In the Homeric legends it is narrated that when Ulysses left the land of the Cyclops, after blinding the one-eyed giant Polyphemus, that unfortunate old giant would sit each morning by the entrance to his cave and from a heap of pebbles would pick up one pebble for each ewe that he let pass out of the cave. Then, in the evening, when the ewes returned, he would drop one pebble for each ewe that he admitted to the cave. In this way, by exhausting the supply of pebbles he had picked up in the morning, he was assured that...

The first geometrical considerations of man must be very ancient and must have subconsciously originated in simple observations stemming from human ability to recognize physical form and to compare shapes and sizes. Certainly one of the earliest geometrical notions to thus impinge itself on even the least reflective mind would be that of distance, in particular the concept that the straight line is the shortest path connecting two points; for most animals seem instinctively to realize this. Another early notion that would gradually emerge from the subconscious to the conscious mind would be that of simple rectilinear forms, such as...

It was about 600 B.C. that geometry entered a third stage of development. Historians of mathematics are unanimous in accrediting this further advancement to the Greeks of the period, and the earliest pioneering efforts to Thales of Miletus, one of the “seven wise men” of antiquity. Thales, it seems, spent the early part of his life as a merchant, becoming wealthy enough to devote much of his later life to study and some travel. He visited Egypt and brought back with him to Miletus knowledge of Egyptian accomplishments in geometry. His many-sided genius won him a reputation as a statesman,...

One of the most attractive, and certainly one of the most famous and most useful, theorems of elementary geometry is the so-calledPythagorean theorem,which asserts that “in any right triangle the square on the hypotenuse is equal to the sum of the squares on the two legs.” If there is a theorem whose birth merits inclusion as a GREAT MOMENT IN MATHEMATICS, the Pythagorean theorem is probably the prime candidate, for it is perhaps the first truly great theorem in mathematics. But when we come to consider the origin of the theorem, we find ourselves treading on anything but...

The first numbers we encounter as we grow up from early childhood are the so-callednatural numbers,orpositive integers:1, 2, 3, .... These numbers are abstractions that arise from the process of counting finite collections of objects. Somewhat later we realize that the needs of daily life require us, in addition to counting individual objects, to measure various quantities, such as length, weight, and time. To satisfy these simple measuring needs,fractionsare required, for seldom will a length, to take an example, appear to contain an exact integral number of a prechosen linear unit. For some measurements,...

The discovery of irrational numbers and of incommensurable magnitudes caused considerable consternation in the Pythagorean ranks. First of all, it seemed to deal a mortal blow to the Pythagorean philosophy that all depends upon the whole numbers—after all, how does an irrational number, like$\sqrt 2 $, depend on the whole numbers if it cannot be written as the ratio of two such numbers? Next, it seemed contrary to common sense, for it was strongly felt intuitively that any magnitude could be expressed bysomerational number. The geometric counterpart was equally startling, for, again contrary to intuition, there exist line...

The Greeks accomplished a great deal in mathematics during the three hundred years following Thales in 600 B.C. Not only did the Pythagoreans and others develop a considerable body of elementary geometry and number theory, but there also evolved notions concerning infinitesimals and summation processes that later, in the seventeenth century, blossomed into the calculus. Also, much higher geometry (that is, the geometry of curves other than the straight line and circle and of surfaces other than the plane and sphere) was developed. Curiously, a great deal of this higher geometry originated in vain attempts to solve three famous challenging...

Following the death of Alexander the Great in 323 B.C., the vast Macedonian Empire was divided into three parts, and the part containing Egypt came under the able governance of Alexander’s talented general, Ptolemy Soter, who shortly assumed kingship of the region. Ptolemy chose Alexandria, only a few miles removed from the mouth of the Nile River, as his capital, and about 300 B.C. he opened the doors of the famous University of Alexandria. In the galaxy of scholars invited to staff the new institution was the mathematician Euclid, probably a one-time attendant at the Platonic Academy in Athens.

One...

It is surprising how many of the great developments of mathematics of modern times find their origins in work done two millennia earlier by the ancient Greeks. As Julian Lowell Coolidge, the eminent Harvard geometer of the early twentieth century, was fond of saying: “There were giants in the land then.” Beyond any doubt the very greatest of all those giants was the famous Archimedes of Syracuse, a man of incredible mathematical talent. In this lecture we shall see that Archimedes, in finding, with the limited means at his disposal, areas of certain curvilinear plane figures and areas and volumes...

The beginnings of trigonometry are obscure. So far as the pre-Hellenic period is concerned, there are some problems in the Rhind papyrus (ca. 1650 B.C.) that involve the cotangent of the dihedral angle at the base of a regular square pyramid, and there is the Babylonian cuneiform tablet known as Plimpton 322* (1900 to 1600 B.C.), which essentially contains a remarkable table of secants of fifteen angles ranging between 45° and 30°. It may well be that further studies into the mathematics of ancient Mesopotamia will disclose a substantial development of practical trigonometry. Babylonian astronomers had amassed a considerable collection...

