MATHEMATICS

Its Contents, Methods and Meaning

THREE VOLUMES BOUND AS ONE

Edited by

A. D. Aleksandrov,
A. N. Kolmogorov,
M. A. Lavrent’ev

Translation edited by

S. H. Gould

DOVER PUBLICATIONS, INC.
Mineola, New York

VOLUME ONE

PART 1

CHAPTER I

A GENERAL VIEW OF MATHEMATICS

An adequate presentation of any science cannot consist of detailed information alone, however extensive. It must also provide a proper view of the essential nature of the science as a whole. The purpose of the present chapter is to give a general picture of the essential nature of mathematics. For this purpose there is no great need to introduce any of the details of recent mathematical theories, since elementary mathematics and the history of the science already provide a sufficient foundation for general conclusions.

§1.   The Characteristic Features of Mathematics

1. Abstractions, proofs, applications. With even a superficial knowledge of mathematics, it is easy to recognize certain characteristic features: its abstractness, its precision, its logical rigor, the indisputable character of its conclusions, and finally, the exceptionally broad range of its applications.

The abstractness of mathematics is easy to see. We operate with abstract numbers without worrying about how to relate them in each case to concrete objects. In school we study the abstract multiplication table, that is, a table for multiplying one abstract number by another, not a number of boys by a number of apples, or a number of apples by the price of an apple.

Similarly in geometry we consider, for example, straight lines and not stretched threads, the concept of a geometric line being obtained by abstraction from all other properties, excepting only extension in one direction. More generally, the concept of a geometric figure is the result of abstraction from all the properties of actual objects except their spatial form and dimensions.

Abstractions of this sort are characteristic for the whole of mathematics. The concept of a whole number and of a geometric figure are only two of the earliest and most elementary of its concepts. They have been followed by a mass of others, too numerous to describe, extending to such abstractions as complex numbers, functions, integrals, differentials, functionals, n-dimensional, and even infinite-dimensional spaces, and so forth. These abstractions, piled up as it were on one another, have reached such a degree of generalization that they apparently lose all connection with daily life and the “ordinary mortal” understands nothing about them beyond the mere fact that “all this is incomprehensible.”

In reality, of course, the case is not so at all. Although the concept of n-dimensional space is no doubt extremely abstract, yet it does have a completely real content, which is not very difficult to understand. In the present book it will be our task to emphasize and clarify the concrete content of such abstract concepts as those mentioned earlier, so that the reader may convince himself that they are all connected with actual life, both in their origin and in their applications.

But abstraction is not the exclusive property of mathematics; it is characteristic of every science, even of all mental activity in general. Consequently, the abstractness of mathematical concepts does not in itself give a complete description of the peculiar character of mathematics.

The abstractions of mathematics are distinguished by three features. In the first place, they deal above all else with quantitative relations and spatial forms, abstracting them from all other properties of objects. Second, they occur in a sequence of increasing degrees of abstraction, going very much further in this direction than the abstractions of other sciences. We will illustrate these two features in detail later, using as examples the fundamental notions of number and figure. Finally, and this is obvious, mathematics as such moves almost wholly in the field of abstract concepts and their interrelations. While the natural scientist turns constantly to experiment for proof of his assertions, the mathematician employs only argument and computation.

It is true that mathematicians also make constant use, to assist them in the discovery of their theorems and methods, of models and physical analogues, and they have recourse to various completely concrete examples. These examples serve as the actual source of the theory and as a means of discovering its theorems, but no theorem definitely belongs to mathematics until it has been rigorously proved by a logical argument. If a geometer, reporting a newly discovered theorem, were to demonstrate it by means of models and to confine himself to such a demonstration, no mathematician would admit that the theorem had been proved. The demand for a proof of a theorem is well known in high school geometry, but it pervades the whole of mathematics. We could measure the angles at the base of a thousand isosceles triangles with extreme accuracy, but such a procedure would never provide us with a mathematical proof of the theorem that the angles at the base of an isosceles triangle are equal. Mathematics demands that this result be deduced from the fundamental concepts of geometry, which at the present time, in view of the fact that geometry is nowadays developed on a rigorous basis, are precisely formulated in the axioms. And so it is in every case. To prove a theorem means for the mathematician to deduce it by a logical argument from the fundamental properties of the concepts occurring in that theorem. In this way, not only the concepts but also the methods of mathematics are abstract and theoretical.

The results of mathematics are distinguished by a high degree of logical rigor, and a mathematical argument is conducted with such scrupulousness as to make it incontestable and completely convincing to anyone who understands it. The scrupulousness and cogency of mathematical proofs are already well known in a high school course. Mathematical truths are in fact the prototype of the completely incontestable. Not for nothing do people say “as clear as two and two are four.” Here the relation “two and two are four” is introduced as the very image of the irrefutable and incontestable.

But the rigor of mathematics is not absolute; it is in a process of continual development; the principles of mathematics have not congealed once and for all but have a life of their own and may even be the subject of scientific quarrels.

In the final analysis the vitality of mathematics arises from the fact that its concepts and results, for all their abstractness, originate, as we shall see, in the actual world and find widely varied application in the other sciences, in engineering, and in all the practical affairs of daily life; to realize this is the most important prerequisite for understanding mathematics.

The exceptional breadth of its applications is another characteristic feature of mathematics.

In the first place we make constant use, almost every hour, in industry and in private and social life, of the most varied concepts and results of mathematics, without thinking about them at all; for example, we use arithmetic to compute our expenses or geometry to calculate the floor area of an apartment. Of course, the rules here are very simple, but we should remember that in some period of antiquity they represented the most advanced mathematical achievements of the age.

Second, modern technology would be impossible without mathematics. There is probably not a single technical process which can be carried through without more or less complicated calculations; and mathematics plays a very important role in the development of new branches of technology.

Finally, it is true that every science, to a greater or lesser degree, makes essential use of mathematics. The “exact sciences,” mechanics, astronomy, physics, and to a great extent chemistry, express their laws, as every schoolboy knows, by means of formulas and make extensive use of mathematical apparatus in developing their theories. The progress of these sciences would have been completely impossible without mathematics. For this reason the requirements of mechanics, astronomy, and physics have always exercised a direct and decisive influence on the development of mathematics.

In other sciences mathematics plays a smaller role, but here too it finds important applications. Of course, in the study of such complicated phenomena as occur in biology and sociology, the mathematical method cannot play the same role as, let us say, in physics. In all cases, but especially where the phenomena are most complicated, we must bear in mind, if we are not to lose our way in meaningless play with formulas, that the application of mathematics is significant only if the concrete phenomena have already been made the subject of a profound theory. In one way or another, mathematics is applied in almost every science, from mechanics to political economy.

Let us recall some particularly brilliant applications of mathematics in the exact sciences and in technology.

The planet Neptune, one of the most distant in the Solar System, was discovered in the year 1846 on the basis of mathematical calculations. By analyzing certain irregularities in the motion of Uranus, the astronomers Adams and Leverrier came to the conclusion that these irregularities were caused by the gravitational attraction of another planet. Leverrier calculated on the basis of the laws of mechanics exactly where this planet must be, and an observer to whom he communicated his results caught sight of it in his telescope in the exact position indicated by Leverrier. This discovery was a triumph not only for mechanics and astronomy, and in particular for the system of Copernicus, but also for the powers of mathematical calculation.

Another example, no less impressive, was the discovery of electromagnetic waves. The English physicist Maxwell, by generalizing the laws of electromagnetic phenomena as established by experiment, was able to express these laws in the form of equations. From these equations he deduced, by purely mathematical methods, that electromagnetic waves could exist and that they must be propagated with the speed of light. On the basis of this result, he proposed the electromagnetic theory of light, which was later developed and deepened in every direction. Moreover, Maxwell’s results led to the search for electromagnetic waves of purely electrical origin, arising for example from an oscillating charge. These waves were actually discovered by Hertz. Shortly afterwards, A. S. Popov. by discovering means for exciting, transmitting, and receiving electromagnetic oscillations made them available for a wide range of applications and thereby laid the foundations for the whole technology of radio. In the discovery of radio, now the common possession of everyone, an important role was played by the results of a purely mathematical deduction.

So from observation, as for example of the deflection of a magnetic needle by an electric current, science proceeds to generalization, to a theory of the phenomena, and to formulation of laws and to mathematical expression of them. From these laws come new deductions, and finally, the theory is embodied in practice, which in its turn provides powerful new impulses for the development of the theory.

It is particularly remarkable that even the most abstract constructions of mathematics, arising within that science itself, without any immediate motivation from the natural sciences or from technology, nevertheless have fruitful applications. For example, imaginary numbers first came to light in algebra, and for a long time their significance in the actual world remained uncomprehended, a circumstance indicated by their very name. But when about 1800 a geometrical interpretation (see Chapter IV, §2) was given to them, imaginary numbers became firmly established in mathematics, giving rise to the extensive theory of functions of a complex variable, i.e., of a variable of the form . This theory of “imaginary” functions of an “imaginary” variable proved itself to be far from imaginary, but rather a very practical means of solving technological problems. Thus, the fundamental results of N. E. Jukovski concerning the lift on the wing of an airplane are proved by means of this theory. The same theory is useful, for example, in the solution of problems concerning the oozing of water under a dam, problems whose importance is obvious during the present period of construction of huge hydroelectric stations.

Another example, equally impressive, is provided by non-Euclidean geometry,* which arose from the efforts, extending for 2000 years from the time of Euclid, to prove the parallel axiom, a problem of purely mathematical interest. N. I. Lobačevskiĭ himself, the founder of the new geometry, was careful to label his geometry “imaginary,” since he could not see any meaning for it in the actual world, although he was confident that such a meaning would eventually be found. The results of his geometry appeared to the majority of mathematicians to be not only “imaginary” but even unimaginable and absurd. Nevertheless, his ideas laid the foundation for a new development of geometry, namely the creation of theories of various non-Euclidean spaces; and these ideas subsequently became the basis of the general theory of relativity, in which the mathematical apparatus consists of a form of non-Euclidean geometry of four-dimensional space. Thus the abstract constructions of mathematics, which at the very least seemed incomprehensible, proved themselves a powerful instrument for the development of one of the most important theories of physics. Similarly, in the present-day theory of atomic phenomena, in the so-called quantum mechanics, essential use is made of many extremely abstract mathematical concepts and theories, as for example the concept of infinite-dimensional space.

There is no need to give any further examples, since we have already shown with sufficient emphasis that mathematics finds widespread application in everyday life and in technology and science; in the exact sciences and in the great problems of technology, applications are found even for those theories which arise within mathematics itself. This is one of the characteristic peculiarities of mathematics, along with its abstractness and the rigor and conclusiveness of its results.

2. The essential nature of mathematics. In discussing these special features of mathematics we have been far from explaining its essence; rather we have merely pointed out its external marks. Our task now is to explain the essential nature of these characteristic features. For this purpose it will be necessary to answer, at the very least, the following questions:

What do these abstract mathematical concepts reflect? In other words, what is the actual subject matter of mathematics?

Why do the abstract results of mathematics appear so convincing, and its initial concepts so obvious? In other words, on what foundation do the methods of mathematics rest ?

Why, in spite of all its abstractness, does mathematics find such wide application and does not turn out to be merely idle play with abstractions ? In other words, how is the significance of mathematics to be explained ?

Finally, what forces lead to the further development of mathematics, allowing it to unite abstractness with breadth of application? What is the basis for its continuing growth ?

In answering these questions we will form a general picture of the content of mathematics, of its methods, and of its significance and its development; that is, we will understand its essence.

Idealists and metaphysicists not only fall into confusion in their attempts to answer these basic questions but they go so far as to distort mathematics completely, turning it literally inside out. Thus, seeing the extreme abstractness and cogency of mathematical results, the idealist imagines that mathematics issues from pure thought.

In reality, mathematics offers not the slightest support for idealism or metaphysics. We will convince ourselves of this as we attempt, in general outline, to answer the listed questions about the essence of mathematics. For a preliminary clarification of these questions, it is sufficient to examine the foundations of arithmetic and elementary geometry, to which we now turn.

§2.  Arithmetic

1. The concept of a whole number. The concept of number (for the time being, we speak only of whole positive numbers), though it is so familiar to us today, was worked out very slowly. This can be seen from the way in which counting has been done by various races who until recent times have remained at a relatively primitive level of social life. Among some of them, there were no names for numbers higher than two or three; among others, counting went further but ended after a few numbers, after which they simply said “many” or “countless.” A stock of clearly distinguished names for numbers was only gradually accumulated among the various peoples.

At first these peoples had no concept of what a number is, although they could in their own fashion make judgments about the size of one or another collection of objects met with in their daily life. We must conclude that a number was directly perceived by them as an inseparable property of a collection of objects, a property which they did not, however, clearly distinguish. We are so accustomed to counting that we can hardly imagine this state of affairs, but it is possible to understand it.*

At the next higher level a number already appears as a property of a collection of objects, but it is not yet distinguished from the collection as an “abstract number,” as a number in general. not connected with concrete objects. This is obvious from the names of numbers among certain peoples, as “hand” for five and “wholeman” for twenty. Here five is to be understood not abstractly but simply in the sense of “as many as the fingers on a hand,” twenty is “as many as the fingers and toes on a man” and so forth. In a completely analogous way, certain peoples had no concept of “black,” “hard,” or “circular.” In order to say that an object is black, they compared it with a crow for example, and to say that there were five objects, they directly compared these objects with a hand. In this way it also came about that various names for numbers were used for various kinds of objects; some numbers for counting people, others for counting boats, and so forth, up to as many as ten different kinds of numbers. Here we do not have abstract numbers, but merely a sort of “appellation,” referring only to a definite kind of objects. Among other peoples there were in general no separate names for numbers, as for example, no word for “three,” although they could say “three men” or “in three places,” and so forth.

Similarly among ourselves, we quite readily say that this or that object is black but much more rarely speak about “blackness” in itself, which is a more abstract concept. *

The number of objects in a given collection is a property of the collection, but the number itself, as such, the “abstract number,” is a property abstracted from the concrete collection and considered simply in itself, like “blackness” or “hardness.” Just as blackness is the property common to all objects of the color of coal, so the number “five” is the common property of all collections containing as many objects as there are fingers on a hand. In this case the equality of the two numbers is established by simple comparison: We take an object from the collection, bend one finger over, and count in this way up to the end of the collection. More generally, by pairing off the objects of two collections, it is possible, without making any use of numbers at all, to establish whether or not the collections contain the same number of objects. For example, if guests are taking their places at the table they can easily, without any counting, make it clear to the hostess that she has forgotten one setting, since one guest will be without a setting.

In this way it is possible to give the following definition of a number: Each separate number like “two,” “five,” and so forth, is that property of collections of objects which is common to all collections whose objects can be put into one-to-one correspondence with one another and which is different for those collections for which such a correspondence is impossible. In order to discover this property and to distinguish it clearly, that is, in order to form the concept of a definite number and to give it a name “six,” “ten,” and so forth, it was necessary to compare many collections of objects. For countless generations people repeated the same operation millions of times and in that way discovered numbers and the relations among them.

2. Relations among the whole numbers. Operations with numbers arose in their turn as a reflection of relations among concrete objects. This is observable even in the names of numbers. For example, among certain American Indians the number “twenty-six” is pronounced as “above two tens I place a six,” which is clearly a reflection of a concrete method of counting objects. Addition of numbers corresponds to placing together or uniting two or more collections, and it is equally easy to see the concrete meaning of subtraction, multiplication, and division. Multiplication in particular arose to a great extent, it seems clear, from the habit of counting off equal collections: that is, by twos, by threes, and so forth.

In the process of counting, men not only discovered and assimilated the relations among the separate numbers, as for example that two and three are five, but also they gradually established certain general laws. By practical experience, it was discovered that a sum does not depend on the order of the summands and that the result of counting a given set of objects does not depend on the order in which the counting takes place, a fact which is reflected in the essential identity of the “ordinal” and “cardinal” numbers: first, second, third, and one, two, three. In this way the numbers appeared not as separate and independent, but as interrelated with one another.

Some numbers are expressed in terms of others in their very names and in the way they are written. Thus, “twenty” denotes “two (times) ten”; in French, eighty is “four-twenties” (quatre-vingt), ninety is “four-twenties-ten”; and the Roman numerals VIII, IX denote that 8 = 5 + 3, 9 = 10 – 1.

In general, there arose not just the separate numbers but a system of numbers with mutual relations and rules.

The subject matter of arithmetic is exactly this, the system of numbers with its mutual relations and rules. * The separate abstract number by itself does not have tangible properties, and in general there is very little to be said about it. If we ask ourselves, for example, about the properties of the number six, we note that 6 = 5 + 1, 6 = 3 · 2, that 6 is a factor of 30 and so forth. But here the number 6 is always connected with other numbers; in fact, the properties of a given number consist precisely of its relations with other numbers.† Consequently, it is clear that every arithmetical operation determines a connection or relation among numbers. Thus the subject matter of arithmetic is relations among numbers. But these relations are the abstract images of actual quantitative relations among collections of objects; so we may say that arithmetic is the science of actual quantitative relations considered abstractly, that is, purely as relations. Arithmetic, as we see, did not arise from pure thought, as the idealists represent, but is the reflection of definite properties of real things; it arose from the long practical experience of many generations.

3. Symbols for the numbers. As social life became more extensive and complicated, it posed broader problems. Not only was it necessary to take note of the number of objects in a set and to tell others about it, a necessity which had already led to formulation of the concept of number and to names for the numbers, but it became essential to learn to count increasingly larger collections, of animals in a herd, of objects for exchange, of days before a fixed date, and so forth, and to communicate the results of the count to others. This situation absolutely demanded improvement in the names and also in the symbols for numbers.

The introduction of symbols for the numbers, which apparently occured as soon as writing began, played a great role in the development of arithmetic. Moreover, it was the first step toward mathematical signs and formulas in general. The second step, consisting of the introduction of signs for arithmetical operations and of a literal designation for the unknown (x), was taken considerably later.

The concept of number, like every other abstract concept, has no immediate image; it cannot be exhibited but can only be conceived in the mind. But thought is formulated in language, so that without a name there can be no concept. The symbol is also a name, except that it is not oral but written and presents itself to the mind in the form of a visible image. For example, if I say “seven,” what do you picture to yourself? Probably not a set of seven objects of one kind or another, but rather the symbol “7,” which forms a sort of tangible framework for the abstract number “seven.” Moreover, a number 18273 is considerably harder to pronounce than to write and cannot be pictured with any accuracy in the form of a set of objects. In this way it came about, though only after some lapse of time, that the symbols gave rise to the conception of numbers so large that they could never have been discovered by direct observation or by enumeration. With the appearance of government, it was necessary to collect taxes, to assemble and outfit an army, and so forth, all of which required operations with very large numbers.

Thus the importance of symbols for the numbers consists, in the first place, in their providing a simple embodiment of the concept of an abstract number.* This is the role of mathematical designations in general: They provide an embodiment of abstract mathematical concepts. Thus + denotes addition, x denotes an unknown number, a an arbitrary given number, and so forth. In the second place the symbols for numbers provide a particularly simple means of carrying out operations on them. Everyone knows how much easier it is “to calculate on paper” than “in one’s head.” Mathematical signs and formulas have this advantage in general: They allow us to replace a part of our arguments with calculations, with something that is almost mechanical. Moreover, if a calculation is written down, it already possesses a definite authenticity; everything is visible, everything can be checked, and everything is defined by exact rules. As examples one might mention addition by individual columns or any algebraic transformation such as “taking over to the other side of the equation with change of sign.” From what has been said, it is clear that without suitable symbols for the numbers arithmetic could not have made much progress. Even more is it true that contemporary mathematics would be impossible without its special signs and formulas.

It is obvious that the extremely convenient method of writing numbers that is in use today could not have been worked out all at once. From ancient times there appeared among various peoples, from the very beginnings of their culture, various symbols for the numbers, which were very unlike our contemporary ones not only in their general appearance but also in the principles on which they were chosen. For example, the decimal system was not used everywhere, and among the ancient Babylonians there was a system that was partly decimal and partly sexagesimal. Table 1 gives some of the symbols for numbers among various peoples. In particular, we see that the ancient Greeks, and later also the Russians, made use of letters to designate numbers. Our contemporary “Arabic” symbols and, more generally, our method of forming the numbers, were brought from India to Europe by the Arabs in the 10th century and became firmly rooted there in the course of the next few centuries.

Table

SYMBOLS FOR THE NUMBERS

1

AMONG VARIOUS PEOPLES

The first peculiarity of our system is that it is a decimal system. But this is not a matter of great importance, since it would have been quite possible to use, for example, a duodecimal system by introducing special symbols for ten and eleven. The most important peculiarity of our system of designating numbers is that it is “positional”; that is, that one and the same number has a different significance depending upon its position. For example, in 372 the number 3 denotes the number of hundreds and 7 the numbers of tens. This method of writing is not only concise and simple but makes calculations very easy. The Roman numerals were in this respect much less convenient, the same number 372 being written in the form CCCLXXII; it is a very laborious task to multiply together two large numbers written in Roman numerals.

Positional writing of numbers demands that in one way or another we take note of any category of numbers that has been omitted, since if we do not do this, we will confuse, for example, thirty-one with three-hundredand-one. In the position of the omitted category we must place a zero, thereby distinguishing 301 and 31. In a rudimentary form, zero already appears in the late Babylonian cuneiform writings, but its systematic introduction was an achievement of the Indians:* It allowed them to proceed to a completely positional system of writing just as we have it today.

But in this way zero also became a number and entered into the system of numbers. By itself zero is nothing; in the Sanskrit language of ancient India, it is called exactly that: “empty” ; but in connection with other numbers, zero acquires content, and well-known properties; for example, an arbitrary number plus zero is the same number, or when an arbitrary number is multiplied by zero it becomes zero.

4. The theory of numbers as a branch of pure mathematics. Let us return to the arithmetic of the ancients. The oldest texts that have been preserved from Babylon and Egypt go back to the second millennium B.C. These and later texts contain various arithmetical problems with their solutions, among them certain ones that today belong to algebra, such as the solution of quadratic and even cubic equations or progressions; all this being presented, of course, in the form of concrete problems and numerical examples. Among the Babylonians we also find certain tables of squares, cubes, and reciprocals. It is to be supposed that they were already beginning to form mathematical interests which were not immediately connected with practical problems.

In any case arithmetic was well developed in ancient Babylon and Egypt. However, it was not yet a mathematical theory of numbers but rather a collection of solutions for various problems and of rules of calculation. It is exactly in this way that arithmetic is taught up to the present time in our elementary schools and is understood by everyone who is not especially interested in mathematics. This is perfectly legitimate, but arithmetic in this form is still not a mathematical theory. There are no general theorems about numbers.

The transition to theoretical arithmetic proceeded gradually.

As was pointed out, the existence of symbols allows us to operate with numbers so large that it is impossible to visualize them as collections of objects or to arrive at them by the process of counting in succession from the number one. Among primitive tribes special numbers were worked out up to 3, 10, 100 and so forth, but after these came the indefinite “many.” In contrast to this situation the use of symbols for numbers enabled the Chinese, the Babylonians, and the Egyptians to proceed to tens of thousands and even to millions. It was at this stage that the possibility was noticed of indefinitely extending the series of numbers, although we do not know how soon this possibility was clearly perceived. Even Archimedes (287–212 B.C.) in his remarkable essay “The Sand Reckoner” took the trouble to describe a method for naming a number greater than the number of grains of sand sufficient to fill up the “sphere of the fixed stars.” So the possibility of naming and writing such a number still required at his time a detailed explanation.

By the 3rd century B.C., the Greeks had clearly recognized twoimportant ideas: first, that the sequence of numbers could be indefinitely extended and second, that it was not only possible to operate with arbitrarily given numbers but to discuss numbers in general, to formulate and prove general theorems about them. This idea represents the generalization of an immense amount of earlier experience with concrete numbers, from which arose the rules and methods for general reasoning about numbers. A transition took place to a higher level of abstraction: from separate given (though abstract) numbers to number in general, to any possible number.

From the simple process of counting objects one by one, we pass to the unbounded process of formation of numbers by adding one to the number already formed. The sequence of numbers is regarded as being indefinitely continuable, and with it there enters into mathematics the notion of infinity. Of course, we cannot in fact, by the process of adding one, proceed arbitrarily far along the sequence of numbers: Who could reach as far as a million-million, which is almost forty times the number of seconds in a thousand years? But that is not the point; the process of adding ones, the process of forming arbitrary large collections of objects is in principle unlimited, so that the possibility exists of continuing the sequence of numbers beyond all limits. The fact that in actual practice counting is limited is not relevant; an abstraction is made from it. It is with this indefinitely prolonged sequence that general theorems about numbers have to deal.

General theorems about any property of an arbitrary number already contain in implicit form infinitely many assertions about the properties of separate numbers and are therefore qualitatively richer than any particular assertions that could be verified for specific numbers. It is for this reason that general theorems must be proved by general arguments proceeding from the fundamental rule for the formation of the sequence of numbers. Here we perceive a profound peculiarity of mathematics: Mathematics takes as its subject not only given quantitative relationships but all possible quantitative relationships and therefore infinity.

In the famous “Elements” of Euclid, written in the 3rd century B.C., we already find general theorems about whole numbers, in particular, the theorem that there exist arbitrarily large prime numbers.*

Thus arithmetic is transformed into the theory of numbers. It is already removed from particular concrete problems to the region of abstract concepts and arguments. It has become a part of “pure” mathematics. More precisely, this was the moment of the birth of pure mathematics itself with the characteristic features discussed in our first section. We must, of course, take note of the fact that pure mathematics was born simultaneously from arithmetic and geometry and that there were already to be found in the general rules of arithmetic some of the rudiments of algebra, a subject which was separated from arithmetic at a later stage. But we will discuss this later.

