Percy Ludgate

Если хочешь победить весь мир, победи себя.
Фёдор Михайлович Достоевский

Percy Edwin Ludgate (1883-1922)
Percy Edwin Ludgate (1883-1922)

The modest Irish accountant Percy Ludgate (1883-1922) is a very interesting figure in the world of calculating machines. Starting some time about 1903 to work on his hobby (calculating machines) nightly, until the small hours, and initially completely independently from great Charles Babbage, he managed to design an extremely interesting and original universal calculating machine (it was the second analytical engine in history). It is no chance that the British computer scientist Brian Randell wrote “One must wonder just how much more he might have achieved, if he had had but a modest fraction of the resources, available to Babbage (to say nothing of Aiken!), and had no succumbed to pneumonia at such a tragically early age.”

There are only two sources of information for Ludgate’s machine—the initial description of his analytical engine in Scientific Proceedings of the Royal Dublin Society of April, 1909 (it was suubmitted at the end of 1908, reviewed by the British physicist Charles Vernon Boys, and communicated by A. W. Conway, Prof. Mathematical Physics) (see the description below) and a short article by Charles V. Boys in the July 1909 number of Nature (A New Analytical Engine, Nature 81, 1909, pages 14–15).

These two publications must have been noticed by the scientific society in Britain, because Ludgate was appointed to write the article (Automatic calculating machines) in the book Napier tercentenary celebration: Handbook of the exhibition of Napier relics and of books, instruments, and devices for facilitating calculation from 1914. No records have been found however on any attempts to patent his machine, or to obtain financial backing for its construction from the government (in 1914 Ludgate stated: Complete descriptive drawings of the machine exist, as well as a description in manuscript, but I have not been able to take any steps to have the machine constructed).

Ludgate Store

The analytical engine of Ludgate has three main components: store, arithmetic unit and sequencing mechanism.
The store (see the nearby conjectural drawing of the store from D. Riches, 1973) used a “shuttle” for each variable. Each shuttle acted as a carrier for a set of protruding metal rods, there being one rod for the sign, and for each of the 20 decimal digits, comprising a number. The current value of each digit of the number currently stored in the shuttle was represented by the lateral position of the corresponding rod, i.e. by the length of rod protruding from the shuttle. The shuttles were to be held in “two coaxial cylindrical shuttle-boxes”. A particular number could be brought to the arithmetic unit by rotating the appropriate shuttle box through an appropriate angle. There was also to be provision for tables of constants, represented by sets of holes, of depth from one to nine units, drilled into the surface of one or more special cylinders. This method would appear to have considerable advantages over that used by Babbage (i.e. columns of toothed disks, each capable of being connected by a train of gear wheels to the arithmetic unit. Ludgate mentions a further advantage, i.e. “that the shuttles are quite independent of the machine, so that new shuttles, representing new variables can be introduced at any time”.

The arithmetic unit is a direct or partial product multiplying machine. In Ludgate’s machine however is used a logarithmic method of multiplication. Each digit of one operand is translated into the corresponding index number (or Irish logarithm). This set of index numbers is then added to the index number form of one of the digits of the other operand. The additions are performed concurrently by simple concatenation of lateral displacements. Then a reverse translation is performed to obtain the set of two-digit partial products. The set of partial products so obtained for each digit in the second operand are then accumulated using a mill, which is presumably a fairly conventional set of coaxial toothed wheels, incorporating a carrying mechanism. Ludgate claims, that he designed his own version of the anticipating carriage of Babbage, i.e. mechanism for assimilation, in a single step, of all the carry digits, produced during the addition of two numbers.
Ludgate was equally unconventional in his scheme for division, which is based upon a table of reciprocals of the integers 100-999, and a rapidly convergent series for (1+x)-1, where |x|‹10-3. The calculation of the series was controlled by what we would now call a built-in subroutine.

The arithmetic unit of Ludgate's machine (conjectural drawing from D. Riches, 1973)
The arithmetic unit of Ludgate’s machine (conjectural drawing from D. Riches, 1973)

The sequencing mechanism of Ludgate was to be controlled by a perforated paper tape, termed a formula paper, on which row of perforations defined a complete instruction. Each instruction specified two operands, the type of arithmetic operation to be performed, and the location (or pairs of locations) which was to receive the result. Thus the mechanism of Ludgate is much simpler that Babbage’s, and all that was necessary was to arrange for the appropriate angle of rotation of the shuttle-box, containing the shuttle representing the required variable. Ludgate clearly agreed with Babbage as to the fundamental importance of conditional branching, although he doesn’t indicate how it was to be done—presumably, following Babbage, he intended that mechanism that read the formula paper could be directed to skip a specified number of rows, either forwards or backwards.
Another feature of the sequencing mechanism was the provision of built-in subroutines. The operation code for division, e. g. caused control to pass temporarily to a sequence of instructions represented by rows of perforations on a permanent dividing cylinder. Another cylinder provided a logarithm subroutine, and Ludgate mentions the possibility of indefinite expansion of the set of such auxiliary cylinders.

The analytical engine of Ludgate had 192×20-digits of memory, could multiply in 10 seconds and take logarithms in 120 seconds, could input and store data and programs, had a printer, and even a fledgling operating system, could be stopped at any stage to add new variables, and could execute subroutines. It was designed to be motor driven and would be a compact and portable 60-cm cube.

Biography of Percy Ludgate

Percy Edwin Ludgate was born on 2 August 1883, in Skibbereen, County Cork, Ireland. He was the son of Michael Edward Ludgate (1839-1923) (born in Mallow, County Cork, to Robert Ludgate, a farmer, and Susanna Willis) and Mary Ann Ludgate (1840-1936) from Iden, Sussex, England (the daughter of Thomas McMahon, a soldier, and Frances Reed), who married on 15 Aug 1863 in Winchester, Hampshire, England. Michael Ludgate served in the army (in Ireland, England and India) for 19 years until 1876, when he was pensioned. Percy was the youngest of five (surviving) children, boys—Thomas Edward (1865-1951), Frederick (1875-1921), Alfred Ernest (1881-1953) and our hero-Percy Edwin, and a girl—Augusta (1871-1954).

By 1890 the family had moved to Dublin, where Michael worked as a clerk and shorthand teacher. In 1899 he was imprisoned (6 weeks) for debt, and his wife Mary and sons Frederick, Alfred and Percy moved to Drumcondra, Dublin. On release Michael appears to have lived apart from his family in Balbriggan, County Dublin.

Percy was educated at St George’s Infants School and at North Strand Parish School, in Dublin, Ireland. From 1898 he was earning in an income as Civil Servant National Education (Boy Copyist) in Dublin, and studied for assistant clerk, as in 1903 he competed successfully for more senior graded clerkship but failed the medical examination. Later he studied accountancy at Rathmines College of Commerce, Dublin, and was awarded a gold medal by the Corporation of Accountants on the occasion of his final examination in 1917.

In the 1911 census, Percy is also in Dublin, working as a commercial clerk at a corn merchant. Around 1915 Ludgate joined the firm of Kevans and Son in Dublin, where he worked as an accountant and auditor until his death.

Percy Ludgate never married and was described by his contemporaries as a “very gentle, a modest simple man” who “possessed the characteristics one usually associates with genius, and he was so regarded by his colleagues on the staff…”. In the period 1914-1918 Ludgate worked for a committee, set up by the War Office, to provide supply for the cavalry divisions of the army and was much prized for the major role, that he played. In the summer of 1922 Ludgate was on holiday in Lucerne, Switzerland, but shortly after his return developed bronchopneumonia (at the time he was living with his mother and his brother Alfred in Dublin), and died on 16 October 1922 (buried in Mount Jerome Cemetery in Dublin).


The description of Ludgate’s analytical machine in Scientific Proceedings of the Royal Dublin Society of April, 1909

ON A PROPOSED ANALYTICAL MACHINE

by Percy E. Ludgate
Read February 23. Ordered for Publication March 9. Published April 28, 1909

I purpose to give in this paper a short account of the result of about six years’ work, undertaken by me with the object of designing machinery capable of performing calculations, however intricate or laborious, without the immediate guidance of the human intellect.

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

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

The following definitions of an Analytical Engine, written by Babbage’s contemporaries, describe its essential functions as viewed from different standpoints:

‘A machine to give us the same control over the executive which we have hitherto only possessed over the legislative department of mathematics.’*
*Q Babbage: ‘Passages from the Life of a Philosopher’, p. 129.

‘The material expression of any indefinite function of any degree of generality and complexity, such as, for instance: F(xyz, log x, sin y, &c.), which is, it will be observed, a function of all other possible functions of any number of quantities.’†
†R. Taylor’s ‘Scientific Memoirs’, 1843, vol. iii., p. 691.

‘An embodying of the science of operations constructed with peculiar reference to abstract number as the subject of those operations.’*
*loc. cit., p. 694.

‘A machine for weaving algebraical patterns.’†
†loc. cit., p. 696.

These four statements show clearly that an Analytical Machine does not occupy common ground with mere “calculating machines”. It holds a position wholly its own.

In order to prevent misconception, I must state that my work was not based on Babbage’s results—indeed, until after the completion of the first design of my machine, I had no knowledge of his prior efforts in the same direction. On the other hand, I have since been greatly assisted in the more advanced stages of the problem by, and have received valuable suggestions from, the writings of that accomplished scholar. There is in some respects a great resemblance between Babbage’s Analytical Engine and the machine which I have designed—a resemblance which is not, in my opinion, due wholly to chance, but in a great measure to the nature of the investigations, which tend to lead to those conclusions on which the resemblance depends. This resemblance is almost entirely confined to the more general, abstract, or mathematical side of the question; while the contrast between the proposed structure of the two projected machines could scarcely be more marked.

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

‡C. Babbage: ‘Passages from the Life of a Philosopher’, p. 450.

Notwithstanding the complete and masterly treatment of the question to be found in the papers mentioned, it will be necessary for me briefly to outline the principles on which an Analytical Machine is based, in order that my subsequent remarks may be understood.

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

I have prepared many drawings of the machine and its parts; but it is not possible in a short paper to go into any detail as to the mechanism by means of which elaborate formulae can be evaluated, as the subject is necessarily extensive and somewhat complicated; and I must, therefore, confine myself to a superficial description, touching only points of particular interest or importance.

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

In referring to the numbers stored in the machine, the difficulty arises as to whether we refer to them as mere numbers in the restricted arithmetical sense, or as quantities, which, though always expressed in numerals, are capable of practically infinite variation. In the latter case they may be regarded as true mathematical variables. It was Babbage’s custom (and one which I shall adopt) when referring to them in this sense to use the term ‘Variable’ (spelt with capital V), while applying the usual meanings to the words ‘number’ and ‘variable’.

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

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

Though the index is analogous to the slide-rule, it is not divided logarithmically, but in accordance with certain index numbers, which, after some difficulty, I have arranged for the purpose. I originally intended to use the logarithmic method, but found that some of the resulting intervals were too large; while the fact that a logarithm of zero does not exist is, for my purpose, an additional disadvantage. The index numbers (which I believe to be the smallest whole numbers that will give the required results) are contained in the following tables:

Column 1 of Table 1 contains zero and the nine digits, and column 2 of the same Table the corresponding simple index numbers. Column 1 of Table 2 sets forth all partial products (a term applied to the product of any two units), while column 2 contains the corresponding compound index numbers. The relation between the index numbers is such that the sum of the simple index numbers of any two units is equal to the compound index number of their product. Table 3 is really a re-arrangement of Table 2, the numbers 0 to 66 (representing 67 divisions on the index) being placed in column 1, and in column 2, opposite to each number in column 1 which is a compound index number, is placed the corresponding simple product.

 
Table 1
 
UNIT SIMPLE
INDEX NO.
ORDINAL
0 50 9
1 0 0
2 1 1
3 7 4
4 2 2
5 23 7
6 8 5
7 33 8
8 3 3
9 14 6

 
Table 2
 
PARTIAL
PRODUCT
COMP
INDEX
NO.
PARTIAL
PRODUCT
COMP
INDEX
NO.
PARTIAL
PRODUCT
COMP
INDEX
NO.
1 0 15 30 36 16
2 1 16 4 40 26
3 7 18 15 42 41
4 2 20 25 45 37
5 23 21 40 48 11
6 8 24 10 49 66
7 33 25 46 54 22
8 3 27 21 56 36
9 14 28 35 63 47
10 24 30 31 64 6
12 9 32 5 72 17
14 34 35 56 81 28
Comp. index numbers of zero: 50, 51, 52, 53, 57, 58, 64, 73, 83, 100

 
Table 3
 
COMP
INDEX NO.
PARTIAL
PRODUCT
COMP
INDEX NO.
PARTIAL
PRODUCT
0 1 34 14
1 2 35 28
2 4 36 56
3 8 37 45
4 16 38
5 32 39
6 64 40 21
7 3 41 42
8 6 42
9 12 43
10 24 44
11 48 45
12
46 25
13
47 63
14 9 48
15 18 49
16 36 50 0
17 72 51 0
18
52 0
19
53 0
20
54
21 27 55
22 54 56 35
23 5 57 0
24 10 58 0
25 20 59
26 40 60
27
61
28 81 62
29
63
30 15 64 0
31 30 65
32
66 49
33 7

 

Now, to take a very simple example, suppose the machine is supplied with a formula-paper designed to cause it to evaluate x for given values of abc, and d, in the equation ab + cd = x and suppose we wish to find the value of x in the particular case where a = 9247, b = 8132, c = 21893, and d = 823.

