"The Calculating Engine is a Locomotive that lays down its own railway." ^{1} |
In order to understand the genesis of the Analytical Engine, one must trace it back to 1822, to the first stages of Babbage's work on the Difference Engine. As mentioned in Chapter Two, Babbage had from the beginning the idea of a machine to calculate tables by difference methods for functions without a constant difference, and he announced this idea in his "Letter to Sir Humphrey Davy" in July, 1822. ^{2} It was out of this seed, which lay dormant until 1834, that the Analytical Engine grew.
Before exploring just what that idea was, it will be necessary to explain in an elementary way another aspect of the method of finite differences. First, it will be remembered from Chapter Two that the fundamental definition of finite differences is:
Δu_{x} = u_{x+w} - u_{x}and that Δu_{x} is itself a function of x. For convenience, it will be assumed for the rest of this discussion that w is equal to one, that is, that we are tabulating various functions always using an increment of unity. It will be further remembered that for any algebraic equation of order n, the n-th order difference is a constant, which is the same as saying that the differences of higher order than n are all zero, and are not needed in a difference machine. Babbage decided that for the full scale Difference Engine the appropriate compromise between a small machine and wide applicability would be to set n equal to six.
Again, it will be remembered that transcendental functions (in this discussion the sine function will be used as the example) do not have any order of difference constant; that is, to tabulate a sine function precisely would take an infinite number of columns in a difference engine. In practice, this is not necessary, since the higher order differences do become smaller; one can substitute for the transcendental function an algebraic function which approximates it over a given domain, then manually alter the differences to proceed through another domain, and so on. As the number of orders of difference available becomes larger, the width of the domain that can be tabulated without intervention increases also; but as the number of significant figures to be retained goes up, the possible domain shrinks. In practice, it is possible to tabulate a sine function through a whole quadrant only by breaking it up into many small domains. To calculate a sine table at intervals of one minute with eight significant digits, a machine which had a constant fourth difference and fifteen digit capacity would have to be reset about every two degrees ^{3}
This approach Babbage found to be objectionable, for two reasons. First, the necessity of manually altering the setting of the machine several times in an operation greatly increased the possibility of introducing errors into the tables it produced, end this he wanted to avoid at all costs; it also became more difficult to verify the accuracy of a produced table, since it was no longer true that if the first and last values were correct, the whole table was correct. Second, Babbage found this approach to transcendental functions to be theoretically impure and inelegant; it did not satisfy his image of the intimate relation that should be possible between the machine and the mathematics on which it was based.
As it turned out, Babbage found that transcendental functions could often be handled quite simply and elegantly by a somewhat different approach to dealing with the fact that they had no order of difference constant. This can be seen in the following simple development.
Taking again the fundamental definition of the method of finite differences, and assuming we want to tabulate the function sin(x) from zero to ninety degrees for increments of one degree, it follows immediately that:
Δsin(x) = sin(x+l) - sin(x)then taking the elementary trigonometric identity:
sin(a) - sin(b) = 2cos½(a+b)sin½(a-b)we get, combining all the constant factors into Z:
Δsin(x) - Zcos(x+½)It then follows that;
Δ^{2}sin(x) = Z [cos(x + 3/2) - cos (x + ½)]Using the identity:
cos(a) - cos(b) = -2sin½(a+b)sin½(a-b)and combining all the constant factors, we get:
Δ^{2}sin(x) = Ksin(x+l)This equation is, then, an expression of the way in which the value of the second difference of our function varies, one which turns out to have a very simple relation to the function itself.
One's immediate impression is apt to be that this relation is circular, since in order to generate new values of the function by it, a difference engine would have to predict those same values. But this is not in fact the case. In order to generate sin(x+l) from sin(x), one need know only a Δsin(x). Δ^{2}sin(x) is needed only to calculate sin(x+2), by way of calculating Δsin(x+l).
In fact, even though this difference equation applies to the second difference of the function, no second difference column is needed to use it. Instead, as soon as sin(x+n) has-been calculated from Δsin(x+n-1) by adding it to sin(x+n-1), we can simply add Ksin(x+n) to Δsin(x+n-1) to form Δsin(x+n), from which sin(x+n+1) follows, and so on.
Although it is thus possible to tabulate sin(x) with only one difference column (plus, of course, a result column - what Babbage called a table column), it unfortunately is not mechanically simple, since in the above account sin(x+n) must be multiplied by the constant K (which is in fact negative), which must have several digits to preserve accuracy. This negates the great advantage of the difference engine, that it reduces the problem of tabulating any kind of function to the mechanically simple problem of repeated addition; however, functions tractable to simple difference equations could still be handled that way; this new approach was to be used only for tables which the basic difference engine cannot handle well anyway.
But clearly, due to its theoretical priority and its mechanical simplicity, a machine to calculate only from equations of constant differences ought to be built first. This was the course that Babbage adopted.
The first working model of the Difference Engine, completed about the end of May, 1822, as well as the full scale version whose construction was undertaken for the government, were both of the simpler kind. However, the section of the full scale machine which was put together in 1832 had added on to it a special provision for demonstrating the sort of technique by which transcendental functions could be handled. It will clarify Babbage's thinking in 1822 and after if we first consider this later device.
In the 1832 section, there were essentially three vertical columns arranged parallel to one another across the front of the machine to hold the table, first difference and second difference. What Babbage did was to add extra axes in front of the table column and the second difference column; each of these axes had a wheel on it which could be set to engage with any one of the wheels on the column to which it was adjacent; further, the two special axes were also geared together in such a way that they always rotated together. In this way, a single digit of the tabulated result, say the units digit, could be fed back onto the second difference column, and from there be added on to the first difference columns if the special digits in the table and second difference column were initially set equal, they would remain equal throughout the calculation, since the second difference digit would be continuously reset to equal the table digit.
Thus Babbage demonstrated in a primitive way the technique for tabulating transcendental functions. However, this device was of no practical use, for two reasons: it could handle only a single digit from the table column, and even so, it could not multiply it by a special constant before adding it back in, as would be required in a useful computation. Returning now to 1822, it has already been stated that the machine of which Babbage made a first working model was one for computing functions with a constant second difference and very few digits. ^{4} It was clear to Babbage from the first that by using the same general design the machine could be extended to employ greater numbers of differences and digits, but this did not satisfy him. He expressed his dissatisfaction with the limitations of constant differences most clearly in the paper he read to the Astronomical Society on December 13, 1822, in the following terms:
I have already stated to the society, in my former communication, that the first engine I had constructed was solely destined to compute tables having constant differences. From this circumstance it will be apparent that after a certain number of terms of a table are computed, unless, as rarely happens, it has a constant order of differences, we must stop the engine and place in it other numbers, in order to produce the next portion of the table. This operation must be repeated more or less frequently according to the nature of the table. The more numerous the order of differences, the less frequently will this operation become requisite. The chance of error in such computations arises from incorrect numbers being placed in the engine; it therefore becomes desirable to limit this chance as much as possible. In examining the analytical theory of the various differences of the sine of an arc, I noticed the property which it possesses of having any of its even orders of differences equal to the sine of the same are increased by some multiple of its increment multiplied by a constant quantity. With the aid of this principle an engine might be formed which would require but little attendance, and I believe that it might in some cases compute a table of the form A sin θ from the 1st value of θ = 0 up to θ = 90° with only one set of figures being placed in it. It is scarcely necessary to observe what an immense number of astronomical tables are comprised under this form, nor the great accuracy which must result from having reduced to so few a number the preliminary computations which are requisite. ^{5}Although this passage was written in December, 1822, Babbage had been considering the problem from early in the year. In none of the papers in which Babbage discussed a machine to operate without a constant difference did he give any mechanical or mathematical details of what he was planning, but it is clear from the train of thought into which he was led that he had imagined just such a modification of the working model he had built as he was later to add on to the section of the full scale machine assembled in 1832, discussed above.
This emerges in Babbage's Letter to Dr. Brewster, dated November 6, 1822, ^{6} and in the paper delivered to the Astronomical Society on December 13, already quoted. In these papers Babbage wished to discuss - because of. its purely theoretical interest - a new kind of mathematical series he had found while thinking about the Difference Engine, a series "whose analytical laws were unknown." ^{7} The example Babbage gave was the following:
Table | 1st Dif. | Sec. Dif. |
---|---|---|
2 | 0 | 2 |
2 | 2 | 4 |
4 | 6 | 0 |
10 | 6 | 6 |
16 | 12 | 8 |
28 | 20 | 8 |
48 | 28 | 6 |
76 | 34 | 0 |
110 | 34 | 4 |
144 | 38 | . |
. | . | . |
. | . | . |
. | . | . |
The principle of generation of this table was that the second difference for any terms was equal to the units figure of the table value of the next succeeding term, or:
Δ^{2}x_{n} = units figure of x_{n+1}It is very clear that this is just the direction in which Babbage would have been led while thinking about a method of generating sine tables if he had devised only a method for transferring single digits from the table to the second difference, and without multiplication; and this is exactly the kind of special mechanism which he later added on to the 1832 machine. It is clear, however, that in 1822 he did not actually build the special mechanism, for he spoke of having computed by hand the above table "such as would have been formed by the engine had it existed in this new shape." ^{8}
The period during which Babbage was thinking along these lines was that between the completion of the first working model of a simple difference engine and the start of construction of a machine for the government. After that point, Babbage pursued the matter very little. He wrote one paper, "On the Determination of the General Term of a New Class of Infinite Series," read to the Cambridge Philosophical Society on May 3, 1824, exploring the mathematical aspects of the new kind of series. ^{9} Also, the possibility of using the new technique in a machine to calculate astronomical tables was discussed in a paper in the Philosophical Magazine for May, 1824, written by Francis Bailey, but based on conversation with Babbage. ^{10} But aside from this, Babbage let the subject drop while he worked on the full scale machine without these special capabilities.
As has been discussed in Chapter Two, Clement suspended work on the Difference Engine in March, 1833. From then until July, 1834, the drawings and parts of the machine (with the exception of the section which had been assembled in 1832) sat in his workshop idle, with Babbage essentially not having access to them. But on July 15, 1834, the parts and drawings were delivered in Babbage's custody. However, as we have seen, he was reluctant to proceed with construction until some new understanding was reached with the government on mutual rights and responsibilities with respect to future work. While Babbage waited for the Prime Minister, the machine waited as well.
Though Babbage did not proceed with any direct work on the Engine, and felt quite disgusted with the circumstances surrounding the project, he had by no means lost interest in the basic idea of the machine. He therefore occupied himself, while waiting, with re-examining the basic plans. In particular, he tried to design a mechanism which would allow the machine to handle difference equations of the sort discussed in the first section of this chapter.
This work is recorded in a volume entitled the "Great Scribbling Book;" ^{11} unfortunately the relevant pages are generally not dated. One can only say that this re-examination began in July or August, 1834, and the first stage of it was complete by the middle of September, 1834.
This first stage consisted of attempts to develop the method used in the 1832 section for demonstrating the transfer of single digits from the table column to the second difference column. The first item to be found is a group of obscure sketches entitled "Plan for adding from one axis to another a number multiplied by 10^{n}." ^{12} But it is not at all clear from the sketches what Babbage had in mind.
On the next page, under the title "Plan for multiplying any numbers on any Δ axis and adding them to any other" Is the following sketch: ^{13}
This plan is clear, and is a modification of the device used in the 1832 machine. On the right and left are, as examples, the first and sixth difference columns. The object is to shift the top three digits from Δ1 to Δ6, in this case dividing them by 100. Miter gears collect the digits off Δ1 as its wheels are turned and put them onto the vertical racks in the center marked A, A', and A''. Similarly, these racks turn the horizontal axes at the bottom, which put the digits onto the lower wheels of Δ6. Since the pinions called b, b' and b'' can be slid back and forth on their axes, they can connect the wheels of Δ1 with whatever set of vertical racks is desired, effecting multiplication by different powers of ten as the transfer is made.
During the succeeding days or weeks, Babbage sketched several variations on methods of effecting this transfer from one end of the machine to the other, the process which he called "the engine eating its own tail," to describe its recursive character. ^{14} By a series of steps which are not recorded, Babbage was led, shortly before September 18, to the very important step of arranging the columns in a circle around large central wheels. ^{15}
In an account of this period written much later, in November, 1869, Babbage said that he had come to adopt an arrangement of horizontal racks connecting the axes, but then realized that if the columns were circularly arranged, the highest and lowest order columns could communicate directly, since they would be adjacent, and no racks would be needed. He then realized that the advantages of the racks could be retained in the new design if they were replaced by large geared wheels in the center. This account of the development is plausible, but doubtless also somewhat of a simplification. ^{16}
Exactly what this new arrangement was is not clear from the 1834 sketch mentioned above, and there was no written description of it. Apparently the difference columns, soon to be called adding axes, were put around one side of the circle; they were divided vertically into a certain number of "cages," corresponding to their separate digits. Special pinions could engage their wheels with the central wheels as desired; the central wheels were simply very large gears rotating freely on a central axis; there were in fact two central wheels per cage. On the other side of the central wheels were the "multiplying pinions," used to step up numbers on the central wheels, that is, to multiply them by powers of ten by moving their digits up a certain number of cages. A particular digit would be transferred from, say, the lower central wheel in one cage to the higher central wheel in another cage higher or lower by a number corresponding to the power of ten by which it was being multiplied; the number of cages stepped would be determined by the way in which the connecting gears on the multiplying pinions were set. The multiplied number could then be taken off by a desired adding axis from the higher central wheels. ^{17}
This arrangement formed what might be described as the final stage of the Difference Engine design, for in filling in the details and making it workable, Babbage was led to realize that there were much greater possibilities lurking in the plan, ones which eventually led him to the fully elaborated plan of the Analytical Engine.
