Charles F. Kettering
Charles Franklin Kettering sometimes known as Charles "Boss" Kettering was an American inventor, engineer and the holder of 186 patents. He was a founder of Delco, was head of research at General Motors from 1920 to 1947. Among his most used automotive developments were the electrical starting motor and leaded gasoline. In association with the DuPont Chemical Company, he was responsible for the invention of Freon refrigerant for refrigeration and air conditioning systems. At DuPont he was responsible for the development of Duco lacquers and enamels, the first practical colored paints for mass-produced automobiles. While working with the Dayton-Wright Company he developed the "Bug" aerial torpedo, considered the world's first aerial missile, he led the advancement of practical, lightweight two-stroke diesel engines, revolutionizing the locomotive and heavy equipment industries. In 1927, he founded a non-partisan research foundation, he was featured on the cover of Time Magazine on January 9, 1933.
Charles was born in Loudonville, United States, the fourth of five children of Jacob Henry Kettering and Martha Kettering. Poor eyesight gave him headaches in school. After graduation he followed his sister Emma into a teaching position at Bunker Hill School. By all accounts he was an innovative teacher, he attracted students to evening scientific demonstrations on electricity, heat and gravity. He took classes before transferring to The Ohio State University, he was a member of the Delta Upsilon Fraternity. Eye problems forced him to withdraw, he took a job as foreman of a telephone line crew. At first, the termination of his studies caused him to be depressed, he found ways to apply his electrical engineering skills on the job, his spirits revived. He met his future wife, Olive Williams; when his eye condition improved, he was able to return to his studies and graduated from OSU in 1904 with an electrical engineering degree. Kettering was hired directly out of school to head up the research laboratory at National Cash Register.
Kettering invented an easy credit approval system, a precursor to today's credit cards, the electric cash register in 1906, which made ringing up sales physically much easier for sales clerks. Kettering distinguished himself as a practical inventor; as he said, "I didn't hang around much with the executive fellows. I lived with the sales gang, they had some real notion of what people wanted." During his five years at NCR, from 1904 to 1909, Kettering secured 23 patents for NCR. He attributed his success to a good amount of luck but added, "I notice the harder I work, the luckier I get."Beginning in 1907, his NCR colleague Edward A. Deeds convinced Kettering to develop improvements for the automobile, he told Kettering, "There is a river of gold running past us," implying that, with their know-how, all they needed to do was dip into it. Deeds and Kettering invited other NCR engineers, including Harold E. Talbott, to join them nights and weekends in their tinkering at Deeds's barn, they became known as the "Barn Gang," and Kettering was their leader.
Boss Ket, they called him. They set their first task as improving ignition. In 1909, Kettering resigned from NCR to work full-time on automotive developments; the Barn Gang incorporated as Delco. Early automobiles required a hand crank for starting; when the spark lever was not properly set, the hand crank kicked back, causing serious injury: a broken wrist, arm, or shoulder. On a winter night in 1908, the result was much worse. Byron Carter, founder of Cartercar, came across a stalled motorist on Belle Isle in the middle of the Detroit River, he gallantly offered to crank the car for the stranded driver. When she forgot to retard the spark, the crank broke Carter's jaw. Complications developed, Carter died of pneumonia; when Cadillac chief, Henry M. Leland, heard the news, he was distraught. Byron Carter was a friend. "The Cadillac car will kill no more men if we can help it," he told his staff. Leland's engineers were able to build an electric self-starter, but the device was not small enough to be practical.
He called Charles Kettering. The engineers at Delco worked around the clock to get the job done by the February 1911 deadline. Kettering described their work thus: They didn't have a job so much as the job had them. Kettering's key insight lay in devising an electrical system that performed the three purposes it continues to serve in modern cars: starter and, as generator, producer of spark for ignition as well as current for lighting. Leland placed an order for 12,000 self-starters. Delco, the research and development outfit, had to learn the business of production. Kettering's self-starter won a Dewar Trophy in 1913, he was a founder member of the Engineers Club of Dayton in 1914. In 1914, Flxible Sidecar Company was incorporated with the help of Kettering, who became president of the company and joined the board of directors. Kettering provided significant funding for the company in its early years after 1916, when Kettering sold his firm, the Dayton Engineering Laboratories Company, to United Motors for $2.5 million.
