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.
Eventually, the concept of numbers became concrete and familiar enough for counting to arise, at times with sing-song mnemonics to teach sequences to others. All the known languages have words for at least "one" and "two" (although this is disputed: see Piraha language), and even some animals like the blackbird can distinguish a surprising number of items.
Advances in the numeral system and mathematical notation eventually led to the discovery of mathematical operations such as addition, subtraction, multiplication, division, squaring, square root, and so forth. Eventually the operations were formalized, and concepts about the operations became understood well enough to be stated formally, and even proven. See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers.
By the High Middle Ages, the positional Hindu-Arabic numeral system had reached Europe, which allowed for systematic computation of numbers. During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm (for use in multiplication and division) and the trigonometric functions. By the time of Isaac Newton's research, paper or vellum was an important computing resource, and even in our present time, researchers like Enrico Fermi would cover random scraps of paper with calculation, to satisfy their curiosity about an equation. Even into the period of programmable calculators, Richard Feynman would unhesitatingly compute any steps which overflowed the memory of the calculators, by hand, just to learn the answer.
AbacusThe 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. 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, the mechanical geared astrolabe by Abū Rayhān al-Bīrūnī, and the torquetum by Jabir ibn Aflah. Muslim mathematicians also made important advances in cryptography, such as the development of cryptanalysis and frequency analysis by Alkindus. Programmable machines were also invented by Muslim engineers, such as the automatic flute player by the Banū Mūsā brothers, and Al-Jazari's humanoid robots and castle clock, which is considered to be the first programmable analog computer.
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. 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, 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.
David Packard and Bill Hewlett in their Palo Alto, California Garage
Hewlett-Packard is Founded. David Packard and Bill Hewlett found Hewlett-Packard in a Palo Alto, California garage. Their first product was the HP 200A Audio Oscillator, which rapidly becomes a popular piece of test equipment for engineers. Walt Disney Pictures ordered eight of the 200B model to use as sound effects generators for the 1940 movie “Fantasia.”