It remains now to summarize our conclusions up to this point, since we have now traced out, though in very hurried fashion, the process whereby theoretical arithmetic arose from the concept of number.

5. The essential nature of arithmetic. Since the birth of theoretical arithmetic is part of the birth of mathematics, we may reasonably expect that our conclusions about arithmetic will throw light on our earlier questions concerning mathematics in general. Let us recall these questions, particularly in their application to arithmetic.

1. How did the abstract concepts of arithmetic arise and what do they reflect in the actual world ?

This question is answered by the earlier remarks about the birth of arithmetic. Its concepts correspond to the quantitative relations of collections of objects. These concepts arose by way of abstraction, as a result of the analysis and generalization of an immense amount of practical experience. They arose gradually; first came numbers connected with concrete objects, then abstract numbers, and finally the concept of number in general, of any possible number. Each of these concepts was made possible by a combination of practical experience and preceding abstract concepts. This, by the way, is one of the fundamental laws of formation of mathematical concepts: They are brought into being by a series of successive abstractions and generalizations, each resting on a combination of experience with preceding abstract concepts. The history of the concepts of arithmetic shows how mistaken is the idealistic view that they arose from “pure thought,” from “innate intuition,” from “contemplation of a priori forms,” or the like.

2. Why are the conclusions of arithmetic so convincing and unalterable ?

History answers this question too for us. We see that the conclusions of arithmetic have been worked out slowly and gradually; they reflect experience accumulated in the course of unimaginably many generations and have in this way fixed themselves firmly in the mind of man. They have also fixed themselves in language: in the names for the numbers, in their symbols, in the constant repetition of the same operations with numbers, in their constant application to daily life. It is in this way that they have gained clarity and certainty. The methods of logical reasoning also have the same source. What is essential here is not only the fact that they can be repeated at will but their soundness and perspicuity, which they possess in common with the relations among things in the actual world, relations which are reflected in the concepts of arithmetic and in the rules for logical deduction.

This is the reason why the results of arithmetic are so convincing; its conclusions flow logically from its basic concepts, and both of them, the methods of logic and the concepts of arithmetic, were worked out and firmly fixed in our consciousness by three thousand years of practical experience, on the basis of objective uniformities in the world around us.

3. Why does arithmetic have such wide application in spite of the abstractness of its concepts ?

The answer is simple. The concepts and conclusions of arithmetic, which generalize an enormous amount of experience, reflect in abstract form those relationships in the actual world that are met with constantly and everywhere. It is possible to count the objects in a room, the stars, people, atoms, and saforth. Arithmetic considers certain of their general properties, in abstraction from everything particular and concrete, and it is precisely because it considers only these general properties that its conclusions are applicable to so many cases. The possibility of wide application is guaranteed by the very abstractness of arithmetic, although it is important here that this abstraction is not an empty one but is derived from long practical experience. The same is true for all mathematics, and for any abstract concept or theory. The possibilities for application of a theory depend on the breadth of the original material which it generalizes.

At the same time every abstract concept, in particular the concept of number, is limited in its significance as a result of its very abstractness. In the first place, when applied to any concrete object it reflects only one aspect of the object and therefore gives only an incomplete picture of it. How often it happens, for example, that the mere numerical facts say very little about the essence of the matter. In the second place, abstract concepts cannot be applied everywhere without certain limiting conditions; it is impossible to apply arithmetic to concrete problems without first convincing ourselves that their application makes some sense in the particular case. If we speak of addition, for example, and merely unite the objects in thought, then naturally no progress has been made with the objects themselves. But if we apply addition to the actual uniting of the objects, if we in fact put the objects together, for example by throwing them into a pile or setting them on a table, in this case there takes place not merely abstract addition but also an actual process. This process does not consist merely of the arithmetical addition, and in general it may even be impossible to carry it out. For example, the object thrown into a pile may break; wild animals, if placed together, may tear one another apart; the materials put together may enter into a chemical reaction: a liter of water and a liter of alcohol when poured together produced not 2, but 1.9 liters of the mixture as a result of partial solution of the liquids; and so forth.

If other examples are needed they are easy to produce.

To put it briefly, truth is concrete; and it is particularly important to remember this fact with respect to mathematics, exactly because of its abstractness.

4. Finally, the last question we raised had to do with the forces that led to the development of mathematics.

For arithmetic the answer to this question also is clear from its history. We saw how people in the actual world learned to count and to work out the concept of number, and how practical life, by posing more difficult problems, necessitated symbols for the numbers. In a word, the forces that led to the development of arithmetic were the practical needs of social life. These practical needs and the abstract thought arising from them exercise on each other a constant interaction. The abstract concepts provide in themselves a valuable tool for practical life and are constantly improved by their very application. Abstraction from all nonessentials uncovers the kernel of the matter and guarantees success in those cases where a decisive role is played by the properties and relations picked out and preserved by the abstraction; namely, in the case of arithmetic, by the quantitative relations.

Moreover, abstract reflection often goes farther than the immediate demands of a practical problem. Thus the concept of such large numbers as a million or a billion arose on the basis of practical calculations but arose earlier than the practical need to make use of them. There are many such examples in the history of science; it is enough to recall the imaginary numbers mentioned earlier. This is just a particular case of a phenomenon known to everyone, namely the interaction of experience and abstract thought, of practice and theory.

§3.  Geometry

1. The concept of a geometric figure. The history of the origin of geometry is essentially similar to that of arithmetic. The earliest geometric concepts and information also go back to prehistoric times and also result from practical activity.

Early man took over geometric forms from nature. The circle and the crescent of the moon, the smooth surface of a lake, the straightness of a ray of light or of a well-proportioned tree existed long before man himself and presented themselves constantly to his observation. But in nature itself our eyes seldom meet with really straight lines, with precise triangles or squares, and it is clear that the chief reason why men gradually worked out a conception of these figures is that their observation of nature was an active one, in the sense that, to meet their practical needs, they manufactured objects more and more regular in shape. They built dwellings, cut stones, enclosed plots of land, stretched bowstrings in their bows, modeled their clay pottery, brought it to perfection and correspondingly formed the notion that a pot is curved, but a stretched bowstring is straight. In short, they first gave form to their material and only then recognized form as that which is impressed on material and can therefore be considered in itself, as an abstraction from material. By recognizing the form of bodies, man was able to improve his handiwork and thereby to work out still more precisely the abstract notion of form. Thus practical activity served as a basis for the abstract concepts of geometry. It was necessary to manufacture thousands of objects with straight edges, to stretch thousands of threads, to draw upon the ground a large number of straight lines, before men could form a clear notion of the straight line in general, as that quality which is common to all these particular cases. Nowadays we learn early in life to draw a straight line, since we are surrounded by objects with straight edges that are the result of manufacture, and it is only for this reason that in our childhood we already form a clear notion of the straight line. In exactly the same way the notion of geometric magnitudes, of length, area, and volume, arose from practical activity. People measured lengths, determined distances, estimated by eye the area of surfaces and the volumes of bodies, all for their practical purposes. It was in this way that the simplest general laws were discovered, the first geometric relations: for example, that the area of a rectangle is equal to the product of the lengths of its sides. It is useful for a farmer to be aware of such a relation, in order that he may estimate the area he has sowed and consequently the harvest he may expect.

So we see that geometry took its rise from practical activity and from the problems of daily life. On this question the ancient Greek scholar, Eudemus of Rhodes, wrote as follows: “Geometry was discovered by the Egyptians as a result of their measurement of land. This measurement was necessary for them because of the inundations of the Nile, which constantly washed away their boundaries.* There is nothing remarkable in the fact that this science, like the others, arose from the practical needs of men. All knowledge that arises from imperfect circumstances tends to perfect itself. It arises from sense impressions but gradually becomes an object of our contemplation and finally enters the realm of the intellect.”

Of course, the measurement of land was not the only problem that led the ancients toward geometry. From the fragmentary texts that have survived, it is possible to form some idea of various problems of the ancient Egyptians and Babylonians and of their methods for solving them. One of the oldest Egyptian texts goes back to 1700 B.C. This is a manual of instruction for “secretaries” (royal officers), written by a certain Ahmes. It contains a collection of problems on calculating the capacity of containers and warehouses, the area of shares of land, the dimensions of earthworks, and so forth.

The Egyptians and Babylonians were able to determine the simplest areas and volumes, they knew with considerable exactness the ratio of the circumference to the diameter of a circle, and perhaps they were even able to calculate the surface area of a sphere; in a word, they already possessed a considerable store of geometrical knowledge. But so far as we can tell, they were still not in possession of geometry as a theoretical science with theorems and proofs. Like the arithmetic of the time, geometry was basically a collection of rules deduced from experience. Moreover, geometry was in general not distinguished from arithmetic. Geometric problems were at the same time problems for calculation in arithmetic.

In the 7th century B.C., geometry passed from Egypt to Greece, where it was further developed by the great materialist philosophers, Thales, Democritus, and others. A considerable contribution to geometry was also made by the successors of Pythagoras, the founders of an idealistic religiophilosophical school.

The development of geometry took the direction of compiling new facts and clarifying their relations with one another. These relations were gradually transformed into logical deductions of certain propositions of geometry from certain others. This had two results: first, the concept of a geometrical theorem and its proof; and second, the clarification of those fundamental propositions from which the others may be deduced, namely, the axioms.

In this way geometry gradually developed into a mathematical theory.

It is well known that systematic expositions of geometry appeared in Greece as far back as the 5th century B.C., but they have not been preserved, for the obvious reason that they were all supplanted by the “Elements” of Euclid (3rd century B.C.). In this work, geometry was presented as such a well-formed system that nothing essential was added to its foundations until the time of N. I. Lobačevskiĭ, more than two thousand years later. The well-known school text of Kiselev, like school books over the whole world, represented in its older editions, nothing but a popular reworking of Euclid. Very few other books in the world have had such a long life as the “Elements” of Euclid, this perfect creation of Greek genius. Of course, mathematics continued to advance, and our understanding of the foundations of geometry has been considerably deepened; nevertheless the “Elements” of Euclid became, and to a great extent remains, the model of a book on pure mathematics. Bringing together the accomplishments of his predecessors, Euclid presented the mathematics of his time as an independent theoretical science; that is, he presented it essentially as it is understood today.

2. The essential nature of geometry. The history of geometry leads to the same conclusions as that of arithmetic. We see that geometry arose from practical life and that its transformation to a mathematical theory required an immense period of time.

Geometry operates with “geometric bodies” and figures; it studies their mutual relations from the point of view of magnitude and position. But a geometric body is nothing other than an actual body considered solely from the point of view of its spatial form,* in abstraction from all its other properties such as density, color, or weight. A geometric figure is a still more general concept, since in this case it is possible to abstract from spatial extension also; thus a surface has only two dimensions, a line, only one dimension, and a point, none at all. A point is the abstract concept of the end of a line, of a position defined to the limit of precision so that it no longer has any parts. It is in this way that all these concepts are defined by Euclid.

Thus geometry has as its object the spatial forms and relations of actual bodies, removed from their other properties and considered from the purely abstract point of view. It is just this high level of abstraction that distinguishes geometry from the other sciences that also investigate the spatial forms and relations of bodies. In astronomy for example, the mutual positions of bodies are studied, but they are the actual bodies of the sky; in geodesy it is the form of the earth that is studied, in crystallography, the form of crystals, and so forth. In all these other sciences, the form and the position of concrete bodies are studied in their dependence on other properties of the bodies.

This abstraction necessarily leads to the purely theoretical method of geometry; it is no longer possible to set up experiments with breadthless straight lines, with “pure forms.” The only possibility is to make use of logical argument, deriving some conclusions from others. A geometrical theorem must be proved by reasoning, otherwise it does not belong to geometry; it does not deal with “pure forms.”

The self-evidence of the basic concepts of geometry, the methods of reasoning and the certainty of their conclusions, all have the same source as in arithmetic. The properties of geometric concepts, like the concepts themselves, have been abstracted from the world around us. It was necessary for people to draw innumerable straight lines before they could take it as an axiom that through every two points it is possible to draw a straight line; they had to move various bodies about and apply them to one another on countless occasions before they could generalize their experience to the notion of superposition of geometric figures and make use of this notion for the proof of theorems, as is done in the well-known theorems about congruence of triangles.

Finally, we must emphasize the generality of geometry. The volume of a sphere is equal to 4/3π R³ quite independently of whether we are speaking of a spherical vessel, of a steel sphere, of a star, or of a drop of water. Geometry can abstract what is common to all bodies, because every actual body does have more or less definite form, dimensions, and position with respect to other bodies. So it is no cause for wonder that geometry finds application almost as widely as arithmetic. Workmen measuring the dimensions of a building or reading a blueprint, an artillery man determining the distance to his target, a farmer measuring the area of his field, an engineer estimating the volume of earthworks, all these people make use of the elements of geometry. The pilot, the astronomer, the surveyor, the engineer, the physicist, all have need of the precise conclusions of geometry.

A clear example of the abstract-geometrical solution of an important problem in physics is provided by the investigations of the well-known crystallographer and geometer, E. S. Fedorov. The problem he set himself of finding all the possible forms of symmetry for crystals is one of the most fundamental in theoretical crystallography. To solve this problem, Fedorov made an abstraction from all the physical properties of a crystal, considering it only as a regular system of geometric bodies “in place of a system of concrete atoms.” Thus the problem became one of finding all the forms of symmetry which could possibly exist in a system of geometric bodies. This purely geometrical problem was completely solved by Fedorov, who found all the possible forms of symmetry, 230 in number. His solution proved to be an important contribution to geometry and was the source of many geometric investigations.

In this example, as in the whole history of geometry, we detect the prime moving force in the development of geometry. It is the mutual influence of practical life and abstract thought. The problem of discovering possible symmetries originated in physical observation of crystals but was transformed into an abstract problem and so gave rise to a new mathematical theory, the theory of regular systems, or of the so-called Fedorov groups.* Subsequently this theory not only found brilliant confirmation in the practical observation of crystals but also served as a general guide in the development of crystallography, giving rise to new investigations, both in experimental physics and in pure mathematics.

§4.  Arithmetic and Geometry

1. The origin of fractions in the interrelation of arithmetic and geometry. Up to now we have considered arithmetic and geometry apart from each other. Their mutual relation, and consequently the more general interrelation of all mathematical theories, has so far escaped our attention. Nevertheless this relation has exceptionally great significance. The interaction of mathematical theories leads to advances in mathematics itself and also uncovers a rich treasure of mutual relations in the actual world reflected by the these theories.

Arithmetic and geometry are not only applied to each other but they also serve thereby as sources for further general ideas, methods, and theories. In the final analysis, arithmetic and geometry are the two roots from which has grown the whole of mathematics. Their mutual influence goes back to the time when both of them had just come into being. Even the simple measurement of a line represents a union of geometry and arithmetic. To measure the length of an object we apply to it a certain unit of length and calculate how many times it is possible to do this; the first operation (application) is geometric, the second (calculation) is arithmetical. Everyone who counts off his steps along a road is already uniting these two operations.

In general, the measurement of any magnitude combines calculation with some specific operation which is characteristic of this sort of magnitude. It is sufficient to mention measurement of a liquid in a graduated container or measurement of an interval of time by counting the number of strokes of a pendulum.

But in the process of measurement it turns out, generally speaking, that the chosen unit is not contained in the measured magnitude an integral number of times, so that a simple calculation of the number of units is not sufficient. It becomes necessary to divide up the unit of measurement in order to express the magnitude more accurately by parts of the unit; that is, no longer by whole numbers but by fractions. It was in this way that fractions actually arose, as is shown by an analysis of historical and other data. They arose from the division and comparison of continuous magnitudes; in other words, from measurement. The first magnitudes to be measured were geometric, namely lengths, areas of fields, and volumes of liquids or friable materials, so that in the earliest appearance of fractions we see the mutual action of arithmetic and geometry. This interaction leads to the appearance of an important new concept, namely of fractions, as an extension of the concept of number from whole numbers to fractional numbers (or as the mathematicians say, to rational numbers, expressing the ratio of whole numbers). Fractions did not arise, and could not arise, from the division of whole numbers, since only whole objects are counted by whole numbers. Three men, three arrows, and so forth, all these make sense, but two-thirds of a man and even two-thirds of an arrow are senseless concepts; even three separate thirds of an arrow will not kill a deer, for this it is necessary to have a whole arrow.

2. Incommensurable magnitudes. In the development of the concept of number, arising from the mutual action of arithmetic and geometry, the appearance of fractions was only the first step. The next was the discovery of incommensurable intervals. Let us recall that intervals are called incommensurable if no interval exists which can be applied to each of them a whole number of times or, in other words, if their ratio cannot be expressed by an ordinary fraction; that is, by a ratio of whole numbers.

At first people simply did not think about the question whether every interval can be expressed by a fraction. If in dividing up or measuring an interval they came upon very small parts, they merely discarded them; in practice, it made no sense to speak of infinite precision of measurement. Democritus even advanced the notion that geometrical figures consist of atoms of a particular kind. This notion, which to our view seems quite strange, proved very fruitful in the determination of areas and volumes. An area was calculated as the sum of rows consisting of atoms, and a volume as the sum of atomic layers. It was in this way, for example, that Democritus found the volume of a cone. A reader who understands the integral calculus will note that this method already forms the prototype of the determination of areas and volumes by the methods of the integral calculus. Moreover, in returning in thought to the times of Democritus, one must attempt to free oneself of the customary notions of today, which have become firmly fixed in our minds by the development of mathematics. At the time of Democritus, geometrical figures were not yet separated from actual ones to the same extent as is now the case. Since Democritus considered actual bodies as consisting of atoms, he naturally also regarded geometrical figures in the same light.

But the notion that intervals consist of atoms comes into contradiction with the theorem of Pythagoras, since it follows from this theorem that incommensurable intervals exist. For example, the diagonal of a square is incommensurable with its side; in other words, the ratio of the two cannot be expressed as the ratio of whole numbers.

We shall prove that the side and the diagonal of a square are in fact incommensurable. If a is the side and b is the diagonal of a square, then according to the theorem of Pythagoras b² = a² + a² = 2a² and therefore

But there is no fraction such that its square is equal to 2. In fact, if we suppose that there is, let p and q be whole numbers for which

where we may assume that p and q have no common factor, since otherwise we could simplify the fraction. But if (p/q)² = 2, then p² = 2q², and therefore p² is divisible by 2. In this case p² is also divisible by 4, since it is the square of an even number. So p² = 4q₁; that is, 2q² = 4q₁, and q² = 2q₁. From this it follows that q must also be divisible by 2. But this contradicts the supposition that p and q have no common factor. This contradiction proves that the ratio b/a cannot be expressed by a rational number. The diagonal and the side of a square are incommensurable.

This discovery made a great impression on the Greek scientists. Nowadays, when we are accustomed to irrational numbers and calculate freely with square roots, the existence of incommensurable intervals does not disturb us. But in the 5th century B.C., the discovery of such intervals had a completely different aspect for the Greeks. Since they did not have the concept of an irrational number and never wrote a symbol like , the previous result indicated that the ratio of the diagonal and the side of the square was not represented by any number at all.

In the existence of iilcommensurable intervals the Greeks discovered a profound paradox inherent in the concept of continuity, one of the expressions of the dialectical contradiction comprised in continuity and motion. Many important Greek philosophers considered this contradiction; particularly well-known among them, because of his paradoxes, is Zeno the Eleatic.

The Greeks founded a theory of ratios of intervals, or of magnitudes in general, which takes into consideration the existence of incommensurable intervals;* it is expounded in the “Elements” of Euclid, and in simplified form is explained today in high school courses in geometry. But to recognize that the ratio of one interval to another (if the second interval is taken as the unit of length, this ratio is simply the length of the first interval) may also be considered as a number, whereby the very concept of number is generalized, to this idea the Greeks were not able to rise: The concept of an irrational number simply did not originate among them† This step was taken at a later period by the mathematicians of the East; and in general, a mathematically rigorous definition of a real number, not depending immediately on geometry, was given only recently: in the seventies of the last century.* The passage of such an immense period of time after the founding of the theory of ratios shows how difficult it is to discover abstract concepts and give them exact formulation.

3. The real number. In describing the concept of a real number, Newton in his “General Arithmetic” wrote: “by number we mean not so much a collection of units as an abstract ratio of a certain quantity to another quantity taken as the unit.” This number (ratio) may be integral, rational, or if the given magnitude is incommensurable with the unit, irrational.

A real number in its original sense is therefore nothing but the ratio of one magnitude to another taken as a unit; in particular cases this is a ratio of intervals, but it may also be a ratio of areas, weights, and so forth.

Consequently, a real number is a ratio of magnitudes in general, considered in abstraction from their concrete nature.

Just as abstract whole numbers are of mathematical interest only in their relations with one another, so abstract real numbers have content and become an object of mathematical attention only in relation with one another in the system of real numbers.

In the theory of real numbers, just as in arithmetic, it is first necessary to define operations on numbers: addition, subtraction, multiplication, division, and also the relations expressed by such words as “greater than” or “less than.” These o~erations and relations reflect actual connections among the various magnitudes; for example, addition reflects the placing together of intervals. A beginning on operations with abstract real numbers was made in the Middle Ages by the mathematicians of the East. Later came the gradual discovery of the most important property of the system of real numbers, its continuity. The system of real numbers is the abstract image of all the possible values of a continuously varying magnitude.

In this way, as in the similar case of whole numbers, the arithmetic of real numbers deals with the actual quantitative relations of continuous magnitudes, which it studies in their general form, in complete abstraction from all concrete properties. It is precisely because real numbers deal with what is common to all continuous magnitudes that they have such wide application: The values of various magnitudes, a length, a weight, the strength of an electric current, energy and so forth, are expressed by numbers, and the interdependence or relations among these entities are mirrored as relations among their numerical values.

To show how the general concept of real numbers can serve as the basis of a mathematical theory, we must give their mathematical definition in a formal way. This may be done by various methods, but perhaps the most natural is to proceed from the very process of measurement of magnitudes which actually did lead in practical life to this generalization of the concept of number. We will speak about the length of intervals, but the reader will readily perceive that we could argue in exactly the same way about any other magnitudes which permit indefinite subdivision.

Let us suppose that we wish to measure the interval AB by means of the interval CD taken as a unit (figure 1).

FIG. 1.

We apply the interval CD to AB, beginning for example with the point A, as long as CD goes into AB. Suppose this is n₀ times. If there still remains from the interval AB a remainder PB, then we divide the interval CD into ten parts and measure the remainder with these tenths. Suppose that n₁, of the tenths go into the remainder. If after this there is still a remainder, we divide our measure into ten parts again; that is, we divide CD into a hundred parts, and repeat the same operation, and so forth. Either the process of measurement comes to an end, or it continues. In either case we reach the result that in the interval AB the whole interval CD is contained n₀ times, the tenths are contained n₁ times, the hundredths n₂ times and so forth. In a word, we derive the ratio of AB to CD with increasing accuracy: up to tenths, to hundredths, and so forth. So the ratio itself is represented by a decimal fraction with n₀ units, n₁ tenths and so forth

This decimal fraction may be infinite, corresponding to the possibility of indefinite increase in the precision of measurement.

Thus the ratio of two intervals, or of two magnitudes in general, is always representable by a decimal fraction, finite or infinite. But in the decimal fraction there is no longer any trace of the concrete magnitude itself; it represents exactly the abstract ratio, the real number. Thus a real number may be formally defined if we wish, as a finite or infinite decimal fraction. *

Our definition will be complete if we say what we mean by the operations of addition and so forth for decimal fractions. This is done in such a way that the operations defined on decimal fractions correspond to the operations on the magnitudes themselves. Thus, when intervals are put together their lengths are added; that is, the length of the interval AB + BC is equal to the sum of the length AB and BC. In defining the operations on real numbers, there is a difficulty that these numbers are represented in general by infinite decimal fractions, while the well-known rules for these operations refer to finite decimal fractions. A rigorous definition of the operations for infinite decimals may be made in the following way. Suppose, for example, that we must add the two numbers a and b. We take the corresponding decimal fractions up to a given decimal place, say the millionth, and add them. We thus obtain the sum a + b with corresponding accuracy, up to two millionths, since the errors in a and b may be added together. So we are able to define the sum of two numbers with an arbitrary degree of accuracy, and in that sense their sum is completely defined, although at each stage of the calculation it is known only with a a certain accuracy. But this corresponds to the essential nature of the case, since each of the magnitudes a and b is also measured only with a certain accuracy, and the exact value of each of the corresponding infinite fractions is obtained as the result of an indefinitely extended increase in accuracy. The relations “greater than” and “less than” may then be defined by means of addition: a is greater than b if there exists a magnitude c such that a = b + c, where we are speaking, of course, of positive numbers.

The continuity of the sequence of real numbers finds expression in the fact that if the numbers a₁, a₂, … increase and b₁, b₂, … diminish but always remain greater than the ai, then between the one series of numbers and the other there is always a number c. This may be visualized on a straight line if its points are put into correspondence with the numbers (figure 2) according to the well-known rule.

FIG. 2.

Here it is clearly seen that the presence of the number c and of the point corrzsponding to it signify the absence of a break in the series of numbers, which is what is meant by their continuity.