The four given numbers are first transferred to the machine by the key-board hereafter mentioned; and the formula-paper causes them to be inscribed in four shuttles. As the shuttles of the inner and outer co-axial shuttle-boxes are numbered consecutively, we may suppose the given values of a and c to be inscribed in the first and second shuttles respectively of the inner box, and of b and d in the first and second shuttles respectively of the outer box; but it is important to remember that it is a function of the formula-paper to select the shuttles to receive the Variables, as well as the shuttles to be operated on, so that (except under certain special circumstances, which arise only in more complicated formulae) any given formula-paper always selects the same shuttles in the same sequence and manner, whatever be the values of the Variables. The magnitude of a Variable only effects the type carried by its shuttle, and in no way influences the movements of the shuttle as a whole.

The machine, guided by the formula-paper, now causes the shuttle-boxes to rotate until the first shuttles of both inner and outer boxes come opposite to a shuttle-race. The two shuttles are then drawn along the race to a position near the index; and certain slides are released, which move forward until stopped by striking the type carried by the outer shuttle. The slides in question will then have moved distances corresponding to the simple index numbers of the corresponding digits of the Variables b. In the particular case under consideration, the first four slides will therefore move 3, 0, 7, and 1 units respectively, the remainder of the slides indicating zero by moving 50 units (see Table 1). Another slide moves in the opposite direction until stopped by the first type of the inner shuttle, making a movement proportional to the simple index number of the first digit of the multiplier a—in this case 14. As the index is attached to the last-mentioned slide, and partakes of its motion, the relative displacements of the index and each of the four slides are respectively 3 + 14, 0 + 14, 7 + 14, and 1 + 14 units (that is, 17, 14, 21, and 15 units), so that pointers attached to the four slides, which normally point to zero on the index, will now point respectively to the 17th, 14th, 21st and 15th divisions of the index. Consulting Table 3, we find that these divisions correspond to the partial products 72, 9, 27, and 18. In the index the partial products are expressed mechanically by movable blades placed at the intervals shown in column 2 of the third table. Now, the duration of the first movement of any blade is as the unit figure of the partial product which it represents, so that the movements of the blades concerned in the present case will be as the numbers 2, 9, 7, and 8, which movements are conveyed by the pointers to the mill, causing it to register the number 2978. A carriage near the index now moves one step to effect multiplication by 10, and then the blades partake of a second movement, this time transferring the tens’ figures of the partial products (i.e. 7, 0, 2, and 1) to the mill, which completes the addition of the units’ and tens’ figures thus:

2 9 7 8
7 0 2 1

7 3 1 8 8

 

the result being the product of the multiplicand b by the first digit of the multiplier a. After this the index makes a rapid reciprocating movement, bringing its slide into contact with the second type of the inner shuttle (which represents the figure 2 in the quantity a), and the process just described is repeated for this and the subsequent figures of the multiplier a until the whole product ab is found. The shuttles are afterwards replaced in the shuttle-boxes, the latter being then rotated until the second shuttles of both boxes are opposite to the shuttle-race. These shuttles are brought to the index, as in the former case, and the product of their Variables (21893 x 823) is obtained, which, being added to the previous product (that product having been purposely retained in the mill), gives the required value of x. It may be mentioned that the position of the decimal point in a product is determined by special mechanism which is independent of both mill and index. Most of the movements mentioned above, as well as many others, are derived from a set of cams placed on a common shaft parallel to the driving-shaft; and all movements so derived are under the control of the formula-paper.

The ordinals in Table 1 are not mathematically important, but refer to special mechanism which cannot be described in this paper, and are included in the tables merely to render them complete.

The sum of two products is obtained by retaining the first product in the mill until the second product is found—the mill will then indicate their sum. By reversing the direction of rotation of the mill before the second product is obtained, the difference of the products results. Consequently, by making the multiplier unity in each case, simple addition and subtraction may be performed.

In designing a calculating machine it is a matter of peculiar difficulty and of great importance to provide for the expeditious carrying of tens. In most machines the carryings are performed in rapid succession; but Babbage invented an apparatus (of which I have been unable to ascertain the details) by means of which the machine could ‘foresee’ the carryings and act on the foresight. After several years’ work on the problem, I have devised a method in which the carrying is practically in complete mechanical independence of the adding process, so that the two movements proceed simultaneously. By my method the sum of m numbers of n figures would take 9m + n units of time. In finding the product of two numbers of twenty figures each, forty additions are required (the units’ and tens’ figures of the partial products being added separately). Substituting the values 40 and 20 for m and n, we get 9 x 40 + 20 = 380, or 9½ time-units for each addition—the time-unit being the period required to move a figure-wheel through 1/10 revolution. With Variables of 20 figures each the quantity n has a constant value of 20, which is the number of units of time required by the machine to execute any carrying which has not been performed at the conclusion of an indefinite number of additions. Now, if the carryings were performed in succession, the time required could not be less than 9 + n, or 29 units for each addition, and is, in practice, considerably greater.*

*For further notes on the problem of the carrying of tens, see C. Babbage: ‘Passages from the Life of a Philosopher’, p. 114, etc.

In ordinary calculating machines division is accomplished by repeated subtractions of the divisor from the dividend. The divisor is subtracted from the figures of the dividend representing the higher powers of ten until the remainder is less than the divisor. The divisor is then moved one place to the right, and the subtraction proceeds as before. The number of subtractions performed in each case denotes the corresponding figure of the quotient. This is a very simple and convenient method for ordinary calculating machines; but it scarcely meets the requirements of an Analytical Machine. At the same time, it must be observed that Babbage used this method, but found it gave rise to many mechanical complications.

My method of dividing is based on quite different principles, and to explain it I must assume that the machine can multiply, add, or subtract any of its Variables; or, in other words, that a formula-paper can be prepared which could direct the machine to evaluate any specified function (which does not contain the sign of division or its equivalent) for given values of its variables.

Suppose, then, we wish to find the value of p/q for particular values of p and q which have been communicated to the machine. Let the first three figures of q be represented by f, and let A be the reciprocal of f, where A is expressed as a decimal of 20 figures. Multiplying the numerator and denominator of the fraction by A, we have (Ap)/(Aq), where Aq must give a number of the form l00… because Aq = q/f. On placing the decimal point after the unit, we have unity plus a small decimal. Represent this decimal by x: then

p


q

= Ap


1 + x

or Ap(1 + x)-1

 

Expanding by the binomial theorem

(1) p


q

= Ap(1 – x + x2 – x3 + x4 – x5 + etc.),

 

or

(2) p


q

= Ap(1 – x)(1 + x2)(1 + x4)(l + x8),etc.

 

The series (1) converges rapidly, and by finding the sum as far as x10 we obtain the correct result to at least twenty figures; whilst the expression (2) gives the result correctly to at least thirty figures. The position of the decimal point in the quotient is determined independently of these formulae. As the quantity A must be the reciprocal of one of the numbers 100 to 999, it has 900 possible values. The machine must, therefore, have the power of selecting the proper value for the quantity A, and of applying that value in accordance with the formula. For this purpose the 900 values of A are stored in a cylinder—the individual figures being indicated by holes of from one to nine units deep in its periphery. When division is to be performed, this cylinder is rotated, by a simple device, until the number A (represented on the cylinder by a row of holes), which is the reciprocal of the first three figures of the divisor, comes opposite to a set of rods. These rods then transfer that number to the proper shuttle, whence it becomes an ordinary Variable, and is used in accordance with the formula. It is not necessary that every time the process of division is required the dividing formula should be worked out in detail in the formula-paper. To obviate the necessity of so doing the machine is provided with a special permanent dividing cylinder, on which this formula is represented in the proper notation of perforations. When the arrangement of perforations on the formula-paper indicates that division is to be performed, and the Variables which are to constitute divisor and dividend, the formula-paper then allows the dividing cylinder to usurp its functions until that cylinder has caused the machine to complete the division.

It will be observed that, in order to carry out the process of division, the machine is provided with a small table of numbers (the numbers A) which it is able to consult and apply in the proper way. I have extended this system to the logarithmic series, in order to give to that series a considerable convergency; and I have also introduced a logarithmic cylinder which has the power of working out the logarithmic formula, just as the dividing cylinder directs the dividing process. This system of auxiliary cylinders and tables for special formulae may be indefinitely extended.

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

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

*C. Babbage: ‘Passages from the Life of a Philosopher’, p. 131.

The Analytical Machine is under the control of two keyboards, and in this respect differs from Babbage’s Engine. The upper key-board has ten keys (numbered 0 to 9), and is a means by which numbers are communicated to the machine. It can, therefore, undertake the work of the number-paper previously mentioned. The lower key-board can be used to control the working of the machine, in which case it performs the work of a formula-paper. The key-boards are intended for use when the nature of the calculation does not warrant the preparation of a formula-paper or a number-paper, or when their use is not convenient. An interesting illustration of the use of the lower key-board is furnished by a case in which a person is desirous of solving a number of triangles (say) of which he knows the dimensions of the sides, but has not the requisite formula-paper for the purpose. His best plan is to put a plain sheet of paper in the controlling apparatus, and on communicating to the machine the known dimensions of one of the triangles by means of the upper key-board, to guide the machine by means of the lower key-board to solve the triangle in accordance with the usual rule. The manipulations of the lower key-board will be recorded on the paper, which can then be used as a formula-paper to cause the machine automatically to solve the remaining triangles. He can communicate to the machine the dimensions of these triangles individually by means of the upper key-board; or he may, if he prefers so doing, tabulate the dimensions in a number-paper, from which the machine will read them of its own accord. The machine is, therefore, able to ‘remember’, as it were, a mathematical rule; and having once been shown how to perform a certain calculation, it can perform any similar calculation automatically so long as the same paper remains in the machine.

It must be clearly understood that the machine is designed to be quite automatic in its action, so that a person almost entirely ignorant of mathematics could use it, in some respects, as successfully as the ablest mathematician. Suppose such a person desired to calculate the cosine of an angle, he obtains the correct result by inserting the formula-paper bearing the correct label, depressing the proper number-keys in succession to indicate the magnitude of the angle, and starting the machine, though he may be quite unaware of the definition, nature, or properties of a cosine.

While the machine is in use its central shaft must be maintained at an approximately uniform rate of rotation—a small motor might be used for this purpose. It is calculated that a velocity of three revolutions per second would be safe; and such a velocity would ensure the multiplication of any two Variables of twenty figures each in about 10 seconds, and their addition or subtraction in about three seconds. The time taken to divide one Variable by another depends on the degree of convergency of the series derived from the divisor, but l½ minutes may be taken as the probable maximum. When constructing a formula-paper, due regard should therefore be had to the relatively long time required to accomplish the routine of division; and it will, no doubt, be found advisable to use this process as sparingly as possible. The determination of the logarithm of any number would take two minutes, while the evaluation of an (for any value of n) by the exponential theorem, should not require more than l½ minutes longer—all results being of twenty figures. †

† The times given include that required for the selection of the Variables to be operated on.

The machine, as at present designed, would be about 26 inches long, 24 inches broad, and 20 inches high; and it would therefore be of a portable size. Of the exact dimensions of Babbage’s Engine I have no information; but evidently it was to have been a ponderous piece of machinery, measuring many feet in each direction. The relatively large size of this engine is doubtless due partly to its being designed to accommodate the large number of one thousand Variables of fifty figures each, but more especially to the fact that the Variables were to have been stored on columns of wheels, which, while of considerable bulk in themselves, necessitated somewhat intricate gearing arrangements to control their movements. Again, Babbage’s method of multiplying by repeated additions, and of dividing by repeated subtractions, though from a mathematical point of view very simple, gave rise to very many mechanical complications. *

* See Report Brit. Assoc., 1878, p. 100.

To explain the power and scope of an Analytical Machine or Engine, I cannot do better than quote the words of the Countess of Lovelace: ‘There is no finite line of demarcation which limits the powers of the Analytical Engine. These powers are coextensive with the knowledge of the laws of analysis itself, and need be bounded only by our acquaintance with the latter. Indeed, we may consider the engine as the material and mechanical representative of analysis, and that our actual working powers in this department of human study will be enabled more effectually than heretofore to keep pace with our theoretical knowledge of its principles and laws, through the complete control which the engine gives us over the executive manipulations of algebraical and numerical symbols.’ †

† R. Taylor’s ‘Scientific Memoirs’, 1843, vol. iii., p. 696.