What may be called the second stage of the transformation from the Difference Engine to the Analytical Engine began at this point, and continued until the summer of 1836, by which time all the main principles had been laid down, although many details remained to be filled in. The course of development during this period was so complex that only the main outlines can be given here. Further, it will be much easier to understand some aspects of the design when they are explained in the context of the operation of the machine as a whole, so such explanations will be at times postponed. ^{18}
The first subject to which Babbage turned was a method for multiplication and division. Working on multiplication was no real departure from the previous phase, for it was necessary in order to use the kind of recursive difference equations in which Babbage was interested. But the fact that the multiplication pinions could step numbers down as well as up, and the fact that division was the inverse of multiplication, led Babbage to try to make the machine capable of both.
Given the ability to step numbers up, multiplication was fairly simple; the problem was to decide how many times at each level of stepping the multiplicand should be added to a result column. The method first devised was as follows. The multiplier was on one adding column, say A^{1}; the multiplicand was on A^{2}; and the product was to be produced on A^{3}. Attached to each figure wheel on the adding columns was a snail; this was a notched spiral wheel with the steps on it corresponding to the ten digits. When an arm from another column fell on this snail, it could sense the digit at which the figure wheel was set. In multiplication, an arm would first sense the lowest cage on A^{1}, and would set a special wheel to the same digit; then this wheel would be gradually "reduced to zero" by another wheel with a single tooth, which would "gather up" the units of the selected digit, one per revolution. With each revolution of the gathering up wheel, all the digits on A^{2} would be added to the central wheels; the cages of A^{3} being connected in the same way to the central wheels, the multiplier would be successively added onto A^{3}. When the special wheel was reduced to zero (i.e. when the multiplicand had been multiplied by the first digit of the multiplier) the following would happen. The snail opposite the second cage of A^{1} would set the special wheel to equal the second digit of the multiplier; also the multiplying pinions would be set so that as the digits from A^{2} were put onto the central wheels, they would be stepped up one cage before being added to A^{3}. The multiplicand would then be multiplied by the second digit of the multiplier by the same gathering up process, the snails and multiplying pinions would be reset for another cage, and so on, until the digits of the multiplier were exhausted. ^{19}
No irony was intended above in saying that multiplication was simple; compared to division it was. Thus even with a one digit divisor, that digit had to be compared with the first digit of the dividend (requiring two sets of snails and feeler arms and other apparatus); if the latter was equal or larger, the former was subtracted from it, and the fact of subtraction was entered as a unit at the top of the result column; then the comparison was repeated, and so on. If the divisor digit was larger, then apparatus had to be shifted to compare it with the next cage, and the successive subtractions had to be registered on a different cage, and so on. Dividing by a multi-digit number was far more complicated, for if the initial comparison of the top two digits of the divisor and dividend showed that they were equal, the second digit of each had to be compared to the other, and perhaps even the third and subsequent ones, before the machine would know which of the two numbers was larger.
At this point, Babbage's new machine was fearsomely complex, even though its powers were still very limited, and so far only the outlines of the parts needed had been considered. Further, it would have been incredibly slow, for the time required for multiplication and division would have been considerably longer than the sum of all the digits in the multiplier or divisor times the basic addition time. But Babbage considered that this simply meant that he had to shorten the basic cycle time and make the complex processes more efficient.
To shorten the cycle time, the fundamental operations had to be improved; these were: basic addition, that is, transferring digits from one wheel to another, and carriage of tens between the wheels in a given column. In the Difference Engine, addition had been made by bolting the two wheels so that they would gear together, then giving the first wheel a full revolution, unbolting the second wheel from it after the proper number of digits had been passed. In the Analytical Engine, Babbage thought through a number of other possibilities. One involved direct gearing between the two wheels, having a projection in one be pushed up by an inclined plane after the proper part of the revolution had been passed so that they were lifted out of gear. ^{20} Another method was to have snails and feelers to set extra wheels equal to the figure wheels, and then add onto the receiving wheels by a gathering up wheel as in multiplication, but this took more time. ^{21}
The method ultimately adopted was taken from the approach used as early as the beginning of October, 1834, ^{22} for the purpose of clearing a number off a column, or as Babbage called it, reducing it to zero. This worked by having the wheels turn freely on an axis which could be moved up and down relative to them by a small amount; fixed to this axis was a series of projecting arms, one for each cage; on each figure wheel was a stud which would be contacted by the projecting arm when the axis was in its lower position. The number on the column would be reduced to zero when it was disconnected from the rest of the machine and the axis was lowered and rotated through a full turn so that it pushed the studs and thus the wheels around until they all were set to zero. It was only some months later, apparently in January, 1835, ^{23} that Babbage realized that the same technique could be used for transferring numbers from one wheel or column to another (and thus also for addition) if they were simply left connected while the reduction to zero was performed. Naturally this entered the number on the second wheel in an opposite sense of rotation, but since most connections were made through a series of wheels and pinions which could be made odd or even in number at will, ^{24} this did not matter.
The second area of basic operation which required improve went was the carriage mechanism. In the Difference Engine, the carriage had been delayed and sequential; that is, carriage could not be performed as a wheel passed from nine to zero in addition, since this would interfere with the separate addition proceeding simultaneously on the next higher digit; therefore carriage was delayed until addition was complete. Further, in performing a series of carriages onto a column of numbers, it is quite possible that a unit will be carried onto a wheel already standing at nine, thus making necessary a second order carriage, and so on; this problem is solved if the carriages are performed sequentially, that is, first on the lowest wheel, then on the next, and so on. This solution gives rise to its own problem, however, since if a large number of digits are used (and Babbage had in mind thirty or forty for the new machine from quite early on) sequential carriage can take a great deal of time, even considerably more than the basic addition itself.
The first approach Babbage used to speed up carriage was what he called hoarding carriage. ^{25} In this method, carriage was delayed not until a single addition was complete, but until a series of nine additions had been made. In the mean time, the carriages required by each addition had been stored on a special set of wheels; when carriage was to be made the numbers on these wheels were transferred to the result wheels as in an ordinary addition. This technique had two disadvantages, however; these carriages could still require second order carriages, so they would require either an additional mechanism to take care of these, or else to be added separately for each digit; both approaches would require extra time and mechanism. Further, hoarding carriage would be effective only in those cases where a whole string of numbers were being summed together; if the machine were performing isolated or separate additions, it would be no improvement over the old method of carriage.
The second new approach Babbage tried, and the one he eventually adopted (although perfecting it to his satisfaction took more time than any other part of the machine), was what he called anticipating carriage, since it allowed the machine to anticipate that a second order carriage would arise from a first order carriage, and perform then at the same time. This operated by having a special mechanism brought into play when the wheel onto which a first order carry was being made already stood at nine; then the special mechanism would cause a carry to be made onto the next higher wheel at the same time, and so on. Thus all carrys could be performed in one step immediately after addition in a time independent of the number of digits in the column. The details of the many different ways Babbage tried to accomplish this need not be considered here, but the first attempts seem to have been made about the end of October, 1834. ^{26}
Although it was clear to Babbage that anticipating carriage would save a large amount of time, it was equally clear that whatever the details of the version finally worked out, the mechanism for carriage would be immensely complicated in relation to that required for addition by itself. This apparent drawback led Babbage to an important step, namely that of separating the carriage mechanism from the columns on which it was intended to act. Thus given one or two carriage mechanisms which could be connected at will with the central wheels or with any adding column in use, a large amount of mechanism could be saved.
This was a most important step in the evolution of the Analytical Engine, for as the same approach was applied to other aspects of the engine, there gradually emerged the notion of a central processing section in which the various arithmetical functions were to be carried out, distinct from the area where the numbers were stored when not being manipulated. The central processing unit Babbage later came to call the "Mill," and the other number columns he called collectively the "Store," this fundamental distinction of course still exists in current computers.
It is not clear exactly when Babbage devised this way of organizing the machine, along with the specialization of the carriage mechanism from which it flowed, but apparently they developed gradually between November, 1834 and the early spring of 1835. During this time Babbage continued to develop the process of division, with respect to which he made a very important breakthrough in principle, though again the exact dates of the different steps are not clear. We can, however, say that the following had been thought through by early in March, 1835. ^{27}
The problem with division was still that of deciding whether the divisor was smaller than the remainder at any given point in the operation, so that subtraction should occur. Babbage's first approach, that of comparing as many digits as necessary by using snails (discussed above), required too complicated a mechanism. His next approach was to eliminate most of this mechanism by having the machine assume that the divisor was larger than the remainder if the two numbers had the same first digit; thus the divisor would be stepped down a cage and also the successive subtractions would be registered on a lower cage, and subtraction would proceed. The difficulty with this approach was that on those occasions when the first digits of divisor and remainder were equal but yet the remainder was larger, the machine would perform ten subtractions where it could have performed one, although it would still produce the right answer; thus this method was too time consuming.
The neat approach Babbage used was most ingenious. The numbers on the divisor and remainder axes were copied onto two subsidiary axes, and the first was subtracted from the second; if the result was negative, that is, if the highest wheel changed from zero to nine and tried to carry down from the (non-existent) wheel above it, then the divisor (back on its regular axis) had to be stepped down a cage; if the result was positive; that is if the top wheel did not change, then the subtraction could proceed back on the main axes without stepping. Thus all the snails and feelers and comparisons of digit to digit were eliminated.
The final step was to reverse the roles of operation and test in this last method. That is, the machine was to assume that the remainder was larger than the divisor and proceed with subtraction; if this assumption were false, it would show up in that the remainder would turn negative in the way indicated above; this would put the machine into a special sequence where it would add the divisor back to the remainder, subtract one from the quotient, and then step down the divisor and the cage in which digits were registered in the quotient before again making a subtraction. This improvement saved time, since in general the total number of subtractions that had to be made was considerably larger than the number of steppings that had to be made.
Again, Babbage had been led for practical reasons to a change in the machine which was to be of far reaching importance; as the Analytical Engine evolved from this point, Babbage applied this method of controlling divisions to other internal decisions the machine had to make, and eventually to general program branching itself.
By early 1835, the number of different kinds of axes which had to be interconnected in different ways depending upon what operation was to be performed had grown so much that a new approach to controlling the machine was required. ^{28} The basic technique was to have connections between different axes made through a series of pinions and possibly the central wheels; the axes on which the pinions were held could be moved up and down a small amount, and the particular position they were in determined what connections were made. The problem of control was thus the problem of governing the position of the pinions. This was to be done by what Babbage called "traveling platforms," which moved up and down repeatedly in the basic cycle of the machine; when a pinion axis was to be raised, it was temporarily locked to a traveling platform; when this reached the proper height, the axis was unlocked from it and locked to some stationary support, so that it would stay in the proper position.
The connections between the pinion axes and the traveling platforms were controlled by the barrels. These were cylinders embedded with many studs arranged in rows; the studs in a particular row would activate levers which made the connections necessary for a particular step. ^{29} Successive steps would be controlled by successive rows of studs, the active row being determined by rotating the barrel.
As this control system had evolved by July, 1835, ^{30} each different variable had its own barrel, which had studs set for the sequences of steps necessary for the different operations. These barrels were all controlled by a central drum, a similar cylinder with studs which could be set by hand for the particular sequence of operations desired. A given row of studs on the drum would specify what operation was to be performed and what variables it was to be performed on; it would turn the barrels so that they would bring about the sequence of steps necessary for that operation; when the operation was complete, control would be returned to the central drum, which would advance a row and call for a new operation. ^{31}
In March and April, 1835, Babbage developed two important improvements in the processes of multiplication and division. First, on March 22, Babbage realized that he could adapt the machine for double precision operation. ^{32} In particular, Babbage realized that if he made it possible for the top of one column to carry and step to and from the bottom of another column, then he could take two numbers of, say, forty digits each (the number of digits he seemed to be planning on at this time), and multiply them together, retaining eighty digits in the product; then when this product was divided by a forty digit divisor, the quotient would still have forty significant digits. On the other hand, it would also be possible to convert the machine generally; thus a machine with sixty variables of twenty digits each could be made into a machine of thirty variables with forty digits each with no real mechanical change. ^{33}
The second change in the way multiplication was to be done was devised on March 30, and was called multiplication by table. ^{34} In this scheme, the machine would start a multiplication by forming a table of the multiplicand multiplied by each of the digits from one to nine: this would be placed on nine special axes, and could be formed quite easily by simple addition. Then a snail wheel would sense the first figure of the multiplier, and supply it to a selecting mechanism, which would cause the corresponding multiple in the table to be added to the result axis. Then the result and the multiplier would each be stepped up one place, and the operation would be repeated for the second digit of the multiplier, and so on. In this way, every digit of the multiplier could be taken care of in one step, rather than requiring a number of additions equal to the digit's value. Division could be handled in a similar manner, although the problem of selecting which particular multiple ought to be subtracted from the dividend was considerably more complex. Babbage did not develop the details of division by table as early as those of multiplication.
It is interesting to note that it was at about this same time that Babbage made the first public statement of his invention of the new machine; this was in the form of a letter to the great Belgian statistician L.A.J. Quetelet, who read it to the meeting of the Royal Academy of Sciences at Brussels on May 7-8, 1835. ^{35} Without going into details of the machine or its capabilities, Babbage said "I am myself astonished at the power I have been enabled to give this machine; a year ago I should not have believed this result possible." Babbage also announced, with characteristic optimism, "that the greatest difficulties of the invention have already been surmounted, and that the plans will be finished in a few months."