4. The conflict of opposites: concrete and abstract. Already in the example of the interaction of arithmetic and geometry we can see that the development of mathematics is a process of conflict among the many contrasting elements: the concrete and the abstract, the particular and the general, the formal and the material, the finite and the infinite, the discrete and the continuous, and so forth. Let us try, for example, to trace the contrast between concrete and abstract in the formation of the concept of a real number. As we have seen, the real number reflects an infinitery improvable process of measurement or, in slightly different terms, an absolutely accurate determination of a magnitude. This corresponds to the fact that in geometry we consider ideally precise forms and dimensions of bodies, abstracting altogether from the mobility of concrete objects and from a certain indefiniteness in their actual forms and dimensions; for example, the interval measured (figure 1) was a completely ideal one.

But ideally precise geometric forms and absolutely precise values for magnitudes represent abstractions. No concrete object has absolutely precise form nor can any concrete magnitude be measured with absolute accuracy, since it does not even have an absolutely accurate value. The length of a line segment, for example, has no sense if one tries to make it precise beyond the limits of atomic dimensions. In every case when one passes beyond well-known limits of quantitative accuracy, there appears a qualitative change in the magnitude, and in general it loses its original meaning. For example, the pressure of a gas cannot be made precise beyond the limits of the impact of a single molecule; electric charge ceases to be continuous when one tries to make it precise beyond the charge on an electron and so forth. In view of the absence in nature of objects of ideally precise form, the assertion that the ratio of the diagonal of a square to the side is equal to the not only cannot be deduced with absolute accuracy from immediate measurement but does not even have any absolutely accirate meaning for an actual concrete square.

The conclusion that the diagonal and the side of a square are incommensurable comes, as we have seen, from the theorem of Pythagoras. This is a theoretical conclusion based on a development of the data of experience; it is a result of the application of logic to the original premises of geometry, which are taken from experience.

In this way the concept of incommensurable intervals, and all the more of real numbers, is not a simple immediate reflection of the facts of experience but goes beyond them. This is quite understandable. The real number does not reflect any given concrete magnitude but rather magnitude in general, in abstraction from all concreteness; in other words, it reflects what is common to particular concrete magnitudes. What is common to all of them consists in particular in this, that the value of the magnitude can be determined more and more precisely; and if we abstract from concrete magnitudes, then the limit of this possible increase in precision, which depends on the concrete nature of the magnitude, becomes indefinite and disappears.

In this way a mathematical theory of magnitudes, since it considers magnitudes in abstraction from their individual nature, must inevitably consider the possibility of unlimited accuracy for the value of the magnitude and must thereby lead to the concept of a real number. At the same time, since it reflects only what is common to various magnitudes, mathematics takes no account of the peculiarity of each individual magnitude.

Since mathematics selects only general properties for consideration, it operates with its clearly defined abstractions quite independently of the actual limits of their applicability, as must happen precisely because these limits are different in different particular cases. These limits depend on the concrete properties of the phenomena under consideration and on the qualitative changes that take place in them. So in making an application of mathematics, it is necessary to verify the actual applicability of the theory in question. To consider matter as continuous and to describe its properties by continuous magnitudes is permissible only if we may abstract from its atomic structure, and this is possible only under well-known conditions.

Nevertheless, the real numbers represent a trustworthy and powerful instrument for the mathematical investigation of actual continuous magnitudes and processes. Their theory is based on practice, on an immense field of applications in physics, technology, and chemistry. Consequently, practice shows that the concept of the real number correctly reflects the general properties of magnitudes. But this correctness is not without limits; it is not possible to consider the theory of real numbers as something absolute, allowing an unlimited abstract development in complete separation from reality. The very concept of the real number is continuing to develop and is in fact still far from being complete.

5. The conflict of opposites: discrete and continuous. The role of another of the mentioned contrasts, the contrast between the discrete and the continuous, may also be illustrated by the development of the concept of number. We have already seen that fractions arose from the division of continuous magnitudes.

On this theme of division there is a humorous question which is extraordinarily instructive. Grandmother has bought three potatoes and must divide them equally between two grandsons. How is she to do it? The answer is: make mashed potatoes.

The joke reveals the very essence of the matter. Separate objects are indivisible in the sense that, when divided, the object almost always ceases to be what it was before, as is clear from the example of “thirds of a man” or “thirds of an arrow.” On the other hand, continuous and homogenous magnitudes or objects may easily be divided and put together again without losing their essential character. Mashed potatoes offer an excellent example of a homogeneous object, which in itself is not separated into parts but may nevertheless be divided in practice into as small parts as desired. Lengths, areas, and volumes have the same property. Although they are continuous in their very essence and are not actually divided into parts, nevertheless they offer the possibility of being divided without limit.

Here we encounter two contrasting kinds of objects: on the one hand, the indivisible, separate, discrete objects; and on the other, the objects which are completely divisible and yet are not divided into parts but are continuous. Of course, these contrasting characteristics are always united, since there are no absolutely indivisible and no completely continuous objects. Yet these aspects of the objects have an actual existence, and it often happens that one aspect is decisive in one case and the other in another.

In abstracting forms from their content, mathematics by this very act sharply divides these forms into two classes, the discrete and the continuous.

The mathematical model of a separate object is the unit, and the mathematical model of a collection of discrete objects is a sum of units, which is, so to speak, the image of pure discreteness, purified of all other qualities. On the other hand, the fundamental, original mathematical model of continuity is the geometric figure; in the simplest case, the straight line.

We have before us therefore two contrasts, discreteness and continuity, and their abstract mathematical images: the whole number and the geometric extension. Measurement consists of the union of these contrasts: The continuous is measured by separate units. But the inseparable units are not enough; we must introduce fractional parts of the original unit. In this way the fractional numbers arise and the concept of numbers develops precisely as a result of the union of the mentioned contrasts.

Then, on a more abstract level, appeared the concept of incommensurable intervals, and, as a result, the real number as an abstract image of unlimited increase in accuracy in the determination of a magnitude. This concept was not formed immediately, and the long path of its development led through many a conflict between these same two contrasting elements, the discrete and the continuous.

In the first place, Democritus represented figures as consisting of atoms and in this way reduced the continuous to the discrete. But the discovery of incommensurable intervals led to the abandonment of such a representation. After this discovery continuous magnitudes were no longer thought of as consisting of separate elements, atoms or points, and they were not represented by numbers, since numbers other than the whole numbers and the fractions were not known at that time.

FIG. 3.

The contrast between the continuous and the discrete appeared in mathematics again with renewed force in the 17th century, when the foundations of the differential and integral calculus were being laid. Here it was the infinitesimal that was under discussion. In some accounts the infinitesimal was thought of as a real, “actually” infinitesimal, “indivisible” particle of the continuous magnitude, like the atoms of Democritus, except that now the number of these particles was considered to be infinitely great. Calculation of areas and volumes, or in other words integration, was thought of as summation of an infinite number of these infinitely small particles. An area, for example, was understood as “the sum of the lines from which it is formed” (figure 3). Consequently, the continuous was again reduced to the discrete, but now in a more complicated way, on a higher level. But this point of view also proved unsatisfactory, and, as a counterweight to it, there appeared, on the basis of Newton’s work, the notion of continuous variables, of the infinitesimal as a continuous variable decreasing without limit. This conception finally carried the day at the beginning of the 19th century, when the rigorous theory of limits was founded. An interval was now thought of as consisting not of points or “indivisibles,” but as an extension, as a continuous medium, where it was only possible to fix separate points, separate values of a continuous magnitude. Mathematicians then spoke of “extension.” In the union of the discrete and the continuous, it was again the continuous that dominated.

But the development of analysis demanded further precision in the theory of variable magnitudes and above all in the general definition of a real number as an arbitrary possible value of a variable magnitude. In the seventies of the last century there arose a theory of real numbers which represents an interval as a set of points, and correspondingly the range of variation of a variable as a set of real numbers. The continuous again consisted of separate discrete points and the properties of continuity were again expressed in the structure of the set of points that formed it. This conception led to immense progress in mathematics and became dominant. But again profound difficulties were discovered in it, and these led to attempts to return on a new level to the notion of pure continuity. Other attempts were made to change the concept of an interval as a set of points. New points of view appeared for the concepts of number, variable, and function. The development of the theory is continuing, and we must await its further progress.

6. Further results of the interaction of arithmetic and geometry. The interaction of geometry and arithmetic played a role elsewhere than in the formation of the concept of a real number. The same interaction of geometry with arithmetic, or more accurately with algebra, also showed itself in the formation of negative and complex numbers, that is of numbers of the form . Negative numbers are represented by points of the straight line to the left of the point representing zero. It was exactly this geometric representation which gave imaginary numbers a firm place in mathematics; up to that time they had not been understood. New concepts of magnitude appeared: for example, vectors, which are represented by directed line segments; and tensors, which are still more general magnitudes; in these again algebra is united with geometry.

The union of various mathematical theories has always played a great and sometimes decisive role in the development of mathematics. We shall see this further on in the rise of analytic geometry, differential and integral calculus, the theory of functions of a complex variable, the recent so-called functional analysis, and other theories. Even in the theory of numbers itself, that is in the study of whole numbers, methods are applied with great success which depend on continuity (namely on the infinitesimal analysis) and on geometry. These methods have given rise to extensive chapters in the theory of numbers, the “analytic theory of numbers,” and the “geometry of numbers.”

From a certain well-known point of view, it is possible to regard the foundations of mathematics as the union of concepts arising from geometry and arithmetic; that is to say, of the general concepts of continuity and of algebraic operations (as generalizations of arithmetic operations). But we will not be able to speak here of these difficult theories. The aim of the present chapter has been to give an impression of the general interaction of concepts, of the union and the conflict between contrasting ideas in mathematics, as illustrated by the interaction of arithmetic and geometry in the development of the concept of number.

§5.  The Age of Elementary Mathematics

1. The four periods of mathematics. The development of mathematics cannot be reduced to the simple accumulation of new theorems but includes essential qualitative changes. These qualitative changes take place, however, not in a process of destruction or abolition of already existing theories but in their being deepened and generalized, so as to form more general theories, for which the way has been prepared by preceding developments.

From the most general point of view, we may distinguish in the history of mathematics four fundamental, qualitatively distinct periods. Of course, it is not possible to draw exact boundary lines between these stages, since the essential traits of each period appeared more or less gradually, but the distinctions among the stages and the passages from one to another are completely clear.

The first stage (or period) is the period of the rise of mathematics as an independent and purely theoretical science. It begins in the most ancient times and extends to the 5th century B.C., or perhaps earlier, when the Greeks laid the foundations of’ “pure” mathematics with its logical connection between theorems and proofs (in that century there appeared, in particular, systematic expositions of geometry like the “Elements” of Hippocrates of Chios). This first stage was the period of the formation of arithmetic and geometry, in the form considered earlier. At this time mathematics consisted of a collection of separate rules deduced from experience and immediately connected with practical life. These rules did not yet form a logically unified system, since the theoretical character of mathematics with its logical proof of theorems was formed very slowly, as material for it was accumulated. Arithmetic and geometry were not separated but were closely interwoven with each other.

The second period may be characterized as the period of elementary mathematics, of the mathematics of constant magnitudes; its simple fundamental results now form the content of a high school course. This period extended for almost 2000 years and ended in the 17th century with the rise of “higher” mathematics. It is with this period that we will be concerned in greater detail in the present section. The following sections will be devoted to the third and fourth periods, namely to the founding and development of analysis and to the period of contemporary mathematics.

2. Mathematics in Greece. The period of elementary mathematics may in its turn be divided into two parts, distinguished by their basic content: the period of the development of geometry (up to the 2nd century A.D.) and the period of the predominance of algebra (from the 2nd to the 17th century). With respect to historical conditions it is divided into three parts, which may be called “Greek,” “Eastern,” and “European Renaissance.” The Greek period coincides in time with the general flowering of Greek culture, beginning with the 7th century B.C., reaching its culmination in the 3rd century B.C. at the time of the great geometers of antiquity, Euclid, Archimedes, and Apollonius, and ending in the 6th century A.D. Mathematics, and especially geometry, enjoyed a wonderful development in Greece. We know the names and the results of numerous Greek mathematicians, although only a few genuine works have come down to us. It is to be remarked that Rome gave nothing to mathematics though it reached its zenith in the 1st century A.D. at a time when the science of Greece, which had been conquered by Rome, was still flourishing.

The Greeks not only developed and systematized elementary geometry to the extent to which it is given in the “Elements” of Euclid and is now taught in our secondary schools, but achieved considerably higher results. They studied the conic sections: ellipse, hyperbola, parabola; they proved certain theorems relating to the elements of what is called projective geometry; guided by the needs of astronomy, they worked out spherical geometry (in the 1st century A.D.) and also the elements of trigonometry, and calculated the first tables of sines (Hipparchus, 2nd century B.C. and Claudius Ptolemy, 2nd century A.D.);* they determined the areas and volumes of a number of complicated figures; for example, Archimedes found the area of the segment of a parabola by proving that it is 2/3 of the area of the rectangle containing it (figure 4). The Greeks were also acquainted with the theorem that of all bodies with a given surface area the sphere has the greatest volume, but their proof has not been preserved and was probably not complete. Such a proof is quite difficult and was first discovered in the 19th century, by means of the integral calculus.

FIG. 4.

In arithmetic and in the elements of algebra, the Greeks also made considerable progress. As was mentioned earlier, they laid the foundation for the theory of numbers. Here belong, for example, their investigations on prime numbers (the theorem of Euclid on the existence of an infinite number of prime numbers and the “sieve” of Eratosthenes for finding prime numbers) and the solution of equations in whole numbers (Diophantus about 246–330 A.D.).

We have already said that the Greeks discovered irrational magnitudes but considered them geometrically, as line segments. So the problems that today we deal with algebraically were treated geometrically by the Greeks. It was in this way that they solved quadratic equations and transformed irrational expressions. For example, the equation that we today write in the form x² + ax = b², they stated as follows: Find a segment x such that if to the square constructed on it we add a rectangle constructed on the same segment and on the given segment a, we obtain a rectangle equal in area to a given square. This dominance of geometry lasted a long time after the Greeks. They were also acquainted with (geometric) methods for extracting square roots and cube roots and with the properties of arithmetic and geometric progressions.

In this way the Greeks were already in possession of much of the material of contemporary elementary algebra but not, however, of the following essential elements: negative numbers and zero, irrational numbers abstracted entirely from geometry, and finally a well-developed system of literal symbols. It is true that Diophantus made use of literal symbols for the unknown quantity and its powers and also of special symbols for addition, subtraction, and equality, but his algebraic equations were still written with concrete numerical coefficients.

In geometry the Greeks attained what we now call “higher” mathematics. Archimedes made use of integral calculus for the calculation of areas and volumes and Apollonius used analytic geometry in his investigations on conic sections. Apollonius actually gives the equations of these curves* but expresses them in geometric language. In these equations there does not yet appear the general notion of an arbitrary constant or of a variable magnitude; and the necessary means of expressing such concepts, namely the literal symbols of algebra, appear only at a later age; they alone could convert such investigations into a source of new theories, which would be truly a part of higher mathematics. The founders of these new theories were guided, a thousand years later, by the legacy of the Greek scientists; in fact, the “Geometry” of Descartes (1637), which laid the foundation for analytic geometry, begins with a selection of problems left by the Greeks.

Such is the general rule. The old theories, by giving rise to new and profound problems, outgrow themselves, as it were, and demand for further progress new forms and new ideas. But these forms and ideas may demand new historical conditions for their birth. In ancient society the conditions necessary for the passage to higher mathematics did not and could not exist; they came on the scene with the development of the natural sciences in modern times, a development which in its turn was conditioned in the 16th and 17th centuries by the new demands of technology and of manufacturing and was connected in this way with the birth and development of capitalism.

The Greeks practically exhausted the possibilities of elementary mathematics, which is the explanation of the fact that the brilliant progress of geometry dried up at the beginning of our era and was replaced by trigonometry and algebra in the works of Ptolemy, Diophantus, and others. In fact, one may consider the works of Diophantus as the beginning of the period in which algebra played the leading role. But the society of the ancients, already verging to its decline, was no longer able to advance science in this new direction.

It should be noted that, a few centuries earlier, arithmetic had already reached a high level in China. The Chinese scientists of the 2nd and 1st centuries B.C. described the rules for arithmetical solution of a system of three equations of the first degree. It is here for the first time in history that negative coefficients are made use of and the rules for operating with negative quantities are formulated. But the solutions themselves were sought only in the form of positive numbers, just as later in the works of Diophantus. These Chinese books also include a method for the extraction of square roots and cube roots.

3. The Middle East. With the end of Greek science a period of scientific stagnation began in Europe, the center of mathematical development being shifted to India, Central Asia, and the Arabic countries.* For a period of about a thousand years, from the 5th to the 15th century, mathematics developed chiefly in connection with the demands of computation, particularly in astronomy, since the mathematicians of the East were for the most part also astronomers. It is true that they added nothing of importance to Greek geometry; in this field they only preserved for later times the results of the Greeks. But the Indian, Arabic, and Central Asian mathematicians achieved immense successes in the fields of arithmetic and algebra.*

As has been mentioned in §2, the Indians invented our present system of numeration. They also introduced negative numbers, comparing the contrast between positive and negative numbers with the contrast between property and debt or between the two directions on a straight line. Finally, they began to operate with irrational magnitudes exactly as with rational, without representing them geometrically, in contrast to the Greeks. They also had special symbols for the algebraic operations, including extraction of roots. For the very reason that the Indian and Central Asian scholars were no longer embarrassed by the difference between the irrational and rational magnitudes, they were able to overcome the “dominance” of geometry, which was characteristic of Greek mathematics, and to open up paths for the development of contemporary algebra, free of the heavy geometric framework into which it had been forced by the Greeks.

The great poet and mathematician, Omar Khayyam (about 1048–1 122), and also the Azerbaijanian mathematician, Nasireddin Tusi (1201–1274), clearly showed that every ratio of magnitudes, whether commensurable or incommensurable, may be called a number; in their works we find the same general definition of number, both rational and irrational, as was introduced above in Newton’s formulation, in §4. The magnitude of these achievements becomes particularly clear when we recall that complete recognition of negative and irrational numbers was attained by European mathematicians only very slowly, even after the beginning of the Renaissance of mathematics in Europe. For example, the celebrated French mathematician Viète (1540–1603), to whom algebra owes a great deal, avoided negative numbers, and in England protests against them lasted even into the 18th century. These numbers were considered absurd, since they were less than zero, that is “less than nothing at all.” Nowadays they have become familiar, if only in the form of negative temperature; everyone reads the newspapers and understands what is mean by “the temperature in Moscow is –8°.”

The word “algebra” comes from the name of a treatise of the mathematician and astronomer Mahommed ibn Musa al-Kharizmi (Mahommed, son of Musa, native of Kharizm), who lived in the 9th century. His treatise on algebra was called Al-jebr w’al-muqabala, which means “transposition and removal.” By transposition (al-jebr) is understood the transfer of negative terms to the other side of an equation, and by removal (almuqabala), cancellation of equal terms on both sides.

The Arabic word “al-jebr” became in Latin transcription “algebra” and the word al-muqabala was discarded, which accounts for the modern term “algebra.”*

The origin of this term corresponds very well to the actual content of the science itself. Algebra is basically the doctrine of arithmetical operations considered formally from a general point of view, with abstraction from the concrete numbers. Its problems bring to the fore the formal rules for transformation of expressions and solution of equations. Al-Kharizmi placed on the title page of his book the actual names of two most general formal rules, expressing in this way the true spirit of algebra.

Subsequently, Omar Khayyam defined algebra as the science of solving equations. This definition retained its significance up to the end of the 19th century, when algebra, along with the theory of equations, struck out in new directions, essentially changing its character but not changing its spirit of generality as the science of formal operations.

The mathematicians of Central Asia found methods for calculation, both exact and approximate, of the roots of a number of equations; they discovered the general formula for the “binomial of Newton,” although they expressed it in words; they greatly advanced and systematized the science of trigonometry, and calculated very accurate tables of sines. These tables were computed, for astronomical purposes, by the mathematician Gyaseddin (about 1427) who was working with the famous Uzbek astronomer Ulug Begh; Gyaseddin also invented decimal fractions 150 years before they were reinvented in Europe.

To sum up, in the course of the Middle Ages in India and in Central Asia the present decimal system of numeration (including fractions) was almost completely built up, as were also elementary algebra and trigonometry. During the same period the achievements of Chinese science began to make their way into the neighboring countries; about the 6th century B.C. the Chinese already had methods for the solution of the simplest indeterminate equations, for approximate calculations in geometry, and for the first steps in approximate solution of equations of the third degree. Essentially the only parts of our present high school course in algebra that were not known before the 16th century were logarithms and imaginary numbers. However, there did not yet exist a system of literal symbols: The content of algebra had outdistanced its form. Yet the form was indispensable: The abstraction from concrete numbers and the formulation of general rules demanded a corresponding method of expression; it was essential to have some way of denoting arbitrary numbers and operations on them. The algebraic symbolism is the necessary form corresponding to the content of algebra. Just as in remote antiquity it had been necessary, in order to operate with whole numbers, that symbols should be invented for them, so now, to operate with arbitrary numbers and to give general rules for their use, it was necessary to work out corresponding symbols. This task, begun at the time of the Greeks, was not brought to completion until the 17th century, when the present system of symbols was finally set up in the works of Descartes and others.

4. Renaissance Europe. At the time of the Renaissance the Europeans became acquainted with Greek mathematics by way of the Arabic translations. The books of Euclid, Ptolemy, and Al-Kharizmi were translated in the 12th century from Arabic into Latin, the common scientific language of Western Europe, and at the same time, the earlier system of calculation, as derived from the Greeks and Romans, was gradually replaced by the present-day Indian method, which was borrowed by the Europeans from the Arabs.

It was only in the 16th century that European science finally surpassed the achievements of its predecessors. Thus the Italians, Tartaglia and Ferrari, solved the general cubic equation, and later, the general equation of the fourth degree (see Chapter IV). Let us note that although these results are not taught in school, they belong, with respect to the methods employed in them, to elementary algebra. To higher algebra we must however refer the general theory of equations.

During the same period imaginary numbers began for the first time to be used; at first this was done in a purely formal manner, without logical foundation, which came considerably later at the beginning of the 19th century. Our present-day algebraic symbols were also worked out; in particular, literal symbols were used by Vi&te in 1591 not only for unknown quantities but also for given ones.

Many mathematicians took a share in this development of algebra. At the same time decimal fractions appeared in Europe; they were invented by the Dutch scholar Stevin, who wrote about them in 1585.

Finally, Napier in Great Britain invented logarithms as an aid in astronomical calculations and wrote about them in 1614; Briggs calculated the first decimal tables of logarithms, which were published in 1624.*

At the same time there appeared in Europe the “theory of combinations” and the general formula for the “binomial of Newton”;† the progressions being already known, and in this way the structure of elementary algebra was completed. Therewith came to an end, at the beginning of the 17th century, the whole period of the mathematics of constant magnitudes, of elementary mathematics as it is now taught, with a few additions, in our schools. Arithmetic, elementary geometry, trigonometry, and elementary algebra were now essentially complete. There followed a transition to higher mathematics, to the mathematics of variable magnitudes.

It is not to be thought, however, that the development of elementary mathematics ceased at this time; for example, new results were discovered and are being constantly discovered today in elementary geometry. Furthermore, it is precisely because of the subsequent development of higher mathematics that we now understand more clearly the essential nature of elementary mathematics itself. But the leading role in mathematics was now taken over by the concepts of variable magnitude, function, and limit. The problems, that led from elementary mathematics to higher mathematics are nowadays clarified and solved by the concepts and methods of higher mathematics (occasionally they are not solvable at all by elementary methods), and there are other problems which may be stated in terms of elementary mathematics but which serve even today as a source of more general results and even of entire theories. Examples are provided by the earlier mentioned theory of regular systems of figures or by problems of the theory of numbers which are elementary in their formulation but far from elementary in the methods by which they are solved. For further details the reader may consult Chapter X.

§6.   Mathematics of Variable Magnitudes

1. Variable and function. In the 16th century the investigation of motion was the central problem of physics. The physical sciences were led to this problem, and to the study of various others involving interdependence of variable magnitudes, by the demands of practical life and by the whole development of science itself.

As a reflection of the general properties of change, there arose in mathematics the concepts of a variable magnitude and a function, and it was this cardinal extension of the subject matter of mathematics that determined the transition to a new stage, to the mathematics of variable magnitudes.

The law of motion of a body in a given trajectory, for example along a straight line, is defined by the manner in which the distance covered by the body increases with time.

Thus Galileo (1564–1642) discovered the law of falling bodies by establishing that the distance fallen increases proportionally to the square of the time. This fact is expressed in the well-known formula

where g is approximately equal to 9.81 m/sec².

In general, the law of motion expresses the distance covered in the time t. Here the time t and the distance s are respectively the “independent” and the “dependent” variable, and the fact that to each time t there corresponds a definite distance s is what is meant by saying that the distance s is a function of the time t.

The mathematical concepts of variable and function are the abstract generalization of concrete variables (such as time, distance, velocity, angle of rotation, and area of surface traced out) and of the interdependences among them (the distance depends on the time and so forth). Just as the concept of a real number is the abstract image of the actual value of an arbitrary magnitude, so a “variable” is the abstract image of a varying magnitude, which assumes various values during the process under consideration. A mathematical variable x is “something” or, more accurately, “anything” that may take on various numerical values. This is the meaning of a variable in general; in particular, we may understand by it the time, the distance, or any other variable magnitude.

In exactly the same way, a function is the abstract image of the dependence of one magnitude on another. The assertion that y is a function of x means in mathematics only that to each possible value of x there corresponds a definite value of y. This correspondence between the values of y and the values of x is called a function. For example, according to the law of falling bodies, the distance covered corresponds to the time of fall by formula (1). The distance is a function of the time. Let us look at some other examples.