A Committee of the British Association which was appointed to report on Babbage’s Engine stated that, ‘apart from the question of its saving labour in operations now possible, we think the existence of such an instrument would place within reach much which, if not actually impossible, has been too close to the limits of human skill and endurance to be practically available’. ‡

‡ Report Brit. Assoc., 1878, p. 101.

In conclusion, I would observe that of the very numerous branches of pure and applied science which are dependent for their development, record, or application on the dominant science of mathematics, there is not one of which the progress would not be accelerated, and the pursuit would not be facilitated, by the complete command over the numerical interpretation of abstract mathematical expressions, and the relief from the time-consuming drudgery of computation, which the scientist would secure through the existence of machinery capable of performing the most tedious and complex calculations with expedition, automatism, and precision.

Literature
1) B. Randell, “Ludgate’s analytical machine of 1909”, The Computer Journal, Volume 14, Issue 3
2) P. E. Ludgate, “On a Proposed Analytical Machine”, Scientific Proceedings, Royal Dublin Society, Volume 12 Issue 9, April 1909
3) “A New Analytical Engine”, Nature, Volume 81, 1 July 1909

Charles Babbage

There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.
Albert Einstein

Charles Babbage (1791-1871)
Charles Babbage (1791-1871)

Charles Babbage, credited deservedly as Father of the Computer, the world renowned inventor of Differential Engine and Analytical Engine, was born on the 26 December 1791, in the family home at 44 Crosby Row, Walworth Road, London. He was the first child of Benjamin Babbage (1753-1827) and Elizabeth Plumleigh Babbage, née Teape (1759-1844). Benjamin Babbage Sr., Charles’ grandfather, was mayor of Totnes in Devon (in southwest England). His son, also Benjamin (Charles’ father) had started out as a goldsmith in the small town of Totnes, a picturesque port in Devonshire, and later became a successful merchant and banker, who had waited until he was 38 year of age and wealthy before marrying and moving to London in 1790 to join a new banking firm—Bitton Estate in Teignmouth. Elizabeth Teape came from a prominent Devonshire family. After Charles, the family had two other sons (both named Henry), who died in infancy, and a daughter—Mary Anne (born in March 1798). She outlived Charles and the two siblings remained close throughout their lives.
Young Charles was brought up as an Anglican and received his earliest education at home. His childhood was marred by chronic illness and hoping that country living would improve his health, around the age of eight his parents began to sent him to country schools. In 1803, his family returned to Devon, and in improved health, Charles was sent to a small residential school in the village of Enfield near London, where he remained for three years. The teacher at Enfield was Stephen Freeman, an amateur astronomer and namely he awakened Charles’s interest in science and mathematics. His love of investigation, which became the ruling passion of his life, was first evinced at that time, when he made an experiment in order to ascertain wherher or not the Devil could really be raised in a personal form (the result was negative, and that removed a doubt which has obscured his religious belief :-).
Charles then moved to a small school near Cambridge for a couple of years. This may have been to prepare for entrance to the University of Cambridge, but it made little impression on him. At age 16 or 17, Babbage returned to Devon to live with his parents. He learned Latin and Greek with a tutor and also spent much time studying mathematics on his own. By then, he was passionately fond of algebra and devoured every book he could find on the subject.
In 1808, the Babbage family moved into the old Rowdens house in East Teignmouth, and Benjamin Babbage became a warden of the nearby St. Michael’s Church. In October 1810, Charles Babbage began his studies at Trinity College, Cambridge. It meant new perspectives and he found the environment, the books and social life intensely stimulating. Here he was to meet new friends who would remain close to him for the rest of his life. His days were spent in sampling the pleasures of undergraduate existence parties with plenty of good food and drink, Sunday breakfasts with his friends after Church, chess and games of whist and trips on the Cam. There was a servant to take care of the routine chores and make Babbage’s life all the more agreeable. All this was financed with the 300 pounds, a big sum for the time, which Charles received as an annual allowance from his father. Among his new friends, John Frederick William Herschel soon took first place. (He was the son of William Herschel, the outstanding astronomer, who had discovered the planet Uranus in 1781. John followed in his father’s footsteps, and became one of the leading men of science in England during the 1800s.) Together they began to devote themselves to mathematics. Other friends of Babbage were the famous mathematician George Peacock and Edward Ryan, a famous English lawyer. In 1812, Babbage, Peacock, Herschel and some other students had founded a little association called the Analytical Society. Its purpose was to introduce continental mathematical methods into the conservative Cambridge.

Georgiana Babbage (1892-1927)
Georgiana Babbage (1892-1927)

In the spring of 1814, Charles Babbage received his honorary degree without examination from Peterhouse, Cambridge. Shortly afterwards, on 25 July 1814, he married to Georgiana Whitmore (1792-1827) (see the nearby portrait), one of the 8 daughters of a wealthy Shropshire family and in the fall, they moved to London. The marriage was not welcomed by his father Benjamin and it would appear that the relations between father and son were far from harmonious. Benjamin Babbage had no complaints against Georgiana. His attitude was that, like himself, Charles should wait until he was properly established financially. Fortunately, Benjamin continued his £300 annual allowance, to which Georgiana could add £150 of her own. With such an income, the couple could maintain a modest life without lavish entertaining. In the August next year was born his first son—Benjamin Herschel (1815-1878). Charles and Georgiana had eight children, 7 sons and 1 daughter, but only three—the above-mentioned Benjamin Herschel, 5th son, Dugald Bromheald (1823-1901), and 6th son, Henry Prevost (1824-1918) survived to adulthood. All other children (Charles Whitmore (born 1817), Georgiana Whitmore (1818-1834), Edward Stewart (1819-1821), Francis Moore (b. 1821), and Alexander Forbes (b. 1827)) died young.
In 1815 Babbage becomes a member of the Royal Society. For a while, he sought paid employment, to prove to his father that he could make something of himself. In 1816, he applied for the post of math professor at East India College at Hartford, but his hopes were soon dashed when another candidate was chosen instead. In 1819, Babbage once more applied for a professorship, this time at Edinburgh. But despite all his recommendations from prominent French and English mathematicians, Babbage did not gain the position. At the same time, he also applied for a seat on the Board of Longitude but this too ended in failure. In 1820, he made a new application but to no avail.
In 1819 Babbage travels to Paris to visit French scientists. There he gets first inspiration for Difference Engine from Baron Gaspard de Prony’s use of division of labour for calculating tables.

Charles Babbage as young
Charles Babbage as young

At the close of 1820, Charles Babbage by now twenty-nine was still without any profession. For the previous six years, he had tried to find something suitable. He had carried out intensive mathematical research and had published a fair number of articles. He had presented several of his findings in lectures at the Royal Society, among whose illustrious members he had managed to establish himself. He had also, once again, shown his predilection for reform by becoming one of the co-founders of the Astronomical Society in 1820. Despite of his unsuccessful attempts to find a job, the family seemed to manage quite comfortably financially.
In 1814 Babbage made his first step in the field of engineering. He invented a new type of lock which he was interested in having manufactured. This was possibly his first serious excursion into that area of human endeavour, which was with time to cast its spell on him. The real passion about engineering will possess him in 1821, when he will begin his Difference Engine.
Besides his lifetime engagement with the construction of Differential Engine and Analytical Engine, Babbage did make occasional forays into other fields. In 1824, he was invited by some investors to organize a life insurance company. The new challenge intrigued him, and he threw himself into the task of determining the appropriate rates to charge for life insurance policies. Having collected a lot of information, Babbage decided that he would have to make some other use of it. In 1826, he published a book on the life insurance industry, A Comparative View of the Various Institutions for the Assurance of Lives. In fewer than 200 pages, this book provided a very useful consumer’s guide to the life insurance companies in England at that time. Readers could use it to compare companies and make intelligent decisions about which one would suit their particular needs.
In 1827 Babbage decided to publish tables of logarithms. He compared several tables, published since then in England. Wherever they differed, he recalculated the value so that he could produce a table completely free from error. With the help of an army engineer, he directed the work of a number of clerks. The corrected table was published in 1827. This table was reprinted many times, even after 1900. Babbage’s Table of Logarithms of the Natural Numbers from 1 to 108,000 was a paradigm of accuracy and was extensively used into the twentieth century.
The year 1827 was a devastating year for Charles. In February of 1827, his father died in Devon at the age of 73. Old Benjamin left sufficient funds to care for his wife, Betty, who moved to London to live with Charles and his family. Babbage inherited about £100,000, the bulk of his father’s estate, which made him a very rich man. The interest on the investments and the rent on the properties provided a comfortable income for the rest of his life. However, his view of a comfortable life did not last long. In July of the same year, his son Charles Whitmore was struck with a childhood disease and died at the age of 10. Then, in a month later, Charles’s wife Georgiana contracted a serious illness. On the 1st of September, both she and a newborn son Alexander Forbes died, apparently from complications caused by childbirth. These deaths caused Babbage to go into a mental breakdown which delayed the construction of his machines.
In 1828, Charles was elected as Lucasian Professor of Mathematics at Cambridge. This university chair, once held by Isaac Newton, was a great honour, though it carried an annual salary of less than £100. Babbage however did not think it was worth the distraction from his beloved Difference Engine. He held the post for ten years, however, he did not live in Cambridge and seldom lectured there. Nevertheless, he was always grateful for the appointment, which he called the only honour I ever received in my own country.
Filled with sorrow, at the end of 1827 Babbage made a long (one year) trip to Europe, when he met a lot of leading European scientists. After returning to England, his famous charm, wit, and humour had been restored, but he had clearly changed and his family life was gone. From 1829 to 1834, Babbage engaged in electoral politics, promoting candidates and even standing for election himself. In 1830 he published a book—Reflections on the Decline of Science in England, which is the best known of Babbage’s many polemics against the scientific institutions of his day and fuelled much debate at the time and after. In addition to the affairs of his family, he continued with the Difference Engine, and managed to write a book on the economy of manufacturing. He began to also to hold regular Saturday evening parties, initially in order to introduce his teenage children, Herschel and Georgiana, into society. Before long, the Babbage soirées formed an important part of the London social scene. Often, the guest list exceeded 200. They came from all parts of polite society: lawyers and judges, doctors and surgeons, deacons and bishops, and scholars and artists by the score. In the midst of this full bustle of activity during the 1830s, personal tragedy again struck Babbage. In 1834, his beloved daughter Georgiana became ill and died on 1 September, only 16 years old. To deal with his grief, Charles threw himself more deeply into his work.
Babbage was able to turn every experience to advantage. After all his visits to workshops and factories both in England and on the continent, he sought to draw general principles from them. In 1832, Babbage compiled these principles into the more than 30 chapters of his book On the Economy of Machinery and Manufactures. Within three years, there were four editions in England, one in America, and translations into German, French, Italian, Spanish, Swedish, and Russian—a real best-seller.

Charles Babbage at middle age
Charles Babbage at middle age

During the 1830s Babbage became progressively more involved in developing the efficiency of rail transport in England. Charles and his son Herschel even constructed a special device, something like black box, which measured the speed of the train and its degree of vibration.
In 1842 the oldest son of Babbage—Herschel, with his family and his brother Dugald, went off on a railway project in Italy. After other jobs, these two sons went to Australia in 1851 to conduct a geological survey. The third son, Henry, decided to join the Indian army. He took up his post there in 1843. Charles’s mother, Betty, was left alone in the old house, where she died in 1844 in her mid-eighties.
Charles fell into a routine that lasted most of the rest of his life. He devoted mornings and afternoons to writing or work on the Analytical Engine or Differential Engine, and then evenings to dinner, followed by a party, a play, or the opera.
In 1861, at the age of 70, Charles Babbage became more aware of his own mortality. He began to devote part of his time to writing a collection of reminiscences. His autobiography, titled Passages from the Life of a Philosopher, was published in 1864.
Babbage was a quite eccentric and a man of extremes. His friends could do no wrong and his enemies could do no right. He once contacted the poet Alfred Tennyson in response to his poem “The Vision of Sin”. Babbage wrote, “In your otherwise beautiful poem, one verse reads,
Every moment dies a man,
Every moment one is born.
…If this were true, the population of the world would be at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest that the next version of your poem should read:
Every moment dies a man,
Every moment 1 1/16 is born.
Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry.”
Babbage especially hated street music, and once wrote: It is difficult to estimate the misery inflicted upon thousands of persons, and the absolute pecuniary penalty imposed upon multitudes of intellectual workers by the loss of their time, destroyed by organ-grinders and other similar nuisances. It is said that organ grinders were playing deliberately outside his house on the day he died.

