the orders. The dimensions of the algebraic equation being known the number of arithmetical processes necessary for Sturm's theorem is consequently known. A set of operation cards can therefore be prepared. These must be accompanied by a corresponding set of variable cards which will represent the columns in the store on which the several coefficients of the given equation and the various combinations required amongst them are to be placed. The next stage is to find a number greater than the greatest root of the given equation. There are various courses for arriving at such a number. Any one of these being selected, another set of operation and variable cards can be prepared to execute this operation. Now as this second process invariably follows the first, the second set of cards may be attached to the first set and the engine will pass on from the first to the second process and again from the second to the third process. But here a difficulty arises. Successive powers of 10 are to be substituted for x in the equation until a certain event happens. A set of cards may be provided to make the substitution of the highest power of 10 and similarly for the others. But on the occurrence of a certain event, namely the change of a sign from plus to minus, this stage of the calculation is to terminate. Now, at a very early period of the inquiry, I had found it necessary to teach the engine to know when any numbers it might be computing pass through zero or infinity. The passage through zero can be easily ascertained thus. Let the continually decreasing number which is being computed be placed upon a column of wheels in connection with a carrying apparatus. After each process this number will be diminished until at last the number is subtracted from it which is greater than the number expressed on those wheels. Thus let it be five zeros five zeros five zeros zero zero four two three. Subtract five zeros five zeros five zeros zero zero five one one. To result five nines five nines five nines nine nine nine one two. Now in every case of a carriage becoming due a certain lever is transferred from one position to another in the cage next above it. Consequently in the highest cage of all say the fiftieth in the analytical engine an arm will be moved or not moved accordingly as the carriages do or do not run up beyond the highest wheel. This arm can of course make any change which has previously been decided upon. In the instance we have been considering it would order the cards to be turned on to the next set. If we wish to find when any number which is increasing exceeds in the number of its digits the number of wheels on the columns of the machine the same carrying can be employed. Hence any directions may be given which the circumstances require. It will be remarked that this does not actually prove even in the analytical engine of 15 figures that the number computed has passed through infinity but only that it has become greater than any number of 50 places of figures. There are however methods by which any machine made for a given number of figures may be made to compute the same formulae with double or any multiple of its original number. But the nature of this work prevents me from explaining that method. It may here be remarked that in the process the cards employed to make the substitutions of the powers of 10 are operation cards. They are therefore quite independent of the numerical values substituted. Hence the same set of operation cards which order the substitutions 1 times 10 to the power of n will, if backed, order the substitution of 2 times 10 to the power of n etc. We may therefore avail ourselves of a mechanism for backing these cards and call it into action whenever the circumstances themselves require it. The explanation of Monsieur Mosotti's difficulty is this mechanical means have been provided for backing or advancing the operation cards to any extent. There exist means of expressing the conditions under which these various processes are required to be called into play. It is not even necessary that two courses only should be possible. Any number of courses may be possible at the same time and the choice of each may depend upon any number of conditions. General Menabria's description It was during these meetings that my highly valued friend Monsieur Menabria collected the materials for that lucid and admirable description which he subsequently published in the Bibliothèque universelle de Genève, Tom 61 October 1842. The elementary principles on which the analytical engine rests were thus in the first instance brought before the public by General Menabria. The Countess of Loveless's notes Some time after the appearance of his memoir on the subject in the Bibliothèque universelle de Genève the late Countess of Loveless informed me that she had translated the memoir of Menabria. Footnote Ada Augusta, Countess of Loveless, only child of the poet Byron. Footnote I asked why she had not herself written an original paper on a subject with which she was so intimately acquainted. To this lady Loveless replied that the sort had not occurred to her. I then suggested that she should add some notes to Menabria's memoir, an idea which was immediately adopted. We discussed together the various illustrations that might be introduced. I suggested several, but the selection was entirely her own. So also was the algebraic working out of the different problems, except indeed that relating to the numbers of Bernoulli which I had offered to do to save Lady Loveless the trouble. This she sent back to me for an amendment having detected a grave mistake which I had made in the process. The notes of the Countess of Loveless extend to about three times the length of the original memoir. Their author has entered fully into almost all the very difficult and abstract questions connected with the subject. These two memoirs taken together furnish to those who are capable of understanding the reasoning a complete demonstration that the whole of the developments and operations of analysis are now capable of being executed by machinery. Various applications. There are various methods by which these developments are arrived at. One by the aid of the differential and integral calculus. Two by the combinatorial analysis of Hindenburg. Three by the calculus of derivations of Arbogast. Each of these systems professes to expand any function according to any laws. Theoretically each method may be admitted to be perfect, but practically the time and attention required are, in the greater number of cases, more than the human mind is able to bestow. Consequently upon several highly interesting questions relative to the Loona theory some of the ablest and most indefatigable of existing analysts are at variance. The analytical engine is capable of executing the laws prescribed by each of these methods. At one period I examined the combinatorial analysis and also took some pains to ascertain from several of my German friends who had had far more experience of it than myself whether it could be used with greater facility than the differential system. They seemed to think that it was more readily applicable to all the usual wants of analysis. I have myself worked with the system of Arbogast and if I were to decide from my own limited use of the three methods, I should, for the purposes of the analytical engine, prefer the calcul de derivation. As soon as an analytical engine 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? In the drawings I have prepared, I proposed to have a thousand variables upon each of which any number not having more than 50 figures can be placed. This machine would multiply 50 figures by other 50 and