Kettering continued to serve as president of Flxible until he became chairman of the board in 1940, a position that he held until his death in 1958. Delco was sold to General Motors as part of United Motors Company. Delco became the foundation for the General Motors Research Delco Electronics. Kettering became vice-p
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts. Founded in 1861 in response to the increasing industrialization of the United States, MIT adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering; the Institute is a land-grant, sea-grant, space-grant university, with a campus that extends more than a mile alongside the Charles River. Its influence in the physical sciences and architecture, more in biology, linguistics and social science and art, has made it one of the most prestigious universities in the world. MIT is ranked among the world's top universities; as of March 2019, 93 Nobel laureates, 26 Turing Award winners, 8 Fields Medalists have been affiliated with MIT as alumni, faculty members, or researchers. In addition, 58 National Medal of Science recipients, 29 National Medals of Technology and Innovation recipients, 50 MacArthur Fellows, 73 Marshall Scholars, 45 Rhodes Scholars, 41 astronauts, 16 Chief Scientists of the US Air Force have been affiliated with MIT.
The school has a strong entrepreneurial culture, the aggregated annual revenues of companies founded by MIT alumni would rank as the tenth-largest economy in the world. MIT is a member of the Association of American Universities. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a "Conservatory of Art and Science", but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, proposed by William Barton Rogers, was signed by the governor of Massachusetts on April 10, 1861. Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances, he did not wish to found a professional school, but a combination with elements of both professional and liberal education, proposing that: The true and only practicable object of a polytechnic school is, as I conceive, the teaching, not of the minute details and manipulations of the arts, which can be done only in the workshop, but the inculcation of those scientific principles which form the basis and explanation of them, along with this, a full and methodical review of all their leading processes and operations in connection with physical laws.
The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars and laboratories. Two days after MIT was chartered, the first battle of the Civil War broke out. After a long delay through the war years, MIT's first classes were held in the Mercantile Building in Boston in 1865; the new institute was founded as part of the Morrill Land-Grant Colleges Act to fund institutions "to promote the liberal and practical education of the industrial classes" and was a land-grant school. In 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from land sales went toward new buildings in the Back Bay. MIT was informally called "Boston Tech"; the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker.
Programs in electrical, chemical and sanitary engineering were introduced, new buildings were built, the size of the student body increased to more than one thousand. The curriculum drifted with less focus on theoretical science; the fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership. During these "Boston Tech" years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliot's repeated attempts to merge MIT with Harvard College's Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard. In its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding; the MIT Corporation approved a formal agreement to merge with Harvard, over the vehement objections of MIT faculty and alumni. However, a 1917 decision by the Massachusetts Supreme Judicial Court put an end to the merger scheme. In 1916, the MIT administration and the MIT charter crossed the Charles River on the ceremonial barge Bucentaur built for the occasion, to signify MIT's move to a spacious new campus consisting of filled land on a mile-long tract along the Cambridge side of the Charles River.
The neoclassical "New Technology" campus was designed by William W. Bosworth and had been funded by anonymous donations from a mysterious "Mr. Smith", starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of film production and processing, founded Eastman Kodak. Between 1912 and 1920, Eastman donated $20 million in cash and Kodak stock to MIT. In the 1930s, President Karl Taylor Compton and Vice-President Vannevar Bush emphasized the importance of pure sciences like physics and chemistry and reduced the vocational practice required in shops and drafting studios; the Compton reforms "renewed confidence in the ability of the Institute to develop leadership in science as well as in engineering". Unlike Ivy League schools, MIT catered more to middle-class families, depended more on tuition than on endow
A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of called programs; these programs enable computers to perform an wide range of tasks. A "complete" computer including the hardware, the operating system, peripheral equipment required and used for "full" operation can be referred to as a computer system; this term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster. Computers are used as control systems for a wide variety of industrial and consumer devices; this includes simple special purpose devices like microwave ovens and remote controls, factory devices such as industrial robots and computer-aided design, general purpose devices like personal computers and mobile devices such as smartphones. The Internet is run on computers and it connects hundreds of millions of other computers and their users.
Early computers were only conceived as calculating devices. Since ancient times, simple manual devices like the abacus aided people in doing calculations. Early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century; the first digital electronic calculating machines were developed during World War II. The speed and versatility of computers have been increasing ever since then. Conventionally, a modern computer consists of at least one processing element a central processing unit, some form of memory; the processing element carries out arithmetic and logical operations, a sequencing and control unit can change the order of operations in response to stored information. Peripheral devices include input devices, output devices, input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and they enable the result of operations to be saved and retrieved.