Charles Babbage (1791-1871)
Charles Babbage (1791-1871)

There is no doubt, that Babbage was an outstanding genius and he is the only man (if any!), who deserves the title Father of the Computer. His Differential Engine was a sophisticated specialized calculating machine, try to compare this monster with the ubiquitous calculator of Colmar, manufactured at the same time. But Differential Engine was nothing, compared to the Analytical Engine. It’s unbelievable, that Babbage dared to design a universal computer more than 100 years, before such a machine to be produced. And he was capable to make it, if he had the necessary support. 100 years in the field of computers is equal to 1000 years in many other areas. At the same time his striving for perfection and his difficult personality foiled a lot of his great ideas. After 10 years of work and а huge sum of money spent, he was very close to completion of Differential Engine, but his disputes with the engineer Clement, and some other trammels, spoiled his plans.
Despite his many achievements, the failure to construct his calculating machines left Babbage in his declining years a disappointed and embittered man. He died of renal inadequacy, secondary to cystitis at age 79 on 18 October 1871 at his home on Dorset Street, London, and was buried in London’s Kensal Green Cemetery. Babbage’s brain is preserved at the Science Museum in London.
In 1872, the year after his death, Babbage’s scientific library was sold at auction. The auction catalogue, containing over 2000 items on topics such as mathematical tables, cryptography, and calculating machines (and including many rare volumes), may be the first catalogue of a library on computing and its history.
Charles Babbage was a genius of the first order—a mathematician and professor, an engineer and inventor, a politician, a writer, a cryptographer, a founder of scientific organizations, and an expert on industry. His pioneering book, On the Economy of Machinery and Manufactures, was cited repeatedly by Marx in Capital and by John Stuart Mill in Principles of Political Economy. He was a human dynamo who needed only five or six hours of sleep a day and who was driven by a millenarian vision of man and machine that brought him within a hair’s breadth of the invention of the greatest of all machines—the Computer.

Analytical Engine of Charles Babbage

Start by doing what’s necessary; then do what’s possible;
and suddenly you are doing the impossible.
St. Francis of Assisi

Charles Babbage (a painting by Samuel Laurence from 1845, at the National Portrait Gallery in London)
Charles Babbage (a painting by Samuel Laurence from 1845, at the National Portrait Gallery in London)

It seems a real miracle, that the first digital computer in the world, which embodied in its mechanical and logical details just about every major principle of the modern digital computer, was designed as early as in 1830s. This was done by the great Charles Babbage, and the name of the machine is Analytical Engine.
In 1834 Babbage designed some improvements to his first computer—the specialized Differential Engine. In the original design, whenever a new constant was needed in a set of calculations, it had to be entered by hand. Babbage conceived a way to have the differences inserted mechanically, arranging the axes of the Difference Engine circularly, so that the result column should be near that of the last difference, and thus easily within reach of it. He referred this arrangement as the engine eating its own tail or as a locomotive that lays down its own railway. But this soon led to the idea of controlling the machine by entirely independent means, and making it perform not only addition, but all the processes of arithmetic at will in any order and as many times as might be required.
Work on the first Difference Engine was stopped on 10 April 1833, and the first drawing of the Analytical Engine is dated in September, 1834. There exist over two hundred drawings, in full detail, to scale, of the engine and its parts. These were beautifully executed by a highly skilled draftsman and were very costly. There are also over four hundred notations of different parts, using Babbage’s system of mechanical notation (see Laws of mechanical notation).
The object of the machine may shortly be given thus (according to Henry Babbage, the youngest son of the inventor): It is a machine to calculate the numerical value or values of any formula or function of which the mathematician can indicate the method of solution. It is to perform the ordinary rules of arithmetic in any order as previously settled by the mathematician, and any number of times and on any quantities. It is to be absolutely automatic, the slave of the mathematician, carrying out his orders and relieving him from the drudgery of computing. It must print the results, or any intermediate result arrived at.

Punched cards of Analytical Engine
Punched cards of Analytical Engine

Babbage intended to design a machine with a repertoire of the four basic arithmetic functions, in contrast with the Difference Engine, which is using only addition. On the analogy of a modern digital computer, the design principle of the Analytical Engine can be divided to:
1. Input. From 1836 on, punched cards (see the nearby photo) were the basic mechanism for feeding into the machine both numerical data and the instructions on how to manipulate them.
2. Output. Babbage’s basic mechanism was always a printing apparatus, but he had also considered graphic output devices even before he adopted punched cards for output as well as input.
3. Memory. For Babbage this was basically the number axes in the store, though he also developed the idea of a hierarchical memory system using punched cards for additional intermediate results that could not fit in the store.
4. Central Processing Unit. Babbage called this the Mill. Like modern processors it provided for storing the numbers being operated upon most immediately (registers); hardware mechanisms for subjecting those numbers to the basic arithmetic operations; control mechanisms for translating the user-oriented instructions supplied from outside into detailed control of internal hardware; and synchronization mechanisms (a clock) to carry out detailed steps in a carefully timed sequence. The control mechanism of the Analytical Engine must execute operations automatically and it consists of two parts: the lower level control mechanism, controlled by massive drums called barrels, and the higher level control mechanism, controlled by punched cards, developed by Jacquard for pattern-weaving looms and used extensively in the beginning of 1800s.

The barrels of Analytical Engine
The barrels of Analytical Engine

The sequence of smaller operations required to effect an arithmetical operation was controlled by massive drums called barrels (see the nearby figure). The barrels had studs fixed to their outer surface in much the same way as the pins of a music box drum or a barrel organ. The barrels orchestrated the internal motions of the engine and specify in detail how multiplication, division, addition, subtraction, and other arithmetic operations, are to be carried out. The barrel shown in the illustration has only several stud positions in each vertical row. In the actual machine, the barrels were much larger because they controlled and coordinated the interaction of thousands of parts. Each row could contain as many as 200 stud positions, and each barrel could have 50 to 100 separate rows. The overall machine had several different barrels controlling different sections. Naturally, the barrels had to be closely coordinated with one another. As a barrel turned, the studs activated specific motions of the mechanism and the position and arrangement of the studs determined the action and relative timing of each motion. The act of turning the drum thus automatically executed a sequence of motions to carry out the desired higher level operation. The process is internal to the Engine and logically invisible to the user. The technique is what in computing is now called a microprogram (though Babbage never used this term), which ensures that the lower level operations required to perform a function are executed automatically.
For higher level control mechanism, Babbage initially intended to use a big central barrel, to specify the steps of a calculation. This idea however seems impractical, because this will require changing the studs on the super barrel, which could be a cumbersome operation. The task of manually resetting studs in the central drum to tell the machine what to do was too cumbersome and error-prone to be reliable. Worse, the length of any set of instructions would be limited by the size of the drum.
His struggle with the problem of control led Babbage to a real breakthrough on 30 June 1836. He conceived of providing instructions and data to the engine not by turning number wheels and setting studs, but by means of punched card input, by means of cards, similar to these, used in the Jacquard looms. This did not render the central drum obsolete nor replace it. Punched cards provided a new top level of the control hierarchy that governed the positioning of the central drum. The central drum remained, but now with permanent sequences of instructions. It took on the function of micro-programming, as this of other barrels. If there were separate barrels for each operation, and a central barrel for controlling the operations drums, the punched card presents a way of instructing the machine (the central drum) as to which operations we wished to perform and in what order, i.e. high-level programming the Engine.

Jacquard loom
A Jacquard loom

The principle of cards was openly borrowed from the Jacquard loom (a mechanical loom, invented by the Frenchman Joseph Marie Jacquard in the beginning of 1800s, based on earlier inventions of his compatriots Basile Bouchon (1725), Jean Falcon (1728) and Jacques Vaucanson (1740)), which used a string of punched cards to automatically control the pattern of a weave (see the nearby photo).
In the loom, rods were linked to wire hooks, each of which could lift one of the longitudinal threads strung between the frame. The rods were gathered in a rectangular bundle, and the cards were pressed one at a time against the rod ends. If a hole coincided with a rod, then the rod passed through the card and no action was taken. If no hole was present then the card pressed back the rod to activate a hook which lifted the associated thread, allowing the shuttle which carried the cross-thread to pass underneath. The cards were strung together with wire, ribbon or tape hinges, and fan, folded into large stacks to form long sequences. The looms were often massive and the loom operator sat inside the frame, sequencing through the cards one at a time by means of a foot pedal or hand lever. The arrangement of holes on the cards determined the pattern of the weave.

How can be programmed the Analytical Engine?

Federico Luigi Menabrea (1809-1896
Federico Luigi Menabrea (1809-1896

We know little of Babbage’s programming ideas. There is nothing in the surviving papers in which this aspect of the machine is thoroughly discussed, e.g., nothing corresponding to a specification of a user instruction set. This is the more remarkable for it is the only aspect of the design that is discussed at length in a contemporary paper. In 1840, Babbage visited Turin in Italy and gave a series of seminars on the Analytical Engine. An account of these, by the Italian engineer Federico Luigi Menabrea (1809-1896, see the nearby image), who later on will become the Prime-Minister of Italy, was translated into English by Ada Lovelace (she has sometimes been acclaimed as the world’s first programmer, a romantically appealing image without any foundation), who appended extensive notes prepared under Babbage’s close guidance (see Sketch of the Analytical Engine). These deal with the familiar modern ideas of flow of control in programs, particularly the formulation of simple loops and nested loops controlled by counters. However, the paper and notes carefully and deliberately skirt around any discussion of details of the means by which these are to be implemented.
It seems that Babbage did not have a command of the issues raised by the user-level programming of the Analytical Engine. It would be quite wrong to infer that Babbage did not understand programming per se. The microprogramming of the barrels for multiplication and division show command of the basic branching and looping ideas and his skills in the microprogramming of addition and subtraction show complete virtuosity. It was from this base that Babbage explored the ideas of user-level programming. The issues of data structuring simply did not arise at the microprogramming level. There is some evidence to suggest that Babbage’s ideas were moving in the directions now familiar in connection with the control mechanisms for loop counting in user-level programs. Had an Analytical Engine been brought to working order, there can be no doubt that Babbage’s programming ideas would have been developed greatly.
From the hardware point of view, two strings of punched cards were needed to specify a calculation to be performed by the Analytical Engine. One string, the “operation cards,” specified the arithmetic operations to be performed. The second string, the “variable cards,” specified the axes in the store that contained the operands and were to receive the results. These two strings cannot be regarded as separate parts of a single instruction, as are the operation and operand fields of an instruction in an electronic digital computer, because the operation and variable cards were intended to move and loop independently of one another under the direction of separate control mechanisms.
Actually there were four, but no two, different kinds of punched cards with different functions:
1. Number cards were used to specify the value of numbers to be entered into the store, or to receive numbers back from the store for external storage.
2. Variable cards specified which axes in the store should be the source of data fed into the mill or the recipient of data returned from it. In modern parlance, they supplied the memory address of the variables to be used.
3. Operation cards determined the mathematical functions to be performed. The logical content of an operation card might have been like this example: “Take the numbers from the variable axes specified by the next two variable cards, and multiply them in the mill; store the result on the variable axis specified by the third variable card.” This was interpreted by the sensing rods on the operation-card reading apparatus and internally translated like this: “Advance the variable cards by one position, and rotate all the barrels to the starting position for a normal multiply-and-store sequence.”
4. Combinatorial cards controlled how variable cards and operation cards turned backward or forward after specific operations were complete. Thus, an operation card might have a logical content like this: “Move the variable cards ahead 25 positions, and set the operation cards to the start of the set that tells how to extract a square root.”
Babbage planned to intersperse the combinatorial cards with the operation cards they controlled, so the four sets of cards required only three card readers (plus one card punch, for number cards being output from the machine).
Babbage seems to have been led to separate the operation and variable cards on largely philosophical grounds stemming from his belief in the need to distinguish symbols for operation from those for quantity in mathematical notations. These views were probably reinforced when he considered the cards necessary for calculations such as the solution of simultaneous equations. Babbage realized also, that programs or subroutines (certainly not terms that he used) would need to be verified, what we would call debugged. He also knew that it would be valuable to rerun verified programs on new sets of data, and even to share programs across multiple engines. Thus, it was a natural and practical approach to specify the data as being independent of the operations. There the pattern of operations required for carrying out row reductions is very simple and a straightforward loop of operation cards is readily found. No such simple loop structure exists for the variable cards, which can only specify single axes in the store. The loop structures that we now recognize concern rows of the matrix of coefficients of the equations and similar concepts related to the structuring of the data. As Babbage did not have the concept of a variable address in the store, neither was the Analytical Engine able to calculate the location of an operand in the store, there was no way in which the user programs could exploit this higher level structure in the data.
It’s amazing how far went Babbage in his programming conceptualization, keeping in mind, that he had no experience in programming an actual computer. From the current point of view, the series of operation cards provided not a program, in current terms, but a series of subroutines. The combinatorial cards provided terminology, a control-flow program, invoking subroutines with call-by-reference values provided by the variable cards. Babbage’s programming concepts clearly included what we call loops, subroutines, and branches (what later generations of programmers called “if ” or “if-then” instructions). Since he had no experience in programming an actual computer however, it is not surprising that Babbage did not get to the modern concepts of higher level languages, interpreters, or compilers.

