own history of mathematics course to me; as one might expect, it was .. of science, and of the history of mathematics in particular. ronaldweinland.info Fig. Euclid's Elements and its role in the history of Mathematics. 8. Archimedes, Eratosthenes. 9. Curves in the Greek Geometry, Apollonius the Great Geometer. PDF | This is a survey on the recent developments (since ) concerning research on the relations between History and Pedagogy of Mathematics (the HPM.
|Language:||English, Spanish, Dutch|
|Genre:||Academic & Education|
|ePub File Size:||15.64 MB|
|PDF File Size:||16.78 MB|
|Distribution:||Free* [*Register to download]|
The history of mathematics: an introduction / David M. Burton.—7th ed. p. cm. Includes bibliographical references and index. ISBN (alk. A Brief History of Mathematics. What is mathematics? What do mathematicians do? ronaldweinland.info~rpyke/ronaldweinland.info There are many excellent reasons to study the history of mathematics. It helps We also encourage History of Mathematics courses taught to entering students.
Skip to main content. Log In Sign Up. Marsigit Hrd. 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texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. The achievements of prehistoric mathematics and the flowering of Pythagorean have significant evidences to trace the nature of mathematics. While the challenge of non-Euclidean geometry to Euclidean geometry has some impacts to the development of contemporary mathematics.
Around BC, the Akkadians4 invented the abacus as a tool for counting and they developed methods of arithmetic with addition, subtraction, multiplication and division.
Around BC, Sumerians had developed an abstract form of writing based on cuneiform i. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day.
The Babylonians 5 appear to have developed a placeholder symbol that functioned as a zero by the 3rd century BC, but its precise meaning and use is still 3 Reed, I. They had no mark to separate numbers into integral and fractional parts as with the modern decimal point.
The three-place numeral 3 7 30 could represent: Similarly, arithmetic started with the commerce and trade of Phoenician merchants. Some hints 7 about the nature of early Greek practical mathematics are confirmed in the arithmetic problems in papyrus texts from Ptolemaic Egypt.
Greek tradition8 was much like the earlier traditions in Egypt and Mesopotamia; however, its development as a theoretical discipline was a distinctive contribution to mathematics.
Greeks came a proof of a general rule for finding all such sets of numbers called Pythagorean triples Figure. Pythagorean triples. For Mesopotamians it appears to be understood that the sets of such numbers a, b, and c form the sides of right triangles, but the Greeks proved this result by Euclid in Element. The transition from practical to theoretical mathematics initiated by Pythagoras by establishing that all things are number and any geometric measure can be associated with some number.
The Flowering of Pythagoreans Pythagoras of Samos10 is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. Pythagoras held that at its deepest level, reality is mathematical in nature.
Pythagoreans 11 represents a coherent body of mathematical doctrines believed that number rules the universe. They made no distinction between mathematics and physics and concern with the study of properties of counting numbers. They believe all measurements could be expressed in terms of natural numbers, or ratios of natural numbers.
They develop geometric theorems and insist that mathematical ideas required proofs. They think numbers had concrete representations as figures of point e. Posy12 points out three important Pythagorean beliefs: J and Robertson, E. The document elaborates the finding of the breath of a rectangle in which its length and diagonal hold, as the following: What is the breadth?
Its size is not known. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? Jones R. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square see Figure 2.
Commensurability of Pythagoras Pythagoras13 held that all things are number and any geometric measure can be associated with some number as the following examples: Euclidean Geometry Around BC14, Euclid was studying geometry in Alexandria and wrote a thirteen-volume book that compiled all the known and accepted rules of geometry 15 called The Elements. Euclid believes in absolute separation of discrete mathematics and magnitudes.
Of the Element, for example, Books 5 and 6 state the theory of proportion for magnitudes, while Book 7 states the theory of proportion for numbers. In these Elements, Euclid attempted to define all geometrical terms 13 Berggren, J. Because mathematics is a science where every theorem is based on accepted assumptions, Euclid first had to establish some axioms with which to use as the basis of other theorems. Euclid uses five axioms as the 5 assumptions, which he needs to prove all other geometric ideas.
The use and assumption of these five axioms16 is what it called something to be categorized as Euclidean geometry. The first four postulates 18 state about straight line that may be drawn from any two points; any terminated straight line that may be extended indefinitely; a circle that may be drawn with any given center and any given radius; and all right angles that are congruent.
Explicitly, those postulates19 are as follows: In the third postulate, Euclid had in mind drawing the circle with center A and radius r, and this postulate tell us that such a drawing is allowed Figure 3. Figure 3: The fifth or the last postulate listed by Euclid stands out a little bit.
It is a bit less intuitive and a lot more convoluted.
It looks like a condition of the geometry more than something fundamental about it. The fifth postulate is see Figure 4: Postulate V If two straight lines lying in a plane are met by another line, and if the sum of he internal angles on one side is less than two right angles, then the straight lines will meet if the extended on the side on which the sum of the angles is less than two right angles.
The Fifth Postulate of Euclides The Fifth Postulate of Geometry Parallel Postulate means that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles and if the two straight lines are produced indefinitely, they will meet on that side on which the angles less than two right angles.
