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Overview
Beginning with the ancient Near East, the author traces the ideas and techniques developed in Egypt, Babylonia, China, and Arabia, looking into such manuscripts as the Egyptian Papyrus Rhind, the Ten Classics of China, and the Siddhantas of India. He considers Greek and Roman developments from their beginnings in Ionian rationalism to the fall of Constantinople; covers medieval European ideas and Renaissance trends; analyzes 17th- and 18th-century contributions; and offers an illuminating exposition of 19th century concepts. Every important figure in mathematical history is dealt with — Euclid, Archimedes, Diophantus, Omar Khayyam, Boethius, Fermat, Pascal, Newton, Leibniz, Fourier, Gauss, Riemann, Cantor, and many others. For this latest edition, Dr. Struik has both revised and updated the existing text, and also added a new chapter on the mathematics of the first half of the 20th century. Concise coverage is given to set theory, the influence of relativity and quantum theory, tensor calculus, the Lebesgue integral, the calculus of variations, and other important ideas and concepts. The book concludes with the beginnings of the computer era and the seminal work of von Neumann, Turing, Wiener, and others.
"The author's ability as a first-class historian as well as an able mathematician has enabled him to produce a work which is unquestionably one of the best." — Nature Magazine.
Product Details
| ISBN-13: | 9780486138886 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 06/28/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 8 MB |
About the Author
Dirk. J. Struik was born in Rotterdam in 1894 and spent most of his teaching career at MIT; he retired in 1960. His Lectures on Classical Differential Geometry, reprinted by Dover in 1988, is still a highly regarded classic, as is his Concise History of Mathematics, one of the first Dover original books in mathematics and first published by Dover in 1948, which reached its current fourth revised edition in 1987.
Professor Struik died on October 21, 2000, twenty-one days after his 106th birthday. Professor. Thomas F. Banchoff of Brown University, longtime friend and colleague of Dr. Struik and an advisor to Dover for the past 30 years, here tells the story of his friend's memorable 100th birthday celebration:
"Dirk Struik was 97 at the time I asked him what he planned to do on his hundredth birthday. He said that his family always had a party, but I then thought of a bright idea, a public celebration lecture where he would sit in the front row and hear people from his past say laudatory things about his contributions. I blurted out, 'What about a lecture on your hundredth birthday?' Without hesitation, he agreed, and that was the start of a grand event.
"Well over two hundred fifty people attended his lecture, about a third who knew him from his mathematical writings, another third acquainted with his work in history and politics, and, according to one wag, the rest wanting to see a hundred-year-old man stand up for an hour. Joan Richards gave a sterling introduction covering the many aspects of his long career. The talk itself was full of personal reflections about the characteristics of these almost legendary figures in modern mathematics and the audience was most appreciative.
"Dirk Struik went on giving lectures, in the United States and in the Netherlands for the next four years. He was a good friend to many people in his long life, and his books on so many subjects will continue to provide inspiration and encouragement to generations of students and teachers." — Tom Banchoff
Read an Excerpt
A Concise History Of Mathematics
By DIRK JAN STRUIK
Dover Publications, Inc.
Copyright © 1987 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-13888-6
CHAPTER 1
The Beginnings
1
Our first conceptions of number and form date back to times as far removed as the Old Stone Age, the Paleolithic. Throughout the hundreds or more millennia of this period men lived in small groups, under conditions differing little from those of animals, and their main energies were directed toward the elementary process of collecting food wherever they could get it. They made weapons for hunting and fishing, developed a language to communicate with each other, and in later paleolithic times enriched their lives with creative art forms, including statuettes and paintings. The paintings in caves of France and Spain (over 15,000 years old) may have had some ritual significance; certainly they reveal a remarkable understanding of form; mathematically speaking, they reveal understanding of two-dimensional mapping of objects in space.
Little progress was made in understanding numerical values and space relations until the transition occurred from the mere gathering of food to its actual production, from hunting and fishing to agriculture. With this fundamental change, a revolution in which the passive attitude of man toward nature turned into an active one, we enter the New Stone Age, the Neolithic.
