The Difference Engine (Novel)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/The_Difference_Engine

 

The Difference Engine

Cover of first edition (hardcover)
Author	William Gibson and Bruce Sterling
Country	United States
Language	English
Genre	Alternate history, steampunk
Publisher	Victor Gollancz Ltd
Publication date	September 1990
Media type	Print (hardback and paperback)
Pages	383 pp (Paperback – 429 pages)
ISBN	0-575-04762-3
OCLC	21299781

The Difference Engine (1990) is an alternative history novel by William Gibson and Bruce Sterling. It is widely regarded as a book that helped establish the genre conventions of steampunk.

It posits a Victorian era Britain in which great technological and social change has occurred after entrepreneurial inventor Charles Babbage succeeded in his ambition to build a mechanical computer (actually his Analytical Engine rather than the difference engine).

The novel was nominated for the British Science Fiction Award in 1990, the Nebula Award for Best Novel in 1991, and both the John W. Campbell Memorial Award and the Prix Aurora Award in 1992.

Setting

The novel is chiefly set in 1855. The historical background diverges from our timeline around 1824, at which point Charles Babbage completes his Difference Engine and proceeds to develop an Analytical Engine. He becomes politically powerful and at the 1830 general election successfully opposes the Tory Government of the Duke of Wellington. Although Wellington stages a coup d'état in 1830 in an attempt to overturn his defeat and prevent the acceleration of technological change and social upheaval, he is assassinated in 1831. The Industrial Radical Party, led by a Lord Byron who survives the Greek War of Independence, comes to power. The Tory Party and hereditary peerage are eclipsed, and British trade unions assist in the ascendancy of the Industrial Radical Party (much as they aided the Labour Party of Great Britain in the twentieth century in our own world). As a result, Luddite anti-technological working class revolutionaries are ruthlessly suppressed.

By 1855, the Babbage computers have become mass-produced and ubiquitous, and their use emulates the innovations that actually occurred during our information technology and Internet revolutions. Other steam-powered technologies have also developed and so, for example, Gurney steam carriages increasingly common. The novel explores the social consequences of an information technology revolution in the nineteenth century, such as the emergence of "clackers" (a reference to hackers), technologically-proficient people, such as Théophile Gautier, who are skilled at programming the Engines through the use of punched cards.

In the novel, the British Empire is more powerful than in our reality because of the development and the use of extremely-advanced steam-driven technology in industry. In addition, similar military technology has enhanced the capabilities of the armed forces (airships, dreadnoughts, and artillery) and the Babbage computers themselves. Under the Industrial Radical Party, Britain shows the utmost respect for leading scientific and industrial figures such as Isambard Kingdom Brunel and Charles Darwin. Indeed, they are collectively called "savants" and often raised to the peerage on their merits, causing a break with the past as regards social prestige and class distinction. The new patterns are also reflected in the educational sphere: classical studies have lost importance to more practical concerns such as engineering and accountancy.

Britain, rather than the United States, opened Japan to Western trade, in part because the United States became fragmented by interference from a Britain which foresaw the implications of a unified United States on the world stage. Counterpart successor states to our world's United States include a (truncated) United States; the Confederate States of America; the Republic of Texas; the Republic of California; a Communist Manhattan Island commune (with Karl Marx as a leading light); British North America (analogous to Canada, albeit slightly larger in this world); Russian America (Alaska); and terra nullius. Napoleon III's French Empire holds an entente with the British, and Napoleon is even married to a British woman. In the world of The Difference Engine, it occupies Mexico, as it did briefly in reality during the American Civil War. Like Great Britain, it has its own analytical/difference engines (ordinateurs), especially used in the context of domestic surveillance within its police force and intelligence agencies. As for the other world powers, Germany remains fragmented, with no suggestion that Prussia will eventually form the core of a unified nation as it did in our own world in 1871, which may be caused by French sabotage analogous to that pursued in the case of the fragmentation of the United States noted above. Japan is awakening after the British ended its isolation, and looks, as in our world, set to become one of this world's leading industrial and economic powers from the twentieth century onward. The intervention of Lords Byron and Babbage provide famine relief with grain confiscated from the landed aristocracy, the Irish potato famine never occurred, there is no agitation for Irish home rule or Irish independence, and the Irish instead have become enthusiastic supporters of the Radical regime.

Among other historical characters, the novel features "Texian" President Sam Houston, as an exile after a political coup in Texas, a reference to Percy Bysshe Shelley (as a Luddite), John Keats as a kinotropist (an operator of mechanical pixelated screens), and Benjamin Disraeli as a publicist and tabloid writer.