print the product of 100 figures or it would divide any number having 100 figures by any other of 50 figures and print the quotient of 50 figures. Allowing but a moderate velocity for the machine, the time occupied by either of these operations would be about one minute. The whole of the numerical constants throughout the works of Laplace, Planat, Le Verrier, Hansen and other eminent men, whose indefatigable labours have brought astronomy to its present advanced state, might easily be recomputed. They are but the numerical coefficients of the various terms of functions developed according to certain series. In all cases in which these numerical constants can be calculated by more than one method, it might be desirable to compute them by several processes until frequent practice shall have confirmed our belief in the infallibility of mechanism. Errors of Tables The great importance of having accurate tables is admitted by all who understand their uses, but the multitude of errors really occurring is comparatively little known. Dr. Lardner in the Edinburgh Review has made some very instructive remarks on this subject. I shall mention too within my own experience. These are selected because they occurred in works when neither care nor expense was spared on the part of the government to ensure perfect accuracy. It is however but just to the eminent men who presided of the preparation of these works for the press to observe that the real fault lay not in them but in the nature of things. In 1828 I lent the government an original manuscript of the table of logarithmic signs, cosines, etc. computed to every second of the quadrant in order that they might have it compared with Taylor's logarithms, Forreth Thum 1795, of which they possessed a considerable number of copies. Nineteen errors were thus detected, and a list of these errors was published in the nautical almanac for 1832. These may be called 19-errata of the first order 1832. An error being detected in one of these errata in the following nautical almanac we find an erratum of the errata in nautical almanac 1832, 1833. But in this very erratum of the second order a new mistake was introduced larger than any of the original mistakes. In the year next following their order have been found erratum in the erratum of the errata in nautical almanac 1832, 1834. In the Table de la Lune by Monsieur P. A. Hansen, Forreth Thum 1857 published at the expense of the English government under the direction of the Astronomer Royal is to be found a list of errata amounting to 155. In the 21st of these original errata there have been found 3 mistakes. These are duly noted in a newly printed list of errata discovered during computations made with them in the nautical almanac so that we now have the errata of an erratum of the original work. This list of errata from the office of the nautical almanac is larger than the original list. The total number of eras at present 1862 discovered in Hansen's Tables of the Moon amounts to above 350. In making these remarks I have no intentions of imputing the slightest blame to the Astronomer Royal who, like other men cannot avoid submitting to inevitable fate. The only circumstance which is really extraordinary is that when it was demonstrated that all tables are capable of being computed by machinery and even when a machine existed which computed certain tables that the Astronomer Royal did not become the most enthusiastic supporter of an instrument which could render such invaluable service to his own science. In the supplementary notices of the Astronomical Society Number 9, Volume 23, page 259, 1863 there occurs a paper by Monsieur G. de Pontacoulon in which 49 numerical coefficients relative to the longitude, latitude and radius vector of the moon are given as computed by Planar, Diloni and Pontacoulon. The computations of Planar and Pontacoulon agree in 13 cases, those of Diloni and Pontacoulon in two and in the remaining 34 cases they all three differ. Remarks on analysis. I am unwilling to terminate this chapter without reference to another difficulty now arising which is calculated to impede the progress of analytical science. The extension of analysis is so rapid, its domain so unlimited and so many inquirers are entering into its fields that a variety of new symbols have been introduced, formed on no common principles. Many of these are merely new ways of expressing well-known functions. Unless some philosophical principles are generally admitted as the basis of all notation there appears a great probability of introducing the confusion of Babel into the most accurate of all languages. A few months ago I turned back to a paper in the Philosophical Transactions 1844 to examine some analytical investigations of great interest by an author who has thought deeply on the subject. It related to the separation of symbols of operation from those of quantity, a question peculiarly interesting to me since the analytical engine contains the embodiment of that method. There was no ready sufficient and simple mode of distinguishing letters which represented quantity from those which indicated operation. To understand the results the author had arrived at it became necessary to read the whole memoir. Although deeply interested in the subject I was obliged with great regret to give up the attempt for it not only occupied much time but placed too great a strain on the memory. Whenever I am thus perplexed it has often occurred to me that the very simple plan I have adopted in my mechanical notation for lettering drawings might be adopted in analysis. On the geometrical drawings of machinery every piece of matter which represents framework is invariably denoted by an upright letter whilst all letters indicating movable parts are marked by inclined letters. The analogous rule would be let all letters indicating operations or modifications be expressed by upright letters whilst all letters representing quantity should be represented by inclined letters. The subject of the principles and laws of notation is so important that it is desirable before it is too late that the scientific academies of the world should each contribute the results of their own examination and conclusions and that some congress should assemble to discuss them. Perhaps it might be still better if each academy would draw up its own views illustrated by examples and have a sufficient number printed to send to all other academies. End of section 10. Section 11 of Passages from the Life of a Philosopher. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org, recording by Avae in May 2019. Passages from the Life of a Philosopher by Charles Babbage. Chapter 11 of the Mechanical Notation. Soon after I had commenced the difference engine my attention was strongly directed to the imperfection of all known modes of explaining and demonstrating the construction of machinery. It soon became apparent that my progress would be seriously impeded unless I could devise more rapid means of understanding and recalling the interpretation of my own drawings. By a new system of very simple signs I ultimately succeeded in rendering the most complicated machine capable of explanation almost without the aid of words. In order thoroughly to understand the action of any machine we must have full information upon the following subjects and it is of the greatest importance that this information should be acquired in the shortest possible time. 1. The actual shape and relative position of every piece of matter of which the machine is composed. This can be accomplished by the ordinary mechanical drawings. Such drawings usually have letters upon them for the sake of reference in the description of the machine. Hitherto such letters were chosen without any principles and in fact gave no indication of anything except the mere spot upon the paper on which they were written. Rules for lettering I then laid down rules for the selection of letters. I shall only mention one or two of them. 1. All upright letters as in A, C, D, E, capital A, capital B represent framing. 