Early Calculating and Computing Machines: From the Abacus to Babbage

The Abacus

There is a long history detailing the invention of computing and calculating machines. The earliest recorded calculating device is the abacus. Used as a simple computing device for performing arithmetic, the abacus most likely appeared first in Babylonia (now Iraq) over 5000 years ago. Its more familiar form today is derived from the Chinese version pictured below.

The abacus is more of a counting device than a true calculator. (See Figure 1.) Nonetheless, it was used for centuries as a reliable means for doing additions and subtractions.


Most of the details about the life of the eminent mathematician Abu Ja’far Muhammad ibn Musa al-Khwarizmi are not known. (See Figure 2.) We do know that he was born around 780 in Baghdad and died about 850. Al-Khwarizimi was one of the more famous scholars at the House of Wisdom in Baghdad, the first major library since the Alexandria.

In one of his most famous works, al-Khwarizimi offered geometrical demonstrations or numerical proofs that were heretofore unknown to Europeans. The book contained the word "al-jabr" in the title, which meant "transposition." Subsequently, Europeans dubbed this new way of thinking about arithmetic "algebra."

For our purposes, though, it is his Latin name--"Algoritmi"--that is most significant for the history of computing. It became synonymous for a new style of reasoning called an "algorithm." This denotes an intelligible step-by-step process devised to solve some mathematical problem. Thus, the connection between the concepts of calculation and mechanism is indelibly forged.

Raymon Lull

Not all early computing devices were dedicated to calculating numbers. Raymon Lull (1230-1315), a Spanish courtier and later converted monk and apologist, is the first person in history known to have devised a "logic" machine—a machine that computes logical proofs rather than doing arithmetic. Logic, of course, is the science of reasoning. It is primarily concerned with the form of reasoning called inference, i.e., deriving new information from previously known information. Logic seeks to articulate the principles that distinguish warranted from unwarranted inferences. Moreover, it is a formal study that postulates that inferences can be measured using abstract methods that consider properties of the inference distinct of its contents.

The Greek philosopher Aristotle (384-323 B.C.) was the first to recognize explicitly this principle of formalism—that information may be captured faithfully and subsequently explored using methods that depend entirely on the system of symbols themselves. In the Prior Analytics, Aristotle advanced the system of the syllogistic, which is the first recorded attempt to represent the properties of reasoning by way of purely formal methods. The syllogistic was intended to show that we derive or deduce new information from what is already known. employing standard valid forms of inference. According to this view, all human reasoning or logic is a kind of computation.

Aristotle's syllogistic and logic were studied extensively by scholars in Greek, Arabic, and later Western cultures.

Lull (in Latin form, Raimundus Lullus) was likewise steeped in this Aristotelian tradition. He devised a machine that was composed of a series of concentric circles, each circle contained symbols representing various concepts about some subject. (See Figure 3.) The circles could be rotated to align or compute various combinations. Each combination consequently represented a statement about that subject. The basic idea was to generate mechanically all of the possible thoughts or ideas that could be expressed about some given subject. With constructive rules on how the wheels may be rotated, Lull hoped to show how true statements could be derived from the set of all possible statements.

Apart from its eccentricities, Lull's machine is founded on two significant ideas or beliefs. First, language and concepts may be represented sufficiently using physical symbols. Secondly, truths can be generated or computed using mechanical methods. These ideas influenced a number of individuals succeeding him.

John Napier and Napier's Bones

Next, we move ahead several centuries and to Scotland. John Napier was born in 1550 near Edinburgh. Though most of the details of his education are unknown, he apparently attended St. Andrews and Cambridge. Napier's fame as a mathematician was secured with his discovery of logarithms. Tables of logarithms made it easier for astronomers, bankers, and others to reduce the more complex operations of multiplication and division to simpler additions and subtractions. We will return to consider the use of logarithms shortly.

During his lifetime, though, Napier was more widely recognized as the inventor of a calculating tool known as "Napier's Bones." These were a series of rods (often carved from bones) that had squares inscribed in them. Using the rods, one could perform multiplication by looking up partial products and summing them. Division could be performed similarly as a series of lookups and subtractions.

Later the rods were mechanized by replacing them with cylinders that could be rotated into position. For a demonstration of how Napier's Bones work consult

Napier's Demo

The Slide Rule

As mentioned earlier, John Napier had introduced the use of logarithms. Subsequently, he collaborated with fellow mathematician Henry Briggs (1561-1630), converting his original logarithmic calculations to the more familiar base-10 representation used today.

The utility of logs can be seen in the following important results.

a * b = 10 ^ ( log (a) + log (b) ), and
a / b = 10 ^ ( log (a) - log (b) )

[Note: ‘*' means product or "times"; and ‘^' denotes "raised to the power of."]

However, one could not exploit these results without performing some time-consuming tasks. In order to multiply two numbers a and b,

  1. You must look up two logs.
  2. You must add them.
  3. You must look up the corresponding number whose log is their sum.

Edmund Gunter (1581-1626) fashioned a device to help remedy this situation. Called "Gunter's Scale," it plotted a logarithmic scale on a two-foot ruler. By adding and subtracting lengths, it was possible to obtain the results of multiplication and division.

William Oughtred (1574-1660) improved upon Gunter's single ruler in 1630 by combining two circular scales that could be moved relative to one another. The moving scales eliminated the need for a divider and thereby became the early ancestor of the modern slide rule. Whether straight or circular, the slide rule represents an analog calculator because the results of the operations are based on the continuous scale of distances.


©Abernethy and Allen, 2004.
Furman University
Last modified: 2/04


Figure 1. A modern version of the abacus is a frame holding rows of beads suspended on metal rods. Each row represents a power of ten in calculations.
Figure 2. A sketch of al-Khwarizimi.
Figure 3. A drawing of one of Lull's logic machines taken from the Ars Brevis.