Plot summary

The action of the story follows Sybil Gerard, a political courtesan and daughter of an executed Luddite leader; Edward "Leviathan" Mallory, a paleontologist and explorer; and Laurence Oliphant, a historical figure who, as is portrayed in the book, was a travel writer whose work was a cover for espionage activities "undertaken in the service of Her Majesty". Linking all their stories is the trail of a mysterious set of reportedly very powerful computer punch cards and the individuals fighting to obtain them. 

Many characters come to believe that the punch cards are a gambling "modus", a programme that would allow the user to place consistently winning bets. The last chapter reveals that the punched cards represent a program that proves two theorems, which, in reality, would not be discovered until 1931 by Kurt Gödel. Ada Lovelace delivers a lecture on the subject in France.

Defending the cards, Mallory gathers his brothers and Ebenezer Fraser, a secret police officer, to fight the revolutionary Captain Swing, who leads a London riot during "the Stink", a major episode of pollution in which London swelters under an inversion layer (comparable to the London Smog of December 1952).

After the abortive uprising, Oliphant and Sybil Gerard meet at a cafe in Paris. Oliphant informs her that he is aware of her true identity but will not pursue it. However, he wants information that would compromise her seducer, Charles Egremont MP, now regarded as an obstacle to the strategies and political ambitions of Lords Brunel and Babbage. Sybil has longed for an opportunity for vengeance against Egremont, and the resultant political scandal destroys his parliamentary career and aspirations for a merit lordship. Oliphant also encounters a Manhattan-based group of feminist pantomime artists.

After several vignettes that elaborate on the alternate historical origins of the world of The Difference Engine, Lovelace delivers her lecture on Gödel's Theorem, as its counterpart is known in our world. She is chaperoned by Fraser and castigated by Sybil Gerard, who is still unable to forgive Ada's father, the late Lord Byron, for his role in her own father's death.

At the very end of the novel is a depiction of an alternate 1991 from the vantage point of a computer, which is revealed to be the narrator as it achieves self-awareness. 

Characters

• The character Michael Godwin was named after attorney Mike Godwin to thank for his technical assistance in linking Sterling and Gibson's computers, which allowed them to collaborate between Austin and Vancouver.
• The characters of Sybil Gerard; her father, Walter Gerard; Charles Egremont; and Dandy Mick are all borrowed from Benjamin Disrael's novel Sybil
  

Literary criticism and significance

The novel has attracted the attention of scholars, including Jay Clayton, who explores the book's attitude toward hacking, as well as its treatment of Babbage and Ada Lovelace; Herbert Sussman, who demonstrates how the book rewrites Benjamin Disraeli's novel Sybil; and Brian McHale, who relates it to the postmodern interest in finding a "new way of 'doing' history in fiction."

The novel was nominated for the British Science Fiction Award in 1990, the Nebula Award for Best Novel in 1991, and both the John W. Campbell Memorial Award and the Prix Aurora Award in 1992.

Influence

The 1993 video game The Chaos Engine (released as Soldiers of Fortune in the USA) was based on The Difference Engine.

The 2003 grand strategy wargame, Victoria: An Empire Under the Sun features an easter egg in which the Manhattan Commune is a release-able nation. The nation also appears in the 2010 sequel Victoria II.

Difference engine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Difference_engine

 

The London Science Museum's difference engine, the first one actually built from Babbage's design. The design has the same precision on all columns, but when calculating polynomials, the precision on the higher-order columns could be lower.

 

A difference engine, first created by Charles Babbage, is an automatic mechanical calculator designed to tabulate polynomial functions. Its name is derived from the method of divided differences, a way to interpolate or tabulate functions by using a small set of polynomial coefficients. Most mathematical functions commonly used by engineers, scientists and navigators, including logarithmic and trigonometric functions, can be approximated by polynomials, so a difference engine can compute many useful tables of numbers. 

The historical difficulty in producing error-free tables by teams of mathematicians and human "computers" spurred Charles Babbage's desire to build a mechanism to automate the process.

History

Closeup of the London Science Museum's difference engine showing some of the number wheels and the sector gears between columns. The sector gears on the left show the double-high teeth very clearly. The sector gears on the middle-right are facing the back side of the engine, but the single-high teeth are clearly visible. Notice how the wheels are mirrored, with counting up from left-to-right, or counting down from left-to-right. Also notice the metal tab between "6" and "7". That tab trips the carry lever in the back when "9" passes to "0" in the front during the add steps (Step 1 and Step 3).

 

Per Georg Scheutz's third differencine

 

The notion of a mechanical calculator for mathematical functions can be traced back to the Antikythera mechanism of the 2nd century BC, while early modern examples are attributed to Pascal and Leibniz in the 17th century. In 1784 J. H. Müller, an engineer in the Hessian army, devised and built an adding machine and described the basic principles of a difference machine in a book published in 1786 (the first written reference to a difference machine is dated to 1784), but he was unable to obtain funding to progress with the idea.