During these next few months and many more, Babbage did indeed continue developing the machine, and he turned his attention increasingly to less critical parts of the Engine, ones which will not be discussed here in any detail. For example, in the late spring of 1835, Babbage began to reconsider his original basic plan of a circular arrangement of the machines he sketched possible arrangements using straight racks, and also combinations of racks and central wheels. ^{36} He also devoted some attention to the forms of output for the new machines as early as July, 1835, he made some notes about a "curve drawing apparatus" which would turn numerical output from the engine into graphic form. ^{37} In December, 1835, Babbage was making plans for having the printing mechanism completely separate from the rest of the engine, perhaps in another room, with only one rack to convey the data to it, ^{38} he also wished to be able to turn out single copies of results by having the type print through carbon paper, rather than making an impression in a stereotype mold. ^{39}
What must be realized at this point, though, is that while the engine as a whole was becoming more and more complex in some respects, some of its individual parts were getting much simpler, due to the specialization that had been made necessary by the very complexity introduced into the particular processes. Most especially, the separation of the Store from the Mill had made it possible for the Store axes which held individual variables to be very simple indeed, much simpler, for example, than one of the columns in the Difference Engine. This of course made it possible and practicable to have the machine able to handle a very large number of variables; Babbage at one time considered as many as one thousand. ^{40}
But although Babbage could imagine problems which would take advantage of this large capacity and also of the rapidity and sophistication of processing of which the Mill was now capable, the machine would not have been able to solve these problems, because the control mechanism was not able to handle the necessary instructions; the process of placing studs for the different operations in the drums by hand was far too limited, too cumbersome and too inflexible to take advantage of the rest of the engine.
This led Babbage, on June 30, 1836, to adopt punched cards for the control of the engine. ^{41} These he adopted quite directly from the system of control of automatic looms by punched cards which had been perfected at the beginning of the century by J.M. Jacquard of Lyons. ^{42} In principle, these cards were identical to the Hollerith punched cards used in twentieth century computers, in that they consisted of cardboard rectangles with rows and columns of spaces which could be punched out or left solid, conveying information through some kind of sensing mechanism to the machine which they were to control. In practice, Babbage's cards were to be sensed by mechanical levers rather than by electrical brushes, so that they had to be made of thicker cardboard, and their area was also to be larger.
The introduction of punched cards into the new engine was important not only as a more convenient form of control than the drums, or because programs could now be of unlimited extent, and could be stored and repeated without the danger of introducing errors in setting the machine by hand; it was important also because it served to crystalize Babbage's feeling that he had invented something really new, something much more than a sophisticated calculating machine. The first indication of Babbage's vision came in a memorandum in his notebook dated July 10, 1836, sayings "This day I had for the first time a general but very indistinct conception of the possibility of making an engine work out algebraic developments, I mean without any reference to the value of the letters" (that is, the value of the letters in the formulae with which the machine was dealing). ^{43} Unfortunately, the rest of this note is very obscure and sheds no light on what Babbage had in mind, but from some of the notes in the vicinity of the above quotation it appears that one of the things Babbage had in mind was eliminating between variables in high order simultaneous equations.
In any case, by this time, midsummer of 1836, all the main principles of the Analytical Engine had been laid down; if one were obliged to name a single point at which it was "invented," this would be it. From this point on, Babbage's task was essentially to fill in the details, although this phrase cannot do justice to the magnitude of what he did. The character and complexity of those details, and indeed what it was that Babbage had by this time invented, can only be understood in terms of the general structure and operation of the machine as a whole.
No description of the mechanical operation of the Analytical Engine has ever been published, and it is doubtful that anyone other than Babbage has ever understood it fully, and very few have understood it at all. Fortunately, in December, 1837 Babbage wrote a paper "On the Mathematical Powers of the Calculating Engine," of fifty three manuscript pages, which provides a rather full, if sometimes less than lucid, account of the operation of the machine. ^{44}
The following description of the Analytical Engine is drawn largely from this 1837 paper, being quoted directly when appropriate, although the original material has been rearranged and condensed, with some explanations rewritten or added as necessary. ^{45} . The Analytical Engine consisted of two sections, the Store and the Mill. Figure 1 is a plan of the Engine drawn in August, 1840, but basically the same as what Babbage described in 1837. The Mill was the collection of circles and devices surrounding the large central wheels in the middle of the drawing. The Store was the section extending off toward the right from the Mill; not all of it is shown. It can be seen from the scale provided. in the drawing that the length of the part of the Engine shown was about ten feet.
Running down the middle of the Store was a series of long racks, one for each cage; any column in the Store could be engaged with these racks. At their left end, the racks connected with the central wheels of the Mill. The section of the Store beyond the edge of the drawing was simply a continuation of the part shown.
The Store may be considered as the place of deposit in which the numbers and quantities given by the conditions of the question are originally placed, in which all the intermediate results are provisionally preserved and in which at the termination all the required results are found. . . .A number of axes each having forty figure wheels placed in different cages one above another are connected with the rack of the Store. These figure wheels are
each numbered from 0 to 9; they may be turned by hand so that any digit may stand opposite a fixed index. Thus any number of not more than forty places of figures may be put upon the figure wheel of each axis.Above the fortieth cage is another cage containing a wheel similar to a figure wheel and also having its circumference divided into ten parts. These parts have the signs (+) plus and (-) minus alternately engraved upon them. Above this wheel is a fixed character to distinguish each particular axis, or rather the variable number which may be found upon its wheels. These fixed marks are v_{1},v_{2},...v_{32}.... as far as the number of quantities which can be contained in the store. . . .
The number of variable cards which can be contained within the store will depend on the length of the rack and number of figure axes which can be placed round it, and although a large number of variables might with perfect safety be employed yet there is obviously a practical limit arising from the weight of the rack to be moved.
One hundred variables would not give an inconveniently large rack, but still the calculations of such an engine would be limited. The limitation can be entirely removed by another set of cards called Number Cards which will presently be described.
If any of the coefficients contain decimals, or if the result is required with decimals, then all the coefficients must be considered as having the same number of decimals. If an imaginary line is drawn between any two cages - the third end fourth for example - then all below it may be considered as decimals. In order to convey to the Engine this information there exists a wheel with the numbers from 1 to 40 engraved on its edge; this wheel being set at any number the Engine will treat all the numbers put into the Store as having that number of decimals. . . . ^{46}
In deciding on the content of the numbers with which the engine should compute, the first consideration was to look at the number of figures which in the present state of mathematical enquiry are required in the most extensive calculations. It then became desirable to look forward to the probable increase which improved observations might require. Finally the mechanical structure of the engine or its necessary arrangement might put limits to this extent or render the time employed in given calculations longer at certain definite intervals.
But even if it were thus possible to satisfy all practical wants, there would still remain a desideratum to render the mechanism philosophically perfect as to its power of converting algebraic expressions into numbers without any limit as to their magnitude or extent.
The result of my reflections has been that numbers containing more than thirty places of figures will not be required for a long time to come. I have however made the drawings of the Engine for forty places of figures. All additions and subtractions may be made with such numbers, and the products amounting to eighty places may be preserved in the Store and brought back into the mill to be divided by numbers of forty places of figures, thus retaining forty places in the quotient. ^{47}
At the beginning of a computation, the numerical data could be fed into the Store by manually setting each wheel of the columns being used. However, it was also possible, and obviously preferable, to feed the data in by a special set of punched cards, called the "Number Cards." These cards had another function which Babbage considered even more important. It might happen that in some very complex computation, more variables would be needed than there were columns in the Store; in this case the Store could be directed to punch certain cards to hold the overflow data, which could later be recalled and read back into the machine as needed. ^{48}
The Store also contained various output provisions. Single copies could be printed through carbon paper; copper plates were punched when the results were to be published. Most interesting was the Curve Drawing Apparatus:
The discovery of laws from the examination of a multitude of tabulated and reduced observations is greatly assisted by the representation of such tables in the form of curves.As one of the employments of a calculating engine would be to reduce collections of facts by some common formula, I thought that at the time it impressed the computed results it would be desirable that it should mark the point of a corresponding curve upon paper or copper if preferred. The three or four first figures of the table will be expressed by the curve. The contrivances for this purpose are not difficult, and their employment does not lengthen the time of the calculation. ^{49}
However, Babbage considered that it would probably be best to have all output from the Store in the form of punched cards. These would then be fed into a totally separate (off-line) printing mechanism which would produce whatever form of output was desired.
The functioning of the Store was controlled by a set of Variable Cards, whose action was co-ordinated with the Operation Cards which controlled the Mill. The Variable Cards could order a number to be given off from a particular store axis to the ingress axis of the Mill, or for a number to be received from the Mill and put on a particular store axis; the Variable Cards also could order that numbers be read and stored from the Number Cards, or that particular variables be punched on cards. ^{50}
The actual operations were performed in the Mill. It was made up of the following principal components: Figure Axes, Carriage Axes, Table Axes, Digit Counting Apparatus, Selecting Apparatus, Barrels, Reducing Apparatus, Operation Cards, and Combinatorial Cards. These components were described by Babbage as follows (and see Figure 1).
1. Of the Figure Axes The Figure Axes A and 'A are connected with each other without the intervention of the central wheels so that a number on the figure wheels of one axis may be transferred to those of the other. These figure wheels are considerably larger than any others in order to allow of sufficient space on their circumference for placing the pinions by which communications are made with other parts of the mill.
By means of some of these pinions a process called Stepping down and another called Stepping up may be performed. It consists in shifting each digit of a number one cage lower or one cage higher, which processes are equivalent to the arithmetical operations of dividing or multiplying the number by ten. Other pinions are fixed on register axes R and Ri and convey the two highest figures of the dividend to the Selecting apparatus.
The third figure axis, ''A, is placed near the Store and constitutes the egress axis. It is adjacent to the digit counting apparatus, with which it communicates.
2. Carriage Axes. These Axes F, 'F, ''F with their peculiar apparatus are employed to execute the carriage of the tens when numbers are added to or subtracted from each other. The carriages F and 'F can be both connected with the Figure axis A, or one of them with the Figure axis A and the other with 'A, or they may by means of the central wheels be connected with any other part of the Mill. The third carriage, ''F, is connected with both the Mill and the store and may be used with either.
Whenever. the number subtracted is greater than that from which it is taken the resulting carriages would, if effected, and if the mechanism admitted, produce a carriage in the forty first cage. This fact is taken advantage of for many purposes: it is one of very great importance and when it happens a Running up, is said to occur. Connected with this part is a lever on which the running up warning acts, and this lever governs many parts of the engine according as the circumstances demand.
3. Table Figure Axes. These axes are ten in number; nine of them contain the table of nine multiples of one factor in Multiplication and of the Divisor in Division. The tenth contains the complement of the Divisor in the latter operation. They are all connected with the central wheels and the number on each figure wheel can be stepped up or down upon the other figure wheel of the same cage. The figure which at each stepping goes off from the bottom wheel is transferred to the top wheel.
4. Of the Digit Counting Apparatus. This is a mechanism by which the digits of any number brought into the mill may be counted; and certain calculations made as to the position of the decimal point in the result of multiplication and Division. It is also used to limit the number of figures employed when the engine is making successive approximations either to the root of equations or to the values of certain functions. It consists of three distinct systems nearly similar to each other.
5. Of the Selecting Apparatus. When a table of the nine multiples of a multiplicand has been made it becomes necessary in order to effect multiplication to select successively those multiples indicated by the successive digits of the multiplier.
This mechanically is not difficult. But when in the process of division it becomes requisite to select that multiple which is next less than the divident from which it is to be subtracted, the mechanical difficulty is of quite a different order, and hitherto nothing but the most refined artifices have been found for accomplishing it. . . .
[6.] Of the Barrels. The barrels are upright cylinders divided into about seventy rings, the circumference of each ring being divided into about eighty parts. A stud may be fixed on any one or more of these portions of each ring. Thus each barrel presents about eighty vertical columns every one of which contains a different combination of fixed studs. These barrels have two movements. 1st They can advance horizontally by a parallel motion of their axis. 2nd They can turn in either direction and to any extent on their axis.
When the barrels advance horizontally these studs act on levers which cause various movements in the mill: the stud belonging to each ring giving a different order.
Amongst these movements or rather these orders for movements the following may be more particularly noticed.
The advanced of a barrel may order:
- A number with its sign to be received into the mill from the ingress axis;
- A number with its sign to be given off from the mill. This number may thus be altogether obliterated from the mill: or it may at the same time be received on the egress wheels or the number may be given off from the mill to the egress wheel and at the same time be itself retained in the mill.
- A variable card to be turned.
- An operation card to be turned.
- The circular movement of the Barrel itself or of any other barrel to another vertical. This always occurs at every step from the beginning to the end of what are called operations. The barrels when once ordered by the operation cards from their zero point to any given vertical always direct themselves to be turned to another vertical preparatory to their next advance. This circular motion is however occasionally changed by an action arising from another source.
7. Of the Reducing Apparatus. Behind each barrel is placed a reducing apparatus.
It consists of six or eight sectors which can be made to act upon the barrel and give it a rotatory movement so as to make it pass over 1, 2, 3, or any required number of verticals previously to its next advance. The levers which put these sectors into action are acted upon by
The first and third of these sources of action occur most frequently. ^{51}
- The studs on their own barrel..
- The studs on any other barrel.
- The Operation Cards.
- The Running up levers.
8. Of the Operation Cards. Those who are acquainted with the cards of a Jacards [sic] loom will readily understand the functions performed by these cards. To those who are unacquainted with that beautiful contrivance it may be necessary to state that the Cards consist of pieces of thick paste board, tin plate, or sheet zinc, pierced with a number of holes. These cards, being strung together by wire or tape hinges, pass
FIGURE 2: Driving and Directive of the Analytical Engine, August 14, 1841. S.K., Drawing II, 94.over a square prism. The prism is situated in front of a number of levers placed in rows which govern the Reducing Apparatus and consequently the barrels. The faces of the prism are perforated so as to present an opening opposite every lever. If the prism alone is made to advance horizontally against these levers then the levers themselves will enter into the holes of the prism and be partly covered by it; but they will not be moved out of their places.