A table with cards and actions for the formula (ab+c)d
A table with cards and actions for the formula (ab+c)d

Let’s take as an illustration the formula (ab+c)d (see the nearby table). The full detail of the cards of all sorts required, and the order in which they would come into play is this:
The four Number Cards for the “given numbers” a, b, c and d, strung together are placed by hand on the roller, these numbers have to be placed on the columns assigned to them in a part of the machine called “The Store,” where every quantity is first received and kept ready for use as wanted.
We have thus besides the Number Cards, three Operation Cards used, and fourteen Directive Cards. Each set of cards would be strung together and placed on a roller or prism of its own; this roller would be suspended and be moved to and from. Each backward motion would cause the prism to move one face, bringing the next card into play, just as on the loom. It is obvious that the rollers must be made to work in harmony, and for this purpose the levers which make the rollers turn would themselves be controlled by suitable means, or by general Directive Cards, and the beats of the suspended rollers be stopped in the proper intervals.

A general plan of the Analytical Engine from 1840
A general plan of the Analytical Engine from 1840

In the upper general view of Analytical Engine can be seen the basic sections: in the right part is a section of the store including 11 variable axes. In practice, the store would have been much longer, with many more variable axes; Babbage sometimes considered a minimum of 100, and as many as 1000. Each variable axis contained many figure wheels rotating around a central axle, each holding one digit of its variable. Babbage usually planned to have 40 digits per variable. One extra wheel on top recorded whether the value was positive or negative.
Running horizontally between the variable axes were the racks, long strips of metal with gear-toothed edges that carried digits back and forth between the store and the mill. Small movable pinions were positioned either to connect a given variable axis to the racks or to leave it unconnected. If a number was going into the mill, the racks would also be connected to the ingress axis in the mill (labelled I). From there, it would be passed to another appropriate part of the mill. When the mill was finished operating on a number, it would be placed on the egress axis (labelled E). This could then be connected to the racks, which would pass the number along to whatever variable axis had been chosen to hold the result.
The mill is the left section, arranged around the large central wheel that interconnect its parts. For clarity, not all aspects of the engine are shown in this diagram. But this may obscure the machine’s complexity and size. The central wheels alone were about 70 cm across. The mill as a whole was about 150 cm feet in each direction. A store with 100 variable axes would have been about 3 m long. The ingress axis had its own anticipating carriage mechanism; an addition or subtraction could be performed there and then passed directly to the egress axis for storage. If a multiplication was coming up, the first nine multiples would be added on the ingress axis and stored on the table axes, shown as T1 through T9.
The results of a full multiplication or division would be formed on the two columns labelled A to the left of the large central wheel. This made it possible to hold intermediate results in double precision form. That is, if two 40-digit numbers were multiplied together, 80 digits of result could be kept on axes A. A subsequent division by another 40-digit number still allowed 40 digits of precision in the result.
One of the main mechanical problems in the Analytical Engine was the problem with carry. At first, Babbage used the method of delayed sequential carry used in the Difference Engine. In this, the basic addition cycle was followed by a separate carry cycle. The carry cycle first performed any carry needed on the lowest digit, then proceeded to the next higher digit, and so on. This method worked, but it was slow because carries were performed separately for each digit. Babbage considered having 30 or 40 digits in each number column, so the carries might take a lot longer than the addition itself. Thus, a single multiplication might take some hundreds of separate addition steps. It was clear that the carry time had to be shortened.
Babbage tried various approaches to optimize the carries, and within a few months had adopted what he called the anticipating carriage. Additional hardware allowed the carriage mechanism to detect simultaneously where carries were needed and where one or more wheels already at 9 might cause a carry to propagate over a series of digits. All carries could be performed at once, regardless of the number of digits on an axis. Working out the details of anticipating carriage took Babbage many years, longer than any other single aspect of the machine. But it could speed operations greatly, justifying the effort. The mechanism was too complex to allow a carriage mechanism for each adding axis. Babbage was forced to adopt a design where a single anticipating carriage mechanism could be connected at will with any adding column through the central wheels. Until then, multiplication had been provided by specialized hardware, and the carriage function had been removed from the adding axes to more specialized central hardware.
Babbage soon realized that addition itself could be removed from the adding axes, and performed through the central wheels. The adding axes simply stored digits on their individual wheels, and they could be connected or disconnected from the central wheels as needed. Babbage separated the machine into a section of storage axes, which he called the Store, and another section where operations were performed, which he called the Mill.
During the years Babbage made different designs for many of the units of the Engine, most often in the direction of simplification. Not all of them however were in this direction, because Babbage was very anxious to speed up calculation. An example of adding hardware to achieve speed was multiplication by table. In the initial method of multiplication by repeated addition, the number of cycles of addition would be equal to the sum of the digits of the multiplier. Thus, to multiply 198814 by 978, for example, 198814 would be added 24 times (9 + 7 + 8), along with 3 shifts.
Babbage planned to work with numbers having as many as 40 digits. Multiplying two 40-digit numbers together might well take 200 addition cycles. Babbage realized that by devoting a few cycles at the start of a long multiplication to some preparation, he could greatly speed the multiplication itself. He called this multiplication by table. In 9 cycles, he could calculate and place on special table axes in the mill the first 9 integral multiples of the multiplicand. Then he could simply pick one of these for each digit of the multiplier and add it to the accumulating product. Multiplying two 40-digit numbers would then take only 40 addition cycles, plus 9 to form the table, a total of 49 addition cycles rather than some 200. A similar method of division by table could also speed division.
It is considered, that the basic design the Analytical Engine was finished by December 1837, when Babbage finally wrote an extended paper, “Of the Mathematical Powers of the Calculating Engine,” which described the machine. He continued design work for many more years, but this involved refinement of detail and alternatives of implementation, not changes of principle. By 1837, Babbage had devised a machine whose basic organization would remain unchanged through all his subsequent work, and indeed through the entire subsequent development of computer design.
Babbage considered that the Analytical Engine was a universal calculating machine in the sense that, given sufficient time, it could carry out any possible arithmetic calculation. This argument is based on three observations. First, arithmetic operations on numbers of more than forty digits can always be carried through by breaking them into 40-digit segments, so the limited number of digits on any store axis is no fundamental limit. Second, calculations can be specified by strings of operation and variable cards of unlimited extent, so there is no limitation to the size or complexity of programs. Third, numbers from the store can be punched onto number cards and later read back, and this provides a backing store of unlimited extent to overcome the limited number of axes in the store.
Babbage was fully aware of speed limitations of his machine. He knew that the machine would in theory make possible far more extended and precise calculations than had ever been attempted by hand, but they would be possible in practice only with a machine that was highly reliable and quite fast. From his earlier work, he knew that reliability required the gears not to turn too quickly. Overall speed had to be achieved with smart design rather than raw power. This is what motivated the immense ingenuity that Babbage invested in time-saving methods like anticipating carriage and multiplication by table.
In the machine design of the late 1830s, the isolated addition of two 40-digit numbers would have taken about 19 seconds. But a lot of this involved moving numbers around between different sections before or after the actual addition. Babbage figured out how to overlap the different parts of the operation when more than two additions were to be performed in succession. This meant that each extra 40-digit addition took only 3.1 seconds. Multiplication and division were similarly accelerated by clever logical design. The duration depended on the number of digits in the numbers. Take the case of a multiplication of 20 digits by 40 digits (a very high degree of precision even by current standards). With sustained additions at 3.1 seconds each, a straightforward step and add approach would have taken nearly 8 minutes to complete. Babbage was able to reduce this to less than 2 minutes. Today, with microprocessor speed measured in millions of multiplications per second, 2 minutes seems incredibly slow. But it was a remarkable accomplishment more than a century before electronic computation.

A model of the Mill of Analytical Engine, constructed about 1870 (© Science Museum, London)
A model of the Mill of Analytical Engine, constructed about 1870 (© Science Museum, London)

After finishing of the work on the design of the Analytical Engine in 1847, Babbage turned to the design of a Difference Engine №2, exploiting the improved and simplified arithmetic mechanisms developed for the Analytical Engine. In 1857 Babbage returned to the design of the Analytical Engine. In this new phase of work Babbage was actively interested in building an Analytical Engine with his own resources. The logical design was somewhat simplified but, most importantly, far simpler and cheaper methods were proposed to implement the basic mechanisms. Babbage first experimented with sheet metal stamping and pressing for making gear wheels and similar parts. Later, he adopted pressure die casting for making parts—a newly invented technique that did not see extensive commercial use until the end of the nineteenth century. Babbage built many experimental models of mechanisms using these new techniques, and, at the time of his death in 1871, a model of a simple mill and printing mechanism was near completion (see the nearby photo).
Babbage’s calculating machines, as well as all related materials were inherited by his youngest survived son, Major-General Henry Prevost Babbage (1824-1918) (see the nearby image), who had shown a strong interest in his father’s work. Even as teenagers Henry and his older brother Dugald spent time in Babbage’s drawing office and workshop learning workshop skills. Henry later acquired a strong grasp of the Difference Engine and Analytical Engine designs, and came to form a close bond with his father whom he visited on furlough from extended military service in India. Babbage bequeathed his drawings, workshop and the surviving physical relics of the engines to Henry who tried to continue his father’s work and to publicize the engines after Babbage’s death.

A part of the mill and printing mechanism of the Analytical Engine, constructed by Henry Babbage
A part of the mill and printing mechanism of the Analytical Engine, constructed by Henry Babbage

Henry was at his father’s bedside when Babbage died in October of 1871, and from 1872 he continued diligently with his father’s work and then intermittently in retirement in 1875. After assembling some small demonstration pieces for Difference Engine Number 1 (one of them he sent to Harvard. In the 1930s the piece attracted the attention of Howard Aiken, the creator of the Harvard Mark I).
In 1888, Henry demonstrated on a meeting of British Association for the Advance of Science a section of the Mill of the Analytical Engine, working to 29 digits, including the anticipatory carry device.
Then Henry built an experimental four-function calculator for the Mill, completing it in 1910. Henry actually decided not to continue with the original design of the Analytical Engine, but instead to develop a manually operated machine for addition, subtraction, multiplication, and division (a four-function calculator), incorporating the mechanisms planned for the mill and printing mechanism of the Engine (see the nearby photo). Using the assembled Mill of Analytical Engine, he would do simple calculations—in this case produce multiples of π.
Although eventually completed in the beginning of the twentieth century, when Henry was himself an old man, this machine appears never to have worked reliably. Moreover, Henry’s work on the engines was sound, but without the boldness and inspiration of his father, the great Charles Babbage.

Differential Engine of Charles Babbage

Believe only half of what you see and nothing that you hear.
Edgar Allan Poe

Charles Babbage, 1791-1871. Portrait from the Illustrated London News, 4 Nov, 1871
Charles Babbage, 1791-1871. Portrait from the Illustrated London News, 4 Nov, 1871

A numerical table is a tool designed to save the time and labour of those engaged in computing work. The oldest tables which are preserved, were compiled in Babylon in the period 1800-1500 B.C. They were intended to be used for the transformation of units, for multiplication and division, and they were inscribed in cuneiform on pieces of clay. During the first century B.C. Claudius Ptolemy in Alexandria created his theory about the motions of the heavenly bodies in a work which later came to be known by the name of Almagest. They were to form one of the Ancient World’s most important astronomical documents and they contained all the necessary tables for the calculation of eclipses as well as various kinds of ephemeris, that is to say tables which specified the positions of the heavenly bodies during a particular period, e.g. each day for a whole year.
During the first half of the 13th century the Ptolemy’s tables caught the attention of King Alphonso the Wise of Castile. He then gathered together a great number of scholars in Toledo who were given the task of calculating a new collection of astronomical tables. The reason for this endeavor was said to be that King Alphonso, who was interested in astronomy, had discovered many errors in Ptolemy’s tables. The work began some time in the 1240s and took about ten years to complete. The tables produced were later known as the Alphonsine Tables. The vast costs involved were paid for by the king, whose name soon spread with the copies of the tables throughout the European scientific world. Besides the Babylonian tables, Ptolemy’s work and the Alphonsine Tables, a great deal of toil went into the production of many other numerical tables of different kinds during this period.
With the introduction of the art of printing throughout Europe during the latter half of the 15th century, the first tables were printed. The Alphonsine Tables for example, were printed in Venice in 1483. At the end of the sixteenth century, several famous arithmetical and trigonometric tables were published. In order to simplify multiplication work, multiplication tables were published. A real revolution in the table business happened after John Napier’s discovery of logarithms in 1614. With a table of logarithms at hand, the computational effort could be greatly reduced. In 1617 Henry Briggs published the first table of logarithms.
Two hundred years later, at the beginning of the 19th century, numerical tables were still the most important calculating aid in Europe. The sole alternatives were Napier’s Bones and the slide rule. Mechanical calculating machines were extremely rare and at most a handful of very select individuals can ever have used them for serious calculations. Most of them were simply remarkable gadgets illustrating man’s scientific progress, rather than genuine aids to calculation. For the normal calculator or scientist who had to carry out complex computations which demanded great accuracy, Napier’s Rods and the slide rule were of little help. In effect, his tools were pen, paper and tables. There were tables for mathematics, astronomy, navigation, physics, engineering, statistics, trade and finance, in the army and in many other areas. The publication of such tables however required a lot of manual calculating work and the final product was full of errors.