The energy of a falling body is expressed by its mass and its velocity according to the formula

For a given body the energy is a function of the velocity υ.

By a familiar law the quantity of heat generated in a conductor in unit time by the passage of an electric current is expressed by the formula

where I is the magnitude of the current and R is the resistance of the conductor. For a given resistance there corresponds to every current I a definite amount of heat Q, generated in unit time. That is, Q is a function of I.

FIG. 5

The area of a right-angled triangle S with a given acute angle α and corresponding side x (see figure 5) is expressed by the formula

For a given angle α the area is a function of the side x.

All these formulas (1)–(4) may be united in the one

This general formula represents a transition from the concrete variable magnitudes t, s, E, Q, υ and so forth to the general variables x and y, and from the concrete dependences (1), (2), (3), (4) to their general form (5). Mechanics and the theory of electricity have to do with concrete formulas (1), (2), (3), interrelating concrete magnitudes, but the mathematical theory of functions deals with the general formula (5), without associating this formula with any concrete magnitudes.

The next degree of abstraction from the concrete consists in our examining not a given dependence of y on x, like , y = sin x, y = log x and so forth, but the general dependence of y on x expressed in the abstract formula

This formula states that the magnitude y is in general some function of x; that is, to each value assumed by x there corresponds, in some fashion or another, a definite value y. The subject matter of mathematics thus consists not only of certain given functions (, y = sin x, and so forth), but of arbitrary (more accurately, more or less arbitrary) functions. These degrees of abstraction, first from concrete magnitudes and then from concrete functions, are analogous to the degrees of abstraction observed in the formation of the concept of a whole number: First, abstraction from concrete collections of objects led to the concept of whole numbers (1, 3, 12, and so forth), and then a further abstraction led to the concept of an arbitrary whole number in general. This generalization is the result of a profound interraction between analysis and synthesis: analysis of separate interrelations and synthesis, in the form of new concepts, of their common features.

The branch of mathematics devoted to the study of functions is called analysis, or often, infinitesimal analysis, since one of the most important elements in the study of functions is the concept of the infinitesimal (the meaning of this concept and its significance are explained in Chapter II).

Since a function is the abstract image of a dependence of one magnitude on another, we may say that analysis takes as its subject matter dependences between variable magnitudes, not between one concrete magnitude and another but between variables in general, in abstraction from their content. An abstraction of this sort guarantees great breadth of application, since one formula or one theorem contains an infinite number of possible concrete cases. An example of this is given already by our simple formulas (1)–(5). So the complete analogy of analysis with arithmetic and algebra becomes evident. They all originate in definite practical problems and give a general abstract expression to concrete relationships in the actual world.

2. Analytic geometry and analysis. Thus the new period of mathematics, beginning in the 17th century, may be defined as the period of the birth and development of analysis. (This is the third of the three important periods mentioned earlier.) It is to be understood, of course, that no theory arises as a result of the mere formation of new concepts, that analysis could not result from the mere existence of the concepts of variable and function. For the founding of a theory, and all the more of a complete branch of science like mathematical analysis, it is necessary that the new concepts become active, so to speak, that among them there be discovered new relationships, and that they permit the solution of new problems.

But more than that, new concepts can originate and develop, and become more general and precise, only on the basis of the very problems they enable us to solve, only through those theorems of which they form a part. The concepts of variable and function did not arise in complete form in the mind of Galileo, Descartes, Newton, or anybody else. They occurred to many mathematicians (for example Napier in connection with logarithms) and gradually assumed a more or less clear, but still by no means final, form with Newton and Leibnitz, being made still more precise and general in the subsequent development of analysis. Their present-day definition was laid down only in the 19th century, but even it is not absolutely rigorous or altogether final. The development of the concept of a function is continuing even at the present time.

Mathematical analysis was based on material furnished by the new science of mechanics, and on problems of geometry and algebra. The first definite step toward the mathematics of variable magnitudes was the appearance in 1637 of the “geometry” of Descartes, where the foundations were laid for the so-called analytic geometry. The basic ideas of Descartes are as follows.

Suppose we are given, for example, the equation

In algebra x and y were understood as unknowns, and since the given equation does not allow us to determine them, it did not present any essential interest for algebra. But Descartes did not consider x and y as unknowns, to be found from the equation, but as variables; so that the given equation expresses the interdependence of two variables. Such an equation may be written in general form, by taking all its terms to the left-hand side, thus:

FIG. 6.

Further, Descartes introduced into the plane the coordinates x, y which are now called Cartesian (figure 6). In this way, to each pair of values x and y there corresponds a point, and conversely to each point there corresponds a pair of coordinates x, y. Consequently, the equation F(x, y) = 0 determines the geometric locus of those points on the plane whose coordinates satisfy the equation. In general, this will be a curve. For example, equation (6) determines the circumference of a circle of radius a with center at the origin. In fact, as is obvious from figure 7, by the theorem of Pythagoras, x² + y² is the square of the distance from the origin O to the point M with coordinates x and y. So equation (6) represents the geometric locus of those points whose distance from the origin is equal to a, which is the circumference of a circle.

Conversely, a geometric locus of points, given by a geometric condition, may also be given by an equation expressing the same condition in the language of algebra by means of coordinates. For example, the geometric condition defining the circumference of a circle, namely that it is a geometric locus of points equidistant from a given point, may be expressed in algebraic language by equation (6).

FIG. 7.

Thus the general problem and the general method of analytic geometry are as follows: We represent a given equation in two variables by a curve on the plane, and from the algebraic properties of the equation we investigate the geometric properties of the corresponding curve; and conversely, from the geometric properties of the curve we find the equation, and then from the algebraic properties of the equation we investigate the geometric properties of the curve. In this way geometric problems may be reduced to algebraic, and so finally to computation.

The content of analytic geometry will be discussed in detail in Chapter III. We now wish to direct attention to the fact that, as is evident from our short explanation, it originated in a union of geometry, algebra, and the general idea of a variable magnitude. The main geometric content of the early beginnings of analytic geometry was the theory of conic sections, ellipse, hyperbola, and parabola. This theory, as we have pointed out, was developed by the ancient Greeks; the results of Apollonius already contained in geometric form the equations of the conic sections. The union of this geometric content with algebraic form, developed after the time of the Greeks, and with the general idea of a variable magnitude, arising from the study of motion, produced analytic geometry.

Among the Greeks the conic sections were a subject of purely mathematical interest, but by the time of Descartes they were of practical importance for astronomy, mechanics, and technology. Kepler (1571–1630) discovered that the planets move around the sun in ellipses, and Galileo established the fact that a body thrown in the air, whether it is a stone or a cannonball, moves along a parabola (to the first approximation, if we may neglect air resistance). As a result, the calculation of various magnitudes referring to the conic sections became an urgent necessity, and it was the method of Descartes that solved this problem. So the way was prepared for his method by the preceding development of mathematics, and the method itself was brought into existence by the insistent demands of science and technology.

3. Differential and integral calculus. The next decisive step in the mathematics of variable magnitudes was taken by Newton and Leibnitz during the second half of the 17th century, in the founding of the differential and integral calculus. This was the actual beginning of analysis, since the subject matter of this calculus is the properties of functions themselves, as distinct from the subject matter of analytic geometry, which is geometric figures. In fact Newton and Leibnitz only brought to completion an immense amount of preparatory work, shared by many mathematicians and going back to the methods for determining areas and volumes worked out by the ancient Greeks.

Here we shall not explain the fundamental concepts of differential and integral calculus and of the theories of analysis that followed them, since this will be done in the special chapters devoted to these theories. We wish only to draw attention to the sources of the calculus, which were mainly the new problems of mechanics and the old problems of geometry, the latter consisting of drawing a tangent to a given curve and of determining areas and volumes. These geometric problems had already been studied by the ancients (it is sufficient to mention Archimedes), and also by Kepler, Cavalieri, and others at the beginning of the 17th century. But the decisive event was the discovery of the remarkable relation between these two types of problems and the formulation of a general method for solving them; this was the achievement of Newton and Leibnitz.

This relation, allowing us to connect the problems of mechanics with thcse of geometry, was discovered because of the possibility, arising from the method of coordinates, of making a graphical representation of the dependence of one variable on another, or in other words of a function. With the help of this graphical representation, it is easy for us to formulate the earlier mentioned relation, between the problems of mechanics and geometry, which was the source of the differential and integral calculus, and consequently to describe the general content of these two types of calculus.

The differential calculus is basically a method for finding the velocity of motion when we know the distance covered at any given time. This problem is solved by “differentiation.” It turns out that the problem is completely equivalent to that of drawing a tangent to the curve representing the dependence of distance on time. The velocity at the moment t is equal to the slope of the tangent to the curve at the point corresponding to t (figure 8).

The integral calculus is basically a method of finding the distance covered when the velocity is known, or more generally of finding the total result of the action of a variable magnitude. This problem is obviously the converse of the problem of the differential calculus (the problem of finding the velocity); it is solved by “integration.” It turns out that the problem of integration is completely equivalent to that of finding the area under the curve representing the dependence of the velocity on time. The distance covered in the interval of time from the moment t₁ to the moment t₂ is equal to the area under the curve between the straight lines corresponding on the graph to the values t₁ and t₂ (figure 9).

FIG. 8.

FIG. 9.

By abstracting from the mechanical formulation of the problems of the calculus and by dealing with functions rather than with dependence of distance or velocity on time, we obtain the general concept of the problems of differential and integral calculus in abstract form.

Fundamental to the calculus, as to the whole subsequent development of analysis, is the concept of a limit, which was formulated somewhat later than the other fundamental concepts of variable and function. In the early days of analysis the role later played by the limit was taken by the somewhat nebulous concept of an infinitesimal. The methods for actual calculation of velocity, given the distance covered (namely, differentiation), and of distance, given the velocity (integration), were founded on a union of algebra with the concept of limit. Analysis originated in the application of these concepts and methods to the aforementioned problems of mechanics and geometry (and also to certain other problems; for example, problems of maxima and minima). The science of analysis was in turn absolutely necessary for the development of mechanics, in the formulation of whose laws its concepts had already appeared in latent form. For example, the second law of Newton, as formulated by Newton himself, states that “the change in momentum is proportional to the acting force” (more precisely: The rate of change of momentum is proportional to the force). Consequently, if we wish to make any use of this law, we must be able to define the rate of change of a variable, that is, to differentiate. (If we state the law in the form that the acceleration is proportional to the force, the problem remains the same, because acceleration is proportional to rate of change of momentum.) Also, it is perfectly clear that in order to state the law governing a motion when the force is variable (in other words, the motion proceeds with a variable acceleration), we must be able to solve the inverse problem of finding a magnitude given its rate of change; in other words, we must be able to integrate. So one might say that Newton was simply compelled to invent differentiation and integration in order to develop the science of mechanics.

4. Other branches of analysis. Along with the differential and integral calculus, other branches of analysis arose: The theory of series (see Chapter II, §14), the theory of differential equations (Chapters V and VI), and the application of analysis to geometry, which later became a special branch of geometry, called differential geometry and dealing with the general theory of curves and surfaces (Chapter VII). All these theories were brought to life by the problems of mechanics, physics, and technology.

The theory of differential equations, the most important branch of analysis, has to do with equations in which the unknown is no longer a magnitude but a function, or in other words a law governing the dependence of one magnitude on another or on several others. It is easy to understand how such equations arose. In mechanics we seek to determine the whole law of motion of a body under given conditions and not just one value of the velocity or of the distance covered. In the mechanics of fluids it is necessary to find the distribution of velocity over the whole mass of fluid in motion, or in other words to find the dependence of the velocity on all three space coordinates and on time. Analogously, in the theory of electricity and magnetism we must find the tension in the field throughout all space; that is, the dependence of this tension on the same three space coordinates, and similarly in other cases.

Problems of this sort arose continually in the various branches of mechanics, including hydrodynamics and the theory of elasticity, in acoustics, in the theory of electricity and magnetism, and in the theory of heat. From the very moment of its birth, analysis remained in close contact with mechanics and with physics in general, its most important achievements being invariably connected with the solution of problems posed by the exact sciences. Beginning with Newton, the greatest analysts, D. Bernoulli (1700–1782), L. Euler (1707–1783), J. Lagrange (1736–1813), H. Poincaré (1854–1912), M. V. Ostrogradskiĭ (1801–1861) and A. M. Lyapunov (1857–1918), as well as many others who laid new foundations in analysis, started as a rule from the urgent problems of contemporary physics.

In this way new theories arose: In direct connection with mechanics, Euler and Lagrange founded a new branch of analysis, called the calculus of variations (see Chapter VIII), and at the end of the 19th century Poincaré and Lyapunov, starting again from the problems of mechanics, founded the so-called qualitative theory of differential equations (see Chapter V, §7).

In the 19th century analysis was enriched by an important new branch, the theory of functions of a complex variable (see Chapter IX). The rudiments of it are to be found in the works of Euler and certain other mathematicians, but its transformation into a well-formed theory took place in the middle of the 19th century and was carried out to a great extent by the French mathematician Cauchy (1789–1857). This theory rapidly underwent an imposing development with numerous significant results that allowed mathematicians to penetrate more deeply into many of the laws of analysis and found important applications in problems of mathemqtics itself, and of physics and technology.

Analysis developed rapidly; not only did it form the center and the most important part of mathematics but it also penetrated into the older regions: algebra, geometry, and even the theory of numbers. Algebra began to be thought of as basically the doctrine of functions expressed in the form of polynomials of one or several variables.* Analytic and differential geometry began to dominate the field of geometry. As far back as Euler, methods of analysis were introduced into the theory of numbers and formed in this way the beginning of the so-called analytic theory of numbers, which contains some of the most profound achievements of the science of whole numbers.

Through the influence of analysis, with its concepts of variable, function, and limit, the whole of mathematics was penetrated by the idea of motion and change, and therefore of dialectic. In exactly the same way, basically through analysis, mathematics was affected by the exact sciences and technology and in turn played a role in their development, since it was the means of giving exact expression to their laws and of solving their problems. Just as among the Greeks mathematics was basically geometry, one may say that after Newton it was basically analysis. Of course, analysis did not completely absorb the whole of mathematics; in geometry, in the theory of numbers, and in algebra the problems and methods characteristic of these sciences were everywhere continued. Thus in the 17th century there arose, along with analytic geometry, another branch of geometry, namely projective geometry, in which purely geometric methods played a dominant role. It originated chiefly in problems of the representation of objects on a plane (projection), and as a result it is particularly useful in descriptive geometry.

At the same time there was developed an important new branch of mathematics, the theory of probability, which takes as its subject matter the uniformities observable in large masses of phenomena, such as a long series of rifle shots or tosses of a coin. In the succeeding period it acquired a special importance in physics and technology and its development was conditioned by the problems which came to it from those branches of science. The characteristic feature of this theory is that it deals with the laws of “random events,” providing mathematical methods for investigation of the irregularities that necessarily appear in random events. The basic features of the theory of probability will be explained in Chapter XI.

5. Applications of analysis. Analysis in all its branches provided physics and technology with powerful methods for the solution of problems of many different kinds. We have already mentioned the earliest of these: to find the rate of change of a magnitude when we know how the magnitude itself depends on time; to find the area of curvilinear figures and the volumes of solids; and to find the total result of some process or another or the total action of a variable magnitude. Thus, the integral calculus allows us to determine the work done by an expanding gas as the pressure changes according to a well-known law; the same integral calculus allows us to compute, for example, the tension of an electric field with an arbitrarily given system of charges, basing our work on the law of Coulomb which determines the tension of a field resulting from a point charge, and so forth.

Further, analysis provided a method for finding the maximum and the minimum values of a magnitude under given conditions. Thus, with the help of analysis it is easy to determine the shape of a cylindrical cistern which for a given volume will have the smallest surface and consequently will require the smallest outlay of material. It turns out that the cistern will have this property if its height is equal to the diameter of its base (figure 10). Analysis allows us to determine the shape of the curve along which a body must roll in order to fall in the shortest time from one given point to another (this curve is the so-called cycloid; figure 11).

FIG. 10.

FIG. 11.

For the solution of these and other problems the reader may turn to Chapters II and VIII.

Analysis, or more precisely the theory of differential equations, allows us not merely to find separate values for variable magnitudes but also to determine unknown functions; that is, to find laws of dependence of certain variables on others. Thus we have the possibility, on the basis of the general laws of electricity, of computing how the current varies with time in a circuit with arbitrary resistance, capacitance, and self-induction. We can determine laws for the distribution of velocities throughout the whole mass of a fluid under given conditions. We can deduce general laws for the vibration of strings and membranes, and for the propagation of vibrations in various media; here we are referring to sound waves, electromagnetic waves, or elastic vibrations propagated through the Earth by earthquakes or explosions. Parenthetically, we may remark that new methods are thereby provided for searching for useful minerals and for carrying out investigations far below the surface of the Earth. Individual problems of this sort will be found in Chapters V and VI.

Finally, analysis not only provides us with methods for solving special problems; it also gives us general methods for mathematical formulation of the quantitative laws of the exact sciences. As was mentioned, earlier, the general laws of mechanics could not be formulated mathematically without recourse to the concepts of analysis, and without such a formulation we would not be able to solve the problems of mechanics. In exactly the same way the general laws for heat conduction, diffusion through porous materials, propagation of vibrations, the course of chemical reactions, the basic laws of electromagnetism, and many other laws simply could not be given a mathematical formulation without the concepts of analysis. It is only as the result of such a formulation that these laws can be applied to the most varied concrete cases, providing a basis for exact mathematical conclusions in the special problems of heat conduction, vibrations, chemical solution, electromagnetic fields, and other problems of mechanics, astronomy, and all the numerous branches of physics, chemistry, heat engineering, power, machine construction, electrical engineering, and so forth.

6. Critical examination of the foundations of analysis. Just as in the history of geometry among Greeks the rigorous and systematic presentation given by Euclid brought to completion a long previous development, so in the development of analysis there arose the necessity of placing it upon a firmer basis than had been provided by the first creators of its powerful methods: Newton, Euler, Lagrange, and others. As the analysis founded by them grew more extensive, it began, on the one hand, to deal with more profound and difficult problems, and on the other, to require from its very extent a more systematic and carefully reasoned basis. The growth of the theory necessitated a systematization and critical analysis of its foundations. To put a theory on a firm foundation requires examination of its entire development and should by no means be considered as a starting point for the theory itself, since without the theory we would simply have no idea of what it is that we need to provide with a foundation. By the way, certain contemporary formalists forget this fact when they consider it advisable to found and develop a theory starting from axioms that have not been selected on the basis of any analysis of the actual material which they are supposed to summarize. But the axioms themselves require a justification of their content; they only sum up other material and provide a foundation for the logical construction of a theory. *

The necessary period of criticism, systematization, and laying of foundations occurred in analysis at the beginning of the last century. Through the efforts of a number of eminent scientists this important and difficult work was brought to a successful completion. In particular, precise definitions were given for the basic concepts of real number, variable, function, limit, and continuity.

However, as we have already had occasion to mention, none of these definitions may be considered as absolutely rigorous or final. The development of these concepts is continuing. Euclid and all the mathematicians in the course of 2,000 years after him no doubt considered his “Elements” as the practical limit of logical rigor. But to a contemporary view the Euclidean foundations of geometry seem quite superficial. This historical example shows that we ought not to flatter ourselves with any idea of “absolute” or “final” rigor in contemporary mathematics. In a science that is not yet dead and mummified, there is not and cannot be anything perfect. But we can say with confidence that the foundations of analysis as they exist at present correspond in a quite satisfactory way to the contemporary problems of science and the contemporary conception of logical precision; and second, that the continued deepening of these concepts and the discussions that are now taking place about them give us no cause, and will not give us cause, simply to reject them; these discussions will lead us to a new, more precise, and more profound understanding, the results of which it is still difficult to estimate.

Although the establishment of the basic principles of a theory forms a summary of its development, it does not represent the end of the theory; on the contrary, it is conducive to further development. This is exactly what happened in analysis. In connection with the deepening of its foundations there arose a new mathematical theory, created by the German mathematician Cantor in the seventies of the last century, namely the general theory of infinite sets of arbitrary abstract objects, whether numbers, points, functions or any other “elements”. On the basis of these ideas there grew up a new chapter in analysis, the so-called theory of functions of a real variable, whose concepts, along with those of the foundations of analysis and the theory of sets, are explained in Chapter XV. At the same time the general ideas of the theory of sets penetrated every branch of mathematics. But this “set-theoretical point of view” is inseparably connected with a new stage in the development of mathematics, which we will now consider briefly.

§7.  Contemporary Mathematics

1. The more advanced character of present-day mathematics. To the four stages of the develoment of mathematics mentioned in §5 there naturally correspond stages in our mathematical education, the material learned at each stage of our study consisting, to a fair degree of approximation, of the basic content of the corresponding period in the history of mathematics.

The basic results of arithmetic and geometry, obtained in the first period of the development of mathematics, form the subject of primary education and are known to us all. For example, when we determine the quantity of material necessary to carry out a certain job, let us say to cover a floor, we are already making use of these first results of mathematics. The most important achievements of the second period, the period of elementary mathematics, are taught in the high schools. The basic results of the third period, the foundations of analysis, the theory of differential equations, higher algebra, and so forth, form the mathematical instruction of an engineer; they are studied in all the schools of higher education, except those devoted purely to the humanities. In this way the basic ideas and results of the mathematics of that period are widely known, use being made of them to some extent by almost every engineer and scientist.

On the other hand, the ideas and results of the present-day period of mathematics are studied almost exclusively in graduate departments of mathematics and physics. Beside mathematical specialists, they are used by researchers in the fields of mechanics and physics, and in a number of the newer branches of technology. Of course, this does not at all mean that they have no practical application, but since they represent the most recent results of science, they are naturally more complicated. Consequently, as we now pass to a general description of the latest stage in the development of mathematics we can no longer consider that everything which we mention briefly will be altogether clear. We will try to present in a few lines the most general character of the new branches of mathematics; their content will be explained in greater detail in the corresponding chapters of the book.

If the present section seems overly difficult it may be passed over at first reading and taken up again after study of the special chapters.

2. Geometry. The beginning of the present-day development of mathematics is characterized by profound changes in all its basic fields: algebra, geometry, and analysis. This change may perhaps be followed most clearly in the field of geometry. In the year 1826 Lobačevskiĭ, and almost simultaneously with him the Hungarian mathematician Janos Bolyai, developed the new non-Euclidean geometry. The ideas of Lobačevskiĭ were far from being immediately understood by all mathematicians. They were too bold and unexpected. But from this moment there began a fundamental new development of geometry; the very conception of what is meant by geometry was changed. Its subjtct matter and the range of its applications were rapidly extended. The most important step, after Lobačesvskiĭ, in this direction was taken in 1854 by the celebrated German mathematician, Riemann. He clearly formulated the general idea that an unlimited number of “spaces” could be investigated by geometry, and at the same time he indicated their possible significance in the real world. In the new development of geometry two features were characteristic.

In the first place the earlier geometry studied only the spatial forms and relations of the material world, and then only to the extent in which they appear in the framework of Euclidean geometry, but now the subject matter of geometry began to include also many other forms and relations of the actual world, provided only they were similar to the spatial ones and therefore allowed the use of geometric methods. The term “space” thereby took on in mathematics a new meaning, broader and at the same time more special. Simultaneously, the methods of geometry became much richer and more varied. In their turn they provide us with more complete means for learning about the physical world around us, the world from which geometry in its original form was abstracted.

In the second place, even in Euclidean geometry important progress was made: In it were studied the properties of incomparably more complicated figures, even including arbitrary sets of points. Also a fundamentally new attitude appeared toward the properties of the figures under investigation. Separate groups of properties were distinguished, which could be investigated in abstraction from others, and this very abstraction within geometry gave rise to many characteristic branches of the subject, which essentially became independent “geometries.” The development of geometry in all these directions is being continued and more and more new “spaces” and their “geometries” are being studied: the space of Lobačesvskiĭ, projective space, Euclidean and other spaces of various dimensions, in particular four-dimensional space, Riemann spaces, Finsler spaces, topological spaces, and so forth. These theories find important application in mathematics itself, outside of geometry, and also in physics and mechanics; particularly noteworthy are their applications in the theory of relativity of contemporary physics, which is a theory of space, time, and gravitation. From what has been said it is clear that we are dealing here with a qualitative change in geometry.

The ideas of contemporary geometry and some of the elements of the theory of various spaces investigated in it will be explained in Chapters XVII and XVIII.

3. Algebra. Algebra too underwent a qualitative change. In the first half of the 19th century new theories arose, which led to changes in its character, and to an extension of its subject matter and its range of application.

In its original form, as pointed out in §5, algebra dealt with mathematical operations on numbers considered from a formal point of view, in abstraction from given concrete numbers. This abstraction found expression in the fact that in algebra magnitudes are denoted by letters, on which calculations are carried out according to well-known formal rules.

Contemporary algebra retains this basis but widens it in a very extensive way. It now considers “magnitudes” of a much more general nature than numbers, and studies operations on these “magnitudes” which are to some extent analogous in their formal properties to the ordinary operations of arithmetic: addition, subtraction, multiplication, and divison. A very simple example is offered by vector magnitudes, which may be added by the well-known parallelogram rule. But the generalization carried out in contemporary algebra is such that even the very term “magnitude” often loses its meaning and one speaks more generally of “elements” on which it is possible to perform operations similar to the usual algebraic ones. For example, two motions carried out one after the other are evidently equivalent to a certain single motion, which is the sum of the two; two algebraic transformations of a formula may be equivalent to a single transformation that produces the same result, and so forth; and so it is possible to speak of a characteristic “addition” of motions or transformations. All this and much else is studied in a general abstract form in contemporary algebra.