Charles Babbage's difference engines

Charles Babbage began to construct a small difference engine in c. 1819 and had completed it by 1822 (Difference Engine 0). He announced his invention on 14 June 1822, in a paper to the Royal Astronomical Society, entitled "Note on the application of machinery to the computation of astronomical and mathematical tables". This machine used the decimal number system and was powered by cranking a handle. The British government was interested, since producing tables was time-consuming and expensive and they hoped the difference engine would make the task more economical.

In 1823, the British government gave Babbage £1700 to start work on the project. Although Babbage's design was feasible, the metalworking techniques of the era could not economically make parts in the precision and quantity required. Thus the implementation proved to be much more expensive and doubtful of success than the government's initial estimate. In 1832, Babbage and Joseph Clement produced a small working model (one-seventh of the calculating section of Difference Engine No. 1, which was intended to operate on 20-digit numbers and sixth-order differences) which operated on 6-digit numbers and second-order differences. Lady Byron described seeing the working prototype in 1833: "We both went to see the thinking machine (for so it seems) last Monday. It raised several Nos. to the 2nd and 3rd powers, and extracted the root of a Quadratic equation." Work on the larger engine was suspended in 1833. 

By the time the government abandoned the project in 1842, Babbage had received and spent over £17,000 on development, which still fell short of achieving a working engine. The government valued only the machine's output (economically produced tables), not the development (at unknown and unpredictable cost to complete) of the machine itself. Babbage did not, or was unwilling to, recognize that predicament. Meanwhile, Babbage's attention had moved on to developing an analytical engine, further undermining the government's confidence in the eventual success of the difference engine. By improving the concept as an analytical engine, Babbage had made the difference engine concept obsolete, and the project to implement it an utter failure in the view of the government.

The incomplete Difference Engine No. 1 was put on display to the public at the 1862 International Exhibition in South Kensington, London.

Babbage went on to design his much more general analytical engine, but later produced an improved "Difference Engine No. 2" design (31-digit numbers and seventh-order differences), between 1846 and 1849. Babbage was able to take advantage of ideas developed for the analytical engine to make the new difference engine calculate more quickly while using fewer parts.

Scheutzian calculation engine

Inspired by Babbage's difference engine in 1834, Per Georg Scheutz built several experimental models. In 1837 his son Edward proposed to construct a working model in metal, and in 1840 finished the calculating part, capable of calculating series with 5-digit numbers and first-order differences, which was later extended to third-order (1842). In 1843, after adding the printing part, the model was completed.

In 1851, funded by the government, construction of the larger and improved (15-digit numbers and fourth-order differences) machine began, and finished in 1853. The machine was demonstrated at the World's Fair in Paris, 1855 and then sold in 1856 to the Dudley Observatory in Albany, New York. Delivered in 1857, it was the first printing calculator sold. In 1857 the British government ordered the next Scheutz's difference machine, which was built in 1859. It had the same basic construction as the previous one, weighing about 10 cwt (1,100 lb; 510 kg).

Others

Martin Wiberg improved Scheutz's construction (c. 1859, his machine has the same capacity as Scheutz's - 15-digit and fourth-order) but used his device only for producing and publishing printed tables (interest tables in 1860, and logarithmic tables in 1875).

Alfred Deacon of London in c. 1862 produced a small difference engine (20-digit numbers and third-order differences).

American George B. Grant started working on his calculating machine in 1869, unaware of the works of Babbage and Scheutz (Schentz). One year later (1870) he learned about difference engines and proceed to design one himself, describing his construction in 1871. In 1874 the Boston Thursday Club raised a subscription for the construction of a large-scale model, which was built in 1876. It could be expanded to enhance precision and weighed about 2,000 pounds (910 kg).

Christel Hamann built one machine (16-digit numbers and second-order differences) in 1909 for the "Tables of Bauschinger and Peters" ("Logarithmic-Trigonometrical Tables with eight decimal places"), which was first published in Leipzig in 1910. It weighed about 40 kilograms (88 lb).

Burroughs Corporation in about 1912 built a machine for the Nautical Almanac Office which was used as a difference engine of second-order. It was later replaced in 1929 by a Burroughs Class 11 (13-digit numbers and second-order differences, or 11-digit numbers and [at least up to] fifth-order differences).

Alexander John Thompson about 1927 built integrating and differencing machine (13-digit numbers and fifth-order differences) for his table of logarithms "Logarithmetica britannica". This machine was composed of four modified Triumphator calculators.

Leslie Comrie in 1928 described how to use the Brunsviga-Dupla calculating machine as a difference engine of second-order (15-digit numbers). He also noted in 1931 that National Accounting Machine Class 3000 could be used as a difference engine of sixth-order.