From this postulate, we may have a question what is the criterion for line l to be parallel to line m? Criterion for parallel line Euclid suggested drawing a transversal Figure 5 , i. Non-Euclidean Geometry Pythagoras20 was the first who inclined to regard number theory as more basic than geometry. The discovery of in-commensurable ratios presented them with a foundational crisis not fully resolved until the 19th century.
Since Greek21, number theory, which concerns only whole numbers, cannot adequately deal with the magnitudes found in geometry. Because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. For it found the irrational numbers, it can be proved that commensurability is false. To proof that commensurability is false we can use reductio ad absurdum procedure.
Posy, C. This can not be proven from the other axioms. See Figure 6 20 Jones, R. Mark off segment AB in the direction of C Euclid22 may forget when it given a line, there must be two points on it, and there exists at least one point not on the line. He should have stated the "self-evident" assumptions. Modern mathematicians perceive that the Fifth Axiom of Geometry is too complex, and seemed derivable from the other postulates. Wallis, J. In other words, similarity preserves shape.
Size, shape, and location are independent of each other and this simpler postulate allows derivation of Euclid's fifth postulate. Meanwhile, Saccheri and Lambert 24 independently tried proving the parallel postulate by the reductio ad absurdum method. Their proof starts with "Neutral geometry" that is Euclidean geometry that excludes the parallel postulate. There 22 Posy, C. Neutral geometry Possibility one is equivalent to the parallel postulate; possibility two can be shown to be contradictory to Neutral geometry; and possibility three couldn't be proven contradictory.
Saccheri and Lambert25 are repugnant to the nature of the straight line and space; and this was the discovery of non-Euclidean geometry. In the late 18th and early 19th century, three men became interested in the acute angle hypothesis. Bolyai, Gauss, and Zobachevsky took the negation of the parallel postulate as a postulate and added it to neutral geometry; and resulting that nothing contradictory followed.
Negation26 of the Parallel Postulate states that there is at least one line l and one point p outside of l such that through p there are at least two lines which do not intersect l.
This implies through any point outside of l, there are infinitely many lines parallel to l. This new geometry is called 25 Posy, C. Angles in a triangles sum to less than degrees. Angles in a quadrilateral sum to less than degrees.
Rectangles do not exist. If two triangles are similar, then they are congruent-- size and shape are not independent. On the other hand, Riemann 27 develops elliptical geometry where there are no parallel lines.
Klein and Belttrami independently prove that there is no hope of contradiction between neutral geometry and the negation of the parallel postulate. If Euclidean geometry is consistent, it must also be true that no contradiction can occur in non-Euclidean geometry.
Accordingly, one could model non-Euclidean geometry inside Euclidean geometry. The Road to Contemporary Mathematics The 17th century28, the period of the scientific revolution, witnesses the consolidation of Copernican heliocentric astronomy and the establishment of inertial physics in the work of Kepler, Galileo, Descartes, and Newton. This period is also one of intense activity and innovation in mathematics.
Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of the differential and integral calculus resulted in a major 27 Ibid.
By the end of the 17th century a program of research based in analysis had replaced classical Greek geometry at the centre of advanced mathematics.
In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and the adoption of a pragmatic attitude to questions of logical rigor distinguished the new mathematics from traditional geometry.
The Invention of the Calculus In his treatise Geometria Indivisibilibus Continuorum Cavalieri29, formulates a systematic method for the determination of areas and volumes. Cavalieri 30 thinks a plane strip can be thought of as infinitely many parallel indivisibles, etc; and he then described principle that, called then Cavalieri's principle, one can move indivisibles composing a figure independently of each other and thus recreate the figure.
It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as BCE and probably much early.
Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million.
Many of the mathematical tablets are "problem texts: Many of the problems involve geometry; the rest are almost always "word problems" where the context is the calculation of the area of an irregular field, the volume of a ditch, the number of bricks to build a ramp,and etc link: One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty — BC.
For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching contained elements of binary numbers. Mantras from the early Vedic period before BCE invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus ca.
Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of BC, which probably occurred while he was in his prime. Despite this, it is generally agreed that Thales is the first of the seven wise men of Greece.
The two earliest mathematical theorems, Thales' theorem and Intercept theorem are attributed to Thales. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed. Although it is not known whether or not Thales was the one who introduced into mathematics the logical structure that is so ubiquitous today, it is known that within two hundred years of Thales the Greeks had introduced logical structure and the idea of proof into mathematics.
Babylonian mathematics also known as Assyro-Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in BC.
Babylonian mathematical texts are plentiful and well edited. When Chinese, Islamic, and Indian mathematicians had been in ascendancy, and Europe fell in Dark Ages, almost all mathematics and intellectual endeavor stagnated. From the 4th to the 12th century, studies of geometry, arithmetic, and translations was limited to Boethius translations of some words of ancient Greek masters. The Islamic Empire made significant contributions to math, since the 8th century.
Over time, Muslim artists discovered all different types of symmetry that can be done on a 2D surface. An important person in the early 16th century was an Italian Franciscan friar named Luca Pacioli.