2. All inclined letters as A, C, D, E, capital A, capital B represent movable parts. 3. All small letters represent working points. One of the most obvious advantages of these rules is that they enable the attention to be more easily confined to the immediate object sought. By other rules it is rendered possible when looking at a plan of any complicated machine to perceive the relative order of superposition of any number of wheels, arms etc. without referring to the elevation or end view. 2. The actual time and duration of any motion throughout the action of any machine can be ascertained almost instantly by a system of science called the Notations of Periods. It possesses equal facilities for ascertaining every contemporaneous as well as for every successive system of movements. 3. The actual connection of each movable piece of the machine with every other on which it acts. Thus, taking from any special part of the drawing the indicating letter and looking for it on a certain diagram called the Trains the whole course of its movement may be traced up to the prime mover or down to the final result. I have called this system of science the Mechanical Notation. By its application to geometrical drawing it has given us a new demonstrative science namely that of proving that any given machine can or cannot exist and if it can exist that it will accomplish its desired object. It is singular that this addition to human knowledge should have been made just about the period when it was beginning to be felt by those most eminently skilled in analysis that the time has arrived when many of its conclusions rested only on probable evidence. This state of things arose chiefly from the enormous extent to which developments were necessarily carried in the lunar and planetary theories. Astronomical Medal After employing this language for several years it was announced in December 1825 that King William IV had founded two medals of 50 guineas each to be given annually by the Royal Society according to rules to be laid down by the council. On the 26th January 1826 it was resolved that it is the opinion of the council that the medals be awarded for the most important discoveries or series of investigations completed and made known to the Royal Society in the year preceding the day of the award. This rule reduced the number of competitors to a very few although I had had some experiences to the mode in which medals were awarded and therefore valued them accordingly I was simply enough to expect that the council of the Royal Society would not venture upon a fraud on the very first occasion of exercising the Royal Liberality. I had also another motive for taking a ticket in this philosophical lottery of medals. Royal Society Medal In 1824 the Astronomical Society did me the honour to award to me the first gold medal they ever bestowed. It was rendered still more grateful by the address of that eminent men the late Henry Thomas Colebrook, the president, who in a spirit of prophecy anticipated the results of years at that period long future. It may not therefore be deemed too sanguine in anticipation when I expressed the hope that an instrument which in its simpler form attains to the extraction of the roots of numbers and approximates to the roots of equations may, in a more advanced state of improvement rise to the approximate solutions of algebraic equations of elevated degrees. I refer to solutions of such equations proposed by Lagrange and more recently by other analysts which involve operations too tedious and intricate for use and which must remain without efficacy unless some mode be devised of abridging the labour or facilitating the means of performance. Footnote The scores of the president on delivering the first gold medal of the Astronomical Society to Charles Babbage as choir. Memoirs of the Astronomical Society Volume 1, page 509 End footnote I felt therefore that the first royal medal might fairly become an object of ambition whatever might be the worth of subsequent ones. In order to qualify myself for this chance I carefully drew up a paper on a method of expressing by science the action of machinery which I otherwise should not have published at that time. This memoir was read at the Royal Society on the 16th March 1826. To the system of science which it first expounded I afterwards gave the name of mechanical notation. It had been used in England and in Ireland although not taught in its schools. It applies to the description of a combat by sea or by land. It can assist in representing the functions of animal life and I have had both from the continent and from the United States specimens of such applications. Finally, to whatever degree of simplicity I may at last have reduced the analytical engine the course through which I arrived at it was the most entangled and perplexed which probably ever occupied the human mind. Through the aid of the mechanical notation I examined numberless plans and systems of computing and I am sure from the nature of its self-necessary verifications that it is impossible I can have been deceived. On the 16th November 1826 that very council of the Royal Society which had made the law took the earliest opportunity to violate it by awarding the two royal medals. The first to Dalton whose great discovery had been made nearly 20 years before and the other to Ivory for a paper published in their transactions three years before. The history of their proceedings will be found in The Decline of Science in England page 115, 1830 End of section 11 Section 12 of passages from the life of a philosopher This is a LibriVox recording All LibriVox recordings are in the public domain For more information or to volunteer please visit LibriVox.org passages from the life of a philosopher by Charles Babbage Section 12 The Exhibition of 1862 On administration toutes les sorties sont mères Maxime, par M.G. de la vie An abject worship of princes and an unaccountable appetite for knighthood are probably unavoidable results of placing second-rate men in prominent positions. Saturday review, January 16th, 1864 Whose fault is this? But tallow, toys and sweetmeats evidently stand high in the estimation of Her Majesty's commissioners. The Times, August 13th, 1862 Circumstances connected with the exhibition of The Difference Engine No. 1 in the International Exhibition of 1862 When the construction of The Difference Engine No. 1 was abandoned by the government in 1842 I was consulted respecting the place in which it should be deposited while aware of the unrivaled perfection of its workmanship and conscious that it formed the first great step towards reducing the whole science of number to the absolute control of mechanism I wished it to be placed wherever the greatest number of persons could see it daily. Engine No. 1 in King's College With this view I advise that it should be placed in one of the much frequented rooms of the British Museum Another locality was however assigned to it and it was confided by the government to the care of King's College Somerset House It remained in safe custody within its glass case in the museum of that body for twenty years It is remarkable that during that long period no person should have studied its structure and by explaining its nature and use have acquired an amount of celebrity which the