The new algebraic theories in this direction arose in the first half of the 19th century in the investigations of a number of mathematicians, among whom we should particularly mention the French mathematician Galois, (1811–1832). The concepts, methods, and results of contemporary algebra find important applications in analysis, geometry, physics, and crystallography. In particular, the theory mentioned at the end of 53 concerning the symmetry of crystals, which was developed by E. S. Fedorov, is based on a union of geometry with one of the new algebraic theories, the so-called theory of groups.

As we see, we are dealing here with a fundamental, qualitative generalization of the subject matter of algebra with a change in the very concept of what algebra is. The ideas of contemporary algebra and the basic elements of some of its theories will be explained in Chapter XX and XVI.

4. Analysis. Analysis in all its branches also made profound progress. In the first place, as was already mentioned in the preceding section, its foundations were made more precise; in particular, its basic concepts were given exact and general definitions: such concepts as function, limit, integral and finally, the basic concept of a variable magnitude (a rigorous definition was given for the real number). A beginning of the process of putting analysis on a more precise foundation was made by the Czech mathematician Bolzano (1781–1848), the French mathematician Cauchy (1789–1857), and a number of others. This greater precision was gained at the same time as the new developments in algebra and geometry were being made; it was brought to completion in its present well-known form in the eighties of the 19th century by the German mathematicians Weierstrass, Dedekind, and Cantor. As was mentioned at the end of §6, Cantor also laid the foundation for the theory of transfinite sets, which plays such a large role in the development of the newer ideas in mathematics.

The increase in precision in the concepts of variable and function in connection with the theory of sets laid the foundation for a further development of analysis. A transition was made to the study of more general functions; and in the same direction the apparatus of analysis, namely the integral and differential calculus, was also generalized. Thus, on the threshold of the present century, there arose the new branch of analysis already mentioned in §6, the so-called theory of functions of a real variable. The development of this theory is chiefly connected with the French mathematicians, Bore1 and Lebesgue and others, and with N. N. Luzin (1883–1950) and his school. In general, the newer branches of analysis are called modern analysis in contradistinction to the earlier so-called classical analysis.

Other new theories arose in analysis. Thus a special branch was formed by the theory of approximation of functions, which studies questions of the best approximate representation of general functions by various “simple” functions, above all by polynomials, that is by functions of the form

The theory of approximation of functions has great importance, if only for the reason that it lays down general foundations for the practical calculation of functions, for the approximate replacement of complicated functions by simpler ones. The rudiments of this theory go back to the very beginnings of analysis. Its modern direction was given to it by the great Russian mathematician P. L. Čebyšev (1821–1894). This direction was later developed into the so-called constructive theory of functions, chiefly in the works of Soviet mathematicians, particularly S. N. Bernšteĭn (born 1880), to whom belong the most. important results in this field. Chapter XII deals with approximation of functions.

We spoke earlier about the development of the theory of functions of a complex variable. We must stiil mention the so-called qualitative theory of differential equations, originating in the works of Poincaré (1854–1912) and A. M. Lyapunov (1857–1918), about which some ideas will be given in Chapter V, and also the theory of integral equations. These theories have great practical importance in mechanics, physics, and technology. Thus, the qualitative theory of differential equations provides solutions of problems concerning stability of motion, and the action of mechanisms or of vibrating electric systems and the like. Stability of a process means in the most general sense that if small changes are made in the initial data or in the conditions of the motion, then the motion itself during the whole of its course will change only slightly. The technical significance of questions of this sort hardly needs to be emphasized.

5. Functional analysis. On the ground prepared by the development of analysis and mathematical physics, along with the new ideas of geometry and algebra, there has grown up an extensive new division of mathematics, the so-called functional analysis, which plays an exceptionally important role in modern mathematics. Many mathematicians shared in creating it; let us mention, for example, the greatest German mathematician of recent times, Hilbert (1862–1943), the Hungarian mathematician Riesz (1880–1956) and the Polish mathematician Banach (1892–1945). The separate Chapter XIX is devoted to functional analysis.

The essence of this new branch of mathematics consists briefly in the following. In classical analysis the variable is a magnitude, or “number,” but in functional analysis the function itself is regarded as the variable. The properties of the given function are determined here not in themselves but in relation to other functions. What is under study is not a separate function but a whole collection of functions characterized by one property or another; for example, the collection of all continuous functions. Such a collection of functions forms the so-called functional space. This procedure corresponds, for example, to the fact that we may consider the collection of all curves on a surface or of all possible motions of a given mechanical system, thereby defining the properties of the separate curves or motions in their relation to other curves or motions.

The transition from the investigation of separate functions to a variable function is similar to the transition from unknown numbers x, y to variables x, y; that is, it is similar to the idea of Descartes mentioned in a preceding paragraph. On the basis of this idea Descartes produced his well-known union of algebra and geometry, of an equation and a curve, which is one of the most important elements in the rise of analysis. Similarly, the union of the concept of a variable function with the ideas of contemporary algebra and geometry produced the new functional analysis. Just as analysis was necessary for the development of the mechanics of the time, so functional analysis provided new methods for the solution of present-day problems of mathematical physics and produced the mathematical apparatus for the new quantum mechanics of the atom. History repeats itself as usual, but in a new way, on a higher plane. As we have said, functional analysis unites the basic ideasand methods of analysis, of modern algebra, and of geometry and in its turn exercises an influence on the development of these branches of mathematics. The problems arising in classical analysis now find new, more general solutions, often almost at a single step, by means of functional analysis. Here, as at a focus, are gathered together, in a very productive way, the most general and abstract ideas of modern mathematics.

From this short sketch, from this mere enumeration of the new directions of analysis(the theory of functions of a real variable, theory of approximation of functions, qualitative theory of differential equations, theory of integral equations, and functional analysis) it may be seen that we are dealing here in fact with an essentially new stage in the development of analysis.

6. Computational mathematics and mathematical logic. At all periods the technical level of the means of computation has had an essential influence on mathematical methods. But the equipment for carrying out calculations which has been at our disposal up until most recent times has been very limited. The simplest devices, such as the abacus, tables of logarithms and the logarithmic sliderule, the calculating machine, and finally more complicated calculators and the automatic calculating machine, these were the basic implements for computation existing up to the forties of the 20th century. These implements made it possible to carry out more or less quickly the separate operations of addition, multiplication, and so forth. But to carry through to final numerical result the practical problems that arise nowadays requires a colossal number of such operations, following one another in a complicated program that sometimes depends on results obtained during the course of the calculation. The solution of such problems proved to be practically impossible or completely valueless on account of the length of the process of solution. But in the last ten years a radical change has taken place in the whole science of computation. Modern calculating machines, constructed on new principles, allow us to make computations with exceptionally great speed and at the same time to carry out complicated chains of calculations automatically, according to extremely flexible programs arranged in advance. Some of the questions connected with the construction and significance of modern calculating machines will be discussed in Chapter XIV.

The new techniques not only enable us to carry out investigations that were formerly quite impracticable but also lead us to change our estimate of the value of many well-known mathematical results. For example, they have given a special stimulus to the development of approximative methods; that is, methods which allow us, by a chain of elementary operations, to reach a desired numerical result with sufficiently great accuracy. The mathematical methods themselves must now be estimated from the point of view of their suitability for corresponding machines.

In close connection with the development of calculating techniques is the subject of mathematical logic. It was developed primarily as a result of intrinsic difficulties arising in mathematics itself, its subject matter being the analysis of mathematical proof. It is itself a branch of mathematics, and includes those branches of general logic that can be objectively formulated and developed by the mathematical method.

Although on the one hand mathematical logic thus goes back to the very sources and foundations of mathematics, it is closely connected, on the other hand, with the most modern questions of computational technique. Naturally, for example, a proof that leads to the setting up of a definite preassigned process, permitting us to approach a desired result with an arbitrary degree of accuracy, is essentially different from more abstract proofs on the existence of the given result.

There also arises here a characteristic range of questions concerning the degree of generality possible in problems that can be dealt with by a method which is completely defined in advance at every step. Profound results have been reached along these lines in mathematical logic, results that are extremely important from a general epistemological point of view.

It would not be an exaggeration to say that with the development of the new computational techniques and the achievements of mathematical logic a new period has begun in modern mathematics, characterized by the fact that its subject matter is not only the study of one object or another but also all the ways and means by which such an object can be defined; not only certain problems, but also all possible methods of sdving them.

To what has been said it is only necessary to add that also in the older branches of mathematics, the theory of numbers, Euclidean geometry, classical algebra and analysis, and the theory of probability, rapid development has continued throughout the whole period of modern mathematics so that these fields have been enriched by many new fundamental ideas and results; let us mention, for example, the results attained in the theory of numbers and in the geometry of everyday space by the Russian and Soviet mathematicians P. L. Čebyčev, E. S. Fedorov, I. M. Vinogradov, and others. The development on a wide front of the theory of probability has been connected with the extraordinarily important regularities observable in statistical physics and in contemporary problems of technology.

7. Characteristic features of modem mathematics. What are the most general characteristics of modern mathematics as a whole, distinguishing it from the earlier development of geometry, algebra, and analysis?

First of all is the immense extension of the subject matter of mathematics and of its applications. Such an extension of subject matter and range of application represents an enormous quantitative and qualitative growth, brought about by the appearance of powerful new theories and methods that allow us to solve problems completely inaccessible up to now. This extension of the subject matter of mathematics is characterized by the fact that contemporary mathematics conscientiously sets itself the task of studying all possible types of quantitative relationships and spatial forms.

A second characteristic feature of modern mathematics is the formation of general concepts on a new and higher level of abstraction. It is precisely this feature that guarantees preservation of the unity of mathematics, in spite of its immense growth in widely differing branches. Even in parts of mathematics that are extremely far from one another similarities of structure are brought to light by the general concepts and theories of the present day. They guarantee that contemporary mathematical methods will have great generality and breath of application; in particular, they produce a profound interpenetration of the fundamental branches of mathematics: geometry, algebra, and analysis.

As one of the characteristic features of modern mathematics, we must also mention the obvious dominance of the set-theoretical point of view. Of course, this point of view owes its significance to the fact that it summarizes in a certain sense the rich content of all the preceding developments of mathematics. Finally, one of the most characteristic features of modern mathematics is the profound analysis of its foundations, of the mutual influence of its concepts, of the structure of its separate theories, and of the methods of mathematical proof. Without such an analysis of foundations it would not be possible to improve or develop any further the principles and theories that have led to the present generalizations.

The characteristic feature of modern mathematics may be said to be that its subject matter consists not only of given quantitative relations and forms but of all possible ones. In geometry, we speak not only of spatial relations and forms but of all possible forms similar to spatial ones. Tn algebra, we speak of various abstract systems of objects with all possible laws of operation on them. In analysis, not only magnitudes are considered as variables but the very functions themselves. In a functional space all the functions of a given type (all the possible interdependences among the variables) are brought together. Summing up, it is possible to say that while elementary mathematics deals with constant magnitudes, and the next period with variable magnitudes, contemporary mathematics is the mathematics of all possible (in general, variable) quantitative relations and interdependences among magnitudes. This definition is, of course, incomplete, but it does emphasize the characteristic feature of modern mathematics which distinguishes it from the mathematics of preceding ages.*

Suggested Reading

Preliminary remark. The original Russian text of Mathematics: its content, methods, and meaning contains a list of recommended books at the end of each of its twenty chapters. In the present translation these books have been retained only if they have been translated into English. In compensation, the lists given here contain many other, readily available, works in the English language.

Books dealing with mathematics in general

E. T. Bell, The development of matheimatics, 2d ed., McGraw-Hill, New York, 1945.

R. Courant and H. Robbins, What is mathematics ? Oxford University Press, New York, 1941.

H. Eves and C. V. Newsom, An introduction to the foundations and fundamental concepts of mathematics, Rinehart, New York, 1958.

G. H. Hardy, A mathematician’s apology, Macmillan, New York, 1940.

R. L. Wilder, Introduction to the foundations of mathematics, Wiley, New York, 1952.

Books of a historical character

R. C. Archibald, Outline of the history of mathematics, 5th ed., Mathematical Association of America, Oberlin, Ohio, 1941.

F. Cajori, History of mathematics, 2d ed., Macmillan, New York, 1919.

Euclid, The thirteen books of Euclid’s Elements translated with an introduction and commentary by T. L. Heath, 2d ed., 3 vols., Dover, New York, 1956.

O. E. Neugebauer, The exact sciences in antiquity, Princeton University Press, Princeton, N. J., 1952.

D. E. Smith, History of mathematics, Vol. I. General survey of the history of elementary mathematics, Vol. II. Special topics of elementary mathematics, Dover, New York, 1958.

D. J. Struik, A concise history of mathematics, Dover, New York, 1948.

B. L. van der Waerden, Science awakening, P. Noordhoff, Groningen, 1954.

* Here we merely point out this example without further explanation, for which the reader may turn to Chapter XVII.

* In fact, every collection of objects, whether it be a flock of sheep or a pile of firewood, exists and is immediately perceived in all its concreteness and complexity. The distinguishing in it of separate properties and relationships is the result of conscious analysis. Primitive thought does not yet make this analysis, but considers the object only as a whole. Similarly, a man who has not studied music perceives a musical composition without distinguishing in it the details of melody, tonality, and so forth, while at the same time a musician easily analyzes even a complicated symphony.

* In the formation of concepts about properties of objects, such as color or the numerosity of a collection, it is possible to distinguish three steps, which we must not, of course, try to separate too sharply from one another. At the first step the property is defined by direct comparison of objects: like a crow, as many as on a hand. At the second, an adjective appears: a black stone or (the numerical adjective being quite analogous) five trees. At the third step the property is abstracted from the objects and may appear “as such”; for example “blackness,” or the abstract number “five.”

* The word “arithmetic,” meaning the “art of calculation,” is derived from the Greek adjective “arithmetic” formed from the noun “arithmos,” meaning “number.” The adjective modifies a noun “techne” (art, technique), which is here understood.

† This is understandable from the most general considerations. An arbitrary abstraction, removed from its concrete basis (just as a number is abstracted from a concrete collection of objects), has no sense “in itself”; it exists only in its relations with other concepts. These relations are already implicit in any statement about the abstraction, in the most incomplete definition of it. Without them the abstraction lacks content and significance, i.e., it simply does not exist. The content of the concept of an abstract number lies in the rules, in the mutual relations of the system of numbers.

* It is worth remarking that the concept of number, which was worked out with such difficulty in a long period of time, is mastered nowadays by a child with relative ease. Why? The first reason is, of course, that the child hears and sees adults constantly making use of numbers, and they even teach him to do the same. But a second reason, and this is the one to which we wish to draw special attention, is that the child already has at hand words and symbols for the numbers. He first learns these external symbols for number and only later masters the meaning of them.

* The first Indian manuscript in which zero appears comes from the end of the 9th century; in it the number 270 is written exactly as we would write it today. But it is probable that zero was introduced in India still earlier, in the 6th century.

* We recall that a prime number is defined as a positive integer greater than unity which is divisible without remainder only by the number itself and by unity.

* What is meant here is the boundaries between shares of land. Let us note, parenthetically, that geometry means land-measurement (in ancient Greek “ge” is land, and “metron” is measure).

* By form we mean also dimensions.

* Compare Chapter XX.

* This theory is ascribed to the Greek scientist Eudoxus, who lived in the 4th century B.C.

† As a result of the fact that the theory of the measurement of magnitudes did not become part of arithmetic but passed over into geometry, mathematics among the Greeks was engulfed by geometry. Such questions, for example, as the solution of quadratic equations, which today we treat in an algebraic way, they stated and solved geometrically. The “Elements” of Euclid contain a considerable number of such questions, which obviously represented for contemporary mathematicians a summary of the foundations not only of geometry in our sense but of mathematics in general. This domination by geometry continued up to the time when Descartes, on the contrary, subjected geometry to algebra. Traces of the long domination by geometry are preserved, for example, in such names as “square” and “cube” for the second and third powers: “a cubed” is a cube with side a.

* We are speaking here not of a descriptive definition, but of a definition which serves as the immediate basis for proofs of theorems about the properties of real numbers. It is natural that such definitions should arise at a later period, when the development of mathematics, and in particular of the infinitesimal analysis, required a suitable definition of the real number represented by “the variable x.” This definition was given in various forms in the seventies of the last century by the German mathematicians Weierstrass, Dedekind, and Cantor.

* Fractions with the periodic digit nine are not considered here, they are identical with the corresponding fraction without nines according to the well-known rule, which is clear from the example: 0.139999 ··· = 0.140000 ···.

* Ptolemy is widely known as the author of a system in which the Earth is considered as the center of the universe and the motion of the heavenly bodies is described as proceeding around it. This system was supplanted by the Copernician system.

* He gives the “equations” of conic sections referred to a vertex. For example “the equation” of the parabola y² = 2px is formulated thus: The square on the side y is equal in area to the rectangle with sides 2p and x. Of course, in place of the symbols p, x, y he uses the corresponding line segments.

* To give some orientation in the dates we list here the times of some of the out standing mathematicians of the East. From India: Aryabhata, born about 476 A.D.; Brahmagupta, about 598–660; Bhaskara, 12th century; from Kharim: Al-Kharizmi, 9th century; Al-Biruni, 973–1048; from Azerbaijan: Nasireddin Tusi, 1201–74; from Samarkand: Gyaseddin Jamschid, 15th century.

* One should keep in mind that it is wrong to associate the development of mathematics in this period chiefly with the Arabs. The term “Arabic” mathematics came into use chiefly because most of the scholars of the East wrote in the Arabic language, which had been spread abroad by the Arab conquests.

* It is to be noted also that the mathematical term “algorithm,” denoting a method or set of rules for computation, comes from the name of the same al-Kharizmi.

* It is interesting to note that Napier did not define logarithms as they are defined nowadays, when we say that in the formula x = aυ the number y is the logarithm of x to the base a. This definition of logarithms appeared later. Napier’s definition was related to the concepts of a variable magnitude and an infinitesimal and amounted to saying that the logarithm of x is a function y = f(x) whose rate of growth is inversely proportional to x; that is, y′ = c/x (see Chapter II). In this way the basis of the definition was essentially a differential equation, defining the logarithm, although differentials had not yet been invented.

† The formula bears the name of Newton not because he was the first to discover it but because he generalized it from integral exponents to arbitrary fractional and irrational exponents.

* Polynomials are functions of the form y = a₀xn + a₁xn–I + . . . + an. The fundamental problem of the algebra of the period, namely the solution of the equation a₀xn + a₁xn–1 + . . . + an, = 0, simply means the search for values of x for which the function y = a₀xn + a₁xn–1 + . . . + an, is equal to zero. The very existence of a solution, of a root of the equation, which is called the fundamental theorem of algebra, is proved by means of analysis (see Chapter IV, §3).

* This double role of the axioms is sometimes lost from view even in works of a methodological character, which thereby attribute to the construction of axioms a significance which does not at all belong to it, namely that of the total construction of a theory.

* This section is followed in the original Russian text by two sections entitled “The essential nature of mathematics” and “The laws of the development of mathematics.” These sections are omitted in the present translation in view of the fact that they discuss in more detail, and in the more general philosophical setting of dialectical materialism, points of view already stated with great clarity in the preceding sections.

CHAPTER II

ANALYSIS

§1.  Introduction

The rise at the end of the Middle Ages of new conditions of manufacture in Europe, namely the birth of capitalism, which at this time was replacing the feudal system, was accompanied by important geographical discoveries and explorations. In 1492, relying on the idea that the earth is spherical, Columbus discovered the New World. The discovery by Columbus greatly extended the boundaries of the known world and produced a revolution in the minds of men. The end of the 15th century and the beginning of the 16th saw the creative activity of the great artist-humanists Leonardo da Vinci, Raphael, and Michelangelo, which gave new meaning to art. In 1543 Copernicus published his work “On the revolution of the heavenly bodies,” which completely changed the face of astronomy; in 1609 appeared the “New astronomy” of Kepler, containing his first and second laws for the motion of the planets around the sun, and in 1618 his book “Harmony of the world,” containing the third law. Galileo, on the basis of his study of the works of Archimedes and his own bold experiments, laid the foundations for the new mechanics, an indispensable science for the newly arising technology. In 1609 Galileo directed his recently constructed telescope, though still small and imperfect, toward the night sky; the first glance in a telescope was enough to destroy the ideal celestial spheres of Aristotle and the dogma of the perfect form of celestial bodies. The surface of the moon was seen to be covered with mountains and pitted with craters. Venus displayed phases like the Moon, Jupiter was surrounded by four satellites and provided a miniature visual model of the solar system. The Milky Way fell apart into separate stars, and for the first time men felt the staggeringly immense distance of the stars. No other scientific discovery has ever made such an impression on the civilized world.*

The further development of navigation, and consequently of astronomy, and also the new development of technology and mechanics necessitated the study of many new mathematical problems. The novelty of these problems consisted chiefly in the fact that they required mathematical study of the laws of motion in a broad sense of the word.

The state of rest and motionlessness is unknown in nature. The whole of nature, from the smallest particles up to the most massive bodies, is in a state of eternal creation and annihilation, in a perpetual flux, in unceasing motion and change. In the final analysis, every natural science studies some aspect of this motion. Mathematical analysis is that branch of mathematics that provides methods for the quantitative investigation of various processes of change, motion, and dependence of one magnitude on another. So it naturally arose in a period when the development of mechanics and astronomy, brought to life by questions of technology and navigation, had already produced a considerable accumulation of observations, measurements, and hypotheses and was leading science straight toward quantitative investigation of the simplest forms of motion.

The name “infinitesimal analysis” says nothing about the subject matter under discussion but emphasizes the method. We are dealing here with the special mathematical method of infinitesimals, or in its modern form, of limits. We now give some typical examples of arguments which make use of the method of limits and in one of the later sections we will define the necessary concepts.

Example 1. As was established experimentally by Galileo, the distance s covered in the time t by a body falling freely in a vacuum is expressed by the formula

(g is a constant equal to 9.81 m/sec²).† What is the velocity of the falling body at each point in its path?

Let the body be passing through the point A at the time t and consider what happens in the short interval of time of length Δt; that is, in the time from t to t + Δt. The distance covered will be increased by a certain increment Δs. The original distance is s₁ = gt²/2; the increased distance is

From this we find the increment

This represents the distance covered in the time from t to t + Δt. To find the average velocity over the section of the path Δs, we divide Δs by Δt:

Letting Δt approach zerowe obtain an average velocity which approaches as close as we like to the true velocity at the point A. On the other hand, we see that the second summand on the right-hand side of the equation becomes vanishingly small with decreasing Δt, so that the average υav approaches the value gt, a fact which it is convenient to write as follows:

Consequently, gt is the true velocity at the time t.

FIG. 1.

Example 2. A reservoir with a square base of side a and vertical walls of height h is full to the top with water (figure 1). With what force is the water acting on one of the walls of the reservoir?

We divide the surface of the wall into n horizontal strips of height h/n. The pressure exerted at each point of the vessel is equal, by a well-known law, to the weight of the column of water lying above it. So at the lower edge of each of the strips the pressure, expressed in suitable units, will be equal respectively to

We obtain an approximate expression for the desired force P, if we assume that the pressure is constant over each strip. Thus the approximate value of P is equal to

To find the true value of the force, we divide the side into narrower and narrower strips, increasing n without limit. With increasing n the magnitude 1/n in the above formula will become smaller and smaller and in the limit we obtain the exact formula

The idea of the method of limits is simple and amounts to the following. In order to determine the exact value of a certain magnitude, we first determine not the magnitude itself but some approximation to it. However, we make not one approximation but a whole series of them, each more accurate than the last. Then from examination of this chain of approximations, that is from examination of the process of approximation itself, we uniquely determine the exact value of the magnitude. By this method, which is in essence a profoundly dialectical one, we obtain a fixed constant as the result of a process or motion.

The mathematical method of limits was evolved as the result of the persistent labor of many generations on problems that could not be solved by the simple methods of arithmetic, algebra, and elementary geometry.

What were the problems whose solution led to the fundamental concepts of analysis, and what were the methods of solution that were set up for these problems ? Let us examine some of them.

The mathematicians of the 17th century gradually discovered that a large number of problems arising from various kinds of motion with consequent dependence of certain variables on others, and also from geometric problems which had not yielded to former methods, could be reduced to two types. Simple examples of problems of the first type are: find the velocity at any time of a given nonuniform motion (or more generally, find the rate of change of a given magnitude), and draw a tangent to a given curve. These problems (our first example is one of them) led to a branch of analysis that received the name “differential calculus.” The simplest examples of the second type of problem are: find the area of a curvilinear figure (the problem of quadrature), or the distance traversed in a nonuniform motion, or more generally the total effect of the action of a continuously changing magnitude (compare the second of our two examples). This group of problems led to another branch of analysis, the “integral calculus.” Thus two fundamental problems were singled out: the problem of tangents and the problem of quadratures.

In this chapter we will describe in detail the underlying ideas of the solution of these two problems. Particularly important here is the theorem of Newton and Leibnitz to the effect that the problem of quadratures is the inverse, in a well-known sense, of the problem of tangents. For solving the problem of tangents, and problems that can be reduced to it, there was worked out a suitable algorithm, a completely general method leading directly to the solution, namely the method of derivatives or of differentiation.