This great event in the history of mankind occurred perhaps ten thousand years ago, after the ice sheet that covered Europe and Asia had melted and made room for forests and deserts. Here nomadic wandering in search of food came slowly to an end. Fishermen and hunters were in large part replaced by simple farmers. Such farmers, remaining in one place as long as the soil stayed fertile, began to build more permanent dwellings; villages emerged as protection against the climate and against predatory enemies. Many such neolithic settlements have been excavated. The remains show how gradually elementary crafts such as pottery, carpentry, and weaving developed. There were granaries, so that the inhabitants were able to provide against winter and hard times by establishing a surplus. Bread was baked, beer was brewed, and in late neolithic times copper and bronze were smelted and prepared. Inventions appeared, notably the potter's wheel and the wagon wheel; boats and shelters were improved. All these remarkable innovations occurred only within limited areas and did not always spread to other localities. The American Indian, for example, did not learn much about the technical use of the wagon wheel until the coming of the European. Nevertheless, as compared with paleolithic times, the tempo of technical improvement was enormously accelerated.
Between the villages a considerable trade existed, which so expanded that connections can be traced between places hundreds of miles apart. The discovery of the arts of smelting and manufacturing, first copper, then bronze tools and weapons, strongly stimulated this commercial activity. This again promoted the further formation of languages. The words of these languages expressed very concrete things and very few abstractions, but already there was room for some simple numerical terms and for some form relations. Many Australian, American, and African tribes were in this stage at the period of their first contact with Europeans: some tribes are still living in these conditions, so that it is possible to study their habits and forms of expression, and to some extent to understand them if we can strip ourselves of preconceived notions.
2
Numerical terms—expressing some of "the most abstract ideas which the human mind is capable of forming," as Adam Smith has said—came only slowly into use. Their first occurrence was qualitative rather than quantitative, making a distinction only between one (or better "a"—"a man," rather than "one man") and two and many. In the old Fiji Island language ten boats are called bola, ten coconuts koro, and a thousand coconuts saloro. The ancient qualitative origin of numerical conceptions can still be detected in the special dual terms existing in certain languages such as Greek or Celtic. When the number concept was extended, higher numbers were first formed by addition: 3 by adding 2 and 1, 4 by adding 2 and 2, 5 by adding 2 and 3.
Here is an example from some Australian peoples:
Murray River: 1 = enea, 2 = petcheval, 3 = petcheval-enea, 4 = petcheval petcheval
Kamilaroi: 1 = mal, 2 = bulan, 3 = guliba, 4 = bulan bulan, 5 = bulan guliba, 6 = guliba guliba.
The development of the crafts of commerce stimulated this crystallization of the number concept. Numbers were arranged and bundled into larger units, usually by the use of the fingers of the hand or of both hands, a natural procedure in trading. This led to numeration first with five, later with ten as a base, completed by addition and sometimes by subtraction, so that 12 was conceived as 10 + 2, or 9 as 10–1. Sometimes 20, the number of fingers and toes, was selected as a base. Of 307 number systems of primitive American peoples investigated by W. C. Eels, 146 were decimal, 106 quinary and quinary decimal, vigesimal and quinary vigesimal. The vigesimal system in its most characteristic form occurred among the Mayas of Mexico and the Celts in Europe.
Numerical records were kept by means of bundling: strokes on a stick, knots on a string, pebbles or shells arranged in heaps of fives—devices very much like those of the old-time innkeeper with his tally stick. From this method to the introduction of special symbols for 5, 10, 20, etc., was only a step, and we find exactly such symbols in use at the beginning of written history, at the so-called dawn of civilization.
One of the oldest examples of the use of a tally stick dates back to paleolithic times and was found in 1937 in Vestonice (Moravia). It is the bone of a young wolf, 7 inches long, engraved with 55 deeply incised notches, of which the first 25 are arranged in groups of 5. They are followed by a simple notch twice as long which terminates the series; then, starting from the next notch, also twice as long, a new series runs up to 30. Other such marked sticks have been found.
It is therefore clear that the old saying found in Jakob Grimm and often repeated, that "counting started as finger counting," is incorrect. Counting by fingers, that is, counting by fives and tens, came only at a certain stage of social development. Once it was reached, numbers could be expressed with reference to a base, with the aid of which large numbers could be formed; thus a primitive type of arithmetic originated. Fourteen was expressed as 10 + 4, sometimes as 15 - 1. Multiplication began where 20 was expressed not as 10 + 10, but as 2 x 10. Such dyadic operations were used for millennia as a kind of middle road between addition and multiplication, notably in Egypt and in the pre-Aryan civilization of Mohenjo-Daro on the Indus. Division began where 10 was expressed as "half of a body," although conscious formation of fractions remained extremely rare. Among North American tribes, for instance, only a few instances of such formations are known, and this is in almost all cases only of 1/2, although sometimes also of 1/3 or 1/4. A curious phenomenon was the love of very large numbers, a love perhaps stimulated by the all-too-human desire to exaggerate the extent of herds of enemies slain; remnants of this tendency appear in the Bible and in other sacred and not-so-sacred writings.