singularity of that knowledge would undoubtedly have produced The college authorities did justice to their charge They put it in the place of honour in the centre of their museum and would no doubt have given facilities to any of their members or to other persons who might have wished to study it The government ignore it But the system quietly pursued by the government of ignoring the existence of the difference engine and its inventor doubtlessly exercised its deadening influence on those who were inclined by taste or requirements to take such a course Footnote An illustration fell under my notice a few days after this paragraph was printed A new work on geometrical drawing commissioned by the Committee of Council on Education was published by Professor Bradley I have not been able to find in it a single word concerning mechanical notation not even the very simplest portion of that science namely the art of lettering drawings It would seem impossible that any professor of so limited a subject could be ignorant of the existence of such an important addition to its powers End footnote I shall enumerate a few instances 1. In 1850 the government appointed a commission to organise the exhibition of 1851 The name of the author of the Economy of Manufactures was not thought worthy by the government to be placed on that commission 2. In 1851 the commissioners of the International Exhibition did not think proper to exhibit the difference engine although it was the property of the nation They were as insensible to the greatest mechanical as to what has been regarded by some the greatest intellectual triumph of their country 3. When it was decided by the people of the United States to have an exhibition at New York they sent a commissioner to Europe to make arrangements for its success He was authorised to apply for the loan of the difference engine for a few months and was empowered to give any pecuniary guarantee which might be required for its safe return That commissioner on his arrival applied to me on the subject I explained to him the state of the case and advised him to apply to the government whose property it was I added that if his application was successful I would at my own expense put the machine in good working order and give him every information requisite for its safe conveyance and use His application was, however, unsuccessful 4. In 1847 Mr. Dargan nobly undertook at a vast expense to make an exhibition in Dublin to aid in the relief of his starving countrymen It was thought that the exhibition of the difference engine would be a great attraction I was informed at the time that an application was made to the government for its loan but that it was also unsuccessful 5. In 1855 the great French exhibition occurred Previously to its opening our government sent commissioners to arrange and superintend the English department These commissioners reported that the English contribution was remarkably deficient in what in France are termed as through more third place scissor a term which includes a variety of instruments for scientific purposes They recommended that a committee should be appointed who could represent to the producers of philosophical instruments how necessary it was that they should upon an occasion of this kind maintain their credit in the eyes of Europe The government also applied to the Royal Society for advice but neither did the Royal Society advise nor the government proposed to exhibit the difference engine The French exhibition of 1855 was remarkable beyond all former ones for the number and ingenuity of the machines that performed arithmetical operations Preeminently above all others stood the Swedish machine for calculating and printing mathematical tables It is honourable to France that its highest reward was deservedly given to the inventor of that machine Whilst it is somewhat remarkable that the English commissioners appointed to report upon the French exhibition omitted all notice of these calculating machines Mr Gravitt succeeds in exhibiting it in 1862 The appearance of the finished portion of the unfinished difference engine, number one at the exhibition of 1862 is entirely due to Mr Gravitt That gentleman had a few years before paid great attention to the Swedish calculating machine of Mr Scheutz and was the main cause of its success in this country Being satisfied that it was possible to calculate and print all tables by machinery Mr Gravitt became convinced that the time must arrive when no tables would ever be calculated or printed except by machines He felt that it was of great importance to accelerate the arrival of that period more especially as numerical tables which are at present the most expensive kind of printing would then become the cheapest In fervorance of this idea Mr Gravitt wrote to Dr Jelf the principle of King's College, Somerset House to suggest that the difference engine of Mr Babbage which had for so many years occupied a prominent place in the museum should be exhibited in the international exhibition of 1862 He at the same time offered his assistance in the removal and reinstatement of that instrument The authorities of the college readily acceded to this plan On further inquiry it appeared that the difference engine of the government and was only deposited with the college It was then found necessary to make an application to the treasury for permission to exhibit it which was accordingly done by the proper authorities The government granted the permission and referred it to the Board of Works to superintend its placement in the building The Board of Works sent me a copy of the correspondence relative to this matter asking my opinion whether any danger might be apprehended for the safety of the machine during its transport and also inquiring whether I had any other suggestion to make upon the subject Knowing the great strength of the work I immediately answered that I did not anticipate the slightest injury from its transport and that, under the superintendence of Mr Gravert I consider it might be removed with perfect safety The only suggestion I ventured to offer was that as the government possessed in the department of the Registrar General a copy made by English workmen of the Swedish Difference Engine that it should be exhibited by the side of mine and that both the engine should be kept constantly working with a very slow motion Swedish engine not exhibited By a subsequent communication I was informed that the Swedish machine could not be exhibited because it was then in constant use computing certain tables relating to the values of lives I regretted this very much I had intended to alter the handle of my own engine in order to make it movable circularly by the same cat bug which I had hoped might have driven both The tables which the Swedish machine was employed in printing were not of any pressing necessity and their execution could upon such an occasion have been postponed for a few months without loss or inconvenience Besides, if the Swedish engine had as I proposed been placed at work its superintendent might have continued his table making with but little delay and the public would have been highly gratified by the site He could also have given information to the public by occasional explanations of its principles Thus might Her Majesty's commissioners have gratified thousands of her subjects who came with intense curiosity prepared to be pleased and instructed and whom they sent away amazed and disappointed From the experience I had during the first week of the exhibition I am convinced that if a fit place had been provided for the two calculating machines so that the public might have seen them both in constant but slow