The history of the creation and development of analysis and of the role played in its growth by the analytic geometry of Descartes has already been described in Chapter I. We see that in the second half of the 17th century and the first half of the 18th a complete change took place in the whole of mathematics. To the divisions that already existed, arithmetic, elementary geometry, and the rudiments of algebra and trigonometry, were added such general methods as analytic geometry, differential and integral calculus, and the theory of the simplest differential equations. It was now possible to solve problems whose solutions up to now had been quite inaccessible.

It turned out that if the law for the formation of a given curve is not too complicated, then it is always possible to construct a tangent to it at an arbitrary point; it is only necessary to calculate, with the help of the rules of differential calculus, the so-called derivative, which in most cases requires a very short time. Up till then it had been possible to draw tangents only to the circle and to one or two other curves, and no one had suspected the existence of a general solution of the problem.

If we know the distance traversed by a moving point up to any desired instant of time, then by the same method we can at once find the velocity of the point at a given moment, and also its acceleration. Conversely, from the acceleration it is possible to find the velocity and the distance, by making use of the inverse of differentiation, namely integration. As a result, it was not very difficult, for example, to prove from the Newtonian laws of motion and the law of universal gravitation that the planets must move around the sun in ellipses according to the laws of Kepler.

Of the greatest importance in practical life is the problem of the greatest and least values of a magnitude, the so-called problem of maxima and minima. Let us take an example: From a log of wood with circular cross section of given radius we wish to cut a beam of rectangular cross section such that it will offer the greatest resistance to bending. What should be the ratio of the sides? A short argument on the stiffness of beams of rectangular cross section (applying simple concepts from the integral calculus), followed by the solving of a maximum problem (which involves calculating a derivative) provides the answer that the greatest stiffness is produced for a rectangular cross section whose height is in the ratio to its base of . The problems of maxima and minima are solved as simply as those of drawing tangents.

At various points of a curved line, if it is not a straight line or a circle, the curvature is in general different. How can we calculate the radius of a circle with the same curvature as the given line at the given point, the so-called radius of curvature of the curve at the point? It turns out that this is equally simple; it is only necessary to apply the operation of differentiation twice. The radius of curvature plays a great role in many questions of mechanics.

Before the invention of the new methods of calculation, it had been possible to find the area only of polygons, of the circle, of a sector or a segment of the circle, and of two or three other figures. In addition, Archimedes had already invented a way to calculate the area of a segment of a parabola. The extremely ingenious method which he used in this problem was based on special properties of the parabola and consequently gave rise to the idea that every new problem in the calculation of area would very likely require its own methods of investigation, even more ingenious and difficult than those of Archimedes. So mathematicians were greatly pleased when it turned out that the theorem of Newton and Leibnitz, to the effect that the inversion of the problem of tangents would solve the problem of quadrature, at once provided a method of calculating the areas bounded by curves of widely different kinds. It became clear that a general method exists, which is suitable for an infinite number of the most different figures. The same remark is true for the calculation of volumes, surfaces, the lengths of curves, the mass of inhomogeneous bodies, and so forth.

The new method accomplished even more in mechanics. It seemed that there was no problem in mechanics that the new calculations would not clarify and solve.

Not long before, Pascal had explained the increase in the size of the Torricelli vacuum with increasing altitude as a consequence of the decrease in atmospheric pressure. But exactly what is the law governing this decrease? The question is answered immediately by the investigation of a simple differential equation.

It is well known to sailors that they should take two or three turns of the mooring cable around the capstan if one man is to be able to keep a large vessel at its mooring. Why is this? It turned out that from mathematical point of view the problem is almost completely identical with the preceding one and can be solved at once.

Thus, after the creation of analysis, there followed a period of tempestuous development of its applications to the most varied branches of technology and natural science. Since it is founded on abstraction from the special features of particular problems, mathematical analysis reflects the actual deep-lying properties of the material world; and this is the reason why it provides the means for investigation of such a wide range of practical questions. The mechanical motion of solid bodies, the motion of liquids and gases of their particular particles, their laws of flow in the mass, the conduction of heat and electricity, the course of chemical reactions, all these phenomena are studied in the corresponding sciences by means of mathematical analysis.

At the same time as its applications were being extended, the subject of analysis itself was being immeasurably enriched by the creation and development of various new branches, such as the theory of series, applications of geometry to analysis, and the theory of differential equations.

Among mathematicians of the 18th century, there was a widespread opinion that any problem of the natural sciences, provided only that one could find a correct mathematical description of it, could be solved by means of analytic geometry and the differential and integral calculus.

Mathematicians proceeded gradually to more complicated problems of natural science and technology, which demanded further development of their methods. For the solution of such problems it became necessary to create further branches of mathematics: the calculus of variations, the theory of functions of a complex variable, field theory, integral equations, and functional analysis. But all these new methods of calculation were essentially immediate extensions and generalizations of the remarkable methods discovered in the 17th century. The greatest mathematicians of the 18th century, David Bernoulli (1700–1782), Leonard Euler (1707–1783) and Lagrange (1736–1813), who blazed new paths in science, constantly took as their starting point the fundamental problems of the exact sciences. This energetic development of analysis was continued into the 19th century by such famous mathematicians as Gauss (1777–1855), Cauchy (1789–1857), M. V. Ostrogradskiĭ (1801–1861), P. L. Čebyšev (1821–1894), Riemann (1826–1866), Abel (1802–1829), Weierstrass (1815–1897), all of whom made truly remarkable contributions to the development of mathematical analysis.

The Russian mathematical genius, N. I. Lobačevskiĭ, had an influence on the development of certain questions of mathematical analysis, and we should also mention the leading mathematicians who were active at the turn of the 20th century: A. A. Markov (1856-1922), A. M. Lyapunov (1857-1918), H. Poincaré (1854-1912), F. Klein (1849-1925), D. Hilbert (1862-1943).

The second half of the 19th century witnessed a profound critical examination and clarification of the foundations of analysis. The various powerful methods that had accumulated were now put on a uniform systematic basis, corresponding to the advanced level of mathematical rigor. All these methods are the means by which, along with arithmetic, algebra, geometry and trigonometry, we give a mathematical interpretation to the world around us, describe the course of actual events, and solve the important practical problems connected with them.

Analytic geometry, differential and integral calculus, and the theory of differential equations are studied at all technical institutes, so that these branches of mathematics are known to millions of citizens; the elements of these sciences are also taught at many technical schools; there is also some question of their being introduced into the secondary schools.

In most recent times the general use of rapid calculating machines has introduced a new era in mathematics. These machines, in conjunction with the branches of mathematics just mentioned, open up strange new possibilities for mankind.

At the present time, analysis and the branches arising from it represent a widely diversified mathematical science, consisting of several broad independent disciplines closely connected with one another; each of these disciplines is being developed and perfected.

More than ever before, a significant role is being played in analysis by the requirements of daily life, by problems connected with the imposing development of technology. Of great importance are the aerodynamical problems of hypersonic velocities, which are being solved with constant success. The most difficult problems of mathematical physics have now reached the stage where they can be solved in practical numerical form. In contemporary physics such theories as quantum mechanics (which studies the problems peculiar to the microcosm of the atom) not only require the most advanced branches of contemporary mathematical analysis for solving their problems but could not even describe their fundamental concepts without the use of analysis.

The purpose of the present chapter is to give a popular presentation, suitable for a reader acquainted only with elementary mathematics, of the growth and the simplest applications of such basic concepts of analysis as function, limit, derivative, and integral. Since the various special branches of analysis will be dealt with in other chapters of the book, the present chapter has a more elementary character and a reader who has already studied a usual first course in analysis may omit it without harm to his understanding of the rest of the book.

§2.  Function

The concept of a function. The various objects or phenomena that we observe in nature are organically connected with one another; they are interdependent. The simplest relations of this sort have long been known to mankind and information about them has been accumulated and formulated as physical laws. These laws indicate that the various magnitudes characterizing a given phenomenon are so closely related to one another that some of them are completely determined by the values of others. For example, the length of the sides of a rectangle completely determines its area, the volume of a given amount of gas at a given temperature is determined by the pressure, and the elongation of a given metallic rod is determined by its temperature. It was uniformities of this sort that served as the origin of the concept of function.

Consider an algebraic formula which, corresponding to each value of the literal magnitudes occurring in it, allows us to find the value of the magnitude expressed by the formula; the basic idea here is that of a function. Let us consider some examples of functions expressed by such formulas.

1. Let us suppose that at the beginning of a certain period of time a material point was at rest and that subsequently it began to fall as the result of gravity. Then the distance s traced out by the point up to time t is expressed by the formula

where g is the acceleration of gravity.

FIG. 2.

2. From a square of side a we construct an open rectangular box of height x (figure 2). The volume V of the box is calculated from the formula

FIG. 3.

Formula (2) allows us, for every height x under the obvious restriction , to find the volume of the box.

3. Let a pillar (figure 3) be erected at the center of a circular skating rink with a light at height h. The illumination T at the edge of the circle may be expressed by the formula

where r is the radius of the circle, tan α = h/r, and A is a certain magnitude characterizing the power of the light. If we know the height h we can calculate T from formula (3).

4. The roots of the quadratic equation

are given by the formula

The characteristic feature of a formula in general, and of the examples just given in particular, is that the formula enables us, for any given value of one of the variables (the time t, the height x of the box, the height h of the pillar, the coefficient p of the quadratic equation), which is called the independent variable, to calculate the value of the other variable (the distance s, the volume V, the illumination T, the root x of the equation), which is called a dependent variable or a function of the first variable.

Each of the formulas introduced provides an example of a function: the distance s traced by the point is a function to the time t; the volume V of the box is a function of height x; the illumination T of the edge of the rink is a function of the height h of the pillar; the two roots of the quadratic equation (4) are functions of the coefficient p.

It should be remarked that in some cases the independent variable may assume any desired numerical value, as in example 4 where the coefficient p of the quadratic equation (4) may be an arbitrary number. In other cases the independent variable may take an arbitrary value from some set (or collection) of numbers determined in advance; as in example 2, where the volume of the box is a function of its height x, which can take any value from the set of numbers x satisfying the inequality 0 < x < a/2. Similarly, in example 3 the illumination T at the edge of the rink is a function of the height h of the pillar, which theoretically can take any value satisfying the inequality h > 0, but in practice h must satisfy the inequalities , where the magnitude H is determined by the technical facilities at the disposal of the administration of the rink.

Let us introduce other examples of this kind. The formula

determines a real function (expressing a relationship between the real numbers x and y) only for those values of x which satisfy the inequalities , and the formula y = log (1 – x²) only for those x which satisfy the inequalities –1 < x < 1.

So it is necessary to take account of the fact that actual functions may not be defined for all numerical values of the independent variable but only for those values which belong to a certain set, which most often fills out an interval on the x-axis, with or without the end points.

We are now in the position to give the definition of a function accepted in present-day mathematics.

The (dependent) magnitude y is a function of the (independent) magnitude x if there exists a rule whereby to each value of x belonging to a certain set of numbers there corresponds a definite value of y.

The set of values x appearing in this definition is called the domain of the function.

Every new concept gives rise to a new symbolism. The transition from arithmetic to algebra was made possible by the construction of formulas which were valid for arbitrary numbers, and the search for general solutions gave rise to the literal symbolism of algebra.

The problem of analysis is the study of functions, that is of the dependence of one variable on another. Consequently, just as in algebra a transition took place from concrete numbers to arbitrary numbers, denoted by letters, so in analysis there was the corresponding transition from concrete formulas to arbitrary formulas. The phrase “y is a function of x” is conventionally written as

Just as in algebra different letters are used for different numbers, so in analysis different notations are used for different types of dependence, that is for different functions: thus we write y = F(x), y = ϕ(x), ···

Graphs of functions. One of the most fruitful and brilliant ideas of the second half of the 17th century was the idea of the connection between the concept of a function and the geometric representation of a line. This connection can be realized, for example, by means of a rectangular Cartesian system of coordinates, with which the reader is certainly familiar in a general way from his secondary school mathematics.

Let us set up on the plane a rectangular Cartesian system of coordinates. This means that on the plane we choose two mutually perpendicular lines (the axis of abscissas and the axis of ordinates), on each of which we fix a positive direction. Then to each point M of the plane we may assign two numbers (x, y), which are its coordinates, expressing in the given system of measurement the distance, taken with the proper sign,* of the point M from the axis of ordinates and the axis of abscissas respectively.

With such a system of coordinates we may represent functions graphically in the form of certain lines. Suppose we are given a function

This means, as we know, that for every value of x belonging to the domain of definition of the given function, it is possible to determine by some means, for example by calculation, a corresponding value y. Let us give to x all possible numerical values, for each x determine y according to our rule (6), and construct on the plane the point with coordinates x and y. In this way, for every point M′ on the x-axis (figure 4) there will correspond a point M with coordinates x and y = f(x). The set of all points M forms a certain line, which we call the graph of the function y = f(x).

FIG. 4.

FIG. 5.

Thus, the graph of the function f(x) is the geometric locus of the points whose coordinates satisfy equation (6).

In school we became acquainted with the graphs of the simplest functions. Thus the reader probably knows that the function y = kx + b, where k and b are constants, is the graph (figure 5) of a straight line forming the angle α with the positive direction of the x-axis, where tan α = k, and intersecting the y-axis at the point (0, b). This function is called a linear function.

Linear functions occur very frequently in the applications. Let us recall that many physical laws are represented, with considerable accuracy, by linear functions. For example, the length l of a body may be considered with good approximation as a linear function of its temperature

where α is the coefficient of linear expansion, and l₀ is the length of the body for t = 0. If x is the time and y is the distance covered by a moving point, then the linear function y = kx + b obviously expresses the fact that the point is moving with uniform velocity k; and the number b denotes the distance, at time x₀ = 0, of the moving point from the fixed zero-point from which we measure our distances. Linear functions are extremely useful because of their simplicity and because it is possible to consider nonuniform changes as being approximately linear, even if only for small intervals.

But in many cases it is necessary to make use of nonlinear functional dependence. Let us recall for example the law of Boyle-Mariotte

where the magnitudes p and ν are inversely proportional. The graph of such a relation represents a hyperbola (figure 6).

The physical law of Boyle-Mariotte corresponds actually to the case that p and υ are positive; it represents a branch of the hyperbola lying in the first quadrant.

The general class of oscillatory processes includes periodic motions, which are usually described by the familiar trigonometric functions. For example, if we extend a hanging spring from its position of equilibrium, then, so long as we stay within the elastic limits of the spring, the point A will perform vertical oscillations which are quite accurately expressed by the law

where x is the displacement of the point A from its position of equilibrium, t is the time, and the numbers a, p and α are certain constants determined by the material, the dimensions, and the initial extension of the spring.

It should be kept in mind that a function may be defined in various domains by various formulas, determined by the circumstances of the case. For example, the relation Q = f(t) between the temperature t of a gram of water (or ice) and the quantity of heat Q in it, as t varies between –10° and + 10°, is a completely determined function which it is difficult to express in a single formula,* but it is easy to represent this function by two formulas. Since the specific heat of ice is equal to 0.5 and that of water is equal to 1, this function, if we agree that Q = 0 at –10°. is represented by the formula

FIG. 6

FIG. 7

as t varies in the interval and by another formula

as t varies in the interval . For t = 0 this function is indefinite or multiple-valued; for convenience, we may agree that at t = 0 it takes some well-defined value, for example f(0) = 45. The graph of the function Q = f(t) is given in figure 7.

We have introduced many examples of functions given by formulas. The possibility of representing a function by means of formulas is extremely important from the mathematical point of view, since such formulas provide very favorable conditions for investigating the properties of the functions by mathematical methods.

But one must not think that a formula is the only method of defining a function. There are many other methods; for example, the graph of the function, which gives a visual geometric picture of it. The following example gives a good illustration of another method.

To record variation of the temperature of the air during the course of 24 hours, meteorological stations make use of an instrument called the thermograph. A thermograph consists of a drum rotated about its axis by a clockwork mechanism, and of a curved brass framework that is extremely sensitive to changes of temperature. As a result, a pen fastened to the framework by a system of levers rises with rising temperature; and conversely, a fall in the temperature lowers the pen. On the drum is wound a ribbon of graph paper, on which the pen draws a continuous line, forming the graph of the function T = f(t), which expresses the interdependence of the time and the temperature of the air. From this graph we may determine, without calculation, the value of the temperature at any moment of time t.

This example shows that a graph in itself determines a function independently of whether the function is given by a formula or not.

Incidentally, we shall return to this question (see Chapter XII) and shall prove the following important assertion: Every continuous graph can be represented by a formula, or, as it is still customary to say, by an analytic expression. This statement is also true for many discontinuous graphs.*

We remark that the truth of this statement, which is of great theoretical importance, was completely realized in mathematics only in the middle of the past century. Up to that time mathematicians understood by the term “function” only an analytic expression (formula). But they were under the mistaken impression that many discontinuous graphs did not correspond to any analytic expression, since they assumed that if a function was given by a formula, then its graph must possess certain particularly desirable properties in comparison with the other graphs.

But in the 19th century, it was discovered that every continuous graph may be represented by a more or less complicated formula. Thus the exceptional role of the analytic expression as a means of definition of functions was weakened and there came into existence the new, more flexible definition given above for the concept of a function. By this definition a variable y is called a function of a variable x if there exists a rule whereby to every value of x in the domain of definition of the function there corresponds a completely determined value y, independent of the way in which this rule is given: by a formula, a graph, a table or, in any other way.

We may remark here that in the mathematical literature the above definition of a function is often associated with the name of Dirichlet, but it is worth emphasizing that this definition was given simultaneously and independently by N. I. Lobačevskiĭ. Finally we suggest as an exercise that the reader sketch the graphs of the functions x³, √ x, sin x, sin 2x, sin (x + π/4), In x, In(1 + x), |x – 3|, (x + | x |)/2.

We should also note that the graph of a function which for all values of x satisfies the relation

is symmetric with respect to the y-axis and in the case

the graph is symmetric with respect to the origin of coordinates. Consider also how to obtain the graph of a function f(a + x), when a is a constant, from the graph of f(x). Finally, consider how, using the graphs of the functions f(x) and ϕ(x), it is possible to find the values of the composite function y = f[ϕ(x)].

§3.  Limits

In §1 it was stated that modern mathematical analysis uses a special method, which was worked out in the course of many centuries and serves now as its basic instrument. We are speaking here of the method of infinitesimals, or, as is essentially the same, of limits. We shall try to give some idea of these concepts. For this purpose we consider the following example.

We wish to calculate the area bounded by the parabola with equation y = x², by the x-axis and by the straight line x = 1 (figure 8). Elementary mathematics will not furnish us with a means for solving this problem. But here is how we may proceed.

We divide the interval [0, 1] along the x-axis into n equal parts at the points

and on each of these parts construct the rectangle whose left side extends up to the parabola. As a result we obtain the system of rectangles shaded in figure 8, the sum Sn of whose areas is given by

*

Let us express Sn in the following form:

The quantity αn, which depends on n, is admittedly rather unwieldy in appearance, but it possesses a certain remarkable property: If n is increased beyond all bounds, then αn approaches 0. This property may also be expressed as follows: If we are given an arbitrary positive number , then it is possible to choose an integer N sufficiently large that for all n greater than N the number αn will be less than the given in absolute value.†

FIG. 8.

The magnitude αn is an example of an infinitesimal in the sense in which that word is used in modern mathematics.

In figure 8 we see that if we increase the number n beyond all bounds, the sum Sn of the areas of the shaded rectangles will approach the desired area of the curvilinear figure. On the other hand, equation (7), in view of the fact that αn approaches zero as n increases beyond all bounds, shows that the sum Sn at the same time approaches 1/3. From this it follows that the desired area S of the figure is equal to 1/3, and we have solved our problem.

So the method under discussion amounts to this, that in order to find a certain magnitude S we introduce another magnitude Sn, a variable magnitude which approaches S through particular values S₁, S₂, S₃, ···, which depend according to some law on the natural numbers n = 1, 2, ···. Then, from the fact that the variable Sn may be represented as the sum of a constant and an infinitesimal αn, we conclude that Sn approaches and so . In the language of the modern theory of limits we may say that for increasing n the variable magnitude Sn approaches a limit, which is equal to .

Now let us give a precise definition of the concepts introduced here.

If a variable magnitude αn(n = 1, 2, ···) has the property that for every arbitrarily small positive number it is possible to choose an integer N so large that for all n > N we have | αn | < , then we say that αn is an infinitesimal and we write

On the other hand, if a variable xn may be represented as a sum

where a is constant and αn is an infinitesimal, then we say that the variable xn, for n increasing beyond all bounds, approaches the number a and we write

The number a is called the limit of xn. In particular the limit of an infinitesimal is obviously zero.

Let us consider the following examples of variable magnitudes

It is clear that xn, yn, and zn are infinitesimals, the first of them approaching zero through decreasing values, the second through increasing negative values, while the third takes on values which oscillate around zero. Further, un → 1, while υn does not have a limit at all, since with increasing n it does not approach any constant number but continually oscillates, taking on the values 1 and −1.

Another important concept in analysis is that of an infinitely large magnitude, which is defined as a variable xn (n = 1, 2, ···), with the property that after choice of an arbitrarily large positive number M it is possible to find a number N such that for all n > N

The fact that the magnitude xn is infinitely large is written thus

Such a magnitude xn is said to approach infinity. If it is positive (negative) from some value on, this fact is expressed thus: xn → + ∞(xn → − ∞). For example, for n = 1, 2, ···

It is easy to see that if a magnitude αn is infinitely large, then βn = 1/αn is infinitely small, and conversely.

Two variable magnitudes xn and yn may beadded, subtracted, multiplied, and divided the one by the other so as to produce new magnitudes that are in general also variable: namely their sum xn + yn, their difference xn – yn, their product xnyn, and their quotient xn/yn. Correspondingly their particular values will be

It is also possible to prove, as is fairly evident, that if the variables xn and yn approach finite limits, then their sum, difference, product, and quotient also approach limits which are correspondingly equal to the sum, difference, product, and quotient of these limits. This fact may be expressed thus:

However, in the case of the quotient it is necessary to assume that the limit of the denominator (lim yn) is not equal to zero. If lim yn = 0 and lim xn ≠ 0, then the ratio of xn to yn will not have a finite limit but will approach infinity.

Especially interesting, and at the same time important, is the case when the numerator and the denominator simultaneously approach zero. Here it is impossible to state in advance whether the ratio xn/yn will approach a limit, and if it does, what that limit will be, since the answer to this question depends entirely on the character of the approach of xn and yn to zero. For example, if

then

On the other hand, the magnitude

evidently does not approach any limit.

Thus the case when the numerator and the denominator of the fraction both approach zero cannot be dealt with in advance by general theorems, and for each particular fraction of this kind it is necessary to make a special investigation.

We shall see later that the fundamental problem of the differential calculus, which may be considered as the problem of determining the velocity of a nonuniform motion at a given moment, reduces to determining the limit of the ratio of two infinitesimal magnitudes, namely the increase of the distance covered and the increase in the time.

So far we have considered variables xn which take on a sequence of numerical values x₁, x₂, x₃, ···, xn, ···, while the index n runs through the sequence of natural numbers n = 1, 2, 3, ···. But it is also possible to consider the case that n varies continuously, like the time for example, and here also to determine the limit of the variable xn. The properties of such limits are completely analogous to those formulated earlier for discrete (that is, discontinuous) variables. We also note that there is no special significance in the fact that n increases beyond all bounds. It is equally possible to consider the case that, while varying continuously, n approaches a given value n₀.

As an example let us investigate the variation in the magnitude of (sin x)/x as x approaches zero. Table 1 shows the values of this magnitude for certain values of x:

Table 1

x

0.50

0.9589 …

0.10

0.9983 …

0.50

0.9996 …

…

…

(it is assumed that the values of x are given in radian measure).

It is obvious that as xapproacheszero themagnitude (sin x)/x approaches 1, but of course we must still give a rigorous proof of this fact. The proof may be obtained, for example, from the following inequality, which is valid for all nonzero angles in the first quadrant:

If we divide both sides of this inequality by sin x, we obtain

from which follows

But as x decreases to zero cos x approaches 1, so that the magnitude (sin x)/x, being contained in the interval between cos x and 1, also approaches 1, that is

We shall have occasion below to make use of this fact.

Our equation has been proved for the case that x approaches zero through positive values. But by changing the proof in an obvious way, it is possible to obtain the same result when x approaches zero through negative values.

Let us now discuss for a moment the following question. A variable magnitude may or may not have a limit and the question arises whether it is possible to give a criterion for determining the existence of a limit for a variable. We will confine ourselves to an important and sufficiently general case, for which such a criterion can be given. Let us suppose that the variable magnitude xn increases or at least does not decrease; that is, it satisfies the inequalities

and let us also suppose we have determined that none of its values exceeds a certain fixed number M; that is, . If we mark the values of xn and the number M on the x-axis, we see that the variable point xn moves along the axis to the right but constantly remains to the left of the point M. It is rather obvious that the variable point xn must inevitably approach a certain limit point a, situated to the left of M or at most coinciding with M.

So, in the case under consideration, the limit

of our variable exists.

The above argument has an intuitive character but we may consider it as a proof. In a course in modern analysis a complete proof of this fact is given on the basis of the theory of real numbers.

As an example let us consider the variable

The first few values are u₁ = 2, u₂ = 2.25, u₃ ≈ 2.37, u₄ ≈ 2.44, ···, which are seen to increase. From the binomial theorem of Newton it is possible to prove that this increase holds for arbitrary n. Moreover, it is also easy to prove that for all n the inequality un < 3 is valid. Consequently, our variable must have a limit which is not greater than 3. We shall see that this limit plays a very important role in mathematical physics and in a certain sense is the most natural base for logarithms of numbers.

