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  • Ancient mathematical sources
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    The Ishango bone Devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers.

    The earliest counting device was probably a form of tally stick. Later record keeping aids throughout the Fertile Crescent included calculi clay spheres, cones, etc.

    The Roman abacus was developed from devices used in Babylonia as early as BC.

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    Since then, many other forms of reckoning boards or tables have been invented. In a medieval European counting housea checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money.

    The ancient Greek -designed Antikythera mechanismdating between and BC, is the world's oldest analog computer. The Antikythera mechanism is believed to be the earliest mechanical analog "computer", according to Derek J. It was discovered in in the Antikythera wreck off the Greek island of Antikytherabetween Kythera and Creteand has been dated to circa BC.

    Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use. A combination of the planisphere and dioptrathe astrolabe was effectively an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer [7] [8] and gear -wheels was invented by Abi Bakr of IsfahanPersia in The sectora calculating instrument used for solving problems in proportion, trigonometry, multiplication and division, and for various functions, such as squares and cube roots, was developed in the late 16th century and found application in gunnery, surveying and navigation.

    The planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage.

    A slide rule The slide rule was invented around —, shortly after the publication of the concept of the logarithm. It is a hand-operated analog computer for doing multiplication and division.

    As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions. Slide rules with special scales are still used for quick performance of routine calculations, such as the E6B circular slide rule used for time and distance calculations on light aircraft.

    In the s Pierre Jaquet-Droza Swiss watchmakerbuilt a mechanical doll automata that could write holding a quill pen. By switching the number and order of its internal wheels different letters, and hence different messages, could be produced.

    In effect, it could be mechanically "programmed" to read instructions. It used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location.

    The differential analysera mechanical analog computer designed to solve differential equations by integrationused wheel-and-disc mechanisms to perform the integration. In Lord Kelvin had already discussed the possible construction of such calculators, but he had been stymied by the limited output torque of the ball-and-disk integrators. The torque amplifier was the advance that allowed these machines to work.

    Starting in the s, Vannevar Bush and others developed mechanical differential analyzers. First computing device A portion of Babbage's Difference engine. Charles Babbagean English mechanical engineer and polymathoriginated the concept of a programmable computer. Considered the " father of the computer ", [15] he conceptualized and invented the first mechanical computer in the early 19th century.

    After working on his revolutionary difference enginedesigned to aid in navigational calculations, in he realized that a much more general design, an Analytical Enginewas possible. The input of programs and data was to be provided to the machine via punched cardsa 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.

    The Engine incorporated an arithmetic logic unitcontrol flow in the form of conditional branching and loopsand integrated memorymaking it the first design for a general-purpose computer that could be described in modern terms as Turing-complete. Eventually, the project was dissolved with the decision of the British Government to cease funding.

    Babbage's failure to complete the analytical engine can be chiefly attributed to difficulties not only of politics and financing, but also to his desire to develop an increasingly sophisticated computer and to move ahead faster than anyone else could follow.

    Nevertheless, his son, Henry Babbage, completed a simplified version of the analytical engine's computing unit the mill in He gave a successful demonstration of its use in computing tables in Analog computers Sir William Thomson 's third tide-predicting machine design, —81 During the first half of the 20th century, many scientific computing needs were met by increasingly sophisticated analog computerswhich used a direct mechanical or electrical model of the problem as a basis for computation.

    However, these were not programmable and generally lacked the versatility and accuracy of modern digital computers. The differential analysera mechanical analog computer designed to solve differential equations by integration using wheel-and-disc mechanisms, was conceptualized in by James Thomsonthe brother of the more famous Lord Kelvin.

    This built on the mechanical integrators of James Thomson and the torque amplifiers invented by H. A dozen of these devices were built before their obsolescence became obvious. By the s the success of digital electronic computers had spelled the end for most analog computing machines, but analog computers remained in use during the s in some specialized applications such as education control systems and aircraft slide rule.

    Digital computers It has been suggested that this section be split out into another article titled Digital computer. Discuss May Electromechanical Bythe United States Navy had developed an electromechanical analog computer small enough to use aboard a submarine. This was the Torpedo Data Computerwhich used trigonometry to solve the problem of firing a torpedo at a moving target. During World War II similar devices were developed in other countries as well.