Again if a card having as many holes as the prism has, or as there are levers opposite to it, is placed upon the advancing face of the prism, no effect can be produced on the levers by this advance of the prism. But if a card having one hole less than the prism is placed on its face, then when the prism advances the lever opposite that hole will be pushed away and any order given for which that lever was appointed.
Suppose after every order the levers to be replaced, and let the prism be turned one quarter round, then a new card will be presented to the levers, and if one or more holes of this second card are stopped up, a different order will be transmitted through the levers to the Reducing Apparatus and thence to the barrels.
Thus by arranging a string of cards with properly prepared holes any series of orders however arbitrary and however extensive may be given through the intervention of these levers.
The number of the levers acted upon by the operation card is small: they respectively direct the barrels to commence the following operations:
- The Addition of two numbers.
- The Subtraction of one number from another.
- The Multiplication of two numbers.
- The same Multiplication limited to a given number of the first figures.
- The Division of one number by another.
- The same Division limited to a given number of figures in the quotient.
The levers numbered 4 and 6 are rarely used; the extraction of roots being the only case in which they are required.
These cards are called into action by orders from the barrels. What they shall do when acting depends on the nature of each individual card. What repetitions they shall be subject to depends on the orders communicated to them from the Combinatorial Cards and their Counting Apparatus. Many calculations are much simplified by having two sets of Variable Cards.
9. Of the Combinatorial Cards. One or more peculiar cards may be inserted among the operation cards of certain formulae. They are called Combinatorial Cards.
The object of these cards is:
To govern the Repeating Apparatus of the Operation and of the Variable Cards and thus to direct at certain intervals the return of those Cards to given places; and to direct the number and nature of the repetitions which are to be made by those cards.
Whenever Combinatorial Cards are used other cards called index cards must occur amongst those of the formulae. The use of these cards is to compute the numbers which are to serve successively for the indices of the combinatorial cards. At what time the Combinatorial Cards shall act depends on the number of repetitions the last of those cards appointed. What orders each Combinatorial Card shall give depends on the nature of each individual Card. ^{52}
Thus the detailed steps within an arithmetical operation were controlled by the Barrels, while the sequence of operations was determined by the operation cards. The interaction between the two occurred as follows:
In any operation it is only necessary that the operation card ordering its commencement should be advanced. This sets the barrels in action and provides through the variable cards if necessary the first quantity to be operated upon; the barrels there call in the other quantities required at the proper times, until the operation is completed, the Mill cleared of all numbers, and the computed result placed in its intended situation in the Store. The last vertical on the barrels belonging to each operation directs them to move to its zero point.Such being the arrangement, the calculation of numerical quantities resulting from two or more successive operations may be readily performed by placing the operation cards, strung together in their proper order on their revolving prism, and the variable cards marking the numbers in the Store on [sic] which are to be the subjects of those operations and also in the order in which they are required on their own revolving prism, and then putting on the final verticals of one of the barrels a stud which orders the operation cards to advance. ^{53}
The actual operation of addition as controlled by the Barrels was performed by the method of reduction to zero already discussed; that is, the Store axis off which a number was to be read was connected to the Mill axis which was to receive the number, and then the wheels of the former were set mechanically to zero, transferring the number to the Mill. Similarly, carriage was performed by the method of anticipation already discussed, although Babbage never finally settled on the details of the carriage mechanism.
The only new or complex element in the process of addition was the treatment of signs, since addition and subtraction were to be performed alike, and since the machine could now contain both positive and negative numbers in the Store. This was to be done in the following way. Suppose two numbers were to be combined; each would have two signs, one corresponding to addition and subtraction, as specified by the operation cards (Babbage called this the algebraic sign), the other corresponding to whether the number actually in the Store as determined by previous operations was positive or negative (called the accidental sign). As a number came onto the ingress axis of the Mill from the Store, the two signs would be compared, simply by adding the pseudo-numbers on their topmost wheel, or sign wheels as a positive sign corresponded to an even number on this wheel, and a negative sign corresponded to an odd number, the result of the addition of the two digits would correctly indicate the combination of the algebraic and accidental signs. If this combined sign was positive, the number was added to the figure axis where addition was to take place; if it was negative, the number was subtracted, that is, an extra pinion was introduced into the train of gears to reverse the direction of rotation.
The same process was applied to the second number as it came into the Mill, and it would be added or subtracted from the Figure Axis as determined by its combined sign (and the same could be done with a third or subsequent number in the same way). The effect of this addition or subtraction might be to change the. sign of the number on the Figure Axis by making it pass through zero; the number on the Figure Axis would then appear as the complement of its arithmetic value; but the engine would know this because a running up would have taken place, so that before the result was used or returned to the Store it would be changed back to its proper form, along with the finally correct sign. ^{54}
Multiplication The process of multiplication is thus performed by the Engine:
The two factors P and Q with their respective accidental signs being placed on the two sets of Figure wheels No. 1 and No. 2 of the Ingress axis, an operation card is turned. This directs the barrels through the reducing apparatus to move circularly to the vertical which commences multiplication.
At the next turn the barrels advance and order the reception of one of the factors, Q. They also direct themselves to move on their axes to the second vertical belonging to multiplication. At the third turn the barrels advance and order the reception of the other factor, P, from the second set of Ingress wheels. At this turn the factor P is subtracted from the factor Q, which had at the preceding turn been placed upon the figure wheels of 'F.
The result of this subtraction determines which of the two factors is largest, and consequently which factor is to be tabulated. Much time is saved by this decision, for supposing one factor to contain only four places of figures and the other to contain thirty five figures, then if a table of the first nine multiples of the first factor were made, its multiplication by the other factor would require thirty five additions, whilst if the larger factor had been tabulated, only four additions would have been necessary. The mechanical mode by which this knowledge is conveyed to the barrels is thus. The barrels, after ordering the subtraction of P from Q in the axis 'F, direct themselves to move on to another vertical; if P is less than Q no Running up takes place, and the order thus given by the barrels is obeyed; but if P is larger than Q then a Running up takes place, and the order given by the barrels to move to a certain vertical is enlarged, and they are directed by the running up lever to move on to a different vertical. This new vertical directs the tabulation of the second factor, whilst the former vertical appointed that of the first factor..
At the next turn the number which the Mill has decided on tabulating is added to each of the nine table axes and to the carriage F.
At the next turn the same number is added to eight of the Table axes and the carriage axis, then to seven, and so on, until the nine multiples have all been formed on the nine Table axes.
During these turns the number of the digits in the multiplier has been ascertained and placed in the Counting Apparatus No. 1.
The Multiplier having been prepared is now stepped down and acts upon the Selecting Apparatus. Whatever digit exists in the lowest figure of the multiplier, by means of the Selecting Apparatus the corresponding Multiple is added over to the product axis 'A. During this addition each multiple is stepped up one cage higher on its own figure wheels; also the multiplier being stepped down, its next figure is conveyed to the Selecting Apparatus preparatory to the next addition to the product.
The product is thus formed beginning with its lowest figure on the figure wheels of the axis ['A]; but in the process of stepping up, the highest figure on the tables axes may at length reach the top or 40th cage of their axes. At the next stepping the highest figure would of course be stepped off the machine and lost, but this is prevented by a communication with the lowest figure wheel on the same axis to which the number stepped off from the highest is transferred. These figures which are so transferred to the lower cages are not added to the same axis but are by a particular mechanism added over to a set of figure wheels on the axis A and the carriage F on which the head of the product is formed.
If a number of twenty figures only is multiplied by another factor containing an equal number of digits, as the product will not exceed forty digits it will be contained on the first axis. But if both factors contain forty digits then the head or first forty digits of the product will be formed on the axis A and the tail or second forty digits of the product will be formed upon 'A.
The signs which entered with the original factors, having been added together on the sign wheels, will give the resulting sign of the product. For if they were both positive, then each sign being equivalent to an even number, the sum of the two signs would give an even number or its equivalent the positive sign. If on the other hand both signs were negative, then a negative sign being equivalent to an odd number, the sum of two odd [numbers] produces an even number, which is also equivalent to the positive sign which the product ought in this case to possess. Lastly, if the sign of one factor be positive and that of the other negative, then the sum of an even and of an odd number being odd, the negative sign results.
The termination of Multiplication arises from the action of the Counting Apparatus, which at a certain time directs the barrels to order the product thus obtained to be stepped down so that the decimal point may be in its proper place, and then to turn to another vertical the consequence of which is the transfer of the product with its proper sign to the wheels on the egress axis from whence they may be removed by Variable cards and the barrels to any axis in the Store. ^{55}
Division Of the numerous difficulties which the contrivance of an engine to perform arithmetical operations presented, none certainly offered more reasonable grounds of difficulty than that structure of it by which it should be enabled to perform the operation of division. . . .
It has already been stated that when any number which happens to be greater than another is subtracted from that other number, an event which has been termed a Running up takes place, and that in consequence of this event any order which may be desired can be given to any part of the engine. Bearing this fact in mind, the division of one number by another may be conceived to take place in the following manner. . . .
The first direction given by the barrels is to subtract the divisor from the dividend. This order is repeated until . . . it is found that a Running up has taken place, a fact which of course indicates that the divisor has been subtracted once too often. The number of times the divisor has been subtracted is marked by a register axis. Now it is in consequence of this fact of a Running up that the engine knows it has subtracted the divisor once too often, and that knowledge is conveyed by the running up levers to the barrels in the form of an order to move to such verticals that at the succeeding turns they shall direct the erroneous subtraction and registration to be corrected, and the remainder of the dividers, to be stepped up, in order to undergo a similar process for the Discovery of the next digit in the quotient. . . .
The first figure in the quotient being thus obtained, a repetition of the same process will of course produce the succeeding figures.
The complete understanding of this principle is of some importance, for it not merely relates to the process of division, but it is capable of being applied to any tentative arithmetical process. I had applied it to the extraction of roots, but other improvements induced me to dispense with it in this instance. Its application is not limited to existing rules, but I believe I may venture to state that whenever new methods are contrived for overcoming arithmetical difficulties by tentative processes, it will be found available for these new purposes; and that even in cases where the tentative processes relate to algebraic expressions it may yet be useful.
I have given a simple case in order to explain the nature of this principle. Division however is not executed by this tedious process, which would required for its performance a number of turns more than equal to the sum of the digits in the quotient. The process really used may be thus described.
The operation card ordering division having advanced directs the barrels to the proper verticals.
The divisor P is first received from the ingress wheel upon several parts of the mill.
The Dividend B next enters. Several processes are gone through in order to ascertain whether it will be necessary to step the dividend up or down, and how many steppings there ought to be of either kind. This depends upon the number of digits of the quantities operated upon, and is principally executed by the counting apparatus, which also computes. the number of figures which must occur in the Quotient.
At about the eighth turn the formation of the table commences, and also the selecting apparatus is prepared.
If the Stepping down is completed, the actual division may commence about the eighteenth [?] turn.
By means of the selecting apparatus the two first figures of the dividend are compared with the first two figures of each of the nine multiples of the divisor, and that multiple which has its two first figures either equal to or nearest less than the first two figures of the dividend is selected and subtracted from the dividend.
The remainder is then stepped up, and the process repeated until the counting apparatus interferes to finish the process.
In case the two first figures of the dividend are exactly the same as the two first figures of one of the multiples of the divisor, there may arise a doubt whether that multiple or the proceeding one should be subtracted: this will depend on the third figure of each quantity. But in this case the mill always takes for granted that the multiple selected is the true one. If however the contrary happens to be the case, then the subtraction of the multiple which is really too large will cause a running up in the dividend, and in consequence of this, orders will be transmitted to the barrels to direct the addition of the first multiple and the subtraction of unity from the multiple registered in the quotient, by which means, the quotient being set right, the first process is then continued.
After the end of the actual division, the quotient and the remainder, if required, are given off to the egress wheels.
As the result of the multiplication of a number containing forty digits by another of equal extent is a product containing eighty digits, and as this product can be conveyed into the store on two sets of figure wheels, a provision has been made for bringing such numbers back into the mill when they are to become dividends, so that the results of the division of a number eighty by one of forty digits shall have the forty figures of its quotient true to the last figure. ^{56}
This completes the explanation of the outlines of the four basic arithmetical operations. But it will be well to provide, as Babbage did not, a brief account of the mechanical approach of a lower level by which the various connections referred to were controlled. The fundamental notion necessary for understanding this is one that Babbage called Chain. This can be understood from the following sketch: ^{57}
Here the power input is on the shaft N; this turns a wheel holding the cams B and D, which act through a rocker-arm to move the piece E back and forth. E slides along a similar piece, called G; G and E will be connected together if the link C is in position in the slot cut away in them, and in this case the rotation of N will cause G to oscillate. The position of C, which pivots around the axis T, is controlled by the small arm F. The principle involved is that the transmission of power from N to G is controlled through a link which acts rather like a switch, and can be operated with much less power than it is controlling. Thus F might in fact be a feeler which senses whether a hole is punched in a certain position in a card.
The way the principle of Chain was applied in practice in the Analytical Engine can be seen in two crucial cases in Figure 2. In it, a variable card can be seen on the left and a Barrel on the right, with the columns they control near the top. One chain can be seen from the large cam wheel, C, next to the Barrel; the rocker arm H will rock the arm K, which will move the pinion axis that can connect the Rack to the figure axis up or down; but this chain will only be complete if the link B is in the proper position, as determined through a couple of arms by a stud near the top of the Barrel. Other arms forming Chains from the Barrel to other axes or to effect the rotation of the Barrel itself are activated by different studs, but only a few are shown in this drawing.