motion and if the superintendent had occasionally given a short explanation of the principles on which they acted they would have been one of the greatest attractions within the building On Mr. Gravert applying to the commissioners for space it was stated that the engine must be placed alongside philosophical instruments Class 13 English engine poked into a hole The only place offered for its reception was a small hole 4 feet 4 inches in front by 5 feet deep On one side of this was the only passage to the office of the superintendent of the class The opposite side was occupied by a glass case in which I placed specimens of the separate parts of the unfinished engine These, although executed by English workmen above 30 years ago were yet in the opinion of the most eminent engineers unsurpassed by any work the building of 1862 contained The back of this recess was closed in and dark and only allowed a space on the wall of about 5 feet by 4 on which to place the hole of the drawings and illustrations of the difference engine Close above the top of the machine was a flat roof which deprived the drawings and the work itself of much light The public at first flocked to it but it was so placed that only three persons could conveniently see it at the same time When Mr. Gravert kindly explained and set it in motion he was continually interrupted by the necessity of moving away in order to allow access to the numerous persons whose business called them to the superintendent's office At a very early period various representations were made to the commissioners by the jury the superintendent and very strongly by the press of the necessity of having some qualified person to explain the machine to the public I was continually informed by the attendance that hundreds of persons had during my absence asked when they could get an opportunity of seeing the machine in motion admiring the earnestness of purpose and the sagacity with which Mr. Gravert had steadily followed out the convictions of his own mind relative to the abolition of all tables except those made and stereotyped by machinery I offered all the assistance in my power to accelerate the accomplishment of his task I lent him for exhibition numerous specimens of the unfinished portions of the difference engine number one These I had purchased on the determination of the government to abandon its construction in 1842 I proposed also to lend him the mechanical notations of the difference engine which had been made at my own expense and were finished by myself and my eldest son Mr. B. Herschel Babbage I had had several applications from foreigners for some account of my system of mechanical notation and great desire was frequently expressed to see the illustrations of the method itself and of its various applications Footnote One object of the mission of Professor Balzani was to take back with him to Russia such an account of the mechanical notation as might facilitate its teaching in the Russian universities I regret that it was entirely out of my power to assist him End footnote These however were so extensive that it was impossible without very great inconvenience to exhibit them even in my own house The loan of other calculating machines offered I therefore wrote to Mr. Gravich to offer him the loan of the following property for the exhibition One, a small calculating machine of the simplest order for adding together any number of separate sums of money provided the total was under £100,000 by Samuel Morland 1666 Two, a very complete and well executed machine for answering all questions in plain trigonometry by Samuel Morland 1663 Three, an original set of Napier's bones Four, a small arithmetical machine by Viscount Mann afterwards Earl Stanhope without date Five, a larger machine to add, subtract, multiply and divide by Viscount Mann 1775 Eight, another similar machine of a somewhat different construction for the same operations by Viscount Mann 1777 Seven, a small difference engine made in London in consequence of its author having read Dr. Lardner's article in the Edinburgh Review of July 1834 No. 120 List of mechanical notations proposed to be lent for the exhibition One, all the drawings explaining the principles of the mechanical notation Two, the complete mechanical notations of the Swedish calculating engine of Mr. Scheutz These latter drawings have been made and used by my youngest son Major Henry P. Babbage now resident in India in explaining the principles of the mechanical notation at the meeting of the British Association at Glasgow and afterwards in London at a meeting of the Civil Engineers Footnote See proceedings of British Association at Glasgow 1855 page 203 also minutes of proceedings of the Institution of Civil Engineers volume 15 1856 Three the mechanical notations of the difference engine No. 1 These have been made at my own expense and were finished by myself and my eldest son Mr. B. Herschel Babbage now resident in South Australia Four a complete set of the drawings of the difference engine No. 2 for calculating and printing tables with 7 orders of differences and 30 places of figures finished in 1849 Five a complete set of the notations of differentiation in demonstration of difference engine No. 2 finished in 1849 These drawings and notations would have required for their exhibition about seven or eight hundred square feet of wall My letter to Mr. Gravert was forwarded to the commissioners with his own application for space to exhibit them The commissioners declined this offer yet during the first six weeks of the exhibition there was at a short distance from the difference engine an empty space of wall large enough for the greater part of these instructive diagrams The portion of wall was afterwards filled up by a vast oil cloths Other large portions of wall to the amounts of thousands of square feet were given up to other oil cloths and to numberless carpets It is evident the royal commissioners were much better qualified to judge of furniture for the feet than of furniture for the head I was myself frequently asked why I did not employ a person to explain the difference engine In reply to some of my friends I inquired whether, when they purchased a carriage they expected the builder to pay the wages of their coachmen Foreign visitors puzzled But my greatest difficulty was with foreigners No explanation I could devise and I tried many appeared at all to satisfy their minds The things seemed to them entirely incomprehensible that the nation possessing the greatest military and commercial marine in the world the nation which had spent so much in endeavouring to render perfect the means of finding the longitude which had recently caused to be computed and published at considerable expense an entirely new set of lunar tables should not have availed itself at any cost of mechanical means of computing and stereotyping such tables seemed entirely beyond their comprehension At last they asked me whether the commissioners were that I assured them that the only one with whom I was personally acquainted certainly was not When hard pressed by difficult questions I thought it my duty as an Englishman to save my country's character even at the expense of my own So on one occasion I suggested to my unsatisfied friends that commissioners were usually selected from the highest class of society and that possibly four out of five had never heard of my name But here again my generous efforts to save the character of my country and its commissioners entirely failed Several of my foreign friends had known me in their own homes and had seen the estimation in which I was held by their own countrymen and by their own sovereign These were still