    Replica of Zuse 's Z3the first fully automatic, digital electromechanical computer. Early digital computers were electromechanical; electric switches drove mechanical relays to perform the calculation. These devices had a low operating speed and were eventually superseded by much faster all-electric computers, originally using vacuum tubes. The Z2created by German engineer Konrad Zuse inwas one of the earliest examples of an electromechanical relay computer. It was quite similar to modern machines in some respects, pioneering numerous advances such as floating point numbers.

    Rather than the harder-to-implement decimal system used in Charles Babbage 's earlier designusing a binary system meant that Zuse's machines were easier to build and potentially more reliable, given the technologies available at that time.

    The engineer Tommy Flowersworking at the Post Office Research Station in London in the s, began to explore the possible use of electronics for the telephone exchange. Experimental equipment that he built in went into operation five years later, converting a portion of the telephone exchange network into an electronic data processing system, using thousands of vacuum tubes. The German encryption machine, Enigmawas first attacked with the help of the electro-mechanical bombes.

    It had paper-tape input and was capable of being configured to perform a variety of boolean logical operations on its data, but it was not Turing-complete. Colossus Mark I contained 1, thermionic valves tubesbut Mark II with 2, valves, was both 5 times faster and simpler to operate than Mark I, greatly speeding the decoding process. Like the Colossus, a "program" on the ENIAC was defined by the states of its patch cables and switches, a far cry from the stored program electronic machines that came later.

    Once a program was written, it had to be mechanically set into the machine with manual resetting of plugs and switches. It combined the high speed of electronics with the ability to be programmed for many complex problems.

    It could add or subtract times a second, a thousand times faster than any other machine. It also had modules to multiply, divide, and square root. High speed memory was limited to 20 words about 80 bytes. Built under the direction of John Mauchly and J. The machine was huge, weighing 30 tons, using kilowatts of electric power and contained over 18, vacuum tubes, 1, relays, and hundreds of thousands of resistors, capacitors, and inductors.

    Turing proposed a simple device that he called "Universal Computing machine" and that is now known as a universal Turing machine.

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    He proved that such a machine is capable of computing anything that is computable by executing instructions program stored on tape, allowing the machine to be programmable. The fundamental concept of Turing's design is the stored programwhere all the instructions for computing are stored in memory. Von Neumann acknowledged that the central concept of the modern computer was due to this paper. Except for the limitations imposed by their finite memory stores, modern computers are said to be Turing-completewhich is to say, they have algorithm execution capability equivalent to a universal Turing machine.

    Stored programs A section of the Manchester Babythe first electronic stored-program computer Early computing machines had fixed programs. Changing its function required the re-wiring and re-structuring of the machine.

    A stored-program computer includes by design an instruction set and can store in memory a set of instructions a program that details the computation.

    The theoretical basis for the stored-program computer was laid by Alan Turing in his paper. In Turing joined the National Physical Laboratory and began work on developing an electronic stored-program digital computer. His report "Proposed Electronic Calculator" was the first specification for such a device.

    The Mark 1 in turn quickly became the prototype for the Ferranti Mark 1the world's first commercially available general-purpose computer. At least seven of these later machines were delivered between andone of them to Shell labs in Amsterdam. The LEO I computer became operational in April [42] and ran the world's first regular routine office computer job.

    Transistors A bipolar junction transistor The bipolar transistor was invented in From onwards transistors replaced vacuum tubes in computer designs, giving rise to the "second generation" of computers. Compared to vacuum tubes, transistors have many advantages: Silicon junction transistors were much more reliable than vacuum tubes and had longer, indefinite, service life. Transistorized computers could contain tens of thousands of binary logic circuits in a relatively compact space.

    At the University of Manchestera team under the leadership of Tom Kilburn designed and built a machine using the newly developed transistors instead of valves. The idea of the integrated circuit was first conceived by a radar scientist working for the Royal Radar Establishment of the Ministry of DefenceGeoffrey W.

    Produced at Fairchild Semiconductor, it was made of siliconwhereas Kilby's chip was made of germanium. This new development heralded an explosion in the commercial and personal use of computers and led to the invention of the microprocessor. While the subject of exactly which device was the first microprocessor is contentious, partly due to lack of agreement on the exact definition of the term "microprocessor", it is largely undisputed that the first single-chip microprocessor was the Intel[52] designed and realized by Ted HoffFederico Fagginand Stanley Mazor at Intel.