Construction of two working No. 2 difference engines

During the 1980s, Allan G. Bromley, an associate professor at the University of Sydney, Australia, studied Babbage's original drawings for the Difference and Analytical Engines at the Science Museum library in London. This work led the Science Museum to construct a working calculating section of difference engine No. 2 from 1985 to 1991, under Doron Swade, the then Curator of Computing. This was to celebrate the 200th anniversary of Babbage's birth in 1999. In 2002, the printer which Babbage originally designed for the difference engine was also completed. The conversion of the original design drawings into drawings suitable for engineering manufacturers' use revealed some minor errors in Babbage's design (possibly introduced as a protection in case the plans were stolen), which had to be corrected. Once completed, both the engine and its printer worked flawlessly, and still do. The difference engine and printer were constructed to tolerances achievable with 19th-century technology, resolving a long-standing debate as to whether Babbage's design would actually have worked. (One of the reasons formerly advanced for the non-completion of Babbage's engines had been that engineering methods were insufficiently developed in the Victorian era.) 

The printer's primary purpose is to produce stereotype plates for use in printing presses, which it does by pressing type into soft plaster to create a flong. Babbage intended that the Engine's results be conveyed directly to mass printing, having recognized that many errors in previous tables were not the result of human calculating mistakes but from error in the manual typesetting process. The printer's paper output is mainly a means of checking the Engine's performance. 

In addition to funding the construction of the output mechanism for the Science Museum's Difference Engine, Nathan Myhrvold commissioned the construction of a second complete Difference Engine No. 2, which was on exhibit at the Computer History Museum in Mountain View, California from 10 May 2008 until 31 January 2016. It has since been transferred to Intellectual Ventures in Seattle where it is on display just outside the main lobby. 

Operation

The difference engine consists of a number of columns, numbered from 1 to N. The machine is able to store one decimal number in each column. The machine can only add the value of a column n + 1 to column n to produce the new value of n. Column N can only store a constant, column 1 displays (and possibly prints) the value of the calculation on the current iteration. 

The engine is programmed by setting initial values to the columns. Column 1 is set to the value of the polynomial at the start of computation. Column 2 is set to a value derived from the first and higher derivatives of the polynomial at the same value of X. Each of the columns from 3 to N is set to a value derived from the ${\displaystyle (n-1)}$ first and higher derivatives of the polynomial.

Timing

In the Babbage design, one iteration (i.e., one full set of addition and carry operations) happens for each rotation of the main shaft. Odd and even columns alternately perform an addition in one cycle. The sequence of operations for column ${\displaystyle n}$ is thus:

1. Count up, receiving the value from column ${\displaystyle n+1}$ (Addition step)
2. Perform carry propagation on the counted up value
3. Count down to zero, adding to column ${\displaystyle n-1}$
4. Reset the counted-down value to its original value

Steps 1,2,3,4 occur for every odd column, while steps 3,4,1,2 occur for every even column.

While Babbage's original design placed the crank directly on the main shaft, it was later realized that the force required to crank the machine would have been too great for a human to handle comfortably. Therefore, the two models that were built incorporate a 4:1 reduction gear at the crank, and four revolutions of the crank are required to perform one full cycle. 

Steps

Each iteration creates a new result, and is accomplished in four steps corresponding to four complete turns of the handle shown at the far right in the picture below. The four steps are:

• Step 1. All even numbered columns (2,4,6,8) are added to all odd numbered columns (1,3,5,7) simultaneously. An interior sweep arm turns each even column to cause whatever number is on each wheel to count down to zero. As a wheel turns to zero, it transfers its value to a sector gear located between the odd/even columns. These values are transferred to the odd column causing them to count up. Any odd column value that passes from "9" to "0" activates a carry lever.
• Step 2. Carry propagation is accomplished by a set of spiral arms in the back that poll the carry levers in a helical manner so that a carry at any level can increment the wheel above by one. That can create a carry, which is why the arms move in a spiral. At the same time, the sector gears are returned to their original position, which causes them to increment the even column wheels back to their original values. The sector gears are double-high on one side so they can be lifted to disengage from the odd column wheels while they still remain in contact with the even column wheels.
• Step 3. This is like Step 1, except it is odd columns (3,5,7) added to even columns (2,4,6), and column one has its values transferred by a sector gear to the print mechanism on the left end of the engine. Any even column value that passes from "9" to "0" activates a carry lever. The column 1 value, the result for the polynomial, is sent to the attached printer mechanism.
• Step 4. This is like Step 2, but for doing carries on even columns, and returning odd columns to their original values.
  

Subtraction

The engine represents negative numbers as ten's complements. Subtraction amounts to addition of a negative number. This works in the same manner that modern computers perform subtraction, known as two's complement. 

