History of Mathematics
History of Mathematics MATH 2720
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This 4 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2720 at University of Connecticut taught by Gerald Leibowitz in Fall. Since its upload, it has received 5 views. For similar materials see /class/205826/math-2720-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
DIOPHANTUS S Problem 1128 In this problem Diophantus asks for two square numbers whose product added to either results in a square numberi Once again he is intrigued by the extent to which rational numbers can be represented as squaresi His answer is and Letls look at the problem from our viewpointi We are looking for positive rational numbers I and y for which 12f 12 and 12f y2 are squares of rational numbers Thus after factoring the conditions to be met are that 12y2 l and y2z2 l are squaresi This is equivalent to both 12 1 and y2 1 being squares But the condition that 12 l a2 can be transformed If we put I and a over a common denominator then the problem is to nd the numerators and that denominator to nd positive integers u vn so that z una vnu2 n2 112 So in fact this is a problem about nding Pythagorean triplesi The extent to which Diophantus realized that HQS concerned Pythagorean triples isn7t clear If we use the basic triple u 3n 4 v 5 for z and similarly use the triple 72425 for y we get 12 E2 and y2 Diophantusls special solutioni Hellenistic Astronomers and the Origins of Trigonometry a brieflook Trigonometry was developed primarily as a tool for solving problems in Greek astronomy during the period 300 BCE7300 CE The major gures in mathematical astronomy were Aristarchus of Samos 3107230 BCE Hipparchus 1807125 BCE Menelaus of Alexandria circa 100 CE and Claudius Ptolemy 857165 CE The trigonometric functions of today 7 sine cosine tangent and their reciprocals 7 were de ned and studied much later but the seeds were sown by these scientists and some of their contemporaries To Aristarchus the Sun was at the center of the solar system See Thomas L Heath s bookAristarehus of Samos the ancient Copernicus About 260 BCE Aristarchus devised a method for measuring the distance between the Earth and the Sun He couldn t nd an absolute distance but rather he found the ratio of the distances between the Earth and the Sun and the Earth and the Moon Aristarchus observed that when the Moon is half full the angle between the line of sight from Earth to Sun and the line of sight from Earth to Moon was a small amount less than a right angle The corresponding gure is a right triangle M S E where A SEM is close to a right angle and hence A ESM is a small angle Aristarchus estimated A ESM to be onethirtieth of a quadrant in our notation 30 Since trigonometry had not yet been invented Aristarchus couldn t then say that MESE sin 3 and then use a trig table or a calculator to nd the numerical value of csc 3 as his ratio What he did do was to derive inegualities which he used to bound the ratio between two quantities Aristarchus inequalities are expressible in modern terms as saying that sincx or tanor lt i lt sinB 3 tan 3 These are impressive results Observe that they say that on the open interval from 0 to a right angle sin xx is a decreasing function and tan xx is an increasing function Aristarchus then showed that 120 lt sin 3 lt 118 so the Sun is between 18 and 20 times as far from the Earth as the Moon is Today it is known that the Sun is in fact much farther from the Earth than that amount The reason for the discrepancy is not in the mathematics 7 It s because A ESM is smaller than the ancient estimate close to 1639h of a degree If one uses this value and Aristarchus calculations one gets a very good idea of the true value of the relation of the distances Clearly for reasoning such as that of Aristarchus to be useful in solving other problems concerning the motion of the planets comets and stars someone had to devise a metho nding values of the sine function and then do the calculations and make the results available The major step was taken by Hipparchus He did not use the sine or cosine functions however For Hipparchus and the later Greek writers the fundamental trigonometric object was the chord of a circle as a function of the corresponding Measuring the chord in units expressed in terms of the radius of the circle Hipparchus made tables of the chord of an arc In a twelvevolume book he explained his methods of calculation wrote out his table and applied it to derive many facts about speci c stars including the locations of 850 xed stars the length of the lunar month and the year the size of the Moon and other facts in astronomy Unfortunately nearly all of Hipparchus writings are no longer in existence what we know of them comes from much later commentaries about the work of Claudius Ptolemy the later mathematical astronomer If on and 5 are acute angles and if 5 lt 01 then Menelaus studied geometry in a plane and on a sphere A theorem of advanced Euclidean geometry is named after him Along the way he needed to deal with the relation of chord to arc A When the radius OB is extended to a diameter BB the Pythagorean B theorem can be applied to the triangle BAB since an angle inscribed in a B