    These so-called smartphones and tablets run on a variety of operating systems and have become the dominant computing device on the market, with manufacturers reporting having shipped an estimated million devices in 2Q
    Because much of genetics is based on quantitative data, mathematical techniques are used extensively in genetics.

    The laws of probability are applicable to crossbreeding and are used to predict frequencies of specific genetic constitutions in offspring. Geneticists also use statistical methods to determine… Ancient mathematical sources It is important to be aware of the character of the sources for the study of the history of mathematics.

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    The history of Mesopotamian and Egyptian mathematics is based on the extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians.

    The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamia or about its influence on Greek mathematics, but it seems likely that this picture of Mesopotamian mathematics will stand.

    Ancient mathematical sources

    This stands in complete contrast to the situation described above for Egyptian and Babylonian documents. Although, in general outline, the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic methodthe pre-Euclidean theory of ratios, and the discovery of the conic sectionshistorians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount of conjecture.

    Many important treatises from the early period of Islamic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the relationship between early Islamic mathematics and the mathematics of Greece and India. In addition, the amount of surviving material from later centuries is so large in comparison with that which has been studied that it is not yet possible to offer any sure judgment of what later Islamic mathematics did not contain, and therefore it is not yet possible to evaluate with any assurance what was original in European mathematics from the 11th to the 15th century.

    In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the major figures in any detail.

    In addition, there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one is to a given period, the more likely these fashions will look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the subject.

    Berggren Mathematics in ancient Mesopotamia Until the s it was commonly supposed that mathematics had its birth among the ancient Greeks. What was known of earlier traditions, such as the Egyptian as represented by the Rhind papyrus edited for the first time only inoffered at best a meagre precedent.

    This impression gave way to a very different view as historians succeeded in deciphering and interpreting the technical materials from ancient Mesopotamia.

    Existing specimens of mathematics represent all the major eras—the Sumerian kingdoms of the 3rd millennium bce, the Akkadian and Babylonian regimes 2nd millenniumand the empires of the Assyrians early 1st millenniumPersians 6th through 4th century bceand Greeks 3rd century bce to 1st century ce.

    The level of competence was already high as early as the Old Babylonian dynastythe time of the lawgiver-king Hammurabi c. The application of mathematics to astronomy, however, flourished during the Persian and Seleucid Greek periods. The numeral system and arithmetic operations Unlike the Egyptians, the mathematicians of the Old Babylonian period went far beyond the immediate challenges of their official accounting duties.

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    For example, they introduced a versatile numeral system, which, like the modern system, exploited the notion of place value, and they developed computational methods that took advantage of this means of expressing numbers; they solved linear and quadratic problems by methods much like those now used in algebra ; their success with the study of what are now called Pythagorean number triples was a remarkable feat in number theory.

    The scribes who made such discoveries must have believed mathematics to be worthy of study in its own right, not just as a practical tool. The older Sumerian system of numerals followed an additive decimal base principle similar to that of the Egyptians. But the Old Babylonian system converted this into a place-value system with the base of 60 sexagesimal.

    The reasons for the choice of 60 are obscure, but one good mathematical reason might have been the existence of so many divisors 2, 3, 4, and 5, and some multiples of the base, which would have greatly facilitated the operation of division.

    Classical trigonometry

    For numbers from 1 to 59, the symbols for 1 and for 10 were combined in the simple additive manner e. But to express larger values, the Babylonians applied the concept of place value. For example, 60 was written as70 as80 asand so on. In fact, could represent any power of The context determined which power was intended.

    By the 3rd century bce, the Babylonians appear to have developed a placeholder symbol that functioned as a zerobut its precise meaning and use is still uncertain. Furthermore, they had no mark to separate numbers into integral and fractional parts as with the modern decimal point. The four arithmetic operations were performed in the same way as in the modern decimal system, except that carrying occurred whenever a sum reached 60 rather than Multiplication was facilitated by means of tables; one typical tablet lists the multiples of a number by 1, 2, 3,…, 19, 20, 30, 40, and To multiply two numbers several places long, the scribe first broke the problem down into several multiplications, each by a one-place number, and then looked up the value of each product in the appropriate tables.

