Among the bones of the intricate skeletal structure of the foot is one lying in the heel just above the talus bone and known as theastragalus. In man, and in animals with a developed foot, the astragalus is quite irregular, but in the hoofed animals, like sheep, goats, and all kinds of deer, the astragalus has a rough symmetry, being squarish in cross section with two rounded ends, one slightly convex and the other slightly concave. These bones are solid and essentially marrowless, hard and durable, somewhat cubical with edges measuring an inch or less, and, with handling, capable...

The prime stimulus to the invention of new mathematical procedures is the presence of problems whose solutions have evaded known methods of mathematical attack. Indeed, the continual appearance of unsolved problems constitutes the life blood that maintains the health and growth of mathematics. In our previous lecture we saw an example of this—it was an elusive problem, the so-calledproblem of the points,that led to the creation of the field of mathematical probability.

In earlier lectures we have seen that the problem of finding certain areas, volumes, and arc lengths gave rise to summation processes that led to...

It will be recalled that the integral calculus originated in the days of Greek antiquity in efforts to find certain areas, volumes, and arc lengths. The basic idea, in the case of areas, for example, is to consider an area as approximated by the areas of a great many very thin parallel rectangular strips. Using modern terminology, one then attempts to find the area as the limit approached by the sum of the areas of these strips, the number of the strips increasing indefinitely, and the widths of the strips approaching zero. With the ultimate introduction of analytic geometry in...

Students of high-school mathematics encounter arithmetic progressions and geometric progressions in their algebra courses. These two progressions are basic and important examples of the more general concept of asequence, and the indicated summation of the terms of either type of progression is an example of the more general concept of aseries.Let us give formal definitions of these more general concepts.

Asequenceis a succession of terms, usually formed according to some fixed rule or law, and aseriesis the indicated sum of the terms of a sequence. For example,

1, 4, 9, 16, 25

and...

The seventeenth century was a spectacular period in the development of mathematics. Early in the century Napier published his invention of logarithms; Harriot and Oughtred contributed noteworthily to the notation and systematization of algebra; Galileo founded the science of dynamics; and Kepler announced his three famous laws of planetary motion. Later in the century Desargues and Pascal opened the new field of projective geometry; Descartes launched modern analytic geometry; Fermât laid the foundations of modern number theory; Pascal, Fermât, and Huygens created the field of mathematical probability; and the first computing machines were invented by Pascal and Leibniz. Then, toward...

We shall now recount, in turn, two very remarkable and revolutionary mathematical developments that occurred in the first half of the nineteenth century. The first one was the discovery, about 1829, of a self-consistent geometry markedly different from the familiar geometry of Euclid; the second was the discovery, in 1843, of an algebra radically different from the customary algebra of the real number system. Each of these developments merits inclusion in any selection of GREAT MOMENTS IN MATHEMATICS, and to do them justice we shall devote two lectures to each one. In each case our treatment will be strongly historical,...

In the previous lecture we saw that, in spite of considerable and prolonged effort, Saccheri, Lambert, and Legendre were unable to find a contradiction under the hypothesis of the acute angle. It is no wonder they found no contradiction under this hypothesis, for it is now known that the geometry developed from a certain basic set of assumptions plus the acute-angle hypothesis is as consistent as the Euclidean geometry developed from the same basic set of assumptions plus the hypothesis of the right angle. In other words, it is now known that the parallel postulatecannotbe deduced as a...

At the start of LECTURE 26, we commented that there were two very remarkable and revolutionary mathematical developments that occurred in the first half of the nineteenth century. The first one—the discovery of a non-Euclidean geometry in about 1829—has been discussed in the last two lectures. We now come to the second one—the discovery of a nonconventional algebra in 1843. We shall see that, just as the former development liberated geometry from the traditional geometry of Euclid, the second development liberated algebra from the traditional algebra of the real number system. As in the former case, we...

Geometry, as we have seen in LECTURE 27, remained shackled to Euclid's version of the subject until Lobachevsky and Bolyai, in 1829 and 1832, liberated it from its bonds by creating an equally consistent geometry in which one of Euclid's postulates fails to hold. With this accomplishment, a deep-rooted and centuries-old conviction that there can be only the one possible geometry was shattered, and the way was opened for the creation of many new and different geometries.

A similar story can be told of algebra. It seemed inconceivable to the mathematicians of the early nineteenth century that there could exist...

Of the algebraic structures that were developed in the nineteenth century there is one, the so-calledgroupstructure, that in time came to be recognized as of cardinal importance in mathematics. Though the concept of a group received its first extensive study by Augustin-Louis Cauchy (1789–1857) and his successors, under the particular guise of substitution groups, the concept had been informally utilized as early as 1770 by Joseph Louis Lagrange (1736–1813), and had been given a definition and its name in 1830 by Évariste Galois (1811–1832), in his profound researches in the theory of equations. With the...