How emerged the idea of Differential Engine
Some time in 1821, the young english mathematician Charles Babbage got the idea about the mechanical computation. He has provided us with two versions of the origin of his ideas about machines, but the one written in 1822 seems more plausible than the other, which appeared in his autobiography some forty years later.
According to the first story, in 1820 or 1821, the Astronomical Society assigned Babbage and his friend Herschel one of the tasks for improving the tables of the navigational book Nautical Almanac. They constructed the appropriate formulas and assigned the arithmetic to clerks. To diminish errors, they had the calculations performed twice, each by a different clerk. Then they compared the two sets for discrepancies. In the course of their tedious checking, Herschel and Babbage found a number of errors, and at one point Babbage said I wish to God these calculations had been executed by steam. It is quite possible, remarked Herschel.
But in his autobiography Babbage remembered another version of the story, which must have happened either in 1812 or 1813:
“… I was sitting in the rooms of the Analytical Society, at Cambridge, my head leaning forward on the table in a kind of dreamy mood, with a table of logarithms lying open before me. Another member, coming into the room, and seeing me half asleep, called out, “Well, Babbage, what are you dreaming about?” to which I replied “I am thinking that all these tables” (pointing to the logarithms) “might be calculated by machinery.”
Babbage was not the first to suggest a printing calculator, nor was he the first to propose the method of differences as a suitable principle on which to base a mechanized calculation. This distinction goes to the german engineer and master builder Johann Helfrich Müller, who described his dreams for calculating machine, based on the method of differences as early in 1784, but his idea remained only on paper. There is an evidence, that at some point, Babbage learned about Müller and his project, but this most probably happened after 1821, when he has already started his work on Differential Engine.

Start of the work on Differential Engine
Whatever the truth, some time in 1820 or 1821 Babbage started his work on the calculating machine, making several designs for clockwork-like mechanisms, that could be made to control a set of wheels with numbers along their edges that could print on paper. He made a small model consisting of 96 wheels and 24 axes, which he later reduced to 18 wheels and 3 axes. The machine was ready by the end of the spring of 1822 and in June was announced publicly and has been examined by several of the members of the Astronomical Society.
It seems Babbage must have known very little about machine design, mechanical calculating and the history of such machines at that time, because he started by considering the use of sliding rods, instead of the more natural use of wheels in the adding mechanism. This kind of mechanism which was “new” in the history of calculating machines, gives rise to grave difficulties in the process of carrying, a fact which Babbage eventually realized. In fact this seems to have been such a revelation to him, that in November of 1822 he noted very solemnly that he had in future decided always to choose circular motion for this purpose.
The working model had a section of the calculating mechanism, including two orders of difference, but no printing mechanism. He successfully calculated the first thirty values, arising from the formula x2 + x + 41, which was a favorite example of his, because it generates a lot of prime numbers. The machine produced correct results at the rate of 33 digits per minute, so the values were tabulated in two and a half minutes. Later on same year Babbage wrote a note to the Society and an article “On the Theoretical Principles of the Machinery for Calculating Tables” for Brewster’s Journal of Science:
I have contrived methods by which type shall be set up by the machine in the order determined by the calculation. The arrangements are such that… there shall not exist the possibility of error in any printed copy of tables computed by this engine.
Babbage wrote also a letter on the general subject to the president of the Royal Society, Sir Humphry Davy. In this letter, Babbage pointed out the advantages such a machine would have for the Government in producing the lengthy tables for navigation and astronomy, and proposed to construct a machine on an enlarged scale for the Government’s use.
The Astronomical Society received Babbage’s proposal with the highest enthusiasm, and the Royal Society reported favorably on his project for building what he called a Difference Engine, a specialized calculating machine for calculation of tables, using the method of differences.

Table of differences
Table of differences

What is the essence of the method of differences, which underlies Babbage’s first automatic calculating machine. Let’s consider the same formula, used by Babbage: T=x2 + x + 41. It generates a sequence of values for T, which happen to be prime numbers, as seen in the table in the nearby figure, in which with D1 is noticed the first difference column, while with D2—the second difference column. If we take the differences between successive values of T, these so called first differences follow quite a simple rule. If we take the differences between the differences, known as the second differences, the result is even more striking-the second difference, is a constant. With this knowledge, the table can be built up in a very simple way, as shown by the box in the table. Take the second difference, and add it to the first difference to form a new first difference, 4+2=6. The process can be generalized. In our example the second difference is constant because the function T is a quadratic. If the function T were a cubic, such as T = x3, then the second difference would vary, but the third difference, the difference between successive second differences, would be constant. In general a polynomial of degree n will have a constant nth difference and each successive new value of the function can be obtained by n simple additions.
The usefulness of difference techniques is greatly increased by the fact that any section of a well-behaved continuous function can be approximated by a polynomial. The shorter the section and the higher the degree of the polynomial the closer the approximation. So if we wished to tabulate a function, such as a sine or the time of sunset, it is only necessary to divide the function into short enough intervals and find a suitable approximating polynomial for each interval. The method of differences can then be used to tabulate the function throughout the interval. This process is known as sub-tabulation. Babbage realized that a machine could carry out this sub-tabulation process. First, he needed a mechanism for storing, separately, the numbers corresponding to the values of the tabular value, the first difference, the second difference, etc. and a mechanism to add each difference to the value of the preceding difference.
In the process of designing and building his Difference Engine, Babbage required many accurate drawings of the parts. While using these drawings, he felt that they did not fully and adequately describe the mechanism. For a machine with many parts moving in various ways, the static drawings could only show the shape and arrangement of the parts. So Charles devised a system of mechanical notation that would also indicate how the parts moved—their speeds and interconnections. Unlike the usual drawings, the notation did not picture the shapes of the parts. Rather, it was a table of numbers, lines, and symbols to describe the machine’s actions. It was a general system that could be used to describe any machine. Charles published a description of his mechanical notation in the Philosophical Transactions of the Royal Society in 1826 and later in 1851 (see Laws of mechanical notation). However, this mechanical notation did not ever come into widespread use.
In an interview held in 1823 between Babbage and the Chancellor of the Exchequer, a rather vague verbal agreement was made whereby the Government would grant funds for the enterprise which was expected to take three years. His own Astronomical Society was so impressed by the machine, that it awarded him its first gold medal in 1824. In the same year the British government advanced Babbage a fee of £1500, and he began to construct the full Difference Engine. Babbage needed a small factory and competent workers, although initially two rooms in Babbage’s house were converted into workshops and a third into a forge. He hired the very good British engineer Joseph Clement (1779-1844), to maintain the mechanical works in his shop. By 1828, Charles had spent more than £6000 on the construction, and the government had only reimbursed him for £1500. After a supportive report from Charles’s friends in the Royal Society, the government agreed to make up the difference. But the work went rather slowly.
The whole project was taking much longer than anyone had anticipated. While the fabrication of basic parts proceeded, shop patterns had to be drawn for others. The full set of plans was not completed until 1830. By then, Clement’s workers had produced many thousands of parts, but had done little assembly.
Soon, Babbage and the government decided that the plans and assembly should be moved out of Clement’s shop. On Babbage’s property was built a two-store fireproof workshop and a second building for the Difference Engine. Babbage’s intention was to move Clement’s whole operation to these new quarters. However, Clement resisted, because with the funds Babbage had supplied him, he had greatly expanded his own workshop. He now had many machine tools and a number of employees and used them to do other work besides that contracted by Babbage. And by the trade practices of the time, he insisted that the machinery belonged to him, not to either Babbage or the government.

A part of Difference Engine, assembled in 1832
A part of Difference Engine, assembled in 1832

During 1832, Clement’s workers completed the assembly of as much of the engine as they had parts for (some 10000 parts were made). Even though the calculating section was largely complete, but the printing section was not. From this time on, no further work was done. Clement would not move his machinery to Babbage’s shop, and only in 1834 was the engine itself transferred. By then, the government had expended £17000, and Babbage had spent some six thousand pounds own money. The government was unwilling to proceed further, given the need to reorganize the whole project after Clement and Babbage had parted company.
Almost all of the parts of the entire calculating mechanism had been made, but not assembled, when work on the project stopped early in 1833. A portion of the calculating mechanism was assembled in 1832 (see the nearby photo) to demonstrate to a committee of the Royal Society and the Parliament that the project was proceeding satisfactorily, but it was limited to two orders of differences and five figures, suitable for demonstration purposes only.
It is about one-third of the height and one-half of the width, or about one-seventh of the entire calculating mechanism and consists of about 2000 bronze and steel parts. The calculating part alone would have been 7 times larger than the little unit that was assembled. The entire machine was expected to contain some 25000 parts and to weight over 2 tons, with dimensions roughly 260 cm high, 230 cm broad, and 100 cm deep.

Elevation (upper part of the figure) and plan drawings of Difference Engine, from 1830
Elevation (upper part of the figure) and plan drawings of Difference Engine, from 1830

The Design of the Difference Engine
The Difference Engine consisted of two major parts—the calculating mechanism and the printing and control mechanism. In the nearby drawing from 1830 of the elevation (upper part of the figure) and plan drawings (lower part) of Difference Engine they are clearly seen. The calculating mechanism is on the left, the axes of figure wheels for the tabular value (far right) and six differences are clearly visible. The printing mechanism is on the right, and the moving table carrying the stereotype printing plate and the sector carrying the digit-type punches are visible in the center of both drawings.

Digits are represented in the Difference Engine by the rotational position of horizontal gear wheels. A number is made up of a series of these figure wheels rotating about a common vertical axis. The bottommost wheel represents units, the next tens, the next hundreds, and so on. The figure wheels are about 15 centimeters in diameter and are spaced vertically about 7.5 centimeters apart on the axes. Babbage used the term axis to mean a stack of figure wheels that together store a number as a collection of decimal digits. The entire Difference Engine consists of an axis for the tabular value of the function, another axis for the difference, a third axis for the second difference, and so on for as many orders of differences as are desired.

A part of Difference Engine (from the frontispiece of Passages from the Life of a Philosopher, 1864)
A part of Difference Engine (from the frontispiece of Passages from the Life of a Philosopher, 1864)

Each axis served not just as a number store, but also as an adding mechanism. Addition occurred in two steps that will be explained with reference to adding the first difference to the tabular value. Inside each first difference figure wheel there is a mechanism that is rotated through just as many steps as the value stored by the figure wheel. If the units figure wheel stands at 3, the mechanism will move through three steps. This motion is conveyed by gearing to the corresponding figure wheel of the tabular value axis. If the latter stood at 5 initially, it will be moved three steps to stand at 8. This process occurs simultaneously in the tens, hundreds, thousands, and other digit positions.

It may happen that addition to a figure wheel will generate a carry that must be propagated into the next higher digit position. If the units digit of the tabular value were initially 6 and 7 is added, it will move forward seven places and come to stand at three, but a carry must also be propagated into the tens figure wheel of the tabular value. Carry propagation is complicated by the fact that if the tens figure wheel already stands at 9 it will be moved forward by the carry to stand at 0 and a new carry will be propagated into the hundreds figure wheel.

In the Difference Engine these consecutive carries may propagate, as on occasion they must, from the units up through the most significant figure wheel. Each addition, therefore, consists of two distinct steps-the simultaneous addition of all figures of the first difference to the corresponding figures of the tabular value, and the consecutive propagation of carries from the units up to the most significant digits as required.

the tabulation of a cubic function from third differences
The tabulation of a cubic function from third differences

Tabulation of a function involves the repetition of this basic addition process for each of the orders of difference involved. As each axis is also an adding mechanism the tabulation of a cubic function from third differences, for example, requires six steps for each tabular value produced (see the nearby figure):
1. Addition of third difference digits to second difference digits
2. Carry propagation among second difference digits
3. The second difference is added to the first difference
4. Carry propagation among first difference digits
5. The first difference is added to the result column
6. Carrying took place in the result column
Negative numbers may be handled with no additional mechanism by representing them as their ten’s complements.
This scheme is readily extended to higher order differences. It is obvious that the number of steps is doubled number of the power of function, which means that for higher power functions will be required a lot of steps. Babbage found a way to rearrange the calculation so that only four steps were required for each tabular value produced irrespective of the number of differences involved. This is characteristic of the sophisticated logical considerations underlying Babbage’s designs.
Babbage observed that when the fist difference is added to the tabular value, in steps five and six, both the third difference and second difference axes are idle. He could thus add the third difference to the second difference, steps one and two, at the same time as the first difference is added to the tabular value. Steps one and two overlap steps five and six. Thus only four units of time, for steps three to six, are needed for each tabular value produced. In modern terminology we would call the arrangement of hardware to perform a calculation in this way a pipeline.
The overlapping idea can be extended to higher differences and a new tabular value can always be produced in four steps, namely:
1. Odd differences are added to even ones and to the result
2. Carrying takes place in the even differences and in the result
3. Even differences are added to the odd
4. Carrying takes place in the odd differences
Not only does this rearranged form of the calculation save considerable time but it also makes the arrangements for driving the calculating mechanism much simpler.
It appears as if Charles Babbage did not initially determine the mathematical capacity of the engine. He only describes it as being intended as a larger engine. In 1823 the engine was being made to calculate with four orders of differences. The number of digits was not mentioned. In 1829 the machine was said to be able to operate with sixth order differences, 12 digits, and to print 16 digits in the result with speed forty-four digits a minute. At some point Babbage settled for six orders of differences, but the number of digits continued to vary, depending on the author. 18 digits are mentioned in 1834 and as an old man Babbage himself said that the whole engine would have been capable of calculating with 20 places of digits.