The Variable Cards form Chains of a slightly different kind. Again, their function is to raise a column with pinions so that a certain Store axis will be connected to the Store Rack. They act through what were called Travelling Platforms, which were shafts below the main machinery which move up and down a small amount at regular intervals; they are shown to the right of the variable cards in the drawing in cross-sections marked P. Arms activated by the variable cards would connect or disconnect the Travelling Platforms from the pinion axes through special sliding links.
One final and slightly different example of a Chain is the technique used for anticipating carriage. The examples of Chains given above are ones where a power train would be completed if one certain contingency were fulfilled. In the Chain involved in anticipating carriage, there were multiple contingencies of different kinds. Thus the beginning of the chain would be moved if there: was a first order carriage at a certain point in the number column; the link would be activated if the wheel next above stood at nine (in the case of addition); the chain could be as long as the series of adjacent wheels standing at nine, or there could be several, separate partial chains set up in a column at the same time.
One final aspect of the operation of the Analytical Engine must be considered here, both because an understanding of it is important in any assessment of the Engine, and because Babbage himself considered it very important; this is the question of the speed of the Engine.
In his 1837 paper, Babbage proposed several principles which had guided the design of the machine, one of them being "that the several operations shall be executed in the shortest possible time." ^{58} Babbage described the great difficulty this principle had caused him, and why he considered it to be extremely important, in the following remarks, which indeed, serve also to express in a way the fundamental paradox of Babbage's project, and also some of the motivation which drove him on.
In fact the whole history of the invention has been a struggle against time. As soon as any contrivance has been made which was unexceptionable as to the mechanism, the question has always arisen: Can it not be executed in less time by some other contrivance? Thus every advance has but raised up a new object of rivalry, itself to be superceded by some more rapid means; nor can I hope that I have nearly reached the limit. If I have approached it, it is more than I have a right to expect as the pioneer in this difficult career. Another age must be the judge of that as well as of the other questions relating to the Engine. . . .The reader who only looks at the mechanical difficulties in contriving such an engine will doubtless be surprised, when he examines the history of the contrivances, . . .. at the lavish rejection of inventions which has taken place in order to achieve rapidity in execution. But the Analyst, who is aware that the last resource in all our difficulties is the conversion of the most intractible expressions into infinite series, will more readily appreciate the importance of even a small abbreviation of the time, when he is informed that it is one of the objects of this engine to arrive at the numerical values of the coefficients of such series, however complicated the laws by which they are formed; and proceding again from these results as established data, it is proposed to assign the values of such expressions for any magnitudes of the variables. ^{59}
As to the actual time taken, Babbage first defined the unit of time in which to describe the length of operations as that required for a figure wheel to pass over one unit of number, as to go from zero to one. Then the minimum cycle for adding one number would be ten units, nine for units which might be added, and one for another digit which might be carried; this cycle of ten, he thought, was the theoretical minimum which any decimal machine could ever attain. In practice, the cycle had to be longer, for time had to be allowed for moving the connecting pinions into place and for making the lockings and unlockings which prevented the machine from making any mistakes; the extra time required brought the cycle up to fifteen units in the case of plain addition, and twenty units in the case of addition with carriage. ^{60}
Beyond this, there were certain other operations which would take extra time; operation and variable cards had to be read, the Barrels had to be set, and so on. In order to accommodate these operations, it in fact took six cycles to add two numbers. However, the machine was so arranged that in certain circumstances it could perform different operations simultaneously in different sections; in a series of additions, it was possible to add each additional number after the first two in a single extra cycle. In other words, adding (or subtracting) n numbers together in succession, each of forty digits, would require n + 4 cycles of twenty units.
As to the length of the unit of time, wishing to give a conservative estimate of the speed of the Engine, Babbage estimated the maximum practicable velocity of the circumference of the figure wheels as ten feet per minute, or two inches per second; this would give a unit of time about 0.157 seconds long. Thus a cycle of twenty units would be about 3.14 seconds. This would mean that adding two numbers together would take about 18.8 seconds, but that each additional number would require only another 3.14 seconds. ^{61}
In the case of multiplication and division, the time required depended on the number of digits in the numbers being operated on, and Babbage's own calculations of the time needed were incomplete; but seemingly for two numbers of about forty and twenty digits respectively, for example, multiplication would have required a little less than two minutes, with division taking somewhat longer. ^{62}
It will be readily apparent that the above mentioned ten unit cycle is a theoretical minimum for addition only in the case of decimal operation, and that there would be a different cycle length for different numerical bases; this was also apparent to Babbage. His initial concern was that the performance of carriage was - even after all the work on it - still taking a large fraction of the total time. Babbage calculated that carriage would take only about one ninth as long if the Mill were operated on a numerical base of one hundreds but he decided not to adopt this base. ^{63} In general, the length of the basic cycle would vary directly with the magnitude of the base, but the number of cycles needed for multiplication or division would vary inversely with the magnitude of the base used. This general problem led Babbage to consider the question of "whether an arithmetic whose basis were 12, 16 or any other number being adopted, the operations might not be performed in a shorter time, notwithstanding the time consumed in converting the numbers out of the decimal scale and in reconverting the results into that scale." ^{64} Babbage did not record the reasoning which he went through, but he decided to continue his planning on a decimal basis.
The modern reader, concerned with the relation of what Babbage did to modern computers, would be interested primarily in evidence that Babbage considered using the binary system. It must be bone in mind, however, that the binary system is used in modern machines primarily not because of any inherent superiority or logical purity it has in comparison to other bases, but because the components of which modern machines are constructed, being electronic or magnetic, are themselves inherently binary, and in this respect less flexible than Babbage's wheels. In any case there is some evidence that Babbage was considering some use of the binary system in the Engine. While discussing Number and Variable Cards in his December, 1837 paper, he remarked that "fourteen levers and their equivalent fourteen holes will be all that is required . . . for eight thousand variables." ^{65} Unfortunately the context does not make very clear what Babbage was talking about, but the fact that 2^{13} equals 8192 strongly suggests that Babbage was pointing out that fourteen binary bits - represented by fourteen possible holes in a card - could convey numbers up to eight thousand to the machine, with the fourteenth hole serving to convey the sign. Later, however, Babbage planned simply to have a different hole punched out for each of the digits from one to nine. ^{66}
We have thus seen the picture of the internal development and operation of the Analytical Engine that emerges from an examination of Babbage's working notebooks and drawings, and from his December, 1837 paper. By this point Babbage had completed the invention of what we now know as the general purpose digital computer, and thus the changes which he worked on in later years were - by definition - not changes in principle, but rather changes to speed up the operation of the machine, or make it easier to construct, or some other such practical matter. Before proceeding with any further mechanical details, it will be well to consider the Engine in a more general context, in terms both of Babbage's own attitude toward it, and the communication concerning it he had with the outside world, in which that attitude is sometimes clearly revealed.
One aspect of this has already been covered in Chapter Two, namely the communication respecting the Analytical Engine which Babbage had with the government in connection with their support for the construction of the Difference Engine. It will be remembered that in his statement to the Duke of Wellington dated December 23, 1834, Babbage had announced the invention of a new machine with far greater powers than the Difference Engine, but one which did not supersede it, ^{67} 1836 Babbage was saying that the old machine had been rendered wholly obsolete. ^{68}
In the first few years of Babbage's work on the Analytical Engine, he carried on almost no correspondence concerning it with his friends and colleagues, or at least almost none is preserved. One notable exception is a letter written to an unnamed friend (apparently an American), dated August 2, 1835. ^{69} In this letter, Babbage described how, on gaining access to his drawings of the Difference Engine the previous September, a re-examination of them had led him to the invention of a new machine at the same time more powerful and less complex than the old one. He continued:
I have had a draftsman constantly at work since September and I have given up all other pursuits for the sake of this, and the progress I have already made surpasses my expectation, it is indeed more than I ever before made in several years. This arises partly from a more enlarged experience, partly by having a better tempered draftsman, partly by having all my drawings in my own house instead of at a distance of four miles. My intention is to finish a complete set of mechanical drawings of the new engine and a variety of Mechanical Notations to explain its operation. I have already got over all the difficulties of the first order and most of those of the second and feel very confident that in twelve months more, if I can carry it on as I have done, that I shall have completed the drawings so that the invention can not ever be lost. ^{70}Babbage then gave a brief outline of the function of the new machine, mentioning that among other things it could extract the roots of numbers by successive approximations, calculate logarithms and Bernoulli numbers, sum converging series, and reduce large numbers of observations.
Babbage concluded this letter as follows:
You will be able to appreciate the influence of such an engine on the future progress of science. I live in a country which is incapable of estimating it and I am much displeased with the treatment I have experienced . . . I have made large sacrifices and am prepared to make still larger. There is however, a limit at which my duty to my children demands my forbearance and I place that at the completion of my drawings, which is in fact the completion of the invention. These will remain when I am gone, and when some more enlightened country than my own shall at some future period become sensible of the value of the engine or ambitious of the fame of calling it into real existence, those drawings will undoubtedly be sought and from them a machine will be made which shall give to all nations and to all after time, tables at once perfect and enduring, and which shall execute calculations from which all human agents would shrink in hopeless despair. I have thus given you an outline of the work I am employed in; the views that it has opened and which it continually suggests are amongst the most interesting and important I have met with. ^{71}It is notable in this passage that Babbage was still thinking at this time, August 1835, in terms of the production of tables as the primary function of the new machine. This idea was one that Babbage was to continue to hold and express for many years, and this was true also of several other themes laid out in the same passage, for example that the machine could not be properly appreciated in England, and that he himself would not build it, but that it would be built after his death, and have a profound influence upon science.
This last point particularly was amplified and powerfully expressed in the passage with which Babbage concluded his paper of December 26, 1837, from which extensive quotation has already been made; the final paragraph read: ^{72}
Whenever engines of this kind exist in the capitals and universities of the world, it is obvious that all those enquirers who wish to put their theories to the test of number will apply their efforts so to shape the analytical results at which they have arrived that they shall be susceptible of calculation by machinery in the shortest possible time, and the whole course of their analysis will be directed towards this object. Those who neglect the indication will find few who avail themselves of formulae whose computation requires the expense and the error attendant on human aid.In the same paper, Babbage felt for the first time the need to express the opposite idea, namely that the machine had limitations; the form in which this was expressed is particularly interesting, for it came close to speaking directly to the modern worry over whether or not a machine can "think." In the passage in question, Babbage was apologizing in a footnote for the use of expressions like "the engine knows" or "the engine forsees" something. He said:
In Substituting mechanism for the performance of operations hitherto executed by intellectual labor it is continually necessary to speak of contrivances by which certain alterations in parts of the machine enable it to execute or refrain from executing particular functions. The analogy between these acts and operations of mind almost forced upon me the figurative employment of the same terms. They were found at once convenient and expressive, and I prefer continuing their use rather than substituting lengthened circumlocutions. For instance, the expression "the engine knows etc." means that one out of many possible results of its calculations has happened and that a certain change. has taken place by which it is compelled to carry on the next computation in a certain appointed way. ^{73}Despite the large amount of time Babbage was putting into the planning of the new calculating machine in the next couple of years, ^{74} his correspondence still did not discuss the Engine. Yet apparently this was despite a growing commitment to it, for on December 26, 1838, Babbage wrote to the Vice Chancellor of Cambridge University, communicating his desire to resign from the Lucasian Professorship, this sacrifice being necessary for the sake of "the completion of the designs and description of the Calculating engine." ^{75}
Not until December, 1839, can one find another letter of interest concerning the machine; on that date, Babbage wrote to the noted French scientist D.F.J. Arago in Paris, asking Arago to get for him a copy of the portrait of Jacquard woven on the Jacquard loom, an example of which Babbage had recently seen in London. He gave the following account of why he was anxious to get the portrait:
You are aware that the system of Cards which Jacquard invented are the means by which we can communicate to a very ordinary loom orders to weave any pattern that may be desired. Availing myself of the same beautiful invention, I have by similar means communicated to my Calculating engine orders to calculate any formula however complicated. But I have also advanced one stage farther, and without making all the cards, I have communicated through the same means orders to follow certain laws in the use of those cards, and thus the Calculating Engine can solve any equations, eliminate between any number of variables, and perform the highest operations of analysis. Imagine then that I am anxious to possess so singular a portrait of your distinguished countrymen. ^{76}Babbage concluded this letter with the following account of this progress:
I am continuing unintermittedly the Drawings of the machine on which I was engaged when you last visited London. I hope at my own expense and by my own efforts to leave behind me the drawing and full descriptions of an engine which has the power of solving by mere mechanical means most of the great problems of analysis.In the original draft, this last sentence continued with the following words, subsequently struck out:
But it is very improbable that I shall ever possess the pecuniary means to undertake its execution. I have spent many thousands of my private fortune on this pursuit, and when the drawings are completed, the invention can never be lost. ^{77}In 1840, Babbage received an invitation from the Italian mathematician Giovanni Plana to attend the second Riunione degli Scienziati Italiani, to be held in Turin in September, where all the leading Italian scientists would meet. The year before, Babbage had declined an invitation to attend the first meeting, on the grounds that it would hinder the progress of the Analytical Engine. ^{78} In 1840, however, Babbage changed his mind, doubtless in part because of the interest Plana had expressed in his invitation:
M. 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:"Hitherto 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." ^{79}
Babbage was very taken with this expression of the possibility that the Analytical Engine could make aspects of analysis which had previously been of purely theoretical interest become practical and useful. He decided to go to Turin, for the following reasons:
The great object of my visit to Turin was to convey to Plana and to some of the Analysts of Italy the principles on which I had contrived an engine to perform as he has beautifully expressed it "the whole Executive of Analysis." The discovery is so much in advance of my own country and I fear even of the age, that it is very important for its success that the fact should not rest on my own unsupported authority. I therefore selected the meeting at Turin as the time of publication, partly from the celebrity of its Academy and partly from my high estimation of Plana. ^{80}Babbage took with him to Italy "such of my models, drawings and notations as I conceived to be best adapted to give an insight into the principles and modes of operating of the Analytical Engine." At Turin, while the regular scientific meetings were in recess, Babbage met privately with Plana "and others of the most eminent geometers and engineers of Italy." Among these "others" was Luigi Federico Menabrea. Babbage later described these meetings as follows: ^{82}
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.Babbage hoped that as a result of his visit to Turin, Plana would give a report on the Analytical Engine to the Academy of Turin, so that its importance would not rest on his own unsupported authority. However, Plana became ill, and the paper did not get written. In March, 1841, Babbage wrote to Angelo Sismonda, suggesting that one of the other people, including Menabrea, who had heard his exposition could assist in drawing up the report. ^{83}
In about July, 1841, Babbage wrote to Alexander von Humbolt, whom he had known for many years:
I am very desirous to make you acquainted with the object that I have proposed to accomplish and to convince you that the means exist of constructing an engine which as Plana has described it gives us as complete a power over the executive of analysis as we have hitherto possessed over the legislative portion. The greater part of the drawings exist by which an engine might be made which shall reduce to numbers any explicit function whose laws of formation can be assigned. For instance, the equation of finite differences expressing the law of formation of the numbers of Bernoulli being given to it, it will work out and print the successive numbers.This engine is unfortunately far too much in advance of my own country to meet with the least support. I have at an expense of many thousand pounds caused the drawings to be executed, and I have carried on experiments for its perfection. Unless however some country more enlightened than my own should take up the subject, there is no chance of that machine ever being executed during my own life, and I am even doubtful how to dispose of these drawings after its termination. ^{84}
Babbage went on to say that he was most anxious to explain the machine to Humbolt and to get his advice, but that as the material needed for the explanation could not readily be taken abroad, he would have to invite Humbolt to visit London to see it.