Method of differences

Fully operational difference engine at the Computer History Museum in Mountain View, California

 

The principle of a difference engine is Newton's method of divided differences. If the initial value of a polynomial (and of its finite differences) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences. For example, consider the quadratic polynomial

${\displaystyle p(x)=2x^{2}-3x+2\,}$

with the goal of tabulating the values p(0), p(1), p(2), p(3), p(4), and so forth. The table below is constructed as follows: the second column contains the values of the polynomial, the third column contains the differences of the two left neighbors in the second column, and the fourth column contains the differences of the two neighbors in the third column: 

x	p(x) = 2x2 − 3x + 2	diff1(x) = ( p(x + 1) − p(x) )	diff2(x) = ( diff1(x + 1) − diff1(x) )
0	2	−1	4
1	1	3	4
2	4	7	4
3	11	11
4	22

The numbers in the third values-column are constant. In fact, by starting with any polynomial of degree n, the column number n + 1 will always be constant. This is the crucial fact behind the success of the method.

This table was built from left to right, but it is possible to continue building it from right to left down a diagonal in order to compute more values. To calculate p(5) use the values from the lowest diagonal. Start with the fourth column constant value of 4 and copy it down the column. Then continue the third column by adding 4 to 11 to get 15. Next continue the second column by taking its previous value, 22 and adding the 15 from the third column. Thus p(5) is 22 + 15 = 37. In order to compute p(6), we iterate the same algorithm on the p(5) values: take 4 from the fourth column, add that to the third column's value 15 to get 19, then add that to the second column's value 37 to get 56, which is p(6). This process may be continued ad infinitum. The values of the polynomial are produced without ever having to multiply. A difference engine only needs to be able to add. From one loop to the next, it needs to store 2 numbers—in this example (the last elements in the first and second columns). To tabulate polynomials of degree n, one needs sufficient storage to hold n numbers.

Babbage's difference engine No. 2, finally built in 1991, could hold 8 numbers of 31 decimal digits each and could thus tabulate 7th degree polynomials to that precision. The best machines from Scheutz could store 4 numbers with 15 digits each.

Initial values

The initial values of columns can be calculated by first manually calculating N consecutive values of the function and by backtracking, i.e. calculating the required differences.

Col ${\displaystyle 1_{0}}$ gets the value of the function at the start of computation ${\displaystyle f(0)}$. Col ${\displaystyle 2_{0}}$ is the difference between ${\displaystyle f(1)}$ and ${\displaystyle f(0)}$...

If the function to be calculated is a polynomial function, expressed as

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,}$

the initial values can be calculated directly from the constant coefficients a₀, a₁,a₂, ..., a_n without calculating any data points. The initial values are thus:

• Col ${\displaystyle 1_{0}}$ = a₀
• Col ${\displaystyle 2_{0}}$ = a₁ + a₂ + a₃ + a₄ + ... + a_n
• Col ${\displaystyle 3_{0}}$ = 2a₂ + 6a₃ + 14a₄ + 30a₅ + ...
• Col ${\displaystyle 4_{0}}$ = 6a₃ + 36a₄ + 150a₅ + ...
• Col ${\displaystyle 5_{0}}$ = 24a₄ + 240a₅ + ...
• Col ${\displaystyle 6_{0}}$ = 120a₅ + ...
• ${\displaystyle ...}$
  

Use of derivatives

Many commonly used functions are analytic functions, which can be expressed as power series, for example as a Taylor series. The initial values can be calculated to any degree of accuracy; if done correctly the engine will give exact results for first N steps. After that, the engine will only give an approximation of the function.

The Taylor series expresses the function as a sum obtained from its derivatives at one point. For many functions the higher derivatives are trivial to obtain; for instance, the sine function at 0 has values of 0 or ${\displaystyle \pm 1}$ for all derivatives. Setting 0 as the start of computation we get the simplified Maclaurin series

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\ x^{n}}$

The same method of calculating the initial values from the coefficients can be used as for polynomial functions. The polynomial constant coefficients will now have the value

${\displaystyle a_{n}\equiv {\frac {f^{(n)}(0)}{n!}}}$

Curve fitting

The problem with the methods described above is that errors will accumulate and the series will tend to diverge from the true function. A solution which guarantees a constant maximum error is to use curve fitting. A minimum of N values are calculated evenly spaced along the range of the desired calculations. Using a curve fitting technique like Gaussian reduction an N−1th degree polynomial interpolation of the function is found. With the optimized polynomial, the initial values can be calculated as above. 

The difference engine in other works

William Gibson and Bruce Sterling's The Difference Engine is an alternate history novel that looks how society would have progressed had the difference engine worked the way Babbage envisioned it.

The story takes place in Victorian England where technological advancement is on the rise. This is due to the effect of the success of Babbage's analytical machine. The convention of steampunk where Victorian fashion is combined with the technological elements of the Industrial Revolution is seen throughout the story due to technology being so advanced in that era.

