Perfect numbers–that is, those equal to the sum of their integral subdivisors–were, in very olden times, a matter of grave consideration and conjecture. The ancients explained many mysteries of Nature by the magic number 6, presumably the first of the perfect numbers, unless 1, or unity, be so considered.

By 500 b.c., we find the Pythagoreans classifying numbers as excessive, perfect, or defective, in comparison with the sum of their integral subdivisors. In the 3rd century b.c., Euclid discussed perfect numbers and concluded by giving a formula for their selection.

In the Middle Ages, theology adopted perfect numbers,...

Every scientifically-minded person agrees with those who hold the opinion that in the appraisement or estimate of the value of an intellectual performance, merit, as acknowledged by one or more class or classes of critics, should be the deciding factor. There is no other science in which this assertion should appear more evident than in mathematics. And yet history records some strange exceptions to this rule, due to the fact that mathematicians also are frequently very human, just like the rest of their scientific brethren.

In the following lines I propose to discuss three outstanding examples of extremely meritorious papers...

There exists today a strong tendency to pay homage and attention to the heroes of science, to humanize its essential creators, and this book is good evidence of the fact. A. Macfarlane (1916), H. W. Turnbull (1929), and G. Prasad (2 Vols., 1933–1934) have issued collections of mathematical biography, but this one is done on a much larger and more pretentious scale. A parade of the following mathematicians passes in rapid review: Zeno, Eudoxus, Archimedes, Descartes, Fermat, Pascal, Newton, Leibniz, the Bernoulli family, Euler, Lagrange, Laplace, Monge, Fourier, Poncelet, Gauss, Cauchy, Lobachevsky, Abel, Jacobi, Hamilton, Galois, Sylvester, Cayley, Weierstrass,...

Norway and Switzerland are not large countries, but their significance in the history of mathematics is not difficult to assess. One need merely mention for the former, Abel and Sophus Lie; for the latter, Euler and the Bernoulli family. The choice of Oslo as a meeting place for the International Congress of Mathematicians, July 13–18, 1936, proved to be a most happy and satisfactory one. Harald the Hard Hearted, one of the Viking kings, founded Oslo in 1045, and Haakon V built about 1300 a typical medieval fortress or castle called Akershus on a rocky cliff on the Oslo...

In 1908 Norway honored the memory of Niels Henrik Abel by erecting a monument of him in the park in front of the Royal Castle in Oslo. Gustav Vigeland, the leading modern Norwegian sculptor, created this monument. The late Felix Klein in hisVorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert(vol. 1, p. 108) compares Abel and Mozart, speaking in this connection of the beautiful monument to Mozart in Vienna. But of the Abel monument he writes:

“I cannot avoid referring on this occasion to the entirely different sort of monument which has been erected to Abel in...

The historiography of a science usually does not enjoy any too high an esteem among its productive representatives. The reasons for this are several and they are not difficult to recognize, especially in a science like mathematics, which can distinguish with such precision between secured possession and unsolved problem. Mathematical problems and methods are indeed, like every other element of existence, historically conditioned; but that portion of the way already traversed—which for their continuation one must know and as such survey—is a relatively short one. Probably long centuries worked only in closest connection with antiquity, but the great...

After primitive man had learned to count, his next task was to invent ways of representing and recording whole numbers. This can be done in such a vast number of ways that even today the possibilities have not been exhausted and new ways of expressing numbers are being devised each year. Any good history of mathematics gives in detail the story of man’s early attempts at writing numbers. It is not my intention to dwell on this subject from a historical point of view. I merely wish to discuss three types of notation which are in use today.

In any...

Some years ago the writer discovered a generalization of the Weierstrass approximation theorem suggested by an inquiry into certain algebraic properties of the continuous real functions on a topological space [1]. This generalization has since shown itself to be very useful in a variety of similar situations. Interest in it has stimulated several improvements in the proof originally given and has also led to some modifications and extensions of the theorem itself. At the same time many interesting applications to classical problems of analysis have been observed by those working with the generalized approximation theorem. The writer, for instance, has...

The first woman mathematician regarding whom we have positive knowledge is the celebrated mathematician-philosopher Hypatia. The exact date of her birth is not known, but recent studies indicate that she was born about a. d. 370 in Alexandria. This would make her about 45 years of age at her death. Hypatia, it seems, was known by two different names, or at least by two different spellings of the same name; the one, Hypatia; the other, Hyptachia. According to Meyer,¹ there were two women with the same name living at about this time; Hypatia, the daughter of Theon of Alexandria; the...