The stereotyping of the tables
The stereotyping of the tables

The matrices for the stereotyping of the tables would have been produced in the printing unit. The result was to be taken from the result column in the calculating unit and transferred to the printing unit. There, eleven steel punches were supposed to print the result and argument in a copper plate, producing a printout similar to this, shown in the nearby figure.
It is a great pity that the work on the Difference Engine ceased so close to completion. Henry Babbage later estimated that only a further five hundred pounds would have sufficed. Babbage could readily have found the funds, however his feelings and attitudes to both the government and Clement could not allowed him doing so. Besides that, within a year or two, Babbage’s mind had moved a long way towards the much more complex and intellectually rewarding Analytical Engine. There was then no way he would have returned to the original Difference Engine design and brought it to completion, even had events made that feasible.
In late 1860s Babbage said: “I have not finished it [Difference Engine] because in working at it I came on the idea of my Analytical Engine, which would do all that it was capable of doing and much more. Indeed the idea was so much simpler that it would have taken more work to complete the calculating machine than to design and construct the other in its entirety, so I turned my attention to the Analytical Machine.”
Nevertheless it is out of question, that the Difference Engine stood as a great monument over man’s ingenuity and ability to mechanize all kinds of labour. The idea was too important and exciting to be forgotten. Babbage’s exertions brought in their wake considerable publicity, which was an important factor in keeping the idea alive. Another factor was naturally the problem itself. A handful of inventors, all with different backgrounds, were to try during the course of the 19th century to build difference engines according to their own ideas. First of them was the Swede Pehr Georg Scheutz, who managed only with small part of Babbage’s resources to produce a working difference engine in the middle of 19th century.
For some years, Babbage displayed the working section of his Difference Engine in one of his drawing rooms and used the portion of the calculating mechanism to calculate nearly a hundred functions. He even designed some improvements to the original mechanism. In the Difference Engine, whenever a new constant was needed in a set of calculations, it had to be entered by hand. In 1834, Babbage conceived a way to have the differences inserted mechanically, arranging the axes of the Difference Engine circularly, so that the Result column should be near that of the last Difference, and thus easily within reach of it. He called this arrangement the engine eating its own tail. But this soon led to the idea of controlling the machine by entirely independent means, and making it perform not only Addition, but all the processes of arithmetic at will in any order and as many times as might be required. Work on the first Difference Engine was stopped on 10 April 1833, and the first drawing of the Analytical Engine is dated in September, 1834.
After finishing of the work on the design of the Analytical Engine in 1847, Babbage turned to the design of a Difference Engine No 2, exploiting the improved and simplified arithmetic mechanisms developed for the Analytical Engine. The logical design was the same as for the earlier Difference Engine, but he employed simpler mechanisms for storing and adding numbers and carry propagation. The printing mechanism was simplified so that a whole number was impressed on a printing plate as a single action rather than in a digit-by-digit manner. A conventional print copy, using inked rollers, was made simultaneously. The control was arranged by a single barrel in a very straightforward manner. The design and a complete set of drawings was prepared by mid-1848. These Babbage offered to the British government, apparently to satisfy a commitment he felt existed in consequence of the failure of the project to build the first Difference Engine, but the government show no interest on the new design.

Major-General Henry Prevost Babbage (1824–1918)
Major-General Henry Prevost Babbage (1824–1918)

Babbage’s calculating machines and related materials were inherited by his youngest son, Major-General Henry Prevost Babbage (1824–1918), who had shown a strong interest in his father’s work. While as teenager Henry and his older brother Dugald spent time in Babbage’s drawing office and workshop learning workshop skills. Henry later acquired a solid grasp of the Difference (and Analytical) Engine designs, and came to form a close bond with his father whom he visited on furlough from extended military service in India. Babbage bequeathed his drawings, workshop and the surviving physical relics of the engines to Henry who tried to continue his father’s work and to publicize the engines after Babbage’s death.
Henry was at his father’s bedside when Babbage died in 18 October 1871, and from 1872 he continued diligently with his father’s work and then intermittently in retirement in 1875. He assembled some six small demonstration pieces for Difference Engine Number 1 and one of them he sent to Harvard. In the 1930s the piece attracted the attention of Howard Aiken, the creator of the Harvard Mark I, a program-controlled calculator.

Vladimir Lukianov

Vladimir Lukianov (1902-1980)
Vladimir Lukianov (1902-1980)

Vladimir Sergeevich Lukianov (Владимир Сергеевич Лукьянов) was born in Moscow, Russian Empire, on 17 March 1902, in the family of an insurance agent. In 1919 he graduated from a Moscow classical high school and entered the building department of the Moscow State University of Railway Engineering. He graduated in 1925 and was sent to build the railroads Troitsk—Orsk and Kartaly—Magnitnaia. After working for five years on the construction and design of railways, in 1930, Lukianov went on research activities in the Central Institute of Railway Engineers in Moscow, where he worked on the calculation of temperatures in concrete structures.

To ensure the quality and durability of concrete structures, earthworks and concreting were carried out only in the summer, but the cracks in the structures were not always able to be avoided. Lukianov decided to explore the possibility of carrying out construction work in the winter time and find out the reasons for the destruction of concrete structures. He began to study temperatures in concrete cells, but current methods of calculation could not provide quick and accurate solution of complex differential equations that describe the temperature, and the analytic solution required many assumptions.

Searching for a new approach to the problem, Lukianov applied the method of analogies between different physical processes, and in 1934 proposed a fundamentally new way of mechanization of transient calculations—method of hydraulic analogies. The result was the creation of a kind of model of thermal processes—so called Hydraulic Integrator.

The first model (one-dimensional) of the Hydraulic Integrator was created in 1936, the so called ИГ-1 (Интегратор Гидравлический 1). It was a primitive device made of roofing iron, tin and glass tubes, but it successfully resolved the problem of investigating the concrete temperature regimes. Moreover, it was the world’s first water-driven calculating machine, as well as the first computer for solving differential equations in partial derivatives.

The Hydraulic Integrator of Vladimir Lukianov (a three-dimensional version)
The Hydraulic Integrator of Vladimir Lukianov (a three-dimensional version)

Its main node are vertical (main) vessels of sizable capacity (in the right part of the upper photo, in front of the operator), interconnected by means of tubes with variable hydraulic resistance and connected to moving vessels. By raising and lowering them, the operator can change the water flow in the main vessels. Start or stop of the process of the calculation is made by means of cranes with common control.

Solving a problem using the Hydraulic Integrator needed some preparation:
1. Create a design scheme of the test process
2. On the basis of this scheme produce a compound of vessels to identify and choose the values of hydraulic resistances of the tubes
3. Calculate the initial values of the quantity to be searched
4. Draw a graph of the external conditions of the simulated process

After the preparation steps the operator should set the initial values: fixed and mobile cranes in closed vessels must be filled with water to the calculated levels and then noted on graph paper attached for piezometers (measuring tubes)—It turns out a kind of curve. Then all the valves must be opened simultaneously, and the operator must change the height of moving vessels in accordance with the schedule of changes in the external conditions of the simulated process. At the same time the water pressure in the main vessels was changed by the same law as the temperature. The liquid level in the piezometers changed, at the right time valves must be closed, stopping the process, and graph paper will mark a new level positions. These marks have to be plotted, which will be the solution.

The Hydraulic Integrator of Vladimir Lukianov (© Moscow Polytechnic Museum)
The Hydraulic Integrator of Vladimir Lukianov (© Moscow Polytechnic Museum)

In 1941 Lukianov constructed a two-dimensional version of the Hydraulic Integrator. In 1940s the machine was put in production in Moscow Plant of Calculating Machines. When in 1949 was established the State Institute for Calculation Machines, the Hydraulic Integrator was one of the first calculating machines put in series production, and was even exported. The Hydraulic Integrator was successfully used in the construction of power plants, mine building, geology, building thermal physics, metallurgy, rocket science and many other areas. In the middle 1970s Hydraulic Integrators were still used in over 115 production, scientific and educational organizations in USSR and abroad (Czechoslovakia, Poland, Bulgaria and China), and remained in use up to 1990s.

Two Hydraulic Integrators of Vladimir Lukianov are currently presented in the collection of analog machines of Moscow Polytechnic Museum (see the nearby image).

Vladimir Lukianov was a professor and doctor of technical sciences, a holder of the State (Stalin) Award of 1951, Order of the Red Banner of Labor, and other state awards. He died on 17 January 1980 in Moscow.

William Quentell

William Prehn Quentell (1861-1932) was a prolific inventor, a holder of at least 33 US patents (as well as many patents in Canada, Germany, France, and Great Britain), stretching over more than 40 years (the last was granted in 1931, the year before he died). His first patent (and most of the others) was for a typewriter, but Quentell was also a holder of many patents for calculating machines.

Quentell's patent drawing (patent №US888262 from 1908)
Quentell’s patent drawing (US patent №888262 from 1908)

In 1890s Quentell worked together with the inventor of typewriting and calculating machines, De Kerniea James Thomas Hiett, and one of Hiett’s patents for typewriters (US547146 from 1895) was assigned to Quentell. Then from early 1900s Quentell worked together with another inventor of typewriting and calculating machines, Franklin Judge. From the beginning of 20th century, right up to the late 1920s (when he returned to typewriters), Quentell devoted most of his designs to calculators and adding machines. During this period, he worked also for a while with the award-winning American engineer Frederick Arthur Hart, one of the designers of Underwood Combined Typewriting and Calculating Machine.

Quentell’s first two patents for calculating machines were granted in 1908 (patents US881717and US888262). Let’s examine the operation of the machine, using the drawing from second patent (see the nearby drawing).

The operation of the device is readily understood. As a key b guided or sliding on cross rods 39 is depressed the bar d is tilted to free itself from the detent blade or fixed bar g and its spring h then slides or snaps the bar for stud i to lock the stem of key b which is now held depressed. At the same time the pawl s is retracted the desired distance. After the desired keys have been actuated the drive lever m is moved and such lever sliding the bars d forward releases the keys which again rise to the former level. At the same time the pawl mechanism actuates the numbered wheel or wheels and the arm 24 of lever m moves the connection 25 and the receding bar d has set its type line or bar 35. The total is shown at the sight line of wheels 8 and can be transferred to the paper of rollers 31 by the operator writing the separate figures showing the total.

Biography of William Quentell

William Prehn Quentell was born in New Orleans on 8 January 1861, the first son of William (born Wilhelm) Reinhard Conrad Quentell and his wife, Marie Therese Corine Baquie Quentell (born 28 Aug. 1840 in New Orleans-died 1868).

Wilhelm Quentell was born in Bremen, Germany, on 1 March 1823, and came to America in 1830s together with his family—father: Wilhelm (1784-1840), mother: Anna Luise Wilhelmine Conradine (1794–1880), and brothers: Friedrich Theodore (1828-1848), and Franz Carl (1830-1858).

Wilhelm and Marie Therese married on 6 February 1860 (Marie Therese was the widow of his brother, Franz Carl, who died on 23 Oct. 1858, and had one daughter, Marie Louise Wilhelmina (1857-1929), from his first marriage).

When William Prehn was a baby, his family removed to Liverpool, England, where William Quentell Sr. established a merchant company (Quentell William and Co.) There on 26 May 1862, was born William’s brother Carl August (1862-1915). On 9 Dec. 1864 was born another boy, Edward Joseph, but he died only 15 months later, on 6 March 1866. In February, 1866, was born a daughter, Marie Olga (1866-1944). Sadly, in Liverpool on 29 August 1868, only 28 years old, died William’s mother—Marie Therese.

William was educated in Liverpool, and attended Liverpool College. The family returned in New Orleans around 1876, where Wilhelm Quentell died on 12 January 1877.

William Quentell worked for many years as a stockbroker on Wall Street (at a time when a seat on the New York Stock Exchange cost $66,000, a huge sum for the time!). He worked for Atwood Violett and Co. (Atwood Violett, an influential banker and broker from New Orleans, was brother-in-law of William, married to his sister Marie Olga since 1891) between 1901-1903, and also later as a broker on the New York Cotton Exchange for a number of years.