One other point about this letter must be mentioned; it is apparently the first place in which Babbage used the term "analytical engine" although in this instance it does not seem to have taken on the role of a proper name for the machine, it did so shortly after. ^{85} Although there is no direct evidence, it seems likely that Babbage came to use this name because of the phrase from Plana's letter to him of 1840, which he quoted quite frequently, to the effect that the machine would allow man's "legislative" power over analysis to be extended to include "executive" power: this relation was then summarized in the term Analytical Engine.
Nothing came of this attempt to gain Humbolt's backing for the Analytical Engine, but apparently Babbage desired it for two main reasons. First, he felt he was unlikely to get understanding and support for the machine from his own countrymen. Second, he wanted support for his ideas at this time because he was about to re-open negotiations with the government over the Difference Engine, and try to persuade them to underwrite the Analytical Engine. ^{86} This is further revealed in a letter from Babbage to Plana, dated October 3, 1841, in which he discussed the description of the Analytical Engine which had finally been drawn up by Menabrea. Babbage said:
If you had made a report on the subject (of the Analytical engine) to the Academy of Turin during the last year it might have been of essential service to me in the discussion of the question with my own Government. As it is I must be content with the description drawn up by M. Menabrea, with which I am well satisfied because he seems to have penetrated completely the principles on which it rests. ^{87}Menabrea's paper was published in October, 1842 in French, and later in English, and will be discussed below. Evidently, however, Babbage was not as "well satisfied" as he might have been, for on October 7, 1841, having finally succeeded in getting an account of the Analytical Engine written by someone else, he set about writing another of his own.
This 1841 paper, called "Of the Analytical Engine," ^{88} was a fairly straightforward introduction to the Analytical Engine on an elementary level, rather similar to the 1837 paper discussed and quoted extensively earlier in this chapter, though not as polished or complete. It did not, for example, go any further into the principles of the Combinatorial Cards or their mode of action, the cards which would have controlled in detail the flow of instructions into the Mill.
What was valuable about the 1841 paper was its introduction, where Babbage set out in a much clearer and more coherent way his conception of the general character of the Analytical Engine. This introduction began as follows:
The object of this engine is twofold. 1st To convert into numbers and to print the results of any formulae which may be required. 2nd To develop any analytical formulae the laws of whose formation are given.In order to accomplish these objects it is necessary that it should be possible; lst To express in the engine any functions whatever of any number of quantities and their numerical magnitude, together with their numerical coefficients, their indices of operation and their algebraic signs. 2ndly That it should be capable of performing with those numbers the four operations of arithmetic. 3rdly That it should be capable of all those combinations of symbols which the laws of analysis may require.
After stating what the Analytical Engine will do it may perhaps be necessary to state what it cannot accomplish. It cannot invent. It must derive from human intellect those laws which it puts in force in the developments it performs. It cannot in fact do anything more than perform with absolute precision and in much shorter time those series of operations which the hand of man might itself much more imperfectly accomplish. It must however be observed on the other hand that this great abridgement of time does in a certain sense render possible calculations which otherwise might be practically although not physically impossible from their tediousness as well as expense.
Another and an important fact is that it will perform calculations which the hand of man has never yet attempted, such as the elimination of one variable between two equations of the 6th dimension, and that in communicating to it our commands to execute such operations, it is quite unnecessary that we should ourselves go through them even in the first instance. This is accomplished by communicating to it the laws according to which the system of operations is performed. ^{89}
In the rest of the introduction to his 1841 paper, Babbage discussed the possibility of actually constructing a machine with the powers described. Without making any remarks as to his own plans or intentions about building a working machine, he set out three prerequisites for the construction of an analytical engine.
First, it must "be possible to carry on the whole of the executive department of analysis by means of the combination of a few simple arithmetical and algebraical processes." Second, it must be the case that "those elementary principles can be fully effected by means purely mechanical." Babbage said that the fulfillment of these pre-conditions was a matter of demonstration, not of opinion; and although he did not set out to prove them, he clearly did not consider them to be in doubt.
The third condition was of a different sort. "Whether that system of contrivances which has been drawn can in the present state of mechanical art be executed so as to work regularly and exactly is an important point, a point on which opinions may fluctuate until experiment has decided the question." But Babbage expressed his confidence that "no man thoroughly acquainted with the present state of machinery in England and having complete drawings and notations of the engine can or will fail in making it work perfectly." ^{90}
As mentioned above, L.F. Menabrea had, by October, 1841 written up a description of the Analytical Engine as described by Babbage in Turin in September, 1840, and this description was shown to Babbage. For reasons that are not clear, it was not published until an additional year had passed; it is possible that some changes or additions suggested by Babbage delayed the matter. In any case, it appeared under the title "Notions sur la machine analytique de M. Charles Babbage" in the Biblioteque Universelle de Geneve in October, 1842, ^{91} and constituted the first published description of the Analytical Engine.
In the summer of 1843, Menabrea's paper was translated by Ada Augusta, Countess of Lovelace, and only legitimate daughter of Lord Bryon; she composed, in extensive consultation with Babbage, a series of long notes to the paper, which together comprised about three times the length of Menabrea's original version. The whole was published in Richard Taylor's Scientific Memoirs for 1843, ^{92} under the title "Sketch of the Analytical Engine invented by Charles Babbage, Esq;" this was the only extensive paper on the Analytical Engine published in English during Babbage's life, or, indeed, up to the present. Although it is clear that Lady Lovelace was a woman of considerable interest and talent, and it is clear that she understood to a very considerable degree Babbage's ideas about the general character and significance of the Analytical Engine, and expressed them well in her notes to Menabrea's paper, it is equally clear that the ideas were indeed Babbage's and not hers; indeed, she never made any claim to the contrary. She made a considerable contribution to publicizing the Analytical Engine, but there is no evidence that she advanced the design or theory of it in anyway. And she did not even express an interest in learning about the machine until January 5, 1841, ^{93} even as late as June 30, 1843, she apparently knew quite little about the mechanical details of the Engine. ^{94}
All of this is said not to belittle Lady Lovelace, but because a very exaggerated view has been formed by some recent writers of the significance of her contribution to the Engine or of her role in Babbage's life. ^{95}
From the present perspective, the Menabrea paper and the Lovelace notes are more interesting as an actual publication on the Analytical Engine than as a source of additional information. Very little was said of the operation or mechanical details of the Engine, far less than in the earlier Babbage manuscript papers. However, the extent of the powers of the machine, and. its limitations, and thus the question of its significance, were developed more clearly and elegantly than in the earlier works. Menabrea said the following:
Considered under the most general point of view, the essential object of the machine being to calculate, according to the laws dictated to it, the values of numerical coefficients which it is then to distribute appropriately on the columns which represent the variables, it follows that the interpretation of formulae and of results is beyond its province, unless indeed this very interpretation be itself susceptible of expression by means of the symbols which the machine employs. Thus, although it is not itself the being that reflects, it may yet be considered as the being which executes the conception of intelligences. . . .Who can foresee the consequences of such an invention? In truth, how many precious observations remain practically barren for the progress of the sciences, because they are not powers sufficient for computing the results! And what discouragement does the perspective of a long and arid computation cast into the mind of a man of genius, who demands time exclusively for meditation, and who beholds it snatched from him by the material routine of operations! Yet it is by the laborious route of analysis that he must reach truth; but he cannot pursue this unless guided by numbers; for without numbers it is not given us to raise the veil which envelopes the mysteries of nature. Thus the idea of constructing an apparatus capable of aiding human weakness in such researches, is a conception which, being realized, would mark a glorious epoch in the history of the sciences. ^{96}In the notes, these ideas were developed further and cast in more general terms:
The bounds of arithmetic were. . . . outstepped the moment the idea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere "calculating machines." It holds a position wholly its own; and the considerations it suggests are most interesting in their nature. In enabling mechanism to combine together general symbols in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science. A new, a vast, and a powerful language is developed for the future use of analysis, in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind than the means hitherto in our possession have rendered possible. Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connexion with each other. We are not aware of its being on record that anything partaking in the nature of what is so well designated the Analytical Engine has been hitherto proposed, or event thought of, as a practical possibility, any more than the idea of a thinking or of a reasoning machine. ^{97}The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with. This it is calculated to effect primarily and chiefly of course, through its executive faculties; but it is likely to exert an indirect and reciprocal influence on science itself in another manner. For, in so distributing and combining the truths and the formulae of analysis, that they may become most easily and rapidly amenable to the mechanical combinations of the engine, the relations and the nature of many subjects in that science are necessarily thrown into new lights, and more profoundly investigated. ^{98}
The publication of the Menabrea paper and of the much expanded Lovelace translation and notes was quite pleasing to Babbage. The former (which Babbage had gone to Turin in order to bring into being) he considered to be a "lucid and admirable description" of the Analytical Engine; the notes Babbage considered to have "entered fully into almost all the very difficult and abstract questions connected with the subject." The paper and notes taken together, said Babbage, "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." ^{99}
Also highly pleased by the translation and notes was Lady Lovelace, and she was even more pleased with herself for having produced them, despite what she considered to have been the inaccuracy and unreliability of Babbage's revision and suggestions. ^{100} On July 30, 1843 she wrote to Babbage, among other things, "The more I study the more irresistible do I feel my genius to be. I do not believe that my father [Lord Byron] was (or ever could have been) such a Poet as I shall be an Analyst (and Metaphysician)." ^{101}
In another letter to Babbage, on August 11, 1843, Lady Lovelace expressed her goal in life as follows. "I wish to add my mite toward expounding and interpreting the Almighty and His laws and works, for the most effective use of mankind, and certainly I should feel it no small glory if I were enabled to be one of His most noted prophets . . . in this world." Apparently this humble desire was to be fulfilled by Lady Lovelace becoming the general director of the execution of a working Analytical Engine. Babbage would promise to confine his attention entirely to the actual construction of the machine and "to abide wholly by the judgement of myself . . . whenever we may differ on all practical matters relating to whatever can involve relations with any fellow creature or fellow creatures." Exactly what scheme Lady Lovelace had in mind, or how she intended to finance it, she would not say, but in any case the matter was never followed up. ^{102}
Babbage's descriptions of the Analytical Engine written in 1837 and 1841, discussed earlier, were doubtless drafts of intended publications, and in December, 1842, Babbage had sent to a friend for his opinion a collection of papers which he was considering making into a book on the Difference and Analytical Engines; ^{103} however, with the appearance of the Menabrea-Lovelace paper, Babbage evidently felt satisfied, for he published nothing on his own, and did not try to write any other papers on the machines for many years. In 1846, he gave as a further reason for his never having published anything on the Analytical Engine "the maxim of never employing my faculties on any but what I consider the highest objects as long as they are capable of advancing them," the completion of the plans of the Analytical Engine being such a "highest object". ^{104}
We have seen how Babbage's conception of the general character and possible applications of the Analytical Engine developed from 1834 on, and also the measures that he took to bring the new machine into public view, although he preferred that this be done through some other author, and, indeed, that it first be made public in a foreign country. We must now survey the practical work and development that was done during the ten years following the point in 1837 at which the state of the Analytical Engine was described earlier in this chapter.
The principle point that must be kept in mind in considering this development is that Babbage had no plans for actually constructing the Analytical Engine in the immediate future; indeed, as we have seen, he several times expressed the opinion that it would not be constructed during his lifetime. This meant that he felt himself to be under no temporal pressure or restraint, and thus free to pursue the consequences of any conceivable modification in the parts of the Engine to their most remote consequences; usually this was to see if the change would save any time. Further, he would often take up consideration of some contrivance which he had rejected some years before, to see if it would work any better in relation to some other device or change he was now considering.