An unexpected turn in twentieth-century mathematics was the abrupt change in the motivation and objectives of algebra. The change became evident by 1925 at the latest, and in about ten years made some of the algebra of the nineteenth and early twentieth centuries seem rococo and strangely antiquated to algebraists of the younger generation.

The transition from individually developed theories, overloaded with masses of intricate theorems—often the seemingly fortuitous outcome of elaborate calculations carried through with consummate manipulative skill—to the deliberate search for unifying abstract principles was sudden. It did not occur without preparation, of course; but the...

Among the many contributions of mathematics to modern civilization the most valuable are not those which serve the physicist and the engineer but rather those which have fashioned our culture and our intellectual climate. Not enough people are aware of the latter contributions, not even of the role mathematics played in the greatest revolution in the history of human thought—the establishment of the heliocentric theory of planetary motions. It is a fact of history that mathematics forged this theory and was the sole argument for it at the time that it was advanced. No more impressive illustration of the...

The Editor has asked for about 400 words on “what mathematics has meant to me.” Notice the “me”—not someone else. This will account for all the “I” and “me” in what follows, for which I apologise. I am as embarrassed as if I had inadvertently stood up in church to tell the congregation how and why I had been saved. You may be even more embarrassed in witnessing my testimony.

My interest in mathematics began with two school prizes, one in Greek, the other for physical laboratory, both richly bound in full calf. The Greek Prize was Clerk Maxwell’s...

We shall try to reconstruct a Who’s Who in Mathematics around 1850. We have to keep in mind that there were much fewer mathematicians in those days, fewer periodicals, little personal contact, no mathematical societies, no scientific meetings or congresses. Mathematics flourished in France, Germany, and Great Britain; outside of these countries research mathematicians were few and far between. There was interchange of ideas, however, mathematicians did write to each other, young students traveled abroad to study and so forth.

In this survey it is reasonable to start with France where there is a splendid mathematical tradition going back to...

It has been said that mathematics is the science of tautology, which is to say that mathematicians spend their time proving that equal quantities are equal. This statement is wrong on two counts: In the first place, mathematics is not a science, it is an art; in the second place, it is fundamentally the study of inequalities rather than equalities.

I would like today to discuss a number of the basic inequalities of analysis, presenting first an algebraic proof of the inequality between the arithmetic and geometric means, and then a most elegant geometric technique due to Young. In passing...

The reader is probably familiar with the binary system and the decimal system and probably understands the basis for any others of that type, such as the trinary or duodecimal. However, I have developed a system that is based, not on an integer, or even a rational number, but on the irrational numberτ(tau), otherwise known as the “golden section,” approximately 1:618033989 in value, and equal to$(1+\sqrt{5})/2$.

In order to understand this system, one must comprehend two peculiarities of the number τ. They are based on tau’s distinctive property¹ that

τⁿ= τⁿ⁻¹+ τⁿ⁻².

(a) Take...

Since one of the most powerful methods in mathematical research is the process of generalization, it is certainly desirable that young students be introduced to this process as early as possible. The purpose of this article is to call attention to the usually untapped possibilities for generalizing theorems on the triangle to theorems about the tetrahedron. Some of these, of course, do appear in our textbooks on solid geometry; but here I shall describe two situations where the appropriate generalizations seem to be generally unknown. The questions to be answered are: (1) What is the generalization to a tetrahedron of...

${\sqrt{10}}$is a useful number for illustrative purposes when one is discussing irrational numbers at an elementary level. It is the hypotenuse of a right triangle whose other sides are 1 and 3, and it can be shown to be irrational by a most unsophisticated argument.

When the nonzero integerpis squared, the resulting integer has exactly twice as many terminal zero digits as doesp(even ifphas none). Thus, ifpis a nonzero integer,p² ends with an even number of zeros, and, ifqis a nonzero integer, 10q² ends with an odd...

In what follows, I speak as an analyst, not a topologist. In particular, I am not discussing topology as an axiomatic structure, nor yet as a touchstone to bring young minds to life, nor as an organized body of theorems. Rather, I shall confine my remarks largely to the role of topology as something which illuminates topics in analysis, and which provides a more geometric viewpoint. I will give a series of unconnected illustrations.

To “illuminate” can mean to clarify or simplify. Suppose we assume a background of the simplest anatomy of topology—acquaintance with the meaning of open, closed,...

The following short proof is offered for the main result of the paper by John H. Staib and Miltiades S. Demos in Vol. 40, page 210 of thisMagazine, namely that {sinn} is dense on [−1, 1].

Let the points whose polar coordinates (r,θ) are (1, 1), (1, 2), (1, 3),… be called integral points. Then the integral points are dense on the unit circle. For, assume that there exists an arc-length of magnitude ∊ > 0 such that betweenθandθ+ ∊ there are no integral points. Then there is another such gap whose initial point...

