Passages from the Life of a Philosopher. Charles Babbage. The mathematician and engineer Charles Babbage is best remembered for his 'calculating machines', which are considered the forerunner of modern computers. Over the course of his life he wrote a number of books based on his scientific investigations, but in this volume, published in , Babbage writes in a more personal vein. He points out at the beginning of the work that it 'does not aspire to the name of autobiography', though the chapters sketch out the contours of his life, beginning with his family, his childhood and formative years studying at Cambridge, and moving through various episodes in his scientific career.
Passages from the Life of a Philosopher
However, the work also diverges into his observations on other topics, as indicated by chapter titles such as 'Street Nuisances' and 'Wit'. Babbage's colourful recollections give an intimate portrait of the life of one of Britain's most influential inventors. I then explained to them the following very simple means by which that verification was accomplished. Besides the sets of cards which direct the nature of the operations to be performed, and the variables or constants which are to be operated upon, there is another class of cards called number cards.
These are much less general in their uses than the others, although they are necessarily of much larger size. 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: thus—. The above card contains eleven vertical rows for holes, each row having nine or any less number of holes.
In this example the tabular number is 3 6 2 2 9 3 9, whilst its number in the order of the table is 2 3 0 3. In fact, the former number is the logarithm of the latter. Of course the Engine will compute all the Tables which it may itself 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 2 3 0 3. 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 2 3 0 3, 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. The Analytical Engine I propose will have the power of expressing every number it uses to fifty places of figures. It will multiply any two such numbers together, and then, if required, will divide the product of one hundred figures by number of fifty places of figures.
Supposing the velocity of the moving parts of the Engine to be not greater than forty feet per minute, I have no doubt that. In the various sets of drawings of the modifications of the mechanical structure of the Analytical Engines, already numbering upwards of thirty, two great principles were embodied to an unlimited extent. The entire control over arithmetical operations, however large, and whatever might be the number of their digits.
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 mechanician. The difficulties which naturally occur to those capable of examining the question, as far as they relate to arithmetic, are these,—.
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. Since every calculating machine must be constructed for the calculation of a definite number of figures, the first datum must be to fix upon that number. In order to be somewhat in advance of the greatest number that may ever be required, I chose fifty places of figures as the standard for the Analytical Engine.
The intention being that in such a machine two numbers, each of fifty places of figures, might be multiplied together and the resultant product of one hundred places might then be divided by another number of fifty 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 five or five thousand digits. With respect to multiplication and division, the time required is greater:—.
Thus if a. This expression contains four pairs of factors, aa' , ab' , a'b , bb' , each factor of which has less than fifty 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 fifty and one hundred, will be nearly four times as long as that of two such numbers of less than fifty places of figures.
The same reasoning will show that if the numbers of digits of each factor are between one hundred and one hundred and fifty, then the time required for the operation will be nearly nine times that of a pair of factors having only fifty digits. 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 fifty digits, when used as one of a hundred digits, can only contain half the number of variables. The next step is therefore to prove b , viz. 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 pasteboard cards the holes corresponding to those numbers. At another part of the machine a series of number cards, resembling those of Jacquard, but delivered to and computed by the machine itself, can be placed.
These can be called for by the Engine itself in any order in which they may be placed, or according to any law the Engine may be directed to use. Hence the condition b is fulfilled, 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 is the simplicity of its structure of that portion of the Engine. 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 thus punched. 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. 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 fifty-one 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 n , with an index below. In the machine I have designed, n may vary from 0 to 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. I propose to make m vary from 0 to 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.
Passages Life Philosopher, First Edition
It thus appears that the two conditions, e and f , which require that the number of constants and of 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 fulfil.
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. 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 rnay 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. In I received from my friend M. Plana a letter pressing me strongly to visit Turin at the then approaching meeting of Italian philosophers. In that letter M.
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Plana 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:—. Considering the exceedingly limited information which could have reached my friend respecting the Analytical Engine, I was equally surprised and delighted at his exact prevision 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 MacCullagh, 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, Messrs.
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.
Plana 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, M. Menabrea, who had already established his 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. One day Mossotti, who had been unavoidably absent from the previous meeting, when a question of great importance had been discussed, again joined the party.
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