3
It also became necessary to measure the length and contents of objects. The standards were rough and often taken from parts of the human body, and in this way units such as fingers, feet, or hands originated. The names "ell," "fathom," and "cubit" remind us also of this custom. When houses were built, as among the agricultural Indians or the pole-house dwellers of Central Europe, rules were laid down for building along straight lines and at right angles. The word "straight" is related to "stretch," indicating operations with a rope; the word "line" to "linen," showing the connection between the craft of weaving and the beginnings of geometry. This was one way in which interest in mensuration evolved.
GEOMETRICAL PATTERNS DEVELOPED BY AMERICAN INDIANS.
(From Spier, see "Literature," below.)
Neolithic man also developed a keen feeling for geometrical patterns. The baking and coloring of pottery, the plaiting of rushes, the weaving of baskets and textiles, and later the working of metals led to the cultivation of plane and spatial relationships. Dance patterns must also have played a role. Neolithic ornamentation rejoiced in the revelation of congruence, symmetry, and similarity. Numerical relationships might enter into these figures, as in certain prehistoric patterns which represent triangular numbers; others display "sacred" numbers.
Figures 1-Figures 4 below give examples of some interesting geometrical patterns occurring in pottery, weaving, and basketry. The design in Fig. 1 can be found on neolithic pottery in Bosnia and on objects of art in the Mesopotamian Ur period. The motif in Fig. 2 exists on Egyptian pottery of the Predynastic period (c. 4000-3500 B.C.). Fig. 3 shows patterns which were used by pole-house dwellers near Ljubljana (Yugoslavia) in the Hallstatt period (Central Europe, c. 1000–500 B.C.). The designs in Fig. 4, rectangles filled with triangles, triangles filled with circles, are from urns in graves near Sopron in Hungary. They show attempts at the formation of triangular numbers, which played an important role in Pythagorean mathematics of a later period.
Patterns of this kind have remained popular throughout historical times. Beautiful examples can be found on dipylon vases of the Minoan and early Greek periods, in the later Byzantine and Arabian mosaics, and on Persian and Chinese tapestry. Originally there may have been a religious or magic meaning to the early patterns, but their esthetic appeal gradually became dominant. In the religions of the Stone Age we can discern attempts at conforming to the forces of nature, the social structure, and the individual experience. Religious ceremonies were permeated with what we see as magic, and this magical element was incorporated into existing conceptions of number and form as well as in sculpture, music, and drawing. There were magical numbers (such as 3, 4, 7) and magical figures (such as the Pentalpha and the Swastika). Some authors have even considered this aspect of mathematics the determining factor in its growth, but though the social roots of mathematics may have become obscured in modern times, they are fairly obvious during this early period of man's history. "Modern" numerology is a leftover from magical rites dating back to neolithic, and perhaps even to paleolithic, times.
4
Even among peoples with a social structure far removed from our technical civilization we find some reckoning of time, and, closely related, some knowledge of the motion of sun, moon, and stars. This knowledge first attained its more scientific character when farming and trade expanded. The use of a lunar calendar dates very far back into the history of mankind, the changing aspects of vegetation being connected with the changes of the moon. People at an early date also paid attention to the solstices or rising of the Pleiades at dawn. The earliest people with records attributed a knowledge of astronomy to their most remote, prehistoric periods. Other peoples used the constellations as guides in navigation. From this astronomy resulted some knowledge of the properties of the sphere, of angular directions, of circles, and of even more intricate figures.
In recent years, considerable attention has been paid to the possible astronomi- cal and calendrical significance of prehistoric stone monuments, such as Stonehenge in England, dating back to c. 2000 B.C. If they had such significance, then the question arises whether this astronomical, and thus also mathematical, knowledge was transmitted from some center, perhaps Mesopotamia (diffusionism), or had a native, autochthonous origin (an opinion that has gained much support lately). The civilizations in America seem to have developed either independently from those of Eurasia and Africa, or at any rate with little interference from them.