Probability theory is the study of mathematical models for random phenomena. A random phenomenon is an empirical phenomenon whose observation under given circumstances leads to various different outcomes. When a coin is tossed, either heads or tails comes up; but on a given toss we can’t predict which. When a die is tossed, one of the six numbered faces comes up; but again we can’t predict which. In such simple random phenomena the various possible outcomes appear to occur with what is called “statistical regularity”. This means that the relative frequencies of occurrence of the various possible outcomes appear to...

Some time ago, the following problem occurred to me: which fits better, a round peg in a square hole or a square peg in a round hole? This can easily be solved once one arrives at the following mathematical formulation of the problem. Which is larger: the ratio of the area of a circle to the area of the circumscribed square or the ratio of the area of a square to the area of the circumscribed circle? One easily finds that the first ratio isπ/4 and that the second is 2/π. Since the first is larger, we may conclude...

When mathematicians are thought of, who remembers James Smith? Or Daniel West? Few people indeed. Yet these men (and 42 others) performed a valuable service in the middle of the last century: they kept track ofπ_t, the ratio of the circumference of a circle to its diameter at timet. See the table for the results of their calculations, rounded off to five decimal places. The data are mostly from DeMorgan [1] and Gould [2]. Lately, very little has been done in this field; we have letπ_tget away from us.

But perhaps something can be saved. If...

Atrigonometric identityis an equation between two rational functions of trigonometric functions, e.g.\[\frac{\tan x}{\csc x-\cot x}-\frac{\sin x}{\csc x+\cot x}=\sec x+\cos x.\]

Thead hocverification of such identities is a standard exercise in elementary trigonometry (see, for example, [1, Chapter 8], from which the above identity comes) which most students have done, usually not in any systematic way. Thus the following theorem, which fits nicely into an undergraduate abstract algebra course, is of some interest to them:

Theorem 1Every trigonometric identity is a consequence ofsin²x + cos²c = 1.

The proof of the theorem, which will be outlined here, uses only some elementary commutative...

It is well known (see, e.g., [3]) that 30 is the largest integer with the property that all smaller integers relatively prime to it are primes. In this note I will consider a related situation in which the corresponding special number turns out to be 70. (For a while I believed 30 to be the key figure in the new context, too, but E. G. Straus showed me that the correct value was indeed 70.) Following the proof of this special property of 70, I will mention a few related problems, some of which seem to me to be very...

The ordinary complex numbers (a+ib) (or, as they were formerly written,$a+b{\sqrt{-1}}$) are added and multiplied according to definite rules. The rule for multiplication reads as follows:

First multiply according to the rules of high school algebra:

(a+ib)(c+id) =ac+adi+bci+bdi²

and then replacei²by (−1):

(a+ib)(c+id) = (ac−bd) + (ad+bc)i.

Complex numbers can also be defined as couples (a,b). The product of two couples (a,b) and (c,d) is defined as the couple (ac−bd,...

A standard exercise for calculus classes reads:Given a triangle of altitude a and base b, find the dimensions of the rectangle of maximum area which can be inscribed in this triangle with one side along the base.

It at least broadens a student’s perspective if he occasionally sees an alternate solution of such a problem avoiding the calculus. The above problem can be settled using an elementary inequality as follows:

Two essentially different possibilities face us, as shown in Figure 1. In case (i), the vertexCis “above” some point of the base; in case (ii), the vertex...

One of the most important results in combinatorial mathematics is Pólya’s enumeration formula. This formula constructs a generating function for the number of different ways to mark the corners of an unoriented figure using a given set of labels. For example, if the figure were an unoriented cube and one could color the corners black or white, Pólya’s formula would yield the following generating function, called thepattern inventory:

b⁸+b⁷w+ 3b⁶w²+ 3b⁵w³+ 7b⁴w⁴+ 3b³w⁵+ 3b²w⁶+bw⁷+w⁸,

where the coefficient ofbⁱw^jis the number of distinct colorings withiblack...

Originally “logic” meant the same as ”the laws of thought” and logicians studied the subject in the hope that they could discover better ways of thinking and surer ways of avoiding error than their forefathers knew, and in the hope that they could teach these arts to all mankind. Experience has shown, however, that this is a wild-goose chase. A normal healthy human being has built in him all the “laws of thought” anybody has ever invented, and there is nothing that logicians can teach him about thinking and avoiding error. This is not to say that he knowshow...