According to the Oxford English Dictionary, the first known use of the word "computer" was in 1613 in a book called The Yong Mans Gleanings by English writer Richard Braithwait: "I haue read the truest computer of Times, the best Arithmetician that euer breathed, he reduceth thy dayes into a short number." This usage of the term referred to a human computer, a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century. During the latter part of this period women were hired as computers because they could be paid less than their male counterparts. By 1943, most human computers were women. From the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations; the Online Etymology Dictionary gives the first attested use of "computer" in the 1640s, meaning "one who calculates". The Online Etymology Dictionary states that the use of the term to mean "'calculating machine' is from 1897."
The Online Etymology Dictionary indicates that the "modern use" of the term, to mean "programmable digital electronic computer" dates from "1945 under this name. Devices have been used to aid computation for thousands of years using one-to-one correspondence with fingers; the earliest counting device was a form of tally stick. Record keeping aids throughout the Fertile Crescent included calculi which represented counts of items livestock or grains, sealed in hollow unbaked clay containers; the use of counting rods is one example. The abacus was used for arithmetic tasks; the Roman abacus was developed from devices used in Babylonia as early as 2400 BC. Since many other forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, markers moved around on it according to certain rules, as an aid to calculating sums of money; the Antikythera mechanism is believed to be the earliest mechanical analog "computer", according to Derek J. de Solla Price.
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, has been dated to c. 100 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; the planisphere was a star chart invented by Abū Rayhān al-Bīrūnī in the early 11th century. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer and gear-wheels was invented by Abi Bakr of Isfahan, Persia in 1235. Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe, an early fixed-wired knowledge processing machine with a gear train and gear-wheels, c. 1000 AD.
The sector, a calculating instrument used for solving problems in proportion, trigonometry and division, for various functions, such as squares and cube roots, was developed in
University of Dayton
The University of Dayton is a private Roman Catholic research university in Dayton, Ohio. Founded in 1850 by the Society of Mary, it is one of three Marianist universities in the nation and the second-largest private university in Ohio; the university's campus is in the city's southern portion and spans 388 acres on both sides of the Great Miami River. The campus is noted for the University of Dayton Arena; the University operates in China's Suzhou Industrial Park, the University of Dayton China Institute. The university has about 8,000 undergraduate and 2,200 post-graduate students from a variety of religious and geographic backgrounds, drawn from across the United States and more than 40 countries, it offers more than 80 academic programs in arts and sciences, business administration and health sciences, law and, in 2009, was first in the country to offer an undergraduate degree program in human rights. The university's notable alumni include: Ohio Lieutenant Governor Jon Husted. In 1849, on a mission to establish a presence for the Society of Mary in America, the Rev. Leo Meyer, S.
M. journeyed from Alsace in France to Cincinnati. But with a cholera epidemic raging to the north, Bishop John Baptist Purcell of the Cincinnati diocese, sent Father Meyer to Emmanuel parish in Dayton to tend to the sick. In Dayton, Father Meyer met local farmer John Stuart, who had lost his infant daughter Mary Louisa to cholera the year before. Heartbroken and his wife wanted to sell their Dewberry Farm property and return to Europe. On March 19, 1850, Father Meyer, joined by three Marianist brothers — teacher Maximin Zehler, cook Charles Schultz and gardener Andrew Edel — purchased the 125-acre hilltop farm from Stuart and renamed it Nazareth. Stuart accepted a promise of $12,000 at 6 percent interest; the property included vineyards, an orchard, a mansion, various farm buildings, the grave of Stuart's daughter, which Meyer promised to maintain. Just a few months the University of Dayton had its beginnings on July 1, 1850, when St. Mary's School for Boys opened its doors to 14 primary students from Dayton.
In September, the first boarding students arrived and classes moved to the mansion. Five years the school burned to the ground, but classes resumed within months. By 1860, when Brother Zehler became president, the enrollment was nearly 100 students; the Civil War had little direct effect on the school because most of the students were too young to serve. College preparatory classes started in 1861 along with a novitiate and school for Marianist candidates; the school would become Chaminade High School, named after the order's founder William Joseph Chaminade, which has since merged with the all-girls Julienne High school run by the Sisters of Notre Dame de Namur to form the co-educational Chaminade-Julienne High School. The core of the Historic Campus was built during this time, starting in 1865 with Zehler Hall, the iconic Immaculate Conception Chapel in 1869, St. Mary's Hall the tallest building in Dayton in 1870. In 1882 the university was incorporated and empowered to confer collegiate degrees by the State of Ohio.