It is customary to denote this limit by the letter e. It is equal to

A more detailed analysis shows that the number e is not rational.*

It is also possible to show that the limit under consideration exists and is equal to e not only when n → + ∞ but also when n → − ∞. In both cases n may also take on noninteger values.

Let us mention an important application to physics of the concept of a limit. It consists of the remarkable fact that only by using the concept of a limit (passage to the limit) is it possible for us to give a complete definition of many of the concrete magnitudes encountered in physics.

Let us also consider for the moment the following geometric example. In elementary geometry the figures considered first are those bounded by straight line segments. But later there arises the more difficult task of finding the length of the circumference of a circle with given radius.

If we analyze the difficulties connected with the solution of this problem, we find that they reduce to the following.

We must give an answer to the question, what is meant by the length of the circumference; that is, we must give a precise definition of this length. It is essential that the definition should be expressible in terms of the lengths of straight-line segments and also that it should provide us with the possibility of effectively calculating the length of the circumference.

It is understood, of course, that the result of this calculation should be in agreement with practical experience. For example, if we consider a circumference consisting of an actual thread, then, if we cut the thread and stretch it out, we must obtain a segment whose length, within the limits of accuracy of measurement, coincides with our computed length.

As is known from elementary geometry, the solution of this problem reduces to the following definition. The length of a circumference is defined to be the limit approached by the perimeter of a regular† polygon inscribed in it as the number of sides of the polygon increases beyond all bounds. Thus the solution of the problem is based essentially on the concept of a limit.

The length of an arbitrary smooth curve is defined in the same way. In the paragraphs just following, we will meet with a number of examples of geometric and physical magnitudes that can be defined only with the concept of a limit.

The concepts of limit and infinitesimal were given a definitive formulation at the beginning of the last century. The definitions introduced here are connected with the name of Cauchy, before whose time mathematicians operated with concepts that were less clear. The present-day concepts of a limit, of an infinitesimal as a variable magnitude, and of a real number, resulted from the development of mathematical analysis and were at the same time the means of stating and clarifying its many achievements.

§4.  Continuous Functions

Continuous functions form the basic class of functions for the operations of mathematical analysis. The general idea of a continuous function may be obtained from the fact that its graph is continuous; that is, its curve may be drawn without lifting the pencil from the paper.

A continuous function gives the mathematical expression of a situation often encountered in practical life, namely that to a small increase in an independent variable there corresponds a small increase in the dependent variable, or function. Excellent examples of a continuous function are given by the various rules governing the motion of bodies s = f(t), expressing the dependence of the distance s on the time t. Since the time and the distance are continuous, a law of motion of the body s = f(t) sets up between them a definite continuous relation, characterized by the fact that to a small increase in the time corresponds a small increase in the distance.

Mankind arrived at the abstraction of continuity by observing the surrounding so-called dense media, namely solids, liquids, and gases; for example, metals, water, and air. In actual fact, as is well known now, every physical medium represents the accumulation of a large number of separate particles in motion. But these particles and the distances between them are so small in comparison with the dimensions of the media in which the phenomena of microscopic physics take place that many of these phenomena may be studied with sufficient accuracy if we consider the medium as being approximately without interstices, that is as continuously distributed over the occupied space. It is on such an assumption that many of the physical sciences are based, for example, hydrodynamics, aerodynamics, and the theory of elasticity. The mathematical concept of continuity naturally plays a large role in these sciences, and in many others as well.

Let us consider an arbitrary function y = f(x) and some specific value of the independent variable x₀. If our function reflects a continuous process, then to values x which differ only slightly from x₀ will correspond values of the function f(x) differing only slightly from the value f(x₀) at the point x₀. Thus if the increment x – x, of the independent variable is small, then the corresponding increment f(x) – f(x₀) of the function will also be small. In other words if the increment of the independent variable x – x₀ approaches zero, then the increment f(x) – f(x₀) of the function must also approach zero, a fact which may be expressed in the following way:

This relation constitutes the mathematical definition of continuity of the function at the point x₀; namely, the function f(x) is said to be continuous at the point x₀, if equality (8) holds.

Finally, we give the following definition. A function is said to be continuous in a given interval, if it is continuous at every point x₀ of this interval; that is, if at every such point equality (8) is fulfilled.

Thus, in order to introduce a mathematical definition of the property of a function reflected in the fact that its graph is continuous (in the everyday sense of this word), it was necessary first to define local continuity (continuity at the point x₀) and then on this basis to define continuity of the function in the whole interval.

This definition, first introduced at the beginning of the last century by Cauchy, is now generally adopted in contemporary mathematical analysis. The test of many concrete examples has shown that it corresponds very well to the practical notion we have of a continuous function, for instance, as represented by its continuous graph.

As examples of continuous functions, the reader may consider the elementary functions well known to him from school mathematics xn, sin x, cos x, as, log x, arc sin x, arc cos x. All these functions are continuous in the intervals for which they are defined.

If continuous functions are added, subtracted, multiplied, or divided (except for division by zero), the result is also a continuous function. But in the case of division the continuity is usually destroyed for those values x₀ for which the function in the denominator vanishes. The result of the division in that case is a function which is discontinuous at the point x₀.

The function y = 1/x may serve as an example of a function which is discontinuous at the point x = 0. Other discontinuous functions are represented by the graphs in figure 9.

FIG. 9a.

FIG. 9b.

FIG. 9c.

FIG. 9d.

FIG. 9e.

We recommend that the reader examine these graphs carefully. He will notice that the breaks in the functions are of different kinds: In some cases a limit f(x) exists as x approaches the point x₀ where the function suffers a discontinuity, but this limit is different from f(x₀). In other cases, as in figure 9c, the limit simply does not exist. It may also happen that as x approaches x₀ from one side f(x) – f(x₀) → 0, but as x → x₀ from the other side, f(x) – f(x₀) does not approach zero. In this case, of course, the function has a discontinuity, but we may say that at such a point it is “continuous from one side.” All these cases are represented in the graphs of figure 9.

As an exercise we recommend to the reader to consider the question, what value must be given to the functions

at those points where they are not defined (that is, at the points where the denominator is equal to zero), in order thqt they may be continuous at these points. Also, is it possible to find such numbers for the functions

These discontinuous functions in mathematics represent the numerous jumplike processes to be met with in nature. In the case of a sudden blow, for example, the value of the velocity of a body changes in such a jumplike fashion. Many qualitative transitions take place with such jumps. In §2 we introduced the function Q = f(t), expressing the way in which the quantity of heat in a given quantity of water (or ice) depends on the temperature. In the neighborhood of the melting point of ice the quantity of heat Q = f(t) changes in a jumplike fashion with changing t.

Functions with isolated discontinuities are encountered quite often in analysis, along with the continuous functions. But as an example of a more complicated function, where the number of discontinuities is infinite, let us consider the so-called Riemann function, which is equal to zero at all irrational points and equal to 1/q at rational points of the form x = p/q (where p/q is a fraction in its lowest terms). This function is discontinuous at all rational points and continuous at irrational points. By altering it slightly we may easily obtain an example of a function which is discontinuous at all points.* Let us remark by the way that even for such complicated functions modern analysis has discovered many in teresting laws, which are investigated in one of the independent branches of analysis, the theory of functions of a real variable. This theory has developed with extraordinary rapidity during the past 50 years.

§5.  Derivative

The next fundamental concept of analysis is the concept of derivative. Let us consider two problems from which it arose historically.

Velocity. At the beginning of the present chapter we defined the velocity of a freely falling body. To do so we made use of a passage to the limit from the average velocity over short distances to the velocity at the given point and the given time. The same procedure may be used to define the instantaneous velocity for an arbitrary nonuniform motion. In fact, let the function

express the dependence of the distance s covered by the material point in the time t. To find the velocity at the moment t = t₀, let us consider the interval of time from t₀ to t₀ + h(h ≠ 0). During this time the point will cover the distance

The average velocity υav over this part of the path will depend on h

and will represent the actual velocity at the point t₀ with greater and greater accuracy as h becomes smaller. It follows that the true velocity at the time t₀ is equal to the limit

of the ratio of the increase in the distance to the increase in the time, as the latter approaches zero without ever being actually equal to zero. In order to calculate the velocity for different forms of motion, we must discover how to find this limit for various functions f(t).

Tangent. We are led to investigate a precisely analogous limit by another problem, this time a geometric one, namely the problem of drawing a tangent to an arbitrary plane curve.

Let the curve C be the graph of a function y = f(x), and let A be the point on the curve C with abscissa x₀ (figure 10). Which straight line shall we call the tangent to C at the point A? In elementary geometry this question does not arise. The only curve studied there, namely the circumference of a circle, allows us to define the tangent as a straight line which has only one point in common with the curve. But for other curves such a definition will clearly not correspond to our intuitive picture of “tangency.” Thus, of the two straight lines L and M in figure 11, the first is obviously not tangent to the curve drawn there (a sinusoidal curve), although it has only one point in common with it; while the second straight line has many points in common with the curve, and yet it is tangent to the curve at each of these points.

To define the tangent, let us consider on the curve C (figure 10) another point A′, distinct from A, with abscissa x₀ + h. Let us draw the secant AA′ and denote the angle which it forms with the x-axis by α. We now allow the point A′ to approach A along the curve C. If the secant AA′ correspondingly approaches a limiting position, then the straight line T which has this limiting position is called the tangent at the point A. Evidently the angle α formed by the straight line T with the x-axis, must be equal to the limiting value of the variable angle β.

The value of tan β is easily determined from the triangle ABA′ (figure 10):

For the limiting position we must have

FIG. 10.

FIG. 11.

that is, the trigonometric tangent of the angle of inclination of the tangent line is equal to the limit of the ratio of the increase in the function f(x) at the point x₀ to the corresponding increase in the independent variable, as the latter approaches zero without ever being actually equal to zero.

Let us give still another example leading to the calculation of an analogous limit. Let us suppose that a variable electric current is flowing through a conductor. Let us assume that we know the function Q = f(t) expressing the quantity of electricity that has passed through a fixed cross section of the conductor up to time t. In the period from t₀ to t₀ + h, there will flow through this cross section a quantity of electricity ΔQ equal to f(t₀ + h) − f(t₀). The average value of the current will therefore be equal to

The limit of this ratio as h → 0 will give us the value of the current at the time t₀

All the three problems discussed, in spite of the fact that they refer to different branches of science, namely mechanics, geometry, and the theory of electricity, have led to one and the same mathematical operation to be performed on a given function, namely to find the limit of the ratio of the increase of the function to the corresponding increase h of the independent variable as h → 0. The number of such widely different problems could be increased at will, and their solution would lead to the same operation. To it we are led, for example, by the question of the rate of a chemical reaction, or of the density of a nonhomogeneous mass and so forth. In view of the exceptional role played by this operation on functions, it has received a special name, differentiation, and the result of the operation is called the derivative of the function.

Thus, the derivative of the function y = f(x), or more precisely, the value of the derivative at the given point x is the limit* approached by the ratio of the increase f(x + h) − f(x) of the function to the increase h of the independent variable, as the latter approaches zero. We often write h = Δx, and f(x + Δx) − f(x) = Δy, in which case the definition of the derivative is written in the concise form:

The value of the derivative obviously depends on the point x at which it is found. Thus the derivative of a function y = f(x) is itself a function of x. It is customary to denote the derivative thus

Certain other notations are also customary for the derivative:

We should also remark that the notation looks like a fraction, although it is read as a single symbol for the derivative. In the following sections the numerator and the denominator of this “fraction” will take on independent meaning, in such a way that their ratio will coincide with the derivative so that this manner of writing is completely justified.

The results of these examples may now be formulated as follows.

The velocity of a point for which the distance s is a given function of the time s = f(t) is equal to the derivative of this function

More concisely, the velocity is the derivative of the distance with respect to time.

The trigonometric tangent of the angle of inclination of the tangent line to the curve y = f(x) at the point with abscissa x is equal to the derivative of the function f(x) at this point:

The strength of the current I at the time t, if Q = f(t) is the quantity of electricity which up to time t has passed through a cross section of the conductor, is equal to the derivative

Let us make the following remark. The velocity of a nonuniform motion at a given time is a purely physical concept, arising from practical experience. Mankind arrived at it as the result of numerous observations on different concrete motions. The study of nonuniform motion of a body on different parts of its path, the comparison of different motions of this sort taking place simultaneously, and in particular the study of the phenomena of collisions of bodies, all represented an accumulation of practical experience that led to the setting up of the physical concept of the velocity of a nonuniform motion at a given time. But the exact definition of velocity necessarily depended upon the method of defining its numerical value, and to define this value was possible only with the concept of the derivative.

In mechanics the velocity of a body moving according to the rule s = f(t) at the time t is defined as the derivative of the function f(t) for this value of t.

The discussion at the beginning of the present section has shown, on the one hand, the advantages of introducing the operation of finding the derivative, and on the other has given a reasonable justification for the above formulated definition of the velocity at any given moment.

Thus, when we raised the question of finding the velocity of a point in nonuniform motion we had, properly speaking, only an empirical notion of its value but no exact definition. But now, as a result of our analysis, we have reached an exact definition of the value of the velocity at a given moment, namely the derivative of the distance with respect to the time. This result is extremely important from a practical point of view, since our empirical knowledge of the velocity has been greatly enriched by the fact that we can now make an exact numerical calculation.

What has just been said refers equally well, of course, to the strength of a current and to many other concepts expressing the rate of some process, physical, chemical, and so forth.

This situation may serve as an example for numerous others of a similar nature, where practical experience has led to the formation of a concept relating to the external world (velocity, work, density, area, and so forth) and then mathematics has enabled us to define this concept precisely, whereupon we can make use of the concept in practical calculations.

We have already noted at the beginning of the chapter that the concept of a derivative arose chiefly as the result of many centuries of effort directed towards the solving of two problems: drawing a tangent to a curve and finding the velocity of a nonuniform motion. These problems, and also the calculation of areas discussed later, interested mathematicians in ancient times. But until the 16th century the statement and the method of solution for each problem of this sort bore an extremely special character. The accumulation of all this extensive material was reduced to a theoretically complete system in the 17th century in the work of Newton and Leibnitz. An important contribution to the foundations of presentday analysis was also made by Euler.

But it must be said that Newton and Leibnitz and their contemporaries provided very little logical basis for their great mathematical discoveries; in their methods of reasoning and in the concepts with which they operated there was much that is unclear from our point of view. Even at that time the mathematicians themselves were quite conscious of this, as is shown by the embittered discussions to be found in their correspondence with one another. However, these mathematicians of the 17th and 18th centuries carried on their purely mathematical activities in very close association with the research of other investigators, in the various branches of natural science (physics, mechanics, chemistry, technology). The statement of a mathematical problem usually arose from practical needs or from a wish to understand some phenomenon of nature, and as soon as the problem was solved, the solution was submitted in one way or another to a practical test. Consequently, in spite of a certain lack of logical basis, mathematics was able to advance in extremely useful directions.

Examples for the calculation of derivatives. The definition of the derivative as the limit

allows us to calculate the derivative of any given concrete function.

Of course, it must be admitted that cases are possible where the function at one point or another or even at many points simply does not have a derivative; in other words, the ratio

FIG. 12.

as h → 0 does not approach a finite limit. This case obviously occurs at every point of discontinuity of the function f(x), since here the ratio

has a numerator which does not approach zero while the denominator decreases without bound. The derivative may also fail to exist at a point where the function is continuous. A simple example is given by any point where the graph of the function forms an angle (figure 12). At such a point the curve of the graph has no definite tangent, and consequently the function has no derivative. Often at such points the expression (10) approaches different values, depending on whether h approaches zero from the right or from the left, so that if h approaches zero in an arbitrary manner, the ratio (10) simply has no limit. An example of a more complicated function without a derivative is given by

The graph of this function is drawn in figure 13. At the point x = 0 it has no derivative because, as is evident from the graph, the secant OA does not approach any definite position even when A → 0 from one side. In fact, the secant OA oscillates endlessly back and forth between the straight line OM and the straight line OL. The corresponding ratio (10) in this case has no limit, even if h preserves the same sign as it approaches zero.

FIG. 13.

Let us remark finally that it is possible to define, in a purely analytic way by means of a formula, a continuous function which does not have a derivative at any point. An example of such a function was first given by the outstanding German mathematician of the last century, Weierstrass.

Consequently the class of differentiable functions is considerably narrower than that of continuous functions.

Let us pass now to the actual calculation of the derivatives of the simplest functions.

1. y = c, where c is a constant. A constant may be considered as a special case of a function that remains equal to the same number for arbitrary x. Its graph is a straight line parallel to the x-axis at a distance equal to c. This straight line forms with the x-axis an angle α = 0, and obviously the derivative of a constant is identically equal to zero: y′ = (c)′ = 0. From the point of view of mechanics, this equation means that the velocity of a fixed point is equal to zero.

2. y = x²

As h → 0 we obtain* in the limit 2x; consequently

3. y = xn (n a positive integer).

Every addend on the right side, beginning with the second, approaches zero as h → 0; consequently

This formula remains true for arbitrary n positive or negative, fractional or even irrational, although the proof must then be different. We will make use of this fact without proving it. Thus for example

4. y = sin x.

As explained earlier, the first fraction approaches unity as h → 0, and cos (x + h/2) obviously approaches cos x. Thus the derivative of the sine is equal to the cosine

We suggest to the reader that by the same sort of argument he prove that

5. Earlier (Chapter II, §3) we have already noted the existence of the limit

We also remarked that for the calculation of this limit no essential role is played by the fact that n took on only positive integral values. It is important only that the infinitesimal 1/n, which is being added to unity, and the exponent n, which is increasing beyond all bounds, should be reciprocal to each other.

Making use of this assertion, we can easily find the derivative of the logarithm y = loga x

The continuity of the logarithm allows us to replace the quantity under the log sign by its limit, which is equal to e; thus

(in this case the role of n → ∞ is played by the increasing quantity x/h). As a result, we obtain the rule for differentiating a logarithm

This rule becomes particularly simple if as the base of our logarithms we choose the number e. Logarithms taken to this base are called natural logarithms and are denoted by In x. We may write

or again

§6.  Rules for Differentiation

From the examples given earlier it may appear that the calculation of the derivative of every new function demands the invention of new methods. This is not the case. The development of analysis was made possible to no small extent by the discovery of a simple unified method for finding the derivative of an arbitrary “elementary” function (that is, a function which may be expressed by a formula consisting of a finite combination of the fundamental algebraic operations, the trigonometric functions, the operation of raising to a power, and the taking of logarithms). At the basis of this method are the so-called rules of differentiation. They consist of a number of theorems that allow us to reduce more complicated problems to simpler ones.

We will explain here the rules of differentiation and will try to be very brief in deducing them. If the reader wishes to form merely a general idea of analysis, he may omit the present section, remembering only that there exists a means of actually finding the derivative of any elementary function. In this case it will be necessary, of course, for him to take on faith some of the calculations in our later examples.

Derivative of a sum. Assume that y is given as a function of x by the expression

where u = ϕ(x) and v = ψ(x) are known functions of x. We assume moreover that we can find the derivatives of the functions u and v. How then are we to find the derivative of the function y? The answer is simple

In fact, let us give x an increment Δx; then u, v, and y will each receive an increment Δu, Δv, and Δy, connected by the equation

Thus*

and after the passage to the limit for Δx → 0 we at once get formula (11), if, of course, the functions u and v have derivatives.

Analogously we may derive the formula for differentiating the difference of two functions

Derivative of a product. The rule for the differentiation of a product is somewhat more complicated. The derivative of the product of two functions, each of which has a derivative, exists, and is equal to the sum of the product of the first function by the derivative of the second and the product of the second by the derivative of the first; that is

In fact, let us give x an increment Δx. Then the functions u, v and y = uv will receive the increments ϕu, Δv, Δy, satisfying the relation

from which

After passage to the limit for Δx → 0 the first two summands on the right side produce the right side of formula (13) while the third summand vanishes.† Consequently, in the limit we obtain the rule (13).

In the particular case v = c = constant, we have

since the derivative of a constant is equal to zero.

Derivative of a quotient. Let y = u/v, where u and v have a derivative for a given x, with v ≡ 0 for that value of x. Obviously

from which

Here we have again made use of the fact that for a function v which has a derivative we necessarily have Δv → 0, when Δx → 0. Thus

Let us give some examples of the application of these rules

We recommend to the reader to prove for himself the formula

Derivative of the inverse function. Let us consider a function y = f(x), which is continuous and increasing (decreasing) on the interval [a, b]. By increasing (decreasing) we mean that to a greater value of x in the interval [a, b] corresponds a greater (smaller) value of y (figure 14).

Let c = f(a) and d = f(b). In figure 14 it is evident that for each value of y from the interval [c, d] (or [d, c], respectively) there corresponds exactly one value of x from the interval [a, b] such that y = f(x). Thus on the interval [c, d] (or [d, c]) we have a completely determined function x = ϕ(y), which is called the inverse function of y = f(x). In figure 14 it is clear that the function ϕ(y) is continuous, a fact which is proved in modern analysis by strictly analytical methods. Now let Δx and Δy correspond respectively to the increments in x and y. It is evident that

FIG. 14.

In the limit this gives us a simple relation between derivatives of the direct and inverse functions

Let us make use of this relation to find the derivative of the function y = ax. The inverse function is x = logay, which we are already able to differentiate, and so we may write

In particular (ex)′ = ex.

As another example let us take y = arc sin x. The inverse function is x = sin y. Thus

Table of derivatives. Let us tabulate the derivatives of the simplest elementary functions (Table 2).

Table 2

These formulas have been calculated and explained earlier, with the exception of the last two which the reader may, if he wishes, easily derive for himself by using the rule for differentiation of an inverse function.

Calculation of the derivative of a function of a function. It remains to consider the last and most difficult rule for differentiation. The reader in possession of this rule and of a set of tables may with perfect right consider that he is able to differentiate any elementary function.

In order to apply the rule we are about to give, it is necessary to be completely clear about how the function we wish to differentiate is constructed; that is, which operations must we perform on the independent variable x, and in which order, to produce the value of the dependent variable y.

For example, to calculate the function

it is necessary first of all to raise x to the second power and then to take the sine of the magnitude so obtained, a procedure which may be described in the following way: y = sin u, where u = x².

On the other hand, in order to calculate the function

it is necessary first of all to find the sine of x, and then to raise the value so found to the second power, a procedure which may be written thus: y = u², where u = sin x.

Here are some examples:

1. y = (3x + 4)³, y = u³, u = 3x + 4.

2. , , u = 1 − x².

3. y = ekx; y = eu, u = kx.

In more complicated cases we have a chain of simple relations, which may have several links. For example,

4. y = cos³ x²; y = u³; v = cos v = x².

If y is a function of the variable u

and u in its turn is a function of the variable x

then y, being a function of u, is also a certain function of x, which may be denoted as follows

By considering more complicated cases we may form, for example, the function

which is equivalent to the equations

and we could form still longer chains.

We now show how to calculate the derivative of the function F(x) defined by equation (20) if we know the derivative of f(u) with respect to u and the derivative of ϕ(x) with respect to x.

Let us give to x the increment Δx; then by (19) u will receive a certain increment Δu and by (18) y will receive an increment Δy. Thus we may write

Now let Δx approach zero. Then . Furthermore, from the continuity of u, the increase Δu → 0, and therefore (the existence of the derivatives and was assumed).

Thus we have proved the important formula for the derivative of a function of a function*

Let us calculate, from formula (21) and the fundamental table of derivatives given, the derivatives of the functions we have been considering:

1. y = (3x + 4)³ = u³, = 3u² · 3 = 9(3x + 4)².

2.

3. y = ekx = eu, .

If y = f(u), u = ϕ(v), v = f(x), then

It is clear how to generalize this formula for the case of an arbitrary (finite) number of functions in the chain. For example,

4.

In our explanation of how to calculate the derivative of a function of a function, we have introduced intermediate variables u, v, . . . But in fact, after a little practice one may dispense with them, simply keeping in mind the functions they denote.

The elementary functions. To close the present section let us remark that the functions whose derivatives were listed in tabular form (Table 2) may be used to define the so-called elementary functions. These elementary functions are defined as those functions that may be obtained from the preceding simple functions by the four arithmetical operations and the operation of taking a function of a function, each of these operations being performed a finite number of times.

For example, the polynomial x² − 2x² + 3x − 5 is an elementary function since it is obtained by arithmetic operations from a number of functions of the form xk. The function ln is also elementary, since it is obtained from the polynomial u = 1 − x² by the operation v = u1/2, and subsequently the operation ln v.

The rules for differentiation discussed earlier are sufficient to obtain the derivative of any elementary function, as soon as we know the derivatives of the simplest elementary functions.

§7.  Maximum and Minimum; Investigation of the Graphs of Functions

One of the simplest and most important applications of the derivative is in the theory of maxima and minima. Let us suppose that on a certain interval we are given a function y = f(x) which is not only continuous but also has a derivative at every point. Our ability to calculate the derivative enables us to form a clear picture of the graph of the function. On an interval on which the derivative is always positive the tangent to the graph will be directed upward. On such an interval the function will increase; that is, to a greater value of x will correspond a greater value of f(x). On the other hand, on an interval where the derivative is always negative, the function will decrease; the graph will run downward.

FIG. 15.

Maximum and minimum. In figure 15 we have drawn the graph of a function y = f(x) defined on the interval (a, b). Of a special interest are the points of this graph whose abcissas are x₀, x₁, x₃.

At the point x₀ the function f(x) is said to have a local maximum; by this we mean that at this point f(x) is greater than at neighboring points; more precisely for every x in a certain interval around the point x₀.

