Navigation and astronomy

Starting with known special cases, the calculation of logarithms and trigonometric functions can be performed by looking up numbers in a mathematical table, and interpolating between known cases. For small enough differences, this linear operation was accurate enough for use in navigation and astronomy in the Age of Exploration. The uses of interpolation have thrived in the past 500 years: by the twentieth century Leslie Comrie and W.J. Eckert systematized the use of interpolation in tables of numbers for punch card calculation.
In our time, even a student can simulate the motion of the planets, an N-body differential equation, using the concepts of numerical approximation, a feat which even Isaac Newton could admire, given his struggles with the motion of the Moon.

Early computation

Main article: Timeline of computing 2400 BC–1949

The earliest known tool for use in computation was the abacus, and it was thought to have been invented in Babylon circa 2400 BC. Its original style of usage was by lines drawn in sand with pebbles. Abaci, of a more modern design, are still used as calculation tools today. This was the first known computer and most advanced system of calculation known to date - preceding Greek methods by 2,000 years.
In 1115 BC, the South Pointing Chariot was invented in ancient China. It was the first known geared mechanism to use a differential gear, which was later used in analog computers. The Chinese also invented a more sophisticated abacus from around the 2nd century BC known as the Chinese abacus).
In the 5th century BC in ancient India, the grammarian Pāṇini formulated the grammar of Sanskrit in 3959 rules known as the Ashtadhyayi which was highly systematized and technical. Panini used metarules, transformations and recursions.
The Antikythera mechanism is believed to be the earliest known mechanical analog computer.^[2] It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC.
Mechanical analog computer devices appeared again a thousand years later in the medieval Islamic world and were developed by Muslim astronomers, such as the equatorium by Arzachel,^[3] the mechanical geared astrolabe by Abū Rayhān al-Bīrūnī,^[4] and the torquetum by Jabir ibn Aflah.^[5] Muslim mathematicians also made important advances in cryptography, such as the development of cryptanalysis and frequency analysis by Alkindus.^[6][7] Programmable machines were also invented by Muslim engineers, such as the automatic flute player by the Banū Mūsā brothers,^[8] and Al-Jazari's humanoid robots^[9] and castle clock, which is considered to be the first programmable analog computer.^[10]
During the Middle Ages, several European philosophers made attempts to produce analog computer devices. Influenced by the Arabs and Scholasticism, majorcan philosopher Ramon Llull (1232–1315) devoted a great part of his life to define and design several logical machines that, by combining simple and undeniable philosophical truths, could produce all the possible knowledge. This machines were never really built, for they were more of a thought experiment devoted to the production of new knowledge by systematic ways; however they could make simple logical operations, they still needed a human being for interpretation of results. Moreover, they lacked of a versatile architecture, each machine serving only to very concrete purposes. No matter what, Llull's work had a severe impact on Gottfried Leibniz (early 18th century), who redeveloped his ideas further and could build several calculating tools with them.
Indeed, when John Napier discovered logarithms for computational purposes in the early 17th century, there followed a period of considerable progress by inventors and scientists in making calculating tools. The apex of this early era of formal computing can be seen in the difference engine and its successor the Analytical Engine, which was never completely constructed but was designed in detail, both by Charles Babbage. The analytical engine combined concepts from his work and that of others to create a device that if constructed as designed would have possessed many properties of a modern electronic computer. These properties include such features as an internal "scratch memory" equivalent to RAM, multiple forms of output including a bell, a graph-plotter, and simple printer, and a programmable input-output "hard" memory of punch cards which it could modify as well as read. The key advancement which Babbage's devices possessed beyond those created before his was that each component of the device was independent of the rest of the machine, much like the components of a modern electronic computer. This was a fundamental shift in thought; previous computational devices served only a single purpose, but had to be at best dissasembled and reconfigured to solve a new problem. Babbage's devices could be reprogramed to solve new problems by the entry of new data, and act upon previous calculations within the same series of instructions. Ada Lovelace took this concept one step further, by creating a program for the analytical engine to calculate Bernoulli numbers, a complex calculation requiring a recursive algorithm. This is considered to the first example of a true computer program, a series of instructions that act upon data not known in full until the program is run.
Several examples of analog compututation survived into recient times. A planimeter is a device which does integrals, using distance as the analog quantity. Until the 1980s, HVAC systems used air both as the analog quantity and the controlling element. Unlike modern digital computers, analog computers are not very flexible, and need to be reconfigured (i.e., reprogrammed) manually to switch them from working on one problem to another. Analog computers had an advantage over early digital computers in that they could be used to solve complex problems using behavioral analogues while the earliest attempts at digital computers were quite limited.

A Smith Chart is a well-known nomogram. Since computers were rare in this era, the solutions were often hard-coded into paper forms such as nomograms,^[11] which could then produce analog solutions to these problems, such as the distribution of pressures and temperatures in a heating system.
None of the early computational devices were really computers in the modern sense, and it took considerable advancement in mathematics and theory before the first modern computers could be designed.

Concrete devices

Computing is intimately tied to the representation of numbers. But long before abstractions like the number arose, there were mathematical concepts to serve the purposes of civilization. These concepts are implicit in concrete practices such as :

• one-to-one correspondence, a rule to count how many items, say on a tally stick, eventually abstracted into numbers;
• comparison to a standard, a method for assuming reproducibility in a measurement, for example, the number of coins;
• the 3-4-5 right triangle was a device for assuring a right angle, using ropes with 12 evenly spaced knots, for example.