Analytical Engine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Analytical_Engine

 

Trial model of a part of the Analytical Engine, built by Babbage, as displayed at the Science Museum (London)

 

The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage.^[ It was first described in 1837 as the successor to Babbage's difference engine, a design for a simpler mechanical computer.

The Analytical Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory, making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete. In other words, the logical structure of the Analytical Engine was essentially the same as that which has dominated computer design in the electronic era. The Analytical Engine is one of the most successful achievements of Charles Babbage. 

Babbage was never able to complete construction of any of his machines due to conflicts with his chief engineer and inadequate funding. It was not until 1941 that the first general-purpose computer, Z3, was actually built, more than a century after Babbage had proposed the pioneering Analytical Engine in 1837.

Design

Two types of punched cards used to program the machine. Foreground: 'operational cards', for inputting instructions; background: 'variable cards', for inputting data

 

Babbage's first attempt at a mechanical computing device, the Difference Engine, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials. Construction of this machine was never completed; Babbage had conflicts with his chief engineer, Joseph Clement, and ultimately the British government withdrew its funding for the project.

During this project, he realised that a much more general design, the Analytical Engine, was possible. The work on the design of the Analytical Engine started in c. 1833.

The input, consisting of programs ("formulae") and data was to be provided to the machine via punched cards, a method being used at the time to direct mechanical looms such as the Jacquard loom. For output, the machine would have a printer, a curve plotter and a bell. The machine would also be able to punch numbers onto cards to be read in later. It employed ordinary base-10 fixed-point arithmetic.

There was to be a store (that is, a memory) capable of holding 1,000 numbers of 40 decimal digits each (ca. 16.2 kB). An arithmetic unit (the "mill") would be able to perform all four arithmetic operations, plus comparisons and optionally square roots. Initially (1838) it was conceived as a difference engine curved back upon itself, in a generally circular layout, with the long store exiting off to one side. Later drawings (1858) depict a regularised grid layout. Like the central processing unit (CPU) in a modern computer, the mill would rely upon its own internal procedures, to be stored in the form of pegs inserted into rotating drums called "barrels", to carry out some of the more complex instructions the user's program might specify.

The programming language to be employed by users was akin to modern day assembly languages. Loops and conditional branching were possible, and so the language as conceived would have been Turing-complete as later defined by Alan Turing. Three different types of punch cards were used: one for arithmetical operations, one for numerical constants, and one for load and store operations, transferring numbers from the store to the arithmetical unit or back. There were three separate readers for the three types of cards. Babbage developed some two dozen programs for the Analytical Engine between 1837 and 1840, and one program later. These programs treat polynomials, iterative formulas, Gaussian elimination, and Bernoulli numbers.

In 1842, the Italian mathematician Luigi Federico Menabrea published a description of the engine based on a lecture by Babbage in French. In 1843, the description was translated into English and extensively annotated by Ada Lovelace, who had become interested in the engine eight years earlier. In recognition of her additions to Menabrea's paper, which included a way to calculate Bernoulli numbers using the machine (widely considered to be the first complete computer program), she has been described as the first computer programmer.

Construction

Henry Babbage's Analytical Engine Mill, built in 1910, in the Science Museum (London)

 

Late in his life, Babbage sought ways to build a simplified version of the machine, and assembled a small part of it before his death in 1871.

In 1878, a committee of the British Association for the Advancement of Science described the Analytical Engine as "a marvel of mechanical ingenuity", but recommended against constructing it. The committee acknowledged the usefulness and value of the machine, but could not estimate the cost of building it, and were unsure whether the machine would function correctly after being built.

Intermittently from 1880 to 1910, Babbage's son Henry Prevost Babbage was constructing a part of the mill and the printing apparatus. In 1910 it was able to calculate a (faulty) list of multiples of pi. This constituted only a small part of the whole engine; it was not programmable and had no storage. (Popular images of this section have sometimes been mislabelled, implying that it was the entire mill or even the entire engine.) Henry Babbage's "Analytical Engine Mill" is on display at the Science Museum in London. Henry also proposed building a demonstration version of the full engine, with a smaller storage capacity: "perhaps for a first machine ten (columns) would do, with fifteen wheels in each". Such a version could manipulate 20 numbers of 25 digits each, and what it could be told to do with those numbers could still be impressive. "It is only a question of cards and time", wrote Henry Babbage in 1888, "... and there is no reason why (twenty thousand) cards should not be used if necessary, in an Analytical Engine for the purposes of the mathematician".

In 1991, the London Science Museum built a complete and working specimen of Babbage's Difference Engine No. 2, a design that incorporated refinements Babbage discovered during the development of the Analytical Engine. This machine was built using materials and engineering tolerances that would have been available to Babbage, quelling the suggestion that Babbage's designs could not have been produced using the manufacturing technology of his time.

