Difference engine

A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions.

These machines were created to solve a major problem in science and engineering. During the 19th century, researchers needed large mathematical tables for navigation and science. These tables often relied on logarithmic and trigonometric functions. Because these complex functions can be approximated by polynomials, a difference engine could compute them efficiently. The name of the machine comes from the method of finite differences. This is a mathematical way to find new values in a pattern using a small set of polynomial coefficients.

The mechanism of the engine relies on a series of vertical columns.

Each column in the machine is capable of storing one decimal number. To begin a calculation, the user must program the engine by setting initial values in these columns. Column 1 is set to the starting value of the polynomial. Column 2 is set using a value derived from the polynomial's derivatives. The machine then performs addition in a specific sequence. It adds the value of column $n + 1$ to column $n$ to produce a new value for column $n$. This process repeats across the columns. The user powers the entire mechanical process by turning a hand crank.

Different versions of these engines varied significantly in their complexity and capacity. Charles Babbage's original plan for Difference Engine No. 1 was massive. It was designed to use 20-digit numbers and operate by sixth-order differences. This specific design would have required about 25,000 parts and weighed 4 tons. Later, Babbage designed an improved version called Difference Engine No. 2. This improved design could handle 31-digit numbers and seventh-order differences. Other inventors created different types of machines as well. For example, the Scheutzian engine was capable of printing its results.

Martin Wiberg also improved upon these designs to produce logarithmic and interest tables.

The history of the difference engine began with early mechanical ideas. The concept of mechanical calculation traces back to the Antikythera mechanism from the 2nd century BC. In the 17th century, thinkers like Pascal and Leibniz created early modern examples. In 1784, an engineer named J. H. Müller devised an adding machine. He described the principles of a difference machine in a book published in 1786. However, Müller could not find the funding to finish his work. Charles Babbage took up the mantle in the 1820s. He announced his invention to the Royal Astronomical Society on June 14, 1822. This announcement focused on applying machinery to astronomical and mathematical tables.

Babbage's work was marked by both great ambition and significant financial struggle. The British government initially supported him because manual table production was expensive. In 1823, they provided Babbage with £1,700 to start the project. However, the metalworking technology of the 1800s could not produce parts with enough precision. This made the project much more expensive than the government expected. By the time the government abandoned the project in 1842, Babbage had spent over £17,000. Despite this, he had not achieved a fully working large-scale engine. The government focused on the cost of development rather than the potential value of the machine's output.

Notable examples of these machines show how the technology evolved over time. Per Georg Scheutz, a Swedish inventor, was inspired by Babbage in 1834. He built a model that could calculate series using 5-digit numbers. By 1853, he finished a larger machine that used 15-digit numbers. This machine was the first printing calculator ever sold when it went to New York in 1857.

Other machines were built much later, such as the Burroughs Class 11 in 1929. In the 20th century, machines like the one built by Christel Hamann in 1909 were used to publish complex mathematical tables. These machines proved that the concept of automated calculation was vital for large-scale data.

Today, we can study these machines through modern, working replicas.

In the 1980s, researchers at the University of Sydney used Babbage's original drawings to build a working section of Difference Engine No. 2. The Science Museum in London also completed a version that weighs about 5 tons and contains 8,000 parts. These replicas even include Babbage's designed printer. The printer was meant to create stereotype plates for mass printing. Babbage wanted to prevent the errors that occurred during manual typesetting. These modern machines connect us to the history of computing and show how mechanical logic paved the way for digital technology.