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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of topics such as quantity …

Image: Euclid

Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World

Carl Friedrich Gauss, known as the prince of mathematicians

Leonhard Euler, who created and popularized much of the mathematical notation used today

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1. Mathematics – Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2. Euclid – Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost

3. Calipers – A caliper is a device used to measure the distance between two opposite sides of an object. A caliper can be as simple as a compass with inward or outward-facing points. The tips of the caliper are adjusted to fit across the points to be measured, the caliper is then removed and it is used in many fields such as mechanical engineering, metalworking, forestry, woodworking, science and medicine. A plurale tantum sense of the word calipers coexists in natural usage with the regular noun sense of caliper, also existing colloquially but not in formal usage is referring to a vernier caliper as a vernier or a pair of verniers. In imprecise colloquial usage, some extend this even to dial calipers. In machine-shop usage, the caliper is often used in contradistinction to micrometer. In this usage, caliper implies only the factor of the vernier or dial caliper. The earliest caliper has been found in the Greek Giglio wreck near the Italian coast, the ship find dates to the 6th century BC. The wooden piece already featured a fixed and a movable jaw, although rare finds, caliper remained in use by the Greeks and Romans. A bronze caliper, dating from 9 AD, was used for minute measurements during the Chinese Xin dynasty, the caliper had an inscription stating that it was made on a gui-you day at new moon of the first month of the first year of the Shijian guo period. The calipers included a slot and pin and graduated in inches, the modern vernier caliper, reading to thousandths of an inch, was invented by American Joseph R. Brown in 1851. It was the first practical tool for exact measurements that could be sold at a price within the reach of ordinary machinists, the inside calipers are used to measure the internal size of an object. The upper caliper in the image requires manual adjustment prior to fitting, fine setting of this caliper type is performed by tapping the caliper legs lightly on a handy surface until they will almost pass over the object. A light push against the resistance of the pivot screw then spreads the legs to the correct dimension and provides the required. The lower caliper in the image has a screw that permits it to be carefully adjusted without removal of the tool from the workpiece. Outside calipers are used to measure the size of an object. The same observations and technique apply to this type of caliper, with some understanding of their limitations and usage, these instruments can provide a high degree of accuracy and repeatability. They are especially useful when measuring over very large distances, consider if the calipers are used to measure a large diameter pipe, a vernier caliper does not have the depth capacity to straddle this large diameter while at the same time reach the outermost points of the pipes diameter

4. Raphael – Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, together with Michelangelo and Leonardo da Vinci, he forms the traditional trinity of great masters of that period. Raphael was enormously productive, running a large workshop and, despite his death at 37. Many of his works are found in the Vatican Palace, where the frescoed Raphael Rooms were the central, the best known work is The School of Athens in the Vatican Stanza della Segnatura. After his early years in Rome much of his work was executed by his workshop from his drawings and he was extremely influential in his lifetime, though outside Rome his work was mostly known from his collaborative printmaking. Raphael was born in the small but artistically significant central Italian city of Urbino in the Marche region and his poem to Federico shows him as keen to show awareness of the most advanced North Italian painters, and Early Netherlandish artists as well. In the very court of Urbino he was probably more integrated into the central circle of the ruling family than most court painters. Under them, the court continued as a centre for literary culture, growing up in the circle of this small court gave Raphael the excellent manners and social skills stressed by Vasari. Castiglione moved to Urbino in 1504, when Raphael was no longer based there but frequently visited, Raphael mixed easily in the highest circles throughout his life, one of the factors that tended to give a misleading impression of effortlessness to his career. He did not receive a humanistic education however, it is unclear how easily he read Latin. His mother Màgia died in 1491 when Raphael was eight, followed on August 1,1494 by his father, Raphael was thus orphaned at eleven, his formal guardian became his only paternal uncle Bartolomeo, a priest, who subsequently engaged in litigation with his stepmother. He probably continued to live with his stepmother when not staying as an apprentice with a master and he had already shown talent, according to Vasari, who says that Raphael had been a great help to his father. A self-portrait drawing from his teenage years shows his precocity and his fathers workshop continued and, probably together with his stepmother, Raphael evidently played a part in managing it from a very early age. In Urbino, he came into contact with the works of Paolo Uccello, previously the court painter, and Luca Signorelli, according to Vasari, his father placed him in the workshop of the Umbrian master Pietro Perugino as an apprentice despite the tears of his mother. The evidence of an apprenticeship comes only from Vasari and another source, an alternative theory is that he received at least some training from Timoteo Viti, who acted as court painter in Urbino from 1495. An excess of resin in the varnish often causes cracking of areas of paint in the works of both masters, the Perugino workshop was active in both Perugia and Florence, perhaps maintaining two permanent branches. Raphael is described as a master, that is to say fully trained and his first documented work was the Baronci altarpiece for the church of Saint Nicholas of Tolentino in Città di Castello, a town halfway between Perugia and Urbino. Evangelista da Pian di Meleto, who had worked for his father, was named in the commission