In October 2010, John Graham-Cumming started a "Plan 28" campaign to raise funds by "public subscription" to enable serious historical and academic study of Babbage's plans, with a view to then build and test a fully working virtual design which will then in turn enable construction of the physical Analytical Engine. As of May 2016, actual construction had not been attempted, since no consistent understanding could yet be obtained from Babbage's original design drawings. In particular it was unclear whether it could handle the indexed variables which were required for Lovelace's Bernoulli program. By 2017, the "Plan 28" effort reported that a searchable database of all catalogued material was available, and an initial review of Babbage's voluminous Scribbling Books had been completed.

Instruction set

Plan diagram of the Analytical Engine from 1840

 

Babbage is not known to have written down an explicit set of instructions for the engine in the manner of a modern processor manual. Instead he showed his programs as lists of states during their execution, showing what operator was run at each step with little indication of how the control flow would be guided. 

Allan G. Bromley has assumed that the card deck could be read in forwards and backwards directions as a function of conditional branching after testing for conditions, which would make the engine Turing-complete:

...the cards could be ordered to move forward and reverse (and hence to loop)...

The introduction for the first time, in 1845, of user operations for a variety of service functions including, most importantly, an effective system for user control of looping in user programs. There is no indication how the direction of turning of the operation and variable cards is specified. In the absence of other evidence I have had to adopt the minimal default assumption that both the operation and variable cards can only be turned backward as is necessary to implement the loops used in Babbage's sample programs. There would be no mechanical or microprogramming difficulty in placing the direction of motion under the control of the user.

In their emulator of the engine, Fourmilab say:

The Engine's Card Reader is not constrained to simply process the cards in a chain one after another from start to finish. It can, in addition, directed by the very cards it reads and advised by the whether the Mill's run-up lever is activated, either advance the card chain forward, skipping the intervening cards, or backward, causing previously-read cards to be processed once again.

This emulator does provide a written symbolic instruction set, though this has been constructed by its authors rather than based on Babbage's original works. For example, a factorial program would be written as: 

N0 6
N1 1
N2 1
×
L1
L0
S1
–
L0
L2
S0
L2
L0
CB?11

where the CB is the conditional branch instruction or "combination card" used to make the control flow jump, in this case backwards by 11 cards.

Influence

Predicted influence

Babbage understood that the existence of an automatic computer would kindle interest in the field now known as algorithmic efficiency, writing in his Passages from the Life of a Philosopher, "As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise—By what course of calculation can these results be arrived at by the machine in the shortest time?"

Computer science

From 1872 Henry continued diligently with his father's work and then intermittently in retirement in 1875.

Percy Ludgate wrote about the engine in 1914 and published his own design for an Analytical Engine in 1908. It was drawn up in detail, but never built, and the drawings have never been found. Ludgate's engine would be much smaller (about 8 cubic feet (230 L)) than Babbage's, and hypothetically would be capable of multiplying two 20-decimal-digit numbers in about six seconds.

Torres y Quevedo wrote about Babbage's engines in Essays on Automatics (1913). Book contains design for an electromechanical machine capable of calculating completely automatically the value of a function.

Vannevar Bush's paper Instrumental Analysis (1936) included several references to Babbage's work. In the same year started Rapid Arithmetical Machine project to investigate the problems of constructing an electronic digital computer.

Despite this groundwork, Babbage's work fell into historical obscurity, and the Analytical Engine was unknown to builders of electro-mechanical and electronic computing machines in the 1930s and 1940s when they began their work, resulting in the need to re-invent many of the architectural innovations Babbage had proposed. Howard Aiken, who built the quickly-obsoleted electromechanical calculator, the Harvard Mark I, between 1937 and 1945, praised Babbage's work likely as a way of enhancing his own stature, but knew nothing of the Analytical Engine's architecture during the construction of the Mark I, and considered his visit to the constructed portion of the Analytical Engine "the greatest disappointment of my life". The Mark I showed no influence from the Analytical Engine and lacked the Analytical Engine's most prescient architectural feature, conditional branching. J. Presper Eckert and John W. Mauchly similarly were not aware of the details of Babbage's Analytical Engine work prior to the completion of their design for the first electronic general-purpose computer, the ENIAC.

Comparison to other early computers

If the Analytical Engine had been built, it would have been digital, programmable and Turing-complete. It would, however, have been very slow. Luigi Federico Menabrea reported in Sketch of the Analytical Engine: "Mr. Babbage believes he can, by his engine, form the product of two numbers, each containing twenty figures, in three minutes". By comparison the Harvard Mark I could perform the same task in just six seconds. A modern PC can do the same thing in well under a billionth of a second.

