A Concise History of Mathematics

Fourth Revised Edition

Dirk J. Struik

• Verlag: Dover Publications
• Jahr: 2012
• ISBN:
  9780486138886
• Auflage: 4
• Thema: Mathematics
• Thema: History & Philosophy of Mathematics

This compact, well-written history — first published in 1948, and now in its fourth revised edition — describes the main trends in the development of all fields of mathematics from the first available records to the middle of the 20th century. Students, researchers, historians, specialists — in short, everyone with an interest in mathematics — will find it engrossing and stimulating.
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.

CHAPTER I

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.²

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. ³ 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 V

stonice (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 30.⁴ 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 1/4.⁵ 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;⁶ the word “line” to “linen,” showing the connection between the craft of weaving and the beginnings of geometry.⁷ 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.

FIG. 1.

FIG. 2.

FIG. 3.

FIG. 4.

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.⁸ The motif in Fig. 2 exists on Egyptian pottery of the Predynastic period (c. 4000-3500 B.C.).⁹ 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.).¹⁰ The designs in Fig. 4, rectangles filled with trian...