This mode of proceeding on Babbage's part has the unfortunate consequence that the great majority of the work he did after 1837 is very uninteresting; it has the character of neither being a theoretical or practical breakthrough, nor of leading up to or at any point constituting any single complete, coherent and final design for the Analytical Engine, for, as mentioned, Babbage was under no pressure to come up with such a final design. Thus most of this work will not be covered in this thesis. But examples of how this kind of consideration could lead down both fruitful and useless paths will be provided, and a few ideas that are interesting because of their similarities to methods used in modern computers will be described.
One point that needs to be made clear is that the tendency of what might be called Babbage's perfectionist approach was not simply to make the machine more complex; rather there was always a conflicting pull between simplicity and effectiveness, say in speeding up the operation of the machine; for Babbage found simplicity of structure or operation for a device able to do a certain job (which we might call efficiency) to be just as theoretically satisfying as raw capacity in itself; and he always thought of himself as working on a machine that would eventually be built.
Babbage described these conflicting tendencies very clearly in describing the difference in approach to the Great Analytical Engine on which he was working before the trip to Turin in September, 1840, and the simplified Small Analytical Engine which he began after this trip. ^{105} In fact, although the description pictured these as two diverse enterprises, the distinction was not maintained in his own work, and the same two approaches were battling each other in everything he did. As Babbage said:
The principles on which the great Calculating Engine was designed and drawn were:1 That it should execute all its operations in the smallest possible time. This was the great guiding principle and even single turns of the hand were considered.
2nd That although the mechanism employed should be as simple as could be contrived, yet in no case should simplicity be purchased at the expense of time, provided the mechanism was of such a kind that it could in the present state of mechanical art be successfully executed.
3rd Expense of construction was never to be considered; whatever plan was thought to be the shortest in time, possible in execution and durable in structure was always adopted. ^{106}
In the Small Analytical Engine on the other hand, the approach was quite different:
The great object for which this was undertaken was to reduce the cost of construction to such a moderate sum as might render it possible that it might be executed.The guiding principle was to give up all the more complicated mechanism, especially that which only effected small savings of time, and even if necessary some of the larger savings when they required great expense. ^{107}
It may well be that Babbage decided to make the plans more practical as a result of the enthusiastic reception his ideas met with in Turin. But in fact, the distinction between Great and Small engines became one between two phases of his thinking through any given new idea. Babbage would go to extreme lengths to see if the idea would save any time; if it would, he would work on it in an attempt to simplify the mechanism necessary for him to adopt it in the machine. When this was done, he would consider the complexity of the device in relation to the time saved; if it was large, he would reject the device; if it was small, he would adopt it. Frequently, however, the efficiency of the device would be ambiguous, so that Babbage could not decide whether to adopt it or not.
An example of this process, which also illustrates the very complex trains of thought through which Babbage's mind worked over long periods of time, is the technique Babbage called Half Zero Carriage. It will be recalled that in his 1837 paper on the Analytical Engine, Babbage had believed that the theoretical minimum time necessary for adding a decimal digit in the machine would be ten units of time, where the unit was defined as the time needed for a figure wheel to turn from one digit to the next. ^{108} In the 1837 plans, the same minimum time was necessary for clearing a column, that is, reducing all its wheels to zero, by the method described earlier.
The Small Analytical Engine, as was said, was begun in November, 1840. One of the first crucial questions was whether to continue using multiplication and division by table (which required a lot of complex mechanism for the table axes and selection apparatus), or to go back to the earlier and much simpler method of multiplication and division by repeated addition and subtraction; the latter method, however, would make the machine much slower, requiring as much as seventeen minutes for division of sixty figures by thirty figures. ^{109}
The first thing Babbage realized was that he could reduce the number of table axes needed to five by the following method. On five axes were placed the first five multiples of the multiplicand; if a given digit of the multiplier was less than five, the selecting mechanism would enter the corresponding multiple in the usual way. If a digit greater than five was presented to the selecting mechanism, it had to, as Babbage put it, "move back from the fifth multiple by a number equal to the excess of that digit above five," that is, by the tens complement of the specified digit. The multiple corresponding to this complement was then subtracted from the result column, and the next higher digit of the multiplier was increased by one. ^{110}
This new form of multiplication would have allowed Babbage to reduce the number of table axes to five for the Small Analytical Engine, but it would have also made the process as a whole, particularly the problems of selection and control, much more complicated. This did not cause Babbage to abandon it, however, for it began to suggest things more interesting to him than the simplification which he had set out on.
On December 15, 1840, Babbage noticed that in one stage of the multiplication process he had accidentally allowed only five units of time for clearing a carriage column from a previous operation (this clearing was overlapping with another part of the operation), whereas the clearing required ten units. ^{111} Instead of lengthening the cycle by five units, Babbage reflected that it took ten units to clear a wheel only because if it stood at nine, it would have to be turned all the way around to get back to zeros but if it were turned back the shortest way, that is, in the other direction if the wheel stood at a number above five, then it would take only five units. This method Babbage called clearing by "Half Zero," seemingly because the zero stood in the middle of the segment through which the wheel might be turned in clearing it. ^{112} It was also clear that a similar method could be used for transferring a number from one column to another whose wheels all stood at zero; at first he intended to do this by putting the same number on two different axes, each to be turned one way, but this was not necessary. ^{113}
However, these ideas suggested to Babbage a method for a new process of division in which the selection process was not tentative (that is, subject to reversal of a step because of a running up), and would thus be faster; the idea was to place on nine special axes the tens complements of the nine multiples of the divisor, and make selection by adding the dividend (or remainder at any given point) to all of these axes. This addition would cause a running-up on some of the axes; the number of the highest axis to run up would give the appropriate digit in the quotients further, the result of the addition on that same column would give the remainder at that point in the division; when this remainder was again added to the complement axes (after they had been reset), the next digit would be obtained, and so on. ^{114}
This method of division was sufficiently fascinating to Babbage that he lost interest in his efforts to devise a Small Analytical Engine that would be easy to build; also he continued to develop this and related ideas rather than that of half-zero figure wheels. In fact it was not until more than a year later, in April, 1842, that Babbage went beyond the use of half-zero for reducing columns to zero, for at this point he realized that he could use half zero for reducing the time of addition and subtraction themselves. ^{115}
This was to be done by a method quite similar to that used for transferring numbers by half zero (two reducing arms turning opposite ways), except that the column receiving the number would not be standing at zero. The effect of this was that if one number was being added to another, the digits standing at or below five would be added in the usual way, while the digits above five would have their tens complements subtracted; likewise in subtractions, the digits above five would have their complements added, while the lower digits would be handled normally. Again, the intention of all this was to reduce the basic time needed for these operations (exclusive of carriage) from nine to five units of time.
The great difficulty with this approach was that it required a new and much more complicated carriage mechanism, since the conditions which determined whether or not a carriage should take place were much more complicated than in conventional operation. As in the first stage of development of the Analytical Engine, Babbage considered using both hoarding and anticipating carriage, ^{116} and again decided that the latter was preferable.
Another circumstance that was important at this point was that Babbage had decided that as stepping numbers up or down only a single cage at a time took too much time, he wanted to be able to step any number of cages at once. He rejected the method of having a very large number of pinions which could be selectively put in gear (which he had considered back in 1834), since the number of pinions would have to be proportional to the square of the number of significant digits; instead he decided to take a number column and simply move it bodily up or down the desired number of cages. The great difficulty was that with his designs at this point, the columns already were about eight feet high and very heavy, so that it was not practical to move them around with any abandon. ^{117}
As the primary obstacle to reducing the height of. each cage was the size of the carriage mechanism, it became desirable to redesign the carriage mechanism both to reduce its size and to make it possible to use half-zero addition and subtraction.
Babbage went on in the following months and years to work very extensively on the details of an anticipating half-zero carriage, and the requirements and possibilities which his designs brought forth continued to interact with and influence in substantial ways the design of the Engine as a whole. The details of the development will not be discussed here. It should be noted, however, that at times, as on August 1, 1845, the difficulties of half-zero carriage led Babbage to decide to abandon it, ^{118} although a few months later he was back to making plans for it. ^{119}
Rather than plunging further into this morass of detail, it will be well to cover briefly a few other important points or threads of development between 1837 and 1847. One subject in particular calls for further explanation, as the description of it in Babbage's 1837 paper, discussed earlier, leaves much to be desired: this is the function and operation of the Combinatorial Cards. The vague explanation of Combinatorial Cards in the paper dated December 26, 1837, may be accounted for by the fact that the first statement on the subject in the notebooks is dated December 13, 1837, and is itself rather vague. ^{120} Apparently what Babbage envisioned was that there would be a series of steps in a program which would be repeated many times at the beginning, and then some additional steps at the end that would be performed only once apiece; Babbage may have had in mind some such thing as computing a number of terms in a mathematical series, and then forming their sum. In any case, the plan was to have a special set of combinatorial counting wheels; on these would be placed the number of times the special set of operations at the beginning of the program was to be repeated. At the end of this set of operation cards would be a special combinatorial card; when it was reached, unity would be subtracted from the combinatorial figure wheels, and the operation cards would be turned back to the beginning. However, when, after a number of repetitions, the combinatorial figure wheels read zero, the index card at the end of the sequence would be ignored, and the Engine would go on to the next card.
Although by May 10, 1838, this plan had become somewhat more coherent, it had not become more flexible. ^{121} It provided only for backing the operation cards to a pre-determined point, although the number of repetitions could be varied. On February 9, 1839, Babbage mentioned another idea, whereby after a series of cards was completed, they could be turned to a point other than the beginning of the series; this was done by putting the number of cards that were to be turned on a special backing apparatus; when the point was reached where the transfer was to be made, the backing apparatus was geared with the card prism, and then reduced to zero, turning the cards as it went. The number that was to be put on the backing apparatus could be either computed internally or specified in advance by a number card, following the principle which Babbage was to express two years later in these words:
Any fact which is certain and known before execution.. . should be ordered by cards. Any fact which although equally certain is unknown previously to the actual working out of the formula should be governed by Mill or Counting Apparatus, etc. ^{122}In late March, 1843, Babbage mentioned a method by which the operation on a single card could be repeated as many times as desired. Each operation card had punched on it the number of times it was to be repeated. The first time it was read, this number was put onto the card counting apparatus; the card was then used again to set the barrels, but the index on it was ignored, unity being subtracted from the card counting apparatus instead. When the number on it reached zero, the cards were advanced by one, a new index was taken in, and so on. It is not clear what this was supposed to accomplish, but seemingly it simply reduced the number of cards needed when an operation was to be done many times together, for this plan did not provide any increased power or flexibility to the Engine. ^{123}
In November and December of 1843, Babbage spelled this plan out a bit further. Each operation card was to be able to convey to a card counting apparatus any index number less than 1000. In order to save space on the card, this was to be done in the following way. Each digit to be conveyed had corresponding to it holes standing for one, three and nine; the holes for one and three also had holes to indicate if they were negative. By combining the digits +1, + 3, and +9, any digit from zero to nine could be expressed, and thus numbers up to 999 could be represented by fifteen holes, rather than thirty. Also, there were to be several different card counting apparatuses on which the indexes could be placed; one might control the number of times an operation was to be repeated, while another would determine the interval through which the cards were to be moved after the repetition was complete. ^{124}
Although Babbage continued to consider various minor modifications of these schemes, the differences were not important. What can be said is that all the elements necessary for full flexibility in instruction addressing were available in the machine. Yet although Babbage was aware that a variety of repetitions and branchings would be necessary, he never felt compelled by the kinds of programs he got around to considering ^{125} to make explicit or coherent any. but simple examples of the methods of program control.
This discussion of the development of the Analytical Engine beyond the stage described earlier in the chapter may be concluded with a few examples of developments in it which are paralleled by features of modern computers. Perhaps the most interesting is Babbage's thoughts on the use of what we might call external memory, and especially the use of pre-computed tables to be consulted by the machine.
We have already seen how Babbage, as early as March, 1835, had intended to do multiplication (and shortly thereafter division) by constructing internal tables of the first nine multiples of the multiplier on special axes. On August 7, 1837, it occurred to Babbage that he might similarly form a table of the low integral powers of the digits one through nine, and that then the first digit of the corresponding roots of numbers could be found by working backward through this table. ^{126} At first it seems he thought of having this table internal, as with the multiplication table, although they were substantially different, in that the multiplication table was specially computed for each different multiplier. However, in early September, 1837, Babbage was considering some matters connected with having the Engine punch its own cards, and it occurred to him that he could have the Engine form tables on cards for its own use. On September. 7 he considered having a table of one hundred cards corresponding to the digits from one to one hundred, each having also punched in it the first two digits of its square root. Then when the Engine wanted the square root of a number, it would turn through these cards to the one corresponding to its first two digits, and would then obtain the first two digits of the square root. In both these methods subsequent digits would be found by some method of successive approximations. ^{127}
Babbage realized that this technique could have wider applications than the extraction of roots, for on September 9, 1837, he noted that "possibly several systems of punched cards might be employed to reduce Astronomical and barometric observations by." ^{128} At about the same time Babbage also considered forming a table of the first one hundred multiples of a multiplier on cards; the selecting apparatus would then have turned to the card corresponding to a given pair of digits, and the multiple on the card would have been added to the appropriate column. This would have halved the number of separate additions (or subtractions in the case of division) necessary for a multiplication, and eliminated the need for table axes in the Mill. However, on September 19, Babbage rejected the idea, because turning through the string of cards to get to the right one would have required an inordinately large amount of time. ^{129}
The idea was not followed up until May 22, 1842, at which point Babbage described a considerably more elaborate system for having the machine be able to consult logarithmic tables in the course of its computation. The machine would itself have computed log tables, and placed on cards both the argument, with five digits, and its logarithm, with ten. The complete set of cards with this table would be placed in a drawer near the Engine. Then when it wanted a logarithm, it would ring a bell for the attendant and display the number whose log it wanted; he would fetch the corresponding card from the drawer and feed it into the machine. The Engine would then subtract the argument punched on the card from the number it had specified, and check that the first five digits of the result were equal to zero; if they were not, the Engine would ring "a great bell to tell [the] Assistant he had made a mistake." ^{130} Babbage also intended to have punched on the same cards the first and second difference of each logarithm, for use in interpolating to more than five figures in the argument. He also realized that the same method would work equally well for functions other than the logarithm.