5
These few illustrations of the beginnings of mathematics show that the historical growth of a science does not necessarily pass through the stages in which we now develop it in our instruction. Some of the oldest geometrical forms known to mankind, such as knots and patterns, only received full scientific attention in recent years. On the other hand, some of our more elementary branches of mathematics, such as the graphical representation or elementary statistics, date back to comparatively modern times. As A. Speiser has remarked with some asperity (and some exaggeration):
Already the pronounced tendency toward tediousness, which seems to be inherent in elementary mathematics, might plead for its late origin, since the creative mathematician would prefer to pay his attention to the interesting and beautiful problems.
6
This may be a good place to mention the mathematics of three ancient civilizations, interesting in themselves, but of little or no influence on the further course of mathematics: the Minoans-Mycenaeans, the Mayas, and the Incas. Their science is not that of "the beginnings," but belongs rather to the category of the next chapter, on the Ancient Orient.
Mathematical symbols used in administration have been found in the ruins of the Minoan-Mycenaean civilization of Crete and the Greek mainland. They belong to the scripts called Linear A and B and belong to the period of c. 1800-1200 B.C. Numbers are represented, as in Egypt (but with different symbols), by special symbols for 1, 10, 100, 1000 in an additive way. There are also symbols for simple fractions, not all unit fractions. Since the scribes did not bake the clay tablets on which they wrote, only those that were baked in the final conflagration of their cities have been preserved, so that we have inadequate knowledge of the extent of the mathematical knowledge of this civilization; it may have been comparable to that of Egypt. At any rate we know that Homer's heroes had scribes who could do some arithmetic on tablets.
The Mayas of Central America, mainly in what is now Yucatán and Guatemala, established a civilization that lasted for a millennium and a half, but reached its height in the so-called classical period, about 200-900 of our era. The arithmetic of the Mayas, mainly deciphered from inscribed stone monuments, some codices, and Spanish chronicles, and closely related to their astronomy, notably their calendric system, was vigesimal (it still is), represented by dots for the units up to 4, and horizontal bars for the fives up to 15. For larger numbers they used a position system with base 20, powers of 20 being represented by the same symbol as 20, the unit symbol. There were some modifications for calendric purposes. This position system required a symbol for zero, often a kind of shell or half-open-eye sign. This system, with its calendric connections, spread to other peoples of Central America. We think of the famous calendar stone found in Mexico City, dating from the time of the Aztecs, who came to this location at the end of the eleventh century of our era.
The Incas built a large empire in and west of the Andes of South America from the middle of the thirteenth century of our era on, their capital being Cuzco. Its vast bureaucracy, strong in administration, crafts, and engineering, used, for communication and information, no writing, but so-called quipos. The simplest quipo has a main cord of colored cotton or sometimes wool, from which knotted cords are suspended with the knots formed into clusters at some distance from each other. Each cluster has a number of knots from 1 to 9, and a cluster of, say, 4 followed by one of 2 and one of 8 knots represents 428. This is therefore a position system, in which our zero is indicated by a greater distance between the knots. The colors of the cords represent things: sheep, soldiers, etc.; and the position of the cords, as well as additional cords suspended from the cords, could tell a very complicated statistical story to the scribes who could "read" the quipos.
Quipos may have hundreds of knotted cords; the largest so far found has 1,800 pendent cords; it may have indicated the composition of an army or work force. Only some 400 quipos have been discovered, all in graves, since the Spanish destroyed quipos as ungodly.
These quipos teach us that we can have a sophisticated bureaucratic civilization without the art of writing. Is it possible that cultures like that of Stonehenge had similar means of communication and storage of information that are now gone forever? Those quipos that survived were buried in the desert region along the Pacific; those buried in less arid regions have all been lost.
(Continues...)
Excerpted from A Concise History Of Mathematics by DIRK JAN STRUIK. Copyright © 1987 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
INTRODUCTIONI. THE BEGINNINGS
II. THE ANCIENT ORIENT
III. GREECE
IV. THE ORIENT AFTER THE DECLINE OF GREEK SOCIETY
V. THE BEGINNINGS IN WESTERN EUROPE
VI. THE SEVENTEENTH CENTURY
VII. THE EIGHTEENTH CENTURY
VIII. THE NINETEENTH CENTURY
IX. THE FIRST HALF OF THE TWENTIETH CENTURY
INDEX