There are two aspects to number study—the search for relationships among numbers and the development of the art of computing with numbers. To the ancient Greeks the former was known asarithmeticand the latter aslogistic.This classification persisted through the Middle Ages until around the close of the fifteenth century, when texts appeared treating both aspects of number work under the single namearithmetic.It is interesting that todayarithmetichas its original meaning in continental Europe, while in England and the United States the popular meaning ofarithmeticis synonymous with that of ancientlogistic,and...

In contrast to his geometry course, a high school student finds algebra a highly symbolized field of mathematical study. The work bristles with plus signs, minus signs, division symbolism, later letters of the alphabet for unknown quantities and early letters for fixed quantities, various symbols (parentheses, brackets, braces) for signs of aggregation, exponents, subscripts, radicals, equality signs, factorial symbols, combination and permutation symbolism, logarithmic notation, and so on. It is seldom realized by the student that all this symbolism is little more than four hundred years old—indeed, much of it is considerably less than four hundred years old.

It...

Though the people of the world today speak and write in a great confusion of different languages, almost all of them do arithmetic with the same number symbols and utilize (pretty much) the same computing schemes.

All around the world numbers are represented, in a common, familiar way, as positional sequences of the appropriate tendigitsymbols 0, 1,2, 3, 4, 5, 6, 7, 8, 9. The representation neatly lends itself to the formation of succinct schemes, or patterns, called algorithms, for performing the operations of arithmetic. Thus there is an addition algorithm and a subtraction algorithm, a multiplication algorithm...

In the second half of the eleventh century, three Persian youths, each a capable scholar, studied together as pupils of one of the greatest wise men of Khorasan, the Imam Mowaffak of Naishapur. The three youths—Nizam ul Mulk, Hasan Ben Sabbah, and Omar Khayyam—became close friends. Since it was the belief that a pupil of the Imam stood great chance of attaining fortune, Hasan one day proposed to his friends that the three of them take a vow to the effect that, to whomever of them fortune should fall, he would share it equally with the others and...

The dissemination and popularization of the Hindu-Arabic numeral system in western Europe was largely accomplished by the publication of certain books extolling and advocating the new numerals.

The earliest Arabic arithmetic known to us is that of al-Khowârizmiî (ca. 825), which was followed by a number of other Arabic arithmetics by later authors. These arithmetics contained rules, fashioned after the Hindu patterns, for computing with the Hindu numerals. They also gave the process known ascasting out 9’s,used for checking arithmetical calculations, and the rules offalse positionanddouble false position,by which certain algebra problems can be...

In lecture 14 we saw how the Persian poet-mathematician Omar Khayyam solved cubic equations geometrically. In this lecture we shall see how, almost 500 years later, Italian mathematicians finally managed to solve cubic equations, and then, shortly after, quartic equations also, algebraically. These accomplishments constitute spectacular episodes in the history of mathematics and yield two great moments in mathematics, one following closely upon the heels of the other. There are elements of the colorful and of the bizarre in the story, and some of the characters in the tale are among the most unusual in all of mathematics.

Briefly told,...

In 1614, a Scottish nobleman, living in Edinburgh, published the details of a wonderful invention that he had made. News of the invention traveled fast, and, in the following year, after some correspondence, a professor of mathematics left London by horse-drawn carriage for the long trip to Edinburgh so that he might pay his personal respects to the ingenious Scotsman. On the interminable journey over the bumpy dirt roads, the professor recorded some of his thoughts in his diary. How tall a forehead, he ruminated, must the nobleman possess in order to house the brains sufficient for discovering so remarkable...

The mighty Antaeus was the giant son of Neptune (god of the sea) and Ge (goddess of the earth), and his strength was invincible so long as he remained in contact with his Mother Earth. Strangers who came to his country were forced to wrestle to the death with him, and so it chanced one day that Hercules and Antaeus came to grips with one another. But Hercules, aware of the source of Antaeus’ great strength, lifted and held the giant from the earth and crushed him in the air.

There is a parable here for mathematicians. For just as...

In the fourteenth century, the Blessed John Colombini of Siena founded a religious group known as theJesuats,which was in no way related to theJesuits.The order was approved by Pope Urban V in 1367. The original work of the order was the care of those stricken by the Black Death, which raged over Europe at the time, and the burial of the fatally smitten. With the passage of time the Jesuat order diminished, and in 1606 an attempt at a revival was made. But certain abuses later crept into the order, with the result that the group...

One of the most effective methods employed by mathematicians is that known as thetransform-solve-invert technique.The essence of the idea is this. To solve a difficult problem,transformit, by some simplifying procedure, into an easier equivalent problem, solve the easier problem, and theninvertthe simplifying procedure to obtain a solution of the original problem. Let us illustrate the idea with some examples.

Suppose someone should ask us a question in French and that we are not overly proficient in that language. We would firsttransformthe question into an equivalent one in English, a language in which...