Name	First operational	Numeral system	Computing mechanism	Programming	Turing complete	Memory
Difference Engine	Not built until the 1990s	Decimal	Mechanical	Not programmable; initial numerical constants of polynomial differences set physically	No	Physical state of wheels in axes
Analytical Engine	Not yet built	Decimal	Mechanical	Program-controlled by punched cards	Yes	Physical state of wheels in axes
Bombe (Poland, UK, US)	1939 (Polish), March 1940 (British), May 1943 (US)	Character computations	Electro-mechanical	Not programmable; cipher input settings specified by patch cables	No	Physical state of rotors
Zuse Z3 (Germany)	May 1941	Binary floating point	Electro-mechanical	Program-controlled by punched 35 mm film stock	In principle	Mechanical relays
Atanasoff–Berry Computer (US)	1942	Binary	Electronic	Not programmable; linear system coefficients input using punched cards	No	Regenerative capacitor memory
Colossus Mark 1 (UK)	December 1943	Binary	Electronic	Program-controlled by patch cables and switches	No	Thermionic valves (vacuum tubes) and thyratrons
Harvard Mark I – IBM ASCC (US)	May 1944	Decimal	Electro-mechanical	Program-controlled by 24-channel punched paper tape (but no conditional branch)	No	Mechanical relays[44]
Zuse Z4 (Germany)	March 1945 (or 1948)	Binary floating point	Electro-mechanical	Program-controlled by punched 35 mm film stock	Yes	Mechanical relays
ENIAC (US)	July 1946	Decimal	Electronic	Program-controlled by patch cables and switches	Yes	Vacuum tube triode flip-flops
Manchester Baby (UK)	1948	Binary	Electronic	Binary program entered into memory by keyboard (first electronic stored-program digital computer)	Yes	Williams cathode ray tube

In popular culture

• The cyberpunk novelists William Gibson and Bruce Sterling co-authored a steampunk novel of alternative history titled The Difference Engine in which Babbage's Difference and Analytical Engines became available to Victorian society. The novel explores the consequences and implications of the early introduction of computational technology.
• There is also mention of the Analytical Engine (or the Clockwork Ouroboros as it is also known there) in The Book of the War, a Faction Paradox anthology edited by Lawrence Miles. This machine was used to calculate a way into the "Eleven Day Empire". Its use resulted in the destruction of the original Houses of Parliament.
• In the novel Perdido Street Station, by British author China Miéville, engines similar to Babbage's serve as "brains" for the robotic constructs of the city of New Crobuzon. One such engine even develops sentient thought due to a recursive algorithmic loop.
• The British Empire of The Peshawar Lancers by S. M. Stirling features a massive water-powered analytical engine at Oxford, used by two of the main characters. It is noted that most of the engines run on steam, and that an even larger one is under construction at the British Capital in Delhi.
• In the Michael Flynn novel In the Country of the Blind, a secret society calling itself the Babbage Society secretly financed the building of Babbage Engines in the mid-19th century. In the novel, the Society uses the Babbage engines along with a statistical science called Cliology to predict and manipulate future history. In the process, they predict the rise of the Nazis and accidentally start the US Civil War.
• In the Neal Stephenson novel The Diamond Age, ubiquitous molecular nanotechnology is described to make use of "rod logic" similar to that imagined by Babbage's design for the Analytical Engine.
• Moriarty by Modem, a short story by Jack Nimersheim, describes an alternate history where Babbage's Analytical Engine was indeed completed and had been deemed highly classified by the British government. The characters of Sherlock Holmes and Moriarty had in reality been a set of prototype programs written for the Analytical Engine. This short story follows Holmes as his program is implemented on modern computers and he is forced to compete against his nemesis yet again in the modern counterparts of Babbage's Analytical Engine.
• A similar setting is used by Sydney Padua in the webcomic The Thrilling Adventures of Lovelace and Babbage. It features an alternate history where Ada Lovelace and Babbage have built the Analytical Engine and use it to fight crime at Queen Victoria's request. The comic is based on thorough research on the biographies of and correspondence between Babbage and Lovelace, which is then twisted for humorous effect.
• Georgia on My Mind is a novelette by Charles Sheffield which involves two major themes: being widowed and the quest for a legendary Babbage computer.
• Hugh Cook's fantasy novels The Wishstone and the Wonderworkers and The Wazir and the Witch feature an Analytical Engine created by the scientist Ivan Petrov. It is used to calculate income tax.
• The Orion's Arm online project features the Machina Babbagenseii, fully sentient Babbage inspired mechanical computers. Each is the size of a large asteroid, only capable surviving in microgravity conditions, and processes data at 0.5% the speed of a human brain.
• The flying ships in the anime Last Exile are seen to have analytical engines inside of them. Although some have more advanced technology, the common ships use analytical engines, and even some of the advanced ships are seen to have clockwork mechanisms as well.
• A working version of the Analytical Engine, created by fictional inventor Ernest Harding (and based on the Babbage concept) was featured on the Murdoch Mysteries (also called The Artful Detective), in Season 5, Episode 9, Invention Convention.