In 1872, upon appointment to the Philosophical Faculty and the Senate of the University of Erlanger, Felix Klein (1849–1925) delivered, according to custom, an inaugural address in his area of specialty. This address, based upon work by himself and the Norwegian mathematician Sophus Lie (1842–1899) in group theory, set forth a remarkable definition of “a geometry,” which served to codify essentially all the existing geometries of the time and pointed the way to new and fruitful avenues of research in geometry. This address, with the program of geometrical study advocated by it, has become known as the Erlanger...

One fancies that sometime toward the end of the nineteenth century (it is difficult to pinpoint even the precise year) the earth in Western Europe shuddered, and if one had put his ear to the ground he would have heard, rumbling to him in the language of ancient Greece from the far-off grave of the great Pythagoras, “I told you so over two thousand years ago; I told you then thateverythingdepends upon the whole numbers.” For, after a remarkable sequence of investigations, mathematicians of Western Europe had shown, by the late nineteenth century, thatallof mathematics is...

Mathematicians, and philosophers, have wrestled with the concepts of infinity and infinite sets from the days of the ancient Greeks, Zeno’s paradoxes being an early indication of some of the difficulties that were encountered. Some of the Greeks, Aristotle and Proclus among them, accepted the fact that a set can be made larger and larger without bound, but denied the existence of a completed infinite set. Throughout the Middle Ages philosophers argued over this matter of the potential versus the actual infinite. It was noted that the comparison of certain infinite sets leads to paradoxes. For example, the points of...

Though the ancient Greeks contributed enormously to the content of mathematics, perhaps their most outstanding contribution to the subject was their organization of mathematics by the axiomatic method. This earliest form of the axiomatic method has become known asmaterial axiomatics,and a description of its pattern was given in LECTURE 7. In LECTURE 8 we considered Euclid’sElements,which is the first great application of material axiomatics that has come down to us.

The Greek concept of axiomatics persisted, after a period of general neglect, into the nineteenth century, when three cardinal events in the development of mathematics led...

In this lecture we shall clarify, by means of examples, the interesting definitions encountered in the previous lecture. In short, we propose to give a simple example of a branch of pure mathematics followed by three applications of that branch. That is, we shall develop a short discourse by formal axiomatics and then, by appropriate interpretations of the primitive terms of the discourse, obtain three models of the discourse. We first explain some notation that will be employed.

Everyone is familiar with the idea of a dyadic relation as a form of connectivity between a pair of objects, for this...

In building up a branch of pure mathematics, one might think that one can set down an arbitrary collection of symbols for the primitive terms of the discourse and then list an arbitrary collection of statements about these primitive terms for the postulates of the discourse. This is not so. There are certain required and certain desired properties that the system of postulates should possess. This lecture will accordingly be devoted to a brief examination of some of the properties of postulate sets.

There are three distinct levels in axiomatic study. First, there are the concrete axiomatic developments of specific...

In 1931 there appeared, in the journalMonatshefte für Mathematik und Physik, a paper entitled, “Über formal unentscheibare Sätze der Principia Mathematica und verwandter Systeme” (“On formally undecidable propositions of Principia Mathematica and related systems"). The author of the paper was a 25-year-old Austrian mathematician and logician named Kurt Gödel, who was, at the time, at the University of Vienna. When the paper appeared, it received only scattered and scant attention, for it concerned itself with a highly specialized area of study that had not yet attracted many researchers and it used a method of proof that was so technically...

About 1812, the eccentric English mathematician and mechanist, Charles Babbage (1792–1871) began to consider the construction of a machine to aid in the calculation of mathematical tables. According to one story, the idea first came to him when the younger Herschel brought in, for checking, some calculations that had been performed for the Astronomical Society. In the course of the tedious checking, Herschel and Babbage found a number of errors, finally causing Babbage to exclaim, “I wish to God these calculations had been executed by steam.” “It is quite possible,” replied Herschel. From this chance interchange of remarks arose...

In our written lecture sequence of forty-three GREAT MOMENTS IN MATHEMATICS many fine candidates for inclusion had to be passed by. The original oral lecture sequence contained a choice of sixtysome GREAT MOMENTS , but even then there still were many important omissions. We deeply regret the necessity for these exclusions and sincerely apologize for the brevity and incompleteness of the written accounts of those items that were finally chosen. By way of slight amends, and before bidding the patient reader of these lectures adieu, it seems fitting at least to mention, in chronological order and with a word or...