more astonished Chinese inquire about it On another occasion an anecdote was quoted against me to prove that my name was well known even in China It may perhaps amuse the reader A short time after the arrival of Count Streletschy in England I had the pleasure of meeting him at the table of a common friend Many inquiries were made relative to his residence in China Much interest was expressed by several of the party to learn on what subject the Chinese were most anxious to have information Count Streletschy told them that the subject of most frequent inquiry was Babbage's calculating machine On being further asked as to the nature of the inquiries he said that they were most anxious to know whether it would go into the pocket Our host now introduced me to Count Streletschy opposite to whom I was then sitting After expressing my pleasure at the introduction I told the Count that he might safely have friends in the Celestial Empire that it was in every sense of the word an out-of-pocket machine At last the commissioners were moved not to supply the deficiency themselves but to address the government to whom the difference engine belonged to send someone to explain it I received a communication from the Board of Works inquiring whether I could make any suggestions for getting over this difficulty I immediately made inquiries and formally been my ammemenuensis and had under my direction worked out many most intricate problems He possessed very considerable knowledge of mathematics and was willing for the moderate remuneration of six shillings a day to be present daily during nine hours to explain the difference engine I immediately sent this information to the Board of Works with the name and address of the person I recommended This, I have little doubt was directly communicated to the commissioners but they did not avail themselves of his services Commissioners inexplicable It is difficult upon any principle to explain the conduct of the royal commissioners of the exhibition of 1862 They were appointed by the government yet when the government itself became an exhibitor and sent for exhibition a difference engine the property of the nation these commissioners placed it in a small hole in a dark corner where it could, with some difficulty be seen by six people at the same time No remunstrance was of the slightest avail It was Hobson's choice that or none It was represented that all other space was occupied A trophy of children's toys whose merits it is true the commissioners were somewhat more competent to appreciate filled one of the most prominent positions in the building On the other hand a trophy of the workmanship of the engineers executed by machine tools 30 years before and admitted by the best judges to be unsurpassed by any rival was placed in a position not very inappropriate for the authorities themselves who condemned it to that locality But no hired aristocratic agent was employed to excite the slumbering perceptions of the commissioners who might have secured a favourable position for the difference engine of their good nature or by imposing upon their imbecility Footnote See the Times 19 January 1863 and elsewhere End footnote It has been urged in extenuation of the conduct of these commissioners that their duty as guardians of the funds entrusted to them and of the interests of the gun-talls compelled them to practice a rigid economy Rigid economy is to be respected only when it is under the control of judgement not of favouritism If the machinery for making arithmetical calculations which was placed at the disposal of the commissioners had been properly arranged it might have made at once a sort of high gratification to the public and even a profit to the exhibition A court for calculating machines Such a group of calculating machines might have been placed by themselves in a small court capable of holding a limited number of persons Round the wards of this court might have been hung the drawings I had offered to lend containing the whole of those necessary for the difference engine number two as well as a large number of illustrations for the explanation of the mechanical notation The Swedish difference engine and my own might have been slowly making calculations during the whole day This court should have been open to the public generally except at two or three periods of half an hour each during which it should have been accessible only to those who have previously secured tickets at a shilling apiece During each half hour the person whom I had recommended to the commissioners might have given a short popular explanation of the subject This attraction might have been still further increased and additional profit made if a single sheet of paper had been printed containing a woodcut of the Swedish machine an impression from a page of the tables computed and stereotyped by it at Somerset House and also an impression from a stereotype plate of the difference engine exhibited by the government A plate of the Swedish machine is in existence in London I am confident that for such a purpose I could have procured the loan of it for the commissioners and I would willingly have supplied them with a stereotype plate in which the frontispage of the present volume was printed together with from 10 to 20 lines of necessary explanation These illustrations of machinery used for computing and printing tables might have been put up into packets of dozens and half-dozens and also have been sold in single sheets at the rate of one penny each copy There can be no doubt the sale of them would have been very considerable As it was, I found the woodcut representing the difference engine number one in great request and during the exhibition I had numberless applications for it having given away my whole stock of about 800 copies An assistant explaining The calculating court might have held comfortably from 60 to 80 seats Each lecture would have produced say, three pounds This being repeated three times each day together with the sale of the woodcuts would have produced about 10 pounds per day, out of which the commissioners would have had six shillings per day to pay the assistant who gave the required explanations If the dignity of the commissioners would not permit them to make money by such means, they might have announced that the proceeds of the tickets would be given to the distressed population of the Manchester district and there would have been crowds of visitors But the rigid economy of the commissioners who refused to expend six shillings a day although it would most probably have produced a return of several hundred pounds was entirely laid aside when their patronage was to be extended to a brother official Captain Folk an officer of engineers whose high order of architectural talent became afterwards so well known to the public and whose whole time and services were retained and paid for by the country was employed to make a design for the exhibition building the commissioners do a job the commissioners approved of this design which comprised two lofty domes uniting in themselves the three folding convenience of being ugly, useless and expensive they then proceeded to pay him five thousand pounds for the job this system of awarding large sums of money to certain favourite public officers who already paid for their services by liberal salaries seems to be a growing evil at the period of the Irish famine the undersecretary of the treasury condescended to accept two thousand five hundred pounds out of the fund raised to save a famished nation some inquiries even recently were occasionally made whether any similar deduction will be allowed from the liberal contributions to the sufferers by the