Babbage was quite pleased with this method, not only because of its usefulness and relative simplicity, but because a description of the ability of the machine to know when it had been fed the wrong card, without a description of the simple means by which this was accomplished, made it sound as if the Engine had very peculiar powers. Consequently he gave a lengthy description of the technique in his book Passages From the Life of a Philosopher, and added the following very prescient observations:
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 formulae than by having recourse even to its own Tables. ^{131}Remarkably, despite the tremendous alteration in the time scale of computation, this "interesting question" is still relevant today.
One advantage of feeding numbers into the Analytical Engine in this way was that it eliminated the possibility of error when wheels had to be set by hand. Babbage realized that this was equally true for the "formula" (program), since the great complexity possible in it made it especially important that it be fully "verified" (debugged). This made especially valuable the separation between the operation cards and the variable and number cards, since it was possible to verify a formula, and then be confident of its accuracy when at some later time it was run with some new set of data. ^{132} It would thus be possible to build up a "library" of verified formulae which would be used repeatedly. He also realized that not only could the Analytical Engine use formula cards that it had previously verified, or number cards that it had previously punched, but that one such machine could share its cards and programs with another, and that therefore a card copying machine would be useful. He went on to say:
It is desirable that all Analytical Engines should have cards of the same size. But it is very improbable that those first used will be of the best size. Hence in making the card copying machine it will be necessary to contrive means of changing the punches so that Cards may be copied in any given size. ^{133}We may conclude by noting two further minor but yet interesting ideas that emerged during this period. One was that it would be desirable to have the Analytical Engine able to do calculations using pounds, shillings and pence, although it is not clear what sorts of monetary calculations Babbage had in mind. Curiously, he intended to accomplish this by altering the internal hardware of the machine, rather than by having special formulae for converting input and output. ^{134}
Another plan to modify the engine so that it could handle material which required something other than straightforward numerical output, was that of providing two extra kinds of cards to control the printing of results; these served a function essentially identical to that served by FORMAT statements in current FORTRAN programming. One set controlled the horizontal and vertical motion of the paper (or copper plate) on which the results were printed, while the other would insert such explanatory symbols and text between the data as was desired. ^{135}
This kind of development, mixing revision in general plans, refinements of details, and new ideas of peripheral improvements, continued steadily until late in 1846, at which point it was abandoned for just over ten years, and then resumed for the remainder of Babbage's life. The reasons for this hiatus and the character of the later work on the Analytical Engine will be discussed in the next chapter.
FOOTNOTES
#1 S.B., "Vol. XV," p. 101, referring to the Analytical Engine.
#2 B.C.E.. p. 212.
#3 See B.C.E.. pp, 264-69.
#4 See Chapter Two.
#5 B.C.E., p. 220. The property of the even differences of the sine function which Babbage discussed is simply a generalization of the property of its second difference already described above.
#6 B.C.E., pp. 216-19.
#7 B.C.E., p. 216.
#8 B.C.E.. pp. 216-19.
#9 Transactions of the Cambridge Philosophical Society. Vol. III (1824), p. 218.
#10 Reprinted in B.C.E., pp. 225-31.
#11 This volume (henceforth G.S.B.) is related to, but not directly part of, the series of regular Scribbling Books at South Kensington. Two pages spreads will be indicated by the first line number on the verso sheet.
#12 G.S.B., 1. 1231.
#13 G.S.B., 1. 1261.
#14 B.C.E., pp. 268, 331. Apparently the only use of this phrase by Babbage at the time was in the outline of a paper to be called "Philosophy of the Calculating Engine," dated August 30, 1834 (Buxton, Vol. VII); this paper was not written.
#15 The first sketch of the circular arrangement is at G.S.B., 1. 1801; the sketch is not dated, but the next page has the date September 18, 1834.
#16 This account is from Babbage's paper "The Analytical Engine," p. 9, dated November 8, 1869; Buxton, Vol. VII.
#17 This account was reconstructed from very vague material in the vicinity of G.S.B., 1. 1801.
#18 Nowhere is any coherent account of this development given by Babbage, and it has been pieced together from scattered material, mostly in G.S.B. and S.B., Vol. I and Vol. II, plus the drawings from the period. Specific references will be given when possible. Note that S.B., Vol. I is sometimes wrongly called Vol. XIII (see Bibliography).
#19 G.S.B., 1. 1861.
#20 See S.B., Vol. I, pp. 175-167 (working backward) and S.K., Drawings II, 5-7 (this refers to the large drawings in the South Kensington collection, in this case numbers five through seven from Case II. See Bibliography on numbering of drawings).
#21 G.S.B. 1. 1861 and S.B., Vol. I, pp. 175-167.
#22 G.S.B., 1. 1891.
#23 S.K., Drawing II, 14.
#24 G.S.B., 1. 1891.
#25 S.B., Vol. I, p. 172; S.K., Drawing II, 3.
#26 S.B., Vol. I. pp. 175. 4; S.K., Drawing II, 4.
#27 On the following, see especially S.B., Vol. I, pp. 30-31, and G.S.B., 1. 1951, 1981. 2011, and 2301.
#28 See, e.g., S.K., Drawings II, 9 and 16.
#29 First notes on barrels and platforms; S.B., Vol. I, pp. 14-21, February 26, 1835.
#30 S.B., Vol. I, pp. 140-41.
#31 S.K., Drawings II, 29 and 35. .
#32 S.B., Vol. I, p. 40.
#33 S.B., Vol..I, pp. 167-165.
#34 S.B., Vol. I, p. 165.
#35 Reprinted in B.C.E., p. 5.
#36 S.B.. Vol. I, p. 163.
#37 S.B., Vol, I, pp. 119-123.
#38 S.B., Vol. II, p. 19.
#39 S.B., Vol. II, p. 45.
#40 P.L.P., p. 126.
#41 First mention: S.B., Vol. II, p. 146; cf. Vol. I, p. 160.
#42 Curiously, Jacquard (1752-1854) had gotten the idea for his loom from an earlier one made by Jacques de Vaueanson, which he came across while rummaging among some old material at the Conservatoire des Arts et Metiers in Paris. See W.H.G. Armytage,A Social History of Engineering (Cambridge, Mass.. 1961), p. 110: John Timbs, Stories of Inventors and Discoverers (London, 1863), pp. 236-38.
#43 S.B., Vol. II, p. 150.
#44 Buxton, Vol. VII. Unfortunately the Buxton papers are in considerable disarray, and the pages of this paper are not all in order or even together. However, it can be reconstructed in full.
#45 References will be given here to the short title "Powers," with page numbers from Babbage's manuscript.
#46 "Powers," pp. 9-11. Note the limitations imposed by the inflexible treatment of the decimal point, in contrast with the exponential representation in current computers. This is one reason Babbage provided what seems like a very excessive number of wheels per column, since this did not in general yield nearly as many significant digits as it would seem.
#47 "Powers," p. 44.
#48 "Powers," pp. 12-13, S.B., Vol. III, p. 93.
#49 "Powers," p. 14.
#50 "Powers," p. 15.
#51 The relation of a barrel to its reducing apparatus and to the rest of the machine can be seen in Figure 2; also shown is the operation of a variable card. The barrels could also be affected by the counting apparatus. See also S.B., Vol. IV, p. 144.
#52 The preceding section on the Mill is all from "Powers," p. 2-8.
#53 "Powers," p. 40.
#54 "Powers," p. 21.
#55 "Powers," pp. 24-26.
#56 "Powers," pp. 32-36.
#57 Adapted from one by Henry P. Babbage; see B.C.E., p. 338.
#58 "Powers," p. 27.
#59 "Powers,", pp. 27-28.
#60 Ibid.
#61 "Powers," p. 23.
#62 See "Powers," pp. 31-32, 37, 46-53.
#63 "Powers," p. 50.
#64 S.B., Vol. III, p. 17 from back.
#65 "Powers," p. 15.
#66 S.B., Vol. IV, p. 385.
#67 B.M., Vol. VII..ff..525-29.
#68 B.M., Vol. VIII, ff. 292-93.
#69 Held by the Burndy Library, in Norwalk, Connecticut.
#70 MS. pp. 3-4.
#71 MS. pp. 8-9.
#72 "Powers," p. 53.
#73 "Powers," p. 19. It is interesting to compare this passage with similar ones in recent works on computers. A good example will be found in Faster then Thought, ed. B.V. Bowden (New York, 1953), p. 29.
#74 Although he was also writing the Ninth Bridgewater Treatise in 1837, and doing very extensive railway experiments in 1839, among other things.
#75 B.M.. Vol. X. f. 67.
#76 B.M., Vol. X, ff. 287-89.
#77 Ibid.
#78 B.M., Vol. X, f. 246.
#79 P.L.P., p. 129.
#80 B.M., Vol. X, f. 582.
#81 Menabrea, 1809-1896, was an engineer and mathematician; he later served as a general in the Italian army and as prime minister from 1867-69; he became Marquis of Valdora.
#82 P.L.P., p. 130. for quotations above and below.
#83 B.M., Vol. X, f. 582.
#84 B.M., Vol. X, f. 638 this draft is not dated, but internal evidence places it somewhat before August 12, 1841.
#85 It is capitalized on B.M., Vol. X, f. 645.
#86 See Chapter Two.
#87 B.M., Vol. X, f. 645.
#88 Buxton, Vol. VII.
#89 Ibid.
#90 Ibid.
#91 Vol. XLI, New Series, pp. 352-376.
#92 Vol. III, pp. 666-731.
#93 B.M., Vol. X, ff. 532-33
#94 Charles Babbage to Ada Augusta Lovelace, June 30, 1843; in the Lovelace. Papers. Quoted in Maboth Moseley, Irascible Genius (London, 1964), p. 169.
#95 Mosely, Irascible Genius, is the prime example, devoting over a quarter of her biography of Babbage to material on Lady Lovelace. Moseley is also convinced that Babbage served for many years as an intermediary between Lady Lovelace and the bookmakers to whom she became deeply and scandalously in debt, and that the generally quite trivial correspondence between her and Babbage involved some phantasmagorical secret code concerning this illicit activity; however, Moseley presents absolutely no serious evidence for this (See Mosely, Chapter 22, esp. p. 196).
#96 B.C.E., pp. 19-20; the paper and notes are reprinted in B.C.E., pp. 4-50.
#97 B.C.E., p. 25.
#98 B.C.E., p. 44.
#99 P.L.P., pp. 135-36.
#100 See, for example, some of the amazingly arrogant remarks quoted in Moseley, Irascible Genius, Chapter 20.
#101 B,M,, Vol. XI, ff. 407-9.
#102 B.M., Vol. XI, ff. 422-27.
#103 B.M., Vol. II, ff. 205-6. It is not clear just what the papers were.
#104 B.M., Vol. XII, ff. 347-57.
#105 For date of change in emphasis, see S.B., Vol. III, p. 68 from back. See also S.K., Notations 172 and 173, dated November, 1840.
#106 S.B., Vol. III, pp. 368-70.
#107 S.B., Vol. III, pp. 374-77.
#108 "Powers," p. 27.
#109 At this point Babbage was considering thirty digits per column as standard, rather than the earlier forty.
#110 S.B., Vol. IV, pp. 448-51, December 8-9, 1840.
#111 S.B., Vol. IV, p. 460.
#112 S.B., Vol. IV, p. 477.
#113 S.B.. Vol. IV, pp. 475, 501.
#114 S.B., Vol. IV, p. 482, January 3, 1841.
#115 S.B., Vol, V, p. 118, and S.K., Drawing II, 102.
#116 S.B., Vol. V, p. 132.
#117 S.B., Vol. V, pp. 183-89.
#118 S.B., Vol. VI, pp. 32-33.
#119 S.B., Drawing II, 136.
#120 S.B., Vol. III, pp. 161-66.
#121 S.B., Vol. III, pp. 75-76 from back.
#122 S.B., Vol. IV, p. 514. February 2, 1841.
#123 S.B., Vol. V, p. 389.
#124 S.B., Vol, V, pp. 454-59, 469-77, 533-34.
#125 Such as the program to compute Bernoulli numbers, described in Note G to the Menabrea paper, B.C.E., pp. 45-50. This was written by Babbage, not Lady Lovelace: see P.L.P., P. 136.
#126 S.B., Vol. III, p. 90.
#127 S.B., Vol. III, pp. 88-89.
#128 S.B., Vol. III, p. 93.
#129 S.B., Vol. III, p. 100.
#130 S.B., Vol. V, p. 152.
#131 P.L.P. p. 122. Compare Babbage's The Exposition of 1851. (London, 1851), pp. 185-87, where the ability of the Engine to reject wrong cards was given as an example of an apparently impossible power Babbage had given it, without the method being explained. The same mystification is attempted with the fact that the Engine could be made so that wheels being turned by hand at any time would not cause errors. Cf. S.B., Vol. V, pp. 334-35, January 28, 1843.
#132 "Powers," pp. 41-43; P.L.P., p. 119.
#133 S.B., Vol. V, p. 545, October 29, 1844.
#134 S.B., Vol. V, p. 322, January 11, 1843
#135 S.B., Vol. IV, pp. 216-18, September 21, 1839.
Preface Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5