When floodwaters struck the community during the Great Dayton Flood of 1913, refugees fled to St. Mary's College high on a hill south of downtown Dayton. St. Mary's College equipped to provide relief to flood victims; because students had not returned to campus from Easter break, the college was amply stocked with food and other provisions. Due to its location on the hill, electric light and heating plants were not affected, a plentiful clean water supply was available and the college had other essential facilities such as a laundry and infirmary; the college's kitchen provided meals to Miami Valley Hospital and provided 12,000 pounds of provisions to St. Elizabeth Hospital; the first night, 400 refugees took shelter at St. Mary's College. In all, the college assisted 800 refugees. Known at various times as St. Mary's School, St. Mary's Institute and St. Mary's College, the school was incorporated as the University of Dayton in 1920 to reflect its close connection with the city of Dayton as well as to claim an American identity for its Catholic students.
In 1923, the University adopted the "Dayton Flyers" nickname for its athletic teams and adopted a university seal with the motto, "Pro Deo et Patria", Latin for "For God and Country." In the 1930s, women were admitted on an equal basis with men, 40 years before most Catholic universities allowed women. The school expanded its programs in engineering, and the continued to attract the children and grandchildren of Catholic immigrants. The growing Catholic presence in Dayton during the 1920s drew the hostility of the Ku Klux Klan, which focused that hostility toward the university. On December 19, 1923, 12 bombs exploded throughout the campus and an 8-foot cross was set on fire. Several hundred Klansmen were routed by hundreds of neighborhood residents who joined students in chasing them off. In June 2014, The University of Dayton announced it will begin divesting coal and fossil fuels from its $670 million investment pool, it is believed to be the first Catholic university in the nation to take this step.
Starting in the 1960s, the University began acquiring hundreds of single-family homes and duplexes in the neighborhoods adjacent to the campus for student housing, extending the campus to Brown Street. In 2007, the Univ
A patent is a form of intellectual property. A patent gives its owner the right to exclude others from making, using and importing an invention for a limited period of time twenty years; the patent rights are granted in exchange for an enabling public disclosure of the invention. In most countries patent rights fall under civil law and the patent holder needs to sue someone infringing the patent in order to enforce his or her rights. In some industries patents are an essential form of competitive advantage; the procedure for granting patents, requirements placed on the patentee, the extent of the exclusive rights vary between countries according to national laws and international agreements. However, a granted patent application must include one or more claims that define the invention. A patent may include many claims; these claims must meet relevant patentability requirements, such as novelty and non-obviousness. Under the World Trade Organization's TRIPS Agreement, patents should be available in WTO member states for any invention, in all fields of technology, provided they are new, involve an inventive step, are capable of industrial application.
There are variations on what is patentable subject matter from country to country among WTO member states. TRIPS provides that the term of protection available should be a minimum of twenty years; the word patent originates from the Latin patere, which means "to lay open". It is a shortened version of the term letters patent, an open document or instrument issued by a monarch or government granting exclusive rights to a person, predating the modern patent system. Similar grants included land patents, which were land grants by early state governments in the USA, printing patents, a precursor of modern copyright. In modern usage, the term patent refers to the right granted to anyone who invents something new and non-obvious; some other types of intellectual property rights are called patents in some jurisdictions: industrial design rights are called design patents in the US, plant breeders' rights are sometimes called plant patents, utility models and Gebrauchsmuster are sometimes called petty patents or innovation patents.
The additional qualification utility patent is sometimes used to distinguish the primary meaning from these other types of patents. Particular species of patents for inventions include biological patents, business method patents, chemical patents and software patents. Although there is some evidence that some form of patent rights was recognized in Ancient Greece in the Greek city of Sybaris, the first statutory patent system is regarded to be the Venetian Patent Statute of 1474. Patents were systematically granted in Venice as of 1474, where they issued a decree by which new and inventive devices had to be communicated to the Republic in order to obtain legal protection against potential infringers; the period of protection was 10 years.. As Venetians emigrated, they sought similar patent protection in their new homes; this led to the diffusion of patent systems to other countries. The English patent system evolved from its early medieval origins into the first modern patent system that recognised intellectual property in order to stimulate invention.