cotton famine the question was raised and the practice reprobated in the House of Commons by men of opposite party politics Mr Gladstone remarked quote if there was one rule connected with the public service which more than any other ought to be scrupulously observed it was this that the salary of a public officer more especially if he were of high rank ought to cover all the services he might be called upon to render any departure from this rule must be dangerous end quote Hansard volume 101 page 138 1848 supply 14th of August 1848 see also the exposition of 1851 8th volume page 217 the Admiralty refuse the following paragraph appears in the Times a short time since under the head naval intelligence quote a reply has been received to the memorial transmitted to the Admiralty a few days since from the inspectors employed on the iron frigate Achilles building at Chatham dockyard requesting that they may be placed on the same footing as regards to increased pay as the junior officers and mechanics working on the iron frigate for the additional number of hours they are employed in the dockyard the Lords of the Admiralty intimate that they cannot exceed to the wishes of the memorialists who are reminded that as salaried officers of the establishment the whole of their time is at the disposal of the Admiralty this decision has caused considerable dissatisfaction end quote footnote about the 20th of May 1863 end footnote it appears that the Admiralty wisely adopted the principle enunciated by Mr Gladstone it may however not unreasonably have caused dissatisfaction to those who had no interest to back them on finding that such large sums are pocketed by those who are blessed with influential friends in high quarters if the commissioners had really wished to have obtained a suitable building at a fair price their course was simple and obvious they need only have stated the nature and amount of accommodation required and then have selected half a dozen of the most eminent firms amongst our greater contractors who could each have given them an estimate of the plans they respectively suggested the commissioners might have made it one of the conditions that they should not be absolutely bound to give the contract to the author of the plan accepted but in case of not employing him a sum previously stipulated should have been assigned for the use of the design by such means they would have had a choice of various plans and if those plans had previously to the decision of the commissioners been publicly exhibited for a few weeks they might have been enlightened by public criticism such a course would have prevented the gigantic job they afterwards perpetrated it could therefore find no support from the commissioners the present commissioners however are fit successes to those who in 1851 ignored the existence of the author of the economy of manufacturers and his inventions they seem to have been deluded into their belief that they possess the strength and the desire quietly to strangle the difference engine it would be idle to break such butterflies upon its matchless wheels or to give permanence to such names by reflecting them from its diamond graven plates footnote for the purpose of testing the steadiness and truth of those tools employed in forming the gum metal plates I had some dozen of them turned with diamond point the perfect equality of its cut caused the reflected light to be resolved into those beautiful images pointed out by Fraunhofer and also so much admired in the celebrated gold buttons produced by the late Mr Barton the Comptroller of the Mint end footnote though the steam hammer can crack the coating without injuring the kernel of the filbert it drops upon the admirable precision of its gigantic power could never be demonstrated by exhausting its energy upon an empty barrel peace then to their memory aptly enshrined in unknown characters within the penetralia of the temple of oblivion consolation for the commissioners these celebrities may there at last console themselves in the enjoyment of one enviable privilege denied to them during their earthly career exemption from the daily consciousness of being found out it is however not quite admirable although deciphering is a brilliant art that one or other of them may have heard of the dread power of the decipherer having myself had some slight acquaintance with that fascinating pursuit it gives me real pleasure to relieve them of this very natural fear by assuring them that not even the most juvenile decipherer could be so stupid as to apply himself to the interpretation of characters known to be meaningless yet there is one name amongst but not of them a fellow worshipper with myself at far other feigns whose hands like mine have wielded the hammer and whose pen like mine has endeavoured to communicate faithfully to his fellow men the measure of those truths he has himself laboriously extracted from the material world with such endowments it is impossible that he could have had any cognizance of this part of the proceedings of his colleagues footnote I have since learned with real satisfaction that my friend Mr Fairburn was not a member of that incompetent commission end footnote Mr Gravert explains the engine at the commencement of the exhibition Mr Gravert was constantly present and was so kind as to explain to many anxious inquirers the nature and uses of the difference engine this however interfered so much with his professional engagements as a civil engineer that it would have been unreasonable to have expected its continuance in fact as not above half a dozen spectators could see the machine at once it was a great sacrifice of valuable time for a very small result during the early part of my own examination of the exhibition I have many opportunities of conversing with experienced workmen well qualified to appreciate the workmanship of the difference engine these I frequently accompanied to its narrow cell and pointed out to them its use as well as the means by which its various parts had received their destined form occasionally also I explained it to some few of my personal friends when Mr Gravert on myself was thus engaged a considerable crowd was often collected who were anxious to hear about although they could not see the engine itself on these occasions I was insulted by impertinent questions conveyed in a loud voice from a person at a distance in the crowd my taste for music and especially for organs was questioned I was charitable enough to suppose that this was an exceptional case but in less than a week another instance occurred after this experience of course I seldom went near the difference engine Mr Gravert who had generously sacrificed a considerable portion of his valuable time and information and instruction of the public was now imperatively called away by professional engagements and the public had no information whatever upon a subject on which it was really very anxious to be instructed Mr Wilmot Buxton explains the difference engine fortunately however the exhibition took place during the long vacation and a friend of mine Mr Wilmot Buxton of the Chantry Bar very frequently accompanied me in my visits possessing a profound knowledge of the mathematical principles embodied in the mechanism I had frequently pointed out to him his nature and relations these I soon found he so well apprehended that I felt justified in entrusting him with one of my keys of the machine in order that he might have access to it without the necessity of my presence whenever he opened it for his own satisfaction or for the instruction of his friends he was speedily surrounded by a far larger portion of the public than could possibly see it