The Postal typewriter of William Quentell, an early model from 1902
The Postal typewriter of William Quentell, an early model from 1902

Quentell was trying for many years to get the financing in order to manufacture his typewriters. In 1898 he was finally able to form a company in New York, and in 1899 he moved to Harrisburg, Pennsylvania, to oversee production of his Keystone typewriter. However, Quentell’s first typewriter venture was unsuccessful. In 1902, he established another company for manufacturing of typewriters, investing a large part of his own money and that of his partners from the Stock Exchange, like Nathaniel L. Carpenter. Quentell worked as a vice president and general manager of the Postal Typewriter Co. based first in New York and then (1904) in Norwalk, Connecticut, until 1909, when it was closed. The Postal typewriter (see the nearby image) enjoyed some popularity, and in the years the company was in business it produced about 30000 typewriters among eight models. The company employed 2000 salesmen in the US, and the typewriter was exported to Great Britain, Germany, Austria, France, and Russia.

William Prehn Quentell gets a passing mention in most historians mostly as the inventor of the Keystone and Postal typewriters. Besides the numerous patents for typewritng and calculating machines, he was a holder of many other patents for various devices like: can soldering machine (1892), lamp extinguisher (1893), font of type (GlobeGothic from 1895), hand cotton picker (1905), cotton harvester (1906), typesetting machine (1907), printing press (1908), and many others.

On 6 Feb. 1892, William Prehn married in Independence, Jackson County, Missouri, to Agness Roberts (b. 27 Apr. 1863 in Missouri-d. 17 Sep., 1949). They had two daughters, born in New York: Olga Marie (3 May 1900-30 Oct. 1953), and Gladys Agnes (7 Aug. 1901-8 Feb. 1966).

The remarkable inventor and businessman William Prehn Quentell died at Mineral Springs, Moore County, North Carolina, on 23 January 1932, two weeks after his 71st birthday. He was buried in Saint Louis Cemetery, New Orleans.

William Cordingley

In the beginning of 20th century, the Englishman William George Cordingley, a merchant and author of several books for commerce, stock exchange, and business tables and calculations from the end of 19th and beginning of 20th centuries, devised a simple adding machine, which he patented in 1907 (British patent №190715435 from 24 Oct. 1907 and later in 1909—pat. №190901751 from 27 May 1909.

The Computometer of Cordingley
The Computometer of William George Cordingley

The calculating device of Cordingley (see the nearby image) was a stylus operated adder of Pascalene type, which was manufactured and sold under the name Computometer.

The Computometer was a brass (copper, zinc alloy), leather and steel (metal) adding device with seven digital positions, as the leftmost (four or five) wheels are decimal, others are divided according to the english monetary system units from the beginning of the 20th century. Besides the two above-mentioned English patents, Cordingley obtained two other patents for his device—in 1910 he got a French patent (№414055 for Perfectionnements aux machines à calculer), and in 1911 an Austrian patent (№49900 for Zehnerübertragung).

Judah Levin

Judah Leib Levin (1863-1926)
Judah Leib Levin (1863-1926)

At the beginning of 20th century Judah Levin (1863-1926), an Orthodox rabbi from Detroit, Michigan, devised three calculating machines, which he patented from 1902 until 1906. A working model only of third calculator of Levin survived to our time, and it is kept in the collection of National Museum of American History, Washington, D.C. (see the image below).

First (US patent No. 706000 from 1902) and second (US patent No. 727392 from 1903) calculator of Levin are simple adders, with a similar internal mechanism, but with different input mechanism (the first one is using turning handles, while the second is using rotating wheels similarly to Pascaline). The third calculator of Levin (US patent No. 815542, Great Britain patent No. 190606717, French patent No. 364433, Canadian patent No. 106810) is more complex ten-key non-printing manually operated adding machine, which deserves our attention because (although it was not put in serial production), it looks like a well designed and robust device.

The third adding machine of Judah Levin (© National Museum of American History, Washington, D.C.)
The third adding machine of Judah Levin (© National Museum of American History, Washington, D.C.)

The machine is stored in a small black suitcase covered with leather, lined with cloth, and provided with a metal handle on top. It is a steel (for the frame), plastic and paper (for keys), leather and velvet (for the suitcase) made device, with overall measurements: 25 cm x 39.5 cm x 15.4 cm.

The ten digit keys are arranged in two columns (on the patent drawing they are arranged in one column) on the left side of the device. Two rows of nine operating keys across the top indicate the place number of the digit entered. The front row is for addition and the other is for subtraction operations. To enter a number, both the digit key and the place key should be depressed.

By providing two sets of keys, one for determining the digits and the other for determining the value of each digit or its place in the number and also to operate the mechanism, the speed of operation is greatly increased and liability of mistakes lessened, as the keys are operated in the same order in which the person would call or write the number. Thus writing or speaking “4000” the digit “4” is expressed first and the its value or place in the number.

Numbers through 9999999 can be indicated. The metal keys have plastic and paper key tops. The space under the keyboard is covered with green velvet. The result is indicated on a row of red number wheels below these two rows of keys.

Biography of Judah Levin

Judah Leib Levin in 1920
Judah Leib Levin in 1920

Judah (Yehuda) Leib (Leyb) Levin was born Judah Leyb Yoke on 6 April 1862, in Traby, Vilna Province, Russian Empire (now a small town in Belarus—Трабы, Ивьевский район, Гродненская область). Levin’s father was Rabbi Nahum Pinchas, who had a rabbinical degree, but did not accept a pulpit; instead, he conducted business as a landowner in Traby. Judah lost his father when he was eight years old, soon afterwards died his mother, and his uncle rabbi Abraham Abramowitz, an esteemed Talmud scholar, assumed the responsibility of raising him. When Judah was 19, he went to study Talmud in Volozhin and Kovno and received rabbinical ordination.

Shortly after his marriage in 1882 to the daughter of a rabbi from Traby—Esther Rhoda Trauber-Levin (1863-1933), Judah Levin, at age 23, became rabbi at Liškiava, Suwalki Province, now village in Lithuania. The family had four sons: Nathan P., Samuel M. (1888-1975), Isadore (born 1894), and Abraham J. (born 1897).

Levin first immigrated to the eastern United States in 1887, alone, while his family stayed in Russia. In 1888 Levin became the rabbi of Rochester, New York. Two years later he returned to Russia to serve as a rabbi in Kreva (now in Belarus). He left Russia for good in 1892 and found a position in New Haven, Connecticut. The rest of his family joined him a year later. In 1897, Levin accepted an appointment as Chief Rabbi of the United (Orthodox) Jewish Congregations of Detroit, a position he held for the rest of his life. Levin was an ardent supporter of religious Zionism, and he helped support the needs of the Jewish community during a period when it was experiencing tremendous growth. Though not a prolific writer, he published two volumes of sermons and commentary.

Judah Levin died in Detroit on 27 March 1926, at the age of 63, leaving many unpublished manuscripts.

Arthur Postans

The British engineer Arthur James Postans (1867-1940) from South Kensington, London, applied for patent for an adding machine in 1902 in several countries. The British (patent No. 190224868), French (pat. №339441), and German (DE169346) patents were granted in 1904. The next year Postans obtained a US patent also (US patent No. 786839). Later Postans received patents for the same device, or improved versions, in Austria, Canada, Switzerland, Sweden and three more British patens.

The Adder of Postans (Courtesy of Mr. John Wolf)
The Adder of Arthur Postans (Courtesy of Mr. John Wolf)

The patent describes an adding-machine in which the depression of keys having on them the index-numbers moves a series of drums in such a way that the number formed by the position of numerals on their peripheries and appearing through an opening in the casing is increased to the extent of the number appearing on the key that has been depressed.

The machine (see the nearby photo) was manufactured by the Adder Cash Register Syndicate Limited of London, England, since 1908 till 1920s and was sold under the name Adder. The dimensions of Adder are 120 x 180 x 115 mm, weight is 1.71 kg.

The Adder was a single-column (suitable for adding columns of numbers) single-axis adding machine with three results dials and a ten-key setting mechanism. The three dials are mounted side by side behind the window at the top of the machine. The setting keys are numbered from 1 to 10 and operate on the right-hand (units) dial only. An automatic tens-carry mechanism advances the tens and hundreds dials as required. The clearing lever at the left-hand side returns the three dials to zero.

Biography of Arthur Postans

Arthur James Postans was born in 1867 in Samford, Suffolk. He was the son of a maltster and farmer—Henry Draper Postans (1837-1922), from Stoke by Nayland, Suffolk, and Julia Norman (1841-1921), from Colchester, Essex. Henry and Julia married in 1863 in Colchester, and had three children: besides Arthur James, they had another son—John Musgrove (1869-1958), and a daughter—Julia May (1865-1953).

In 1890 Arthur James Postans married in Bloomsbury, London, to Harriet Mary Cocke (Cooke) (1859-1943). They had two sons: Arthur Norman Fleetwood (1891-1952) and Roy William Clivion Lovegrove (1892-1970), and two daughters: Margery Elizabeth Ellison (1895-1979) and Mary Romara (1895-1963).

Arthur became an experimental engineer and starting from middle 1890s he was granted numerous British and foreign patents not only for adding machines, but also for other devices like: driving-gear for velocipedes, electrical ignition apparatus for internal-combustion engines, safety razor, zeroing mechanism, stropping-machine for razors, indicator for cash-registers, printing mechanism for cash-registers or like machines.

Arthur’s brother, John Musgrove, was educated at an Ipswich boarding school before following his father and grandfather’s footsteps as a farmer. He won a gold medal in shooting at 1908 Olympic Games in London.

Arthur James Postans died in 1940 in Alcester, Warwickshire.

Abraham Gancher

The Golden Gem adding machine enjoyed a long sales success through the first half of the 20th century. It was based on the Arithmachine of Heinrich Goldmann (a.k.a as Henry Goldman) from late 1890s, and on the efforts of three inventors—Abraham Gancher (a Russian Jew and emigrant to USA), Nobyoshi Hakrew Kodama (a Japanese emigrant to USA), and Albert Zabriskie.

The Golden Gem adding machine
The Golden Gem adding machine

The Golden Gem was introduced about 1907 by the Automatic Adding Machine Co. of New York, and was based not so on the first similar patent in the USA, taken by Nobyoshi Kodama (patent N. 753586, assigned one-half to Rebecca, the wife of Abraham Gancher), but on the second patent (US pat. No. 816342, taken by Kodama and Gancher). The company’s advertising in 1917 claims over 100000 had been sold by that year. At that point they cost $10 each.

The overall size of the device is 6.9 cm x 13 cm x 10 cm, and it was quite heavy, some 750 g. To operate the Golden Gem, the stylus is inserted into a link corresponding to the desired number and pulled down. As the continuous chain revolves, it advances a number wheel whose value is seen in the window at top. When a wheel revolves from 9 to 0, a tens carry mechanism automatically advances the next wheel by one. (This works well, but advancing the tens carry on multiple digits at once (e.g. from 999 to 1000) requires some extra hand strength!).

Subtraction is possible (via the nines complement method) using the red numbers shown to the right of each chain. Clearing of the result register is achieved by turning the knob at bottom right until all digits show zero.

Automatic Adding Machine Co. also produced a tally counter. It was basically the same as the Golden Gem, except it had five digits, had shorter chains, and instead of the slots in the front there was a lever on the right that incremented the units digit. Gancher also designed a version with a printing mechanism, but this was even less successful than the counter.

Biography of Abraham Gancher

It seems the main driving force behind Golden Gem and Automatic Adding Machine Co. of New York was Abraham Gancher, so let’s see what is known about him.

Abraham Isaac Gancher was a Russian Jew, born on 13 July 1875 somewhere in the Empire. He emigrated to USA in 1892 (most probably due to “pogroms” (anti-Jewish riots) that swept the southern and western provinces of the Russian Empire in 1880s), alongside his parents, Isaac (Chaim Yitzchak) Gancher (1846-1934) and Sarah (nee Berezonsky) Gancher (1850-1919), his brother Jacob (Yaakov) (1882-1958, who became a physician and surgeon), and two sisters: Lizzie Gancher-Bergman (1871-1960) and Fannie Gancher-Husinsky (1881-1966). The family initially settled at Hartford, Connecticut, but soon removed to Waterbury, Conn., where Abraham used to work as a leather salesman in late 1890s.

In 1899 Abraham Gancher married to Rebecca (b. 1876 in New York), and they had a son—Simon. Abraham Gancher became interested in adding machines a few years later and worked in this area more then ten years (he was active in the Automatic Adding Machine Company through at least 1918.) Besides the several patents for adding machines, he got also a patent for appliance for educational, amusement, and advertising purposes (US1075248). Gancher was also a small-handwriting specialist and had, apparently, procured himself a place in Ripley’s Believe it or Not by writing the Bill of Rights on a postage stamp! In 1920s Abraham worked in the family business (industrial surplus) on Broadway, but eventually lost everything in a gamble to buy a seat on the NY Exchange.

Abraham Gancher died on 1 September 1965, in New York.