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    He found the answer to the problem by adding up these intermediate results. These tables also assisted in division, for the values that head them were all reciprocals of regular numbers.

    Regular numbers are those whose prime factors divide the base; the reciprocals of such numbers thus have only a finite number of places by contrast, the reciprocals of nonregular numbers produce an infinitely repeating numeral. In base 10, for example, only numbers with factors of 2 and 5 e. In base 60, only numbers with factors of 2, 3, and 5 are regular; for example, 6 and 54 are regular, so that their reciprocals 10 and 1 6 40 are finite.

    To divide a number by any regular number, then, one can consult the table of multiples for its reciprocal. An interesting tablet in the collection of Yale University shows a square with its diagonals. The scribe thus appears to have known an equivalent of the familiar long method of finding square roots.

    They also show that the Babylonians were aware of the relation between the hypotenuse and the two legs of a right triangle now commonly known as the Pythagorean theorem more than a thousand years before the Greeks used it. A type of problem that occurs frequently in the Babylonian tablets seeks the base and height of a rectangle, where their product and sum have specified values. In the same way, if the product and difference were given, the sum could be found. This procedure is equivalent to a solution of the general quadratic in one unknown.

    Babylonia Code of Hammurabi Crystalinks

    In some places, however, the Babylonian scribes solved quadratic problems in terms of a single unknown, just as would now be done by means of the quadratic formula.

    Although these Babylonian quadratic procedures have often been described as the earliest appearance of algebrathere are important distinctions. The scribes lacked an algebraic symbolism; although they must certainly have understood that their solution procedures were general, they always presented them in terms of particular cases, rather than as the working through of general formulas and identities.

    They thus lacked the means for presenting general derivations and proofs of their solution procedures. Their use of sequential procedures rather than formulas, however, is less likely to detract from an evaluation of their effort now that algorithmic methods much like theirs have become commonplace through the development of computers.

    If one selects values at random for two of the terms, the third will usually be irrationalbut it is possible to find cases in which all three terms are integers: Such solutions are sometimes called Pythagorean triples.

    A tablet in the Columbia University Collection presents a list of 15 such triples decimal equivalents are shown in parentheses at the right; the gaps in the expressions for h, b, and d separate the place values in the sexagesimal numerals: The entries in the column for h have to be computed from the values for b and d, for they do not appear on the tablet; but they must once have existed on a portion now missing.

    In the table the implied values p and q turn out to be regular numbers falling in the standard set of reciprocals, as mentioned earlier in connection with the multiplication tables.

    Scholars are still debating nuances of the construction and the intended use of this table, but no one questions the high level of expertise implied by it. Mathematical astronomy The sexagesimal method developed by the Babylonians has a far greater computational potential than what was actually needed for the older problem texts.

    With the development of mathematical astronomy in the Seleucid period, however, it became indispensable.

    Astronomers sought to predict future occurrences of important phenomena, such as lunar eclipses and critical points in planetary cycles conjunctions, oppositions, stationary points, and first and last visibility. The results were then organized into a table listing positions as far ahead as the scribe chose. Although the method is purely arithmetic, one can interpret it graphically: While observations extending over centuries are required for finding the necessary parameters e.

    Within a relatively short time perhaps a century or lessthe elements of this system came into the hands of the Greeks. Although Hipparchus 2nd century bce favoured the geometric approach of his Greek predecessors, he took over parameters from the Mesopotamians and adopted their sexagesimal style of computation.

    Through the Greeks it passed to Arab scientists during the Middle Ages and thence to Europe, where it remained prominent in mathematical astronomy during the Renaissance and the early modern period. To this day it persists in the use of minutes and seconds to measure time and angles. Aspects of the Old Babylonian mathematics may have come to the Greeks even earlier, perhaps in the 5th century bce, the formative period of Greek geometry.

    There are a number of parallels that scholars have noted. Further, the Babylonian rule for estimating square roots was widely used in Greek geometric computations, and there may also have been some shared nuances of technical terminology.

    Although details of the timing and manner of such a transmission are obscure because of the absence of explicit documentation, it seems that Western mathematics, while stemming largely from the Greeks, is considerably indebted to the older Mesopotamians.

    History of trigonometry

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