The importance of recreational mathematics and the involvement of amateur mathematicians has been dramatically demonstrated recently in connection with the problem of tiling the plane with congruent pentagons. The problem is to describe completely all convex pentagons whose congruent images will tile the plane (without overlaps or gaps). The problem was thought to have been solved by R. B. Kershner, who announced his results in 1968 [18], [19]. In July, 1975, Kershner’s article was the main topic of Martin Gardner’s column, “Mathematical Games” inScientific American. Inspired by the challenge of the problem, at least two readers attempted their own...

It was shown by Cauchy [1] and Dehn [2] that a convex polyhedron made of rigid plates which are hinged at their edges is a rigid structure. However, if the structure is not convex, but still simply connected, there are several possibilities. It may be any of the following cases:

(a) rigid,

(b) infinitesimally movable (shaky),

(c) two or more stable forms (multi-stable),

(d) a continuously movable linkage.

The regular icosahedron of twenty triangular faces is convex and rigid. If six pairs of faces with an edge in common are replaced by other pairs of isosceles faces with their edges...

Born in 1707, Leonhard Euler grew up in the town of Riehen, near Basel, Switzerland. Encouraged by his father, Paulus, a minister, young Leonhard received very early instruction from Johann I Bernoulli, who immediately recognized Euler’s talents. Euler completed his work at the University of Basel at age 15, and at age 19 won a prize in the competition organized by the Academy of Sciences in Paris. His paper discussed the optimal arrangement of masts on sailing ships (Meditationes super problemate nautico…). In 1727 Euler attempted unsuccessfully to obtain a professorship of physics in Basel by submitting a dissertation on...

The purpose of this note is to suggest an unusual approach to the teaching of some standard material about systems of coupled ordinary differential equations. The approach relates the mathematics to a topic that is already on the minds of many college students: the time-evolution of a love affair between two people. Students seem to enjoy the material, taking an active role in the construction, solution, and interpretation of the equations.

The essence of the idea is contained in the following example.

Juliet is in love with Romeo, but in our version of this story, Romeo is a fickle lover....

This article gives a brief sketch of the evolution of group theory. It derives from a firm conviction that the history of mathematics can be a useful and important integrating component in the teaching of mathematics. This is not the place to elaborate on the role of history in teaching, other than perhaps to give one relevant quotation:

Although the study of the history of mathematics has an intrinsic appeal of its own, its chief raison d’être is surely the illumination of mathematics itself. For example the gradual unfolding of the integral concept from the volume computations of Archimedes to...

The common oscillating lawn sprinkler has a hollow curved sprinkler arm, with a row of holes on top, which rocks slowly back and forth around a horizontal axis. Water issues from the holes in a family of streams, forming a curtain of water that sweeps back and forth to cover an approximately rectangular region of lawn. Can such a sprinkler be designed to spread water uniformly on a level lawn?

We break the analysis into three parts:

1. How should the sprinkler arm be curved so that streams issuing from evenly spaced holes along the curved arm will be evenly spaced...

Since this paper was first given to educators, let me start with a classroom experience. It happened in a course in which my students had read some of Euclid’sElements of Geometry. A student, a social science major, said to me, “I never realized mathematics was like this. Why, it’s like philosophy!” That is no accident, for philosophy is like mathematics. When I speak of the centrality of mathematics in western thought, it is this student’s experience I want to recapture—to reclaim the context of mathematics from the hardware store with the rest of the tools and bring it...

Last Sunday I was leisurely reading the May 1984 issue ofMathematics Magazine. Coming to the “Philatelists take note” item (on page 187) about the stamp issued by East Germany to mark the bicentennial of Euler’s death, I remembered that the same stamp was used on a recent request for reprints I received from East Germany. So I retrieved that postcard and washed off the stamp. It turned out that I was lucky, and the stamp was much less marred by cancellation marks than the one reproduced inMathematics Magazine. Contemplating the stamp I noticed that the drawing of the...

Most schools advertise their “average class size,” yet most students find themselves in larger classes most of the time. Here is a typical example.

In the first quarter of the 1980–81 academic year, 111 courses including tutorials, were given at Harvard School of Public Health. These ranged in size from one student to 229. The average class size, from the administration’s and professors’ perspective, was 14.5. The expected class size for a typical student was over 78! This huge discrepancy was due to the existence of a few very large classes. Indeed, only three courses had more than 78...

The theory of knots and links is the analysis of disjoint simple closed curves in ordinary 3-dimensional space. It is the consideration of a collection of pieces of string in 3-space, the two ends of each string having vanished by being fastened together as in a necklace. Many examples can be seen in the diagrams that follow. If the strings can be moved around from one position to another those two positions are the same link or ‘equivalent’ links. Of course, during the movement no part of a string is permitted to pass right through another part in some supernatural...