but who was still attracted by his lucid oral explanation it was fortunate for many of the visitors to the exhibition that this occurred for the demands on his time when present were incessant and hundreds thus acquired from his explanations a popular view of the subject after the close of the exhibition Mr Graverton myself attended to prepare the difference engine for its return to the museum of King's College to our great astonishment we found that it had already been removed to the museum at South Kensington not only the difference engine itself but also the illustrations and all the unfinished portions of exquisite workmanship which I had lent to the exhibition for its explanation were gone on Mr Graverton applying to the board of works it was stated that the difference engine itself was removed to the museum because the authorities of King's College had declined receiving it and immediate instructions were of course given for the restoration of my own property End of section 12 section 13 of passages from the life of a philosopher this is a LibriVox recording or LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org passages from the life of a philosopher by Charles Babbage section 13 the late Prince consort Suum Guike I have had one opportunity of fairly estimating some portion of the character of the late justly lamented Prince consort to this I will now venture to allude in 1842 Count Menstor visited London a few days after I had a note from the late Duke of Wellington in which he informed me that on the previous evening he had met at the palace the Queen's Uncle Count Menstor who had expressed to the Duke his wish to see my calculating engine the Duke then inquired whether I could conveniently make some arrangement for that purpose I immediately wrote to the Duke that if he would appoint an hour on any morning of the ensuing week I should be able of the ensuing week I should have great pleasure in showing and explaining the difference engine to Count Menstor it was afterwards arranged that on the following Tuesday at two o'clock Count Menstor and the Duke should pay me a visit in Dorset Street on Monday morning I received another note from the Duke informing me that Prince Albert had expressed his intention to accompany Count Menstor in the proposed visit and that it would be more convenient if the hour were changed to one instead of two o'clock I must freely admit that I did not greatly rejoice at this addition to the party I resolved however strictly to perform the duties thus thrown upon me as a host as well as all those to which Prince Albert was entitled by his elevated position the woven portrait before I took the Prince into the fire-proof building in which the difference engine was then deposited I asked his Royal Highness to allow me to show him a portrait of Jacquard which was at that time hanging up in my drawing-room as it would greatly assist in explaining the nature of calculating machines when we had arrived in front of the portrait I pointed it out as the object to which I solicited the Prince's attention oh that engraving remarked the Duke of Wellington said Prince Albert to the Duke it is not an engraving I felt for a moment very great surprise but this was changed into a much more agreeable feeling when the Prince instantly added I have seen it before I felt at once that the Prince was a good man and true and I resolved that I would not confine myself to the rigid rules of etiquette but that I would help him with all my heart in whatever line his inquiries might be directed the portrait of Jacquard was in fact a sheet of woven silk framed and glazed but looking so perfectly like an engraving that it had been mistaken for such by two members of the Royal Academy Wilkie's conjecture a short time after I became possessed of this beautiful work of art I met Wilkie and invited him to come and see my recent acquisition he called on me one morning I placed him at a short distance in front of the portrait which he admired greatly I then asked him what he thought it was he answered an engraving on which I asked of what kind to this he replied line engraving to be sure I drew him a little nearer he then mentioned another style of engraving at last having placed Wilkie close to the portrait he said after a considerable pause can it be lithography a splendid collection of arms from Afghanistan recently sent to me from India by Sir Edward Ryan was lying on the tables in one of the rooms we passed through these had attracted the notice of the Prince and on returning the whole party examined them with the greatest interest I now conducted my visitors to the fireproof building in which the difference engine was placed Prince Albert was, I understood sufficiently acquainted with the higher departments of mathematical science to appreciate the influence of such an instrument on its future progress but the circumstance that charmed me was his bearing towards his uncle Count Menstorff it was perfectly natural it could be felt, admired and honoured but not described when the sad fact of the nation's loss became known to me I immediately reverted with some anxiety to a work I had published ten years before on the exhibition of 1851 I feared less in speaking of that event I might have committed some injustice whilst I was indignant at that under which I was myself suffering I willingly reprinted here because it contained no empty words of flattery but analysed the reasons which commanded our respect quote the merit of the original conception of the present exposition 1851 is insignificant in comparison with that of the efforts by which it was carried out and with the importance of its practical results to have seen from afar its effects on the improvement, the wealth and the happiness of the people to have seized the fit moment when, by the right use of the influence of an exalted station it was possible to overcome the deeply rooted prejudices of the upper classes to remove the still more formidable because latent impediments of party generously to have undertaken great responsibility and with indefatigable labour to have endeavoured to make the best out of the only materials at hand these are endowments of no ordinary kind to move in any rank of society an exception to its general rules is a very difficult and if accompanied by the consciousness of the situation a very painful position to a reflecting mind penalties of exalted station whatever may be the cause whether exalted rank unbounded wealth surpassing beauty or unrivaled wit the renown of daring deeds the magic of a worldwide fame to all within those narrow limits the dangers and the penalties are great each exists an isolated spirit each unconsciously imprisoned within its crystal globe perceives the colours of all external objects modified by those tints imparted to them by its own surrounding sphere no change of view can teach it to rectify this partial judgment throughout its earthward course the same undying rainbow attends to the last its parent drop its sympathies rarely indeed can some deep searching mind after long comparison perceive the real colours of those translucent shells which encompass kindred spirits and thus at length enable him to accromotise the medium which surrounds his own to one who has thus rectified the colour-blindness of his intellectual vision how deep the sympathy he feels for those still involved from which he has himself escaped none can so justly appreciate that sense of loneliness that solitude of mind which surrounds unquestioned eminence on its lofty throne none therefore can make so large an allowance for its errors none so skilfully assist in guiding his hazardous career End of section 13