semicircle is a right angle AB2 AB 2 diameter2 Here AB is the chord of arc AB and AB is the chord of semicircle 7 arc AB This is essentially the fundamental identity of modern trigonometry sin2 6 cos2 6 l since the cosine is the complement s sine Slnrt 2 3 C089 Both Menelaus and C Ptolemy used this identity to derive many theorems Claudius Ptolemy wrote major books about geography and astronomy In his cosmology the Earth is at the center of the universe and planetary orbits are slightly perturbed circular paths Ptolemy continued the work of Hipparchus and used new trigonometric identities to make very detailed tables of values of the chord function These included a halfarc formula like our Sin9 2 formula The halfarc formula contains a square root so Ptolemy had to calculate numerous square roots inevitably resulting in truncation or roundoff errors of approximation These trigonometric tables were used in elaborate often elegant astronomical calculations Following Hipparchus or perhaps Hipsicles 180 BCE he divided a circle into 360 equal parts which his predecessors had called degrees He used sexagesimal subdivisions of a part to avoid as he said the embarrassment of fractions So the concept of a fraction apparently still required a splitting into a sum of distinct unit fractions For instance he approximated crd 36 as 37p 4 55 with one chordal part being one sixtieth of a radius since he usually referred to a radius of 60 units this is 37455 units in Neugebauer s notation otherwise it is 037455 ofa radius He approximated the ratio later called pi by 3830 7 which is 377120 or 3141666 Ptolemy discovered spherical as well as planar trigonometric identities Two of Ptolemy s properties of chords of a circle correspond to the formulas for the sine of a sum and a difference of two angles He applied these facts to prove what is now called Ptolemy s Theorem for Cyclic Quadrilaterals which says that given four successive points on a circle the sum of the products of the lengths of the two sets of opposite sides equals the product of the diagonals All of the work mentioned above appeared in his book Syntaxis Mathematica Mathematical Collection some time around 150 CE In his A History of Mathematics Carl Boyer calls it by far the most in uential and signi cant trigonometric work of all antiquity The scientists who read the Syntaxis over the centuries 39 39 39 it to be 39 C called it magiste or greatest and after it was translated into Arabic Moslem scholars called the collectionAl M agiste This linguisitically mixed phrase meant the greatest and gradually the title of Ptolemy s masterpiece became the Almagest This book remained the standard astronomy book for 1300 years Ptolemy wrote an eight volume Geography in which he again applied geometry and trigonometry as well as various methods for projecting the points on a sphere into a plane 7 including the method now known as stereographic projection Boyer remarks The importance of Ptolemy for geography can be gauged from the fact that the earliest maps in the Middle Ages that have come down to us in manuscripts none before the thirteenth century had as prototypes the maps made by Ptolemy more than a thousand years earlier Let us nish this brief summary with a quotation from Claudius Ptolemy himself When I trace at my pleasure the windings to and fro of the heavenly bodies I no longer touch the earth with my feet I stand in the presence of Zeus himself and take my ll of ambrosia food of the gods Trigonometric Functions Some History Trigonometry began in the work of Greek astronomers especially Aristarchus of Samos and was developed further by the Greek mathematical astronomers Claudius Ptolemy and Menelaus in Alexandria Egypt in the first few centuries of the Common Era For these scholars there was one major function the length of the chord of a circle as a function of the length of the circular arc joining its endpoints It was Indian astronomers 500 900 CE who decided to study what we now call the sine and cosine as functions of an angle in a right triangle After this knowledge was brought to the Arab and Persian world in medieval times and developed further it was transmitted to Europe through contacts among merchants and as a result of warfare The word quotsinusquot arose from a misinterpretation of the Arabic word for this concept Thomas Flinck of Flensburg Germany introduced the names quottangentquot and quotsecantquot into trigonometry in his book of 1583 Between 1624 and 1636 Edmund Gunter invented quotcosinequot and quotcotangentquot with the prefix quotco quot meaning quotcomplementquot In a right triangle the cosine of an acute angle is the sine of its complementary angle Inverse trigonometric functions were considered early in the 1700s by Daniel Bernoulli who used quotAsinquot for the inverse sine of a number and in 1736 Euler wrote quotA tquot for the inverse tangent As early as 1772 JL Lagrange used the symbols quotarcsinquot and quotarctanquot These writers were identifying an angle with the arc it subtends when placed at the center of a circle John Herschel introduced the sin4 and tan4 notations in an article in the Philosophical Transactions of London in 1813
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