By the 16th century, the English Crown would habitually abuse the granting of letters patent for monopolies. After public outcry, King James I of England was forced to revoke all existing monopolies and declare that they were only to be used for "projects of new invention"; this was incorporated into the Statute of Monopolies in which Parliament restricted the Crown's power explicitly so that the King could only issue letters patent to the inventors or introducers of original inventions for a fixed number of years. The Statute became the foundation for developments in patent law in England and elsewhere. Important developments in patent law emerged during the 18th century through a slow process of judicial interpretation of the law. During the reign of Queen Anne, patent applications were required to supply a complete specification of the principles of operation of the invention for public access. Legal battles around the 1796 patent taken out by James Watt for his steam engine, established the principles that patents could be issued for improvements of an existing machine and that ideas or principles without specific practical application could legally be patented.
Influenced by the philosophy of John Locke, the granting of patents began to be viewed as a form of intellectual property right, rather than the obtaining of economic privilege. The English legal system became the foundation for patent law in countries with a common law heritage, including the United States, New Zealand and Australia. In the Thirteen Colonies, inventors could obtain patents through petition to a given colony's legislature. In 1641, Samuel Winslow was granted the first patent in North America by the Massachusetts General Court for a new process for making salt; the modern French patent system was created during the Revolution in 1791. Patents were granted without examination. Patent costs were high. Importation patents protected new devices coming from foreign countries; the patent law was revised in 1844 - patent cost was lowered and importation patents were abolished. The first Patent Act of the U. S. Congress was passed on April 10, 1790, titled "An Act to promote the progress of
United States Naval Computing Machine Laboratory
The United States Naval Computing Machine Laboratory was a secret design and manufacturing site for code-breaking machinery located in Building 26 of the National Cash Register company in Dayton and operated by the United States Navy during World War II. It is now on the List of IEEE Milestones, one of its machines is on display at the National Cryptologic Museum; the laboratory was established in 1942 by the Navy and National Cash Register Company to design and manufacture a series of code-breaking machines targeting German Enigma machines, based on earlier work by the British at Bletchley Park. Joseph Desch led the effort. Preliminary designs, approved in September 1942, called for a electronic machine to be delivered by year's end. However, these plans were soon judged infeasible, revised plans were approved in January 1943 for an electromechanical machine, which became the US Navy bombe; these designs were proceeding in parallel with, influenced by, British attempts to build a high-speed bombe for the German 4-rotor Enigma.
Indeed, Alan Turing visited Dayton in December 1942. His reaction was far from enthusiastic: It seems a pity for them to go out of their way to build a machine to do all this stopping if it is not necessary. I am now converted to the extent of thinking that starting from scratch on the design of a Bombe, this method is about as good as our own; the American Bombe program was to produce one for each wheel order. I used to smile inwardly at the conception of Bombe hut routine implied by this program, their test can hardly be considered conclusive as they were not testing for the bounce with electronic stop finding devices. Nobody seems to be told about rods or offiziers or banburismus unless they are going to do something about it; the American approach was, successful. The first two experimental bombes went into operation in May 1943, running in Dayton so they could be observed by their engineers. Designs for production models were completed in April, 1943, with initial operation starting in early June.
All told, the laboratory constructed 121 bombes which were employed for code-breaking in the US Navy's signals intelligence and cryptanalysis group OP-20-G in Washington, D. C.. Construction was accomplished in three shifts per day by some 600 WAVES, 100 Navy officers and enlisted men, a large civilian workforce. 3,000 workers operated the bombes to produce "Ultra" decryptions of German Enigma traffic. According to a contemporary US Navy report, the bombes were used on naval jobs until all daily keys had been run. During the previous six months, about 45% of the bombe time had been devoted to non-naval problems carried out at the request of the British. British production and reliability problems with their own high-speed bombes had recently led to construction of 50 additional Navy units for Army and Air Force keys; the building in Dayton, called Building 26 was an Art Deco design of Dayton firm Schenck & Williams and was located at Patterson Blvd and Stewart Street. The building was demolished by the University of Dayton in January 2008.
History of the Bombe Project John A. N. Lee, Colin Burke, Deborah Anderson, "The US Bombes, NCR, Joseph Desch, 600 WAVES: The First Reunion of the US Naval Computing Machine Laboratory", IEEE Annals of the History of Computing, Volume 22, No. 3 July–September 2000. Dayton Codebreakers IEEE Global History Network IEEE History Center Oral history interview with Robert E. Mumma, Charles Babbage Institute, University of Minnesota. Mumma recounts how NCR's war-time work on cryptanalytic equipment took all the company's effort, how this shaped company policy resisting government contract work after the war. Oral history interview with Carl Rench, Charles Babbage Institute, University of Minnesota. DeBrosse, Jim.
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to