Hipparchus /hɪˈpɑːrkəs/ of Nicaea, or morecorrectly Hipparchos (Greek: Ἵππαρχος,Hipparkhos; c. 190 BC – c. 120 BC), wasa Greek astronomer, geographer, andmathematician of the Hellenistic period. He isconsidered the founder of trigonometrybut is most famous for his incidental discoveryof precession of the equinoxes.Hipparchus was born in Nicaea, Bithynia (now Iznik, Turkey), and probably died on the islandof Rhodes. He is known to have been a working astronomer at least from 162 to127 BC.Hipparchus is considered the greatest ancient astronomical observer and, bysome, the greatest overall astronomer of antiquity. He was the first whose quantitative andaccurate models for the motion of the Sun and Moon survive. For this he certainly made useof the observations and perhaps the mathematical techniques accumulated over centuries bythe Chaldeans from Babylonia. He developed trigonometry andconstructedtrigonometric tables, and he solved several problems of spherical trigonometry.With his solar and lunar theories and his trigonometry, he may have been the first to developa reliable method to predict solar eclipses. His other reputed achievements include thediscovery of Earths precession, the compilation of the first comprehensive star catalog of thewestern world, and possibly the invention of the astrolabe, also of the armillary sphere, whichhe used during the creation of much of the star catalogue. It would be three centuriesbefore Claudius Ptolemaeus synthesis of astronomy would supersede the work ofHipparchus; it is heavily dependent on it in many areas.
Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise onmathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematicsteacher of all time.There is other information about Euclid given by certain authors but it is not thought to be reliable. Two different typesof this extra information exists. The first type of extra information is that given by Arabian authors who state thatEuclid was the son of Naucrates and that he was born in Tyre. It is believed by historians of mathematics that this isentirely fictitious and was merely invented by the authors.The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authorswho first gave this information. In fact there was aEuclid of Megara, who was a philosopher who lived about 100years before the mathematician Euclid of Alexandria. It is not quite the coincidence that it might seem that there weretwo learned men called Euclid. In fact Euclid was a very common name around this period and this is one furthercomplication that makes it difficult to discover information concerning Euclid of Alexandria since there are referencesto numerous men called Euclid in the literature of this period.Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in thedating given. However, although we do not know for certain exactly what reference to Euclid in Archimedeswork Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in Onthe sphere and the cylinder. The obvious conclusion, therefore, is that all is well with the argument of Proclus and thiswas assumed until challenged by Hjelmslev in . He argued that the reference to Euclid was added to Archimedesbook at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give suchreferences, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid andthere is no such reference.Euclids most famous work is his treatise on mathematics The Elements. The book was a compilation of knowledgethat became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were firstproved by Euclid but the organisation of the material and its exposition are certainly due to him. In fact there is ampleevidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number ofdefinitions which are never used such as that of an oblong, a rhombus, and a rhomboid.The Elements begins with definitions and five postulates. The first three postulates are postulates of construction, forexample the first postulate states that it is possible to draw a straight line between any two points. These postulatesalso implicitly assume the existence of points, lines and circles and then the existence of other geometric objects arededuced from the fact that these exist. There are other assumptions in the postulates which are not explicit. Forexample it is assumed that there is a unique line joining any two points. Similarly postulates two and three, onproducing straight lines and drawing circles, respectively, assume the uniqueness of the objects the possibility ofwhose construction is being postulated.The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This mayseem "obvious" but it actually assumes that space in homogeneous - by this we mean that a figure will beindependent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one andonly one line can be drawn through a point parallel to a given line. Euclids decision to make this a postulate led toEuclidean geometry. It was not until the 19thcentury that this postulate was dropped and non-euclideangeometries were studied.There are also axioms which Euclid calls common notions. These are not specific geometrical properties but rathergeneral assumptions which allow mathematics to proceed as a deductive science.
Diophantus of Alexandria (Ancient Greek: Διόφαντορ ὁ Ἀλεξανδπεύρ. b. between A.D. 200 and 214, d. between 284 and 298 at age84), sometimes called "the father of algebra", was anAlexandrian Greek mathematicianand the author of a series of bookscalledArithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica, Pierre deFermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he hadfound "a truly marvelous proof of this proposition," now referred to as Fermats Last Theorem. This led to tremendous advancesin number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remainimportant areas of mathematical research. Diophantus coined the term παρισὀτης to refer to an approximate equality.This termwas rendered as adaequalitat in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima forfunctions and tangent lines to curves. Diophantus was the firstGreek mathematician who recognized fractions as numbers; thus heallowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraicequations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between A.D. 200 and 214 to 284 or 298. Much of ourknowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems(sometimes called his epitaph) states:Here lies Diophantus, the wonder behold.Through art algebraic, the stone tells how old:God gave him his boyhood one-sixth of his life,One twelfth more as youth while whiskers grew rife;And then yet one-seventh ere marriage begun;In five years there came a bouncing new son.Alas, the dear child of master and sageAfter attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years, heended his life.This puzzle implies that Diophantus lived to be 84 years old. However, the accuracy of the information cannot be independently confirmed.In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandoras Box as one of the hardest solving puzzles in the game,which needed to be unlocked by solving other puzzles first.Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called DarkAges, since the study of ancient Greek had greatly declined. The portion of the Greek Arithmetica that survived,however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medievalByzantine scholars. In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In1463 German mathematician Regiomontanus wrote:“No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flowerof the whole of arithmetic lies hidden . . . .”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published.However, Bombelli borrowed many of the problems for his own book Algebra. The editioprinceps of Arithmetica was published in 1575 by Xylander. The best known Latin translation of Arithmetica wasmade by Bachet in 1621 and became the first Latin edition that was widely available.Pierre de Fermat owned a copy,studied it, and made notes in the margins.