A local minimum is defined analogously. For our function a local maximum occurs at the points x₀ and x₃, and a local minimum at the point x₁.

At every maximum or minimum point, if it is inside the interval [a, b], i.e., if it does not coincide with one of the end points a or b, the derivative must be equal to zero.

This last statement, a very important one, follows immediately from the definition of the derivative as the limit of the ratio Δy/Δx. In fact, if we move a short distance from the maximum point, then . Thus for positive Δx the ratio Δy/Δx is nonpositive, and for negative Δx the ratio Δy/Δx is nonnegative. The limit of this ratio, which exists by hypothesis, can therefore be neither positive nor negative and there remains only the possibility that it is zero. By inspection of the diagram it is seen that this means that at maximum or minimum points (it is customary to leave out the word “local,” although it is understood) the tangent to the graph is horizontal. In figure 15 we should remark that at the points x₂, and x₄ also the tangent is horizontal, just as it is at the points x₀, x₁, x₃, although at these points the function has neither maximum nor minimum. In general, there may be more points at which the derivative of the function is equal to zero (stationary points) than there are maximum or minimum points.

Determination of the greatest and least values of a function. In numerous technical questions it is necessary to find the point x at which a given function f(x) attains its greatest or its least value on a given interval.

In case we are interested in the greatest value, we must find x₀ on the interval [a, b] for which among all x on [a, b] the inequality is fulfilled.

But now the fundamental question arises, whether in general there exists such a point. By the methods of modern analysis it is possible to prove the following existence theorem: If the function f(x) is continuous on a finite interval, then there exists at least one point on the interval for which the function attains its maximum (minimum) value on the interval [a, b].

From what has been said already, it follows that these maximum or minimum points must be sought among the “stationary” points. This fact is the basis for the following well-known method for finding maxima and minima.

First we find the derivative of, f(x) and then solve the equation obtained by setting it equal to zero

If x₁, x₂, ···, xn, are the roots of this equation, we then compare the numbers f(x₁, f(x₂), ···, f(xn) with one another. Of course, it is necessary to take into account that the maximum or minimum of the function may be found not within the interval but at the end (as is the case with the minimum in figure 15) or at a point where the function has no derivative (as in figure 12). Thus to the points x₁, x₂, ···, xn, we must add the ends a and b of the interval and also those points, if they exist, at which there is no derivative. It only remains to compare the values of the function at all these points and to choose among them the greatest or the least.

With respect to the stated existence theorem, it is important to add that this theorem ceases, in general, to hold in the case that the function f(x) is continuous only on the interval (a, b); that is, on the set of points x satisfying the inequalities a < x < b. We leave it to the reader to consider the fact that the function 1/x has neither a maximum nor a minimum on the interval (0, 1).

Let us look at some examples.

From a square piece of tin of side a it is required to make a rectangular open box of maximum volume. If from the corners of the original square we take away squares of side x (see §2, example 2) we get a box with the volume

Our problem then becomes to find the value of x for which the function V(x) attains its greatest value on the interval . In accordance with the rule, we find the derivative and set it equal to zero

Solving this equation, we find the two roots

To these we adjoin the left end of the interval (the right end is identical with x₁) and compare the values of the function at these points

Thus the box will have the greatest volume, equal to 2/27 a³, for the height x = a/6.

As a second example, let us examine the problem of the lamp at the skating rink (see §2, example 3). At what height h should we place the lamp in order that the edge of the rink may receive the greatest illumination?

For formula (3) §2, our problem reduces to determining the value of h for T = A sin α/h² + r² takes on its greatest value. Instead of h it is more convenient here to find the angle α (figure 3, Chapter I). We have

so that

Then it is required to find the maximum of the function T(α) among those values of α which satisfy the inequality 0 < α < π2. To do this, we find the derivative and set it equal to zero

This equation splits into two

The first equation has the root α = π2, which coincides with the end of the interval (0, π/2). The second equation may be put in the form

But since 0 < α < π/2, we have the result α ≈ 35°15′. So this is the value for which the function T(α) attains its maximum (at the ends of the interval, T = 0). The desired height h is thus equal to

For best illumination of the edge of the rink the lamp should be placed at a height equal to about 0.7 times the radius.

But now let us suppose that the facilities at our disposal do not allow us to raise the lamp to a height greater than a certain H. Then the angle a may vary not from 0 to π/2 but only within the narrower limits arc tan (H/r). For example, let r = 12 meters and H = 9 meters. In this case, it is in fact possible to raise the lamp to the height , which amounts to somewhat more than 8 meters, so that this is what we ought to do. But if H is less than 8 meters (for example, if we have at our disposal only a pole of length 6 meters), then it turns out that the derivative of the function T(α) in the interval [O, arc tan (H/r)] is nowhere equal to zero. In this case the maximum is attained at the end of the interval, and the lamp should be raised to the greatest possible height H = 6 meters.

Up to now we have considered a function on a finite interval. If the interval is infinite in length, then even a continuous function may fail to attain its greatest or least value but may, for example, continue to grow or to decrease as x approaches infinity.

Thus the functions y = kx + b (see figure 5, Chapter I), y = arc tan x (figure 16a), y = In x (figure 16b) nowhere attain either a maximum or a minimum. The function y = e−x² (figure 16c) attains its maximum at the point x = 1, but nowhere attains a minimum. As for the function y = x/(1 + x²) (figure 16d), it reaches its minimum at the point x = − 1 and its maximum at the point x = 1.

FIG. 16a.

FIG. 16b.

FIG. 16c.

FIG. 16d.

In the case of an interval of infinite length the investigation may be reduced to the ordinary rules. It is only necessary to consider in place of f(a) and f(b) the limits

Derivatives of higher orders. We have just seen how, for closer study of the graph of a function, we must examine the changes in its derivative f′(x). This derivative is a function of x, so that we may in turn find its derivative.

The derivative of the derivative is called the second derivative and is denoted by

Analogously, we may calculate the third derivative

and more generally the nth derivative or, as it is also called, the derivative of nth order

Of course, it must be kept in mind that, for a certain value of x (or even for all values of x) this sequence may break off at the derivative of some order, say the kth; it may happen that f(k)(x) exists but not f(k + 1)(x). Derivatives of arbitrary order will appear later in §9 in connection with the Taylor formula. For the moment we confine ourselves to the second derivative.

Significance of the second derivative; convexity and concavity. The second derivative has a simple significance in mechanics. Let s = f(t) be a law of motion along a straight line; then s′ is the velocity and s″ is the “velocity of the change in the velocity” or more simply the “acceleration” of the point at time t. For example, for a falling body under the force of gravity

that is, the acceleration of falling bodies is constant.

The second derivative also has a simple geometric meaning. Just as the sign of the first derivative determines whether the function is increasing or decreasing, so the sign of the second derivative determines the side toward which the graph of the function will be curved.

Suppose, for example, that on a given interval the second derivative is everywhere positive. Then the first derivative increases and therefore f′(x) = tan α increases and the angle a of inclination of the tangent line itself increases (figure 17). Thus as we move along the curve it keeps turning constantly to the same side, namely upward, and is thus, as they say, “convex downward.”

FIG. 17.

On the other hand, in a part of a curve where the second derivative is constantly negative (figure 18) the graph of the function is “convex upward.”*

FIG. 18.

Criteria for maxima and minima; study of the graphs of curves. If throughout the whole interval over which x varies the curve is convex upward and if at a certain point x₀ of this interval the derivative is equal to zero, then at this point the function necessarily attains its maximum; and its minimum in the case of convexity downward. This simple consideration often allows us, after finding a point at which the derivative is equal to zero, to decide thereupon whether at this point the function has a local maximum or minimum.*

Example 1. Let us study the appearance of the graph of the function

We take its first derivative and set it equal to zero,

The roots of the equation obtained in this way are x₁ = 2, x₂ = 3. The corresponding values of the function are

We then mark these two points on the diagram. Along with these we may also mark the point with coordinates x = 0 and y = f(0) = −2 where the graph intersects the y-axis. The second derivative is f″(x) = 2x − 5. This reduces to zero for , so that

The point

is a point of inflection of the graph. To the left of this point the curve is convex upward, and to the right it is convex downward.

It is now evident that the point x = 2 is a maximum point and the point x = 3 is a minimum point for the function.

FIG. 19.

On the basis of these results we conclude that the graph of the function y = f(x) has the appearance sketched in figure 19. To the right of the point (0, −2) the curve rises with increasing x, is convex upwards, and attains its maximum at the point , after which it begins to fall. At the point , where f″(x) = 0, the convexity changes to concavity. Then at the point the function attains its minimum and from there on rises to infinity. The final statement comes from the fact that the first term of the function, the one containing the highest (third) power of x, approaches infinity faster than the second and third terms. For the same reason the graph of the function approaches − ∞ as x assumes numerically larger negative values.

Example 2. We shall prove the inequality for arbitrary x. For this purpose we consider the function f(x) = ex − x − 1. Its first derivative is f′(x) = ex − 1, which reduces to zero only for x = 0. The second derivative f″(x) = ex > 0 for all x. Consequently the graph of the function f(x) is convex downward. The number f(0) = 0 is a minimum for the function and for all x.

The study of graphs has many different purposes. They often show very clearly, for example, the number of real roots of a given equation. Thus, in order to demonstrate that the equation

has a single real root, we may study the graphs of the functions y = ex and y = 2/x (as sketched in figure 20). It is easy to see that these graphs intersect at only one point, so that the equation ex = 2/x has exactly one root.

The methods of analysis are extensively applied to questions of approximate calculation of the roots of an equation. On this subject, see Chapter IV, §5.

§8.  Increment and Differential of a Function

The differential of a function. Let us consider a function y = f(x) that has a derivative. The increment of this function

corresponding to the increment Δx, has the property that the ratio Δy/Δx, as Δx → 0, approaches a finite limit, equal to the derivative

This fact may be written as an equality

FIG. 20.

where the value of a depends on Δx in such a way that as Δx → 0, α also approaches zero. Thus the increment of a function may be represented in the form

where α → 0, if Δx → 0.

The first summand on the right side of this equality depends on Δx in a very simple way, namely it is proportional to Δx. It is called the diflerentinl of the function, at the point xn corresponding to the given increment Δx, and is denoted by

The second summand has the characteristic property that, as Δx → 0, it approaches zero more rapidly than Δx, as a result of the presence of the factor α. It is therefore said to be an infinitesimal of higher order than Δx and, in case f′;(x) ≠ 0, it is also of higher order than the first summand. By this we mean that for sufficiently small Δx the second summand is small in itself and its ratio to Δx is also arbitrarily small.

FIG. 21.

The decomposition of Δy into two summands, of which the first (the principal part) depends linearly on Δx and the second is negligible for small Δx, may be illustrated by figure 21. The segment BC = Δy, where BC = BD + DC, BD = tan β · Δx = f′(x) Δx = dy, and DC is an infinitesimal of higher order than Δx.

In practical problems the differential is often used as an approximate value for the increment in the function. For example, suppose we have the problem of determining the volume of the walls of a closed cubical box whose interior dimensions are 10 × 10 × 10 cm and the thickness of whose walls is 0.05 cm. If great accuracy is not required, we may argue as follows. The volume of all the walls of the box represents the increment Δy of the function y = x³ for x = 10 and Δx = 0.1. So we find approximately

For symmetry in the notation it is customary to denote the increment of the independent variable by dx and to call it also a differential. With this notation the differential of the function may be written thus:

Then the derivative is the ratio, f′(x) = dy/dx of the differential of the function to the differential of the independent variable.

The differential of a function originated historically in the concept of an “indivisible.” This concept, which from a modern point of view was never very clearly defined, was in its time, in the 18th century, a fundamental one in mathematical analysis. The ideas concerning it have undergone essential changes in the course of several centuries. The indivisible, and later the differential of a function, were represented as actual infinitesimals, as something in the nature of an extremely small constant magnitude, which however was not zero. The definition given in this section is the one accepted in present-day analysis. According to this definition the differential is a finite magnitude for each increment Δx and is at the same time proportional to Δx. The other fundamental property of the differential, the character of its difference from Δy, may be recognized only in motion, so to speak: if we consider an increment Δx which is approaching zero (which is infinitesimal), then the difference between dy and Δy will be arbitrarily small even in comparison with Δx.

This substitution of the differential in place of small increments of the function forms the basis of most of the applications of infinitesimal analysis to the study of nature. The reader will see this in a particularly clear way in the case of differential equations, dealt with in this book in Chapters V and VI.

Thus, in order to determine the function that represents a given physical process, we try first of all to set up an equation that connects this function in some definite way with its derivatives of various orders. The method of obtaining such an equation, which is called a differential equation, often amounts to replacing increments of the desired functions by their corresponding differentials.

As an example let us solve the following problem. In a rectangular system of coordinates Oxyz, we consider the surface obtained by rotation of the parabola whose equation (in the Oyz plane) is z = y². This surface is called a paraboloid of revolution (figure 22). Let v denote the volume of the body bounded by the paraboloid and the plane parallel to the Oxy plane at a distance z from it. It is evident that v is a function of z (z >0).

FIG. 22.

To determine the function v, we attempt to find its differential dv. The increment Δv of the function v at the point z is equal to the volume bounded by the paraboloid and by two planes parallel to the Oxy plane at distances z and z + Δz from it.

It is easy to see that the magnitude of Δv is greater than the volume of the circular cylinder of radius and height Δz but less than that of the circular cylinder with radius and height Δz.

Thus

and so

where θ is some number depending on Δz and satisfying the inequality 0 < θ < 1.

So we have succeeded in representing the increment Δv in the form of a sum, the first summand of which is proportional to Δz, while the second is an infinitesimal of higher order than Δz (as Δz → 0). It follows that the first summand is the differential of the function v

or

since Δz = dz for the independent variable z.

The equation so obtained relates the differentials dv and dz (of the variables v and z) to each other and thus is called a differential equation.

If we take into account that

where v′ is the derivative of v with respect to the variable z, our differential equation may also be written in the form

To solve this very simple differential equation we must find a function of z whose derivative is equal to πz. Problems of this sort are treated in a general way in §§10 and 11, but for the moment we urge the reader to verify that a solution of our equation is given by v = πz²/2 + C, where for C we may choose an arbitrary number.* In our case the volume of the body is obviously zero for z = 0 (see figure 22), so that C = 0. Thus our function is given by v = πz²/2.

The mean value theorem and examples of its application. The differential expresses the approximate value of the increment of the function in terms of the increment of the independent variable and of the derivative at the initial point. So for the increment from x = a to x = b, we have

It is possible to obtain an exact equation of this sort if we replace the derivative f′(a) at the initial point by the derivative at some intermediate point, suitably chosen in the interval (a, b). More precisely: If y = f(x) is a function which is differentiable on the interval , then there exists a point ξ, strictly within this interval, such that the following exact equality holds

The geometric interpretation of this “mean-value theorem” (also called Lagrange’s formula or the finitedifference formula) is extraordinarily simple. Let A, B be the points on the graph of the function f(x) which correspond to x = a and x = b, and let us join A and B by the chord AB (figure 23). Now let us move the straight line AB, keeping it constantly parallel to itself, up or down. At the moment when this straight line cuts the graph for the last time, it will be tangent to the graph at a certain point C. At this point (let the corresponding abscissa be x = ξ), the tangent line will form the same angle of inclination α as the chord AB. But for the chord we have

FIG. 23.

On the other hand at the point C

This equation

is exactly the mean-value theorem. *

Formula (22) has the peculiar feature that the point ξ appearing in it is unknown to us; we know only that it lies “somewhere in the interval (a, b).” But in spite of this indeterminacy, the formula has great theoretical significance and is part of the proof of many theorems in analysis. The immediate practical importance of this formula is also very great, since it enables us to estimate the increase in a function when we know the limits between which its derivative can vary. For example,

Here a, b and ξ are angles, expressed in radian measure; ξ is some value between a and b; ξ itself is unknown, but we know that .

From formula (22) it is clear that a function whose derivative is everywhere equal to zero must be a constant; at no part of the interval can it receive an increment different from zero. Analogously, the reader will easily prove that a function whose derivative is everywhere positive must everywhere increase, and if its derivative is negative, the function must decrease. We give here without proof one of the many generalizations of the mean-value theorem.

For arbitrary functions ϕ(x) and ψ(x) differentiable in the interval [a, b], provided only that ψ′(x) ≠ 0 in (a, b), the following equation† holds

where ξ is some point in the interval (a, b).‡

From this theorem we can derive a general method for calculating the limits of an expression like

if ϕ(0) = ψ(0) = 0. From formula(23) we have

where ξ is between 0 and x, and therefore ξ → 0 together with x. This allows us to calculate the limit

instead of the limit (24), which is in many cases very much easier.*

Example. Let us find the . By making use of the rule three times, we have successively

§9.  Taylor’s Formula

The function

where the coefficients ak are constants, is called a polynomial of degree n. In particular, y = ax + b is a polynomial of the first degree and y = ax² + bx + c is a polynomial of the second degree. Polynomials may be considered as the simplest of all functions. In order to calculate their value for a given x, we require only the operations of addition, subtraction, and multiplication; not even division is needed. Polynomials are continuous for all x and have derivatives of arbitrary order. Also, the derivative of a polynomial is again a polynomial, of degree lower by one, and the derivatives of order n + 1 and higher of a polynomial of degree n are equal to zero.

If to the polynomials we adjoin functions of the form

for the calculation of which we also need division, and also the functions and and, finally, arithmetical combinations of these functions, we obtain essentially all the functions whose values can be calculated by methods learned in the secondary school.

While we were still in school, we formed some notion of a number of other functions, like

But though we became acquainted with the most important properties of these functions, we found no answer in elementary mathematics to the question: How can we calculate them? What sort of operations, for example, is it necessary to perform on x in order to obtain log x or sin x? The answer to this question is given by methods that have been worked out in analysis. Let us examine one of these methods.

Taylor’s formula. On an interval containing the point a, let there be given a function f(x) with derivatives of every order. The polynomial of first degree

has the same value as f(x) at the point x = a and also, as is easily verified, has the same derivative as f(x) as this point. Its graph is a straight line, which is tangent to the graph of f(x) to the point a. It is possible to choose a polynomial of the second degree, namely

which at the point of x = a has with f(x) a common value and a common first and second derivative. Its graph at the point a will follow that of f(x) even more closely. It is natural to expect that if we construct a polynomial which at x = a has the same first n derivatives as f(x) at the same point, then this polynomial will be a still better approximation to f(x) at points x near a. Thus we obtain the following approximate equality, which is Taylor’s formula

The right side of this formula is a polynomial of degree n in (x − a). For each x the value of this polynomial can be calculated if we know the values of f(a), f′(a), ···, f(n)(a).

For functions which have an (n + 1)th derivative, the right side of this formula, as is easy to show, differs from the left side by a small quantity which approaches zero more rapidly than (x − a)n. Moreover, it is the only possible polynomial of degree n that differs from f(x), for x close to a, by a quantity that approaches zero, as x → a, more rapidly than (x − a)n. If f(x) itself is an algebraic polynomial of degree n, then the approximate equality (25) becomes an exact one.

Finally, and this is particularly important, we can give a simple expression for the difference between the right side of formula (25) and the actual value of f(x). To make the approximate equality (25) exact, we must add to the right side a further term, called the “remainder term”

This final supplementary term*

has the peculiarity that the derivative appearing in it is to be calculated in each case not at the point a but at a suitably chosen point ξ, which is unknown but lies somewhere in the interval between a and x.

The proof of equality (26) is rather cumbersome but quite simple in essence. We shall give here a somewhat artificial version of the proof, which has the merit of being concise.

In order to find out by how much the left side in the approximate formula (25) differs from the right, let us consider the ratio of the difference between the two sides in equality (25) to the quantity −(x −a)n + 1

We also introduce the function

of a variable u, taking x to be fixed (constant). Then the numerator in (27) will represent the increase of this function as we pass from u = a to u = x, and the denominator will be the increase over the same interval of the function

We now make use of the generalized mean-value theorem quoted earlier

Differentiating the functions ϕ(u) and ψ(u) with respect to u (it must be recalled that the value of x has been fixed) we find that

The equality of this last expression with the original quantity (27) gives Taylor’s formula in the form (26).

In the form (26) Taylor’s formula not only provides a means of approximate calculation of f(x) but also allows us to estimate the error. Let us consider the simple example

The values of the function sin x and of its derivatives of arbitrary order are known for x = 0. Let us make use of these values to write Taylor’s formula for sin x, choosing a = 0 and limiting ourselves to the case n = 4. We find successively

Therefore

Although the exact value R₅ is unknown, still we can easily estimate it from the fact that . For all values of x between 0 and π/4 we have

Consequently, on the interval [0, π/4] the function sin x may be considered, with accuracy up to as equal to the polynomial of third degree

If we were to take more terms in Taylor’s expansion for sin x, we would obtain a polynomial of higher degree which would approximate sin x still more closely.

The tables for trigonometric and other functions are calculated by similar methods.

The laws of nature, as a rule, can be expressed with good approximation by functions that may be differentiated as often as we like and that in their turn may be approximated by polynomials, the degree of the polynomial being determined by the accuracy desired.

Taylor’s series. If in formula (25) we take a larger and larger number of terms, then the difference between the right side and f(x), expressed by the remainder term Rn + 1(x), may tend to zero. Of course this will not always occur: neither for all functions nor for all values of x. But there exists a broad class of functions (the so-called analytic functions) for which the remainder term Rn + 1(x) does in fact approach zero as n + ∞, at least for all values of x within a certain interval around the point a. For these functions the Taylor formula allows us to calculate f(x) with any desired degree of accuracy. Let us examine such functions more closely.

If Rn + 1(x) + 0 as n → ∞, then from (26) it follows that

In this case we say that f(x) has been expanded in a convergent infinite series

in increasing powers of (x − a). This series is called a Taylor series, and f(x) is said to be the sum of the series. Let us consider some examples

1.

2.

3. .

4. .

5. .

The first of these examples is the famous binomial theorem of Newton, which was obtpined by Newton for all n but completely proved in his time only for integral n. This example served as a model for the establishment of the general Taylor formula. The last two formulas allow us, for x = 1, to calculate with arbitrarily good approximation the numbers e and π.

The Taylor formula, which opens up the way for most of the calculations in applied analysis, is extremely important from the practical point of view.

Many of the laws of nature, physical and chemical processes, the motion of bodies, and the like, are expressed with great accuracy by functions which may be expanded in a Taylor series. The theory of such functions can be formulated in a clearer and more complete way if we consider them as functions of a complex variable (see Chapter IX).

The idea of approximating a function by polynomials or of representing it as the sum of an infinite number of simpler functions underwent farreaching developments in analysis, where it now forms an independent branch, the theory of approximation of functions (see Chapter XII).

§10.  Integral

From Chapter I and from §1 of the present chapter the reader already knows that the concept of the integral, and more generally of the integral calculus, had its historical origin in the need for solving concrete problems, a characteristic example of which is the calculation of the area of a curvilinear figure. The present section is devoted to these questions. In it we will also discuss the aforementioned connection between the problems of the differential and the integral calculus, which was not fully cleared up until the 18th century.

Area. Let us suppose that a curve above the x-axis forms the graph of the function y = f(x). We attempt to find the area S of the segment bounded by the line y = f(x), by the x-axis and by the straight lines drawn through the points x = a and x = b parallel to the y-axis.

To solve this problem we proceed as follows. We divide the interval [a, b] into n parts, not necessarily equal. We denote the length of the first part by Δx₁, of the second by Δx₂, and so forth up to the final part Δxn. In each segment we choose points ξ₁, ξ₂, ···, ξn and set up the sum

The magnitude Sn is obviously equal to the sum of the areas of the rectangles shaded in figure 24.

FIG. 24.

The finer we make the subdivision of the segment [a, b], the closer Sn will be to the area S. If we carry out a sequence of such constructions, dividing the interval [a, b] into successively smaller and smaller parts, then the sums Sn will approach S.

The possibility of dividing [a, b] into unequal parts makes it necessary for us to define what we mean by “successively smaller” subdivisions. We assume not only that n increases beyond all bounds but also that the length of the greatest Δxi in the nth subdivision approaches zero. Thus

The calculation of the desired area has in this way been reduced to finding the limit (29).

FIG. 25.

We note that when we first set up the problem, we had only an empirical idea of what we mean by the area of our curvilinear figure, but we had no precise definition. But now we have obtained an exact definition of the concept of area: It is the limit (29). We now have not only an intuitive notion of area but also a mathematical definition, on the basis of which we can calculate the area numerically (compare the remarks at the end of §3, concerning velocity and the length of a circumference).

We have assumed that . If f(x) changes sign, then in figure 25, the limit (29) will give us the algebraic sum of the areas of the segments lying between the curve y = f(x) and the x-axis, where the segments above the x-axis are taken with a plus sign and those below with a minus sign.

Definite integral. The need to calculate the limit (29) arises in many other problems. For example, suppose that a point is moving along a straight line with variable velocity v = f(t). How are we to determine the distance s covered by the point in the time from t = a to t = b?

Let us assume that the function f(t) is continuous; that is, in small intervals of time the velocity changes only slightly. We divide the interval [a, b] into n parts, of length Δt₁, Δt₂, ···, Δtn. To calculate an approximate value for the distance covered in each interval Δti, we will suppose that the velocity in this period of time is constant, equal throughout to its actual value at some intermediate point ξ₁. The whole distance covered will then be expressed approximately by the sum

and the exact value of the distance s covered in the time from a to b, will be the limit of such sums for finer and finer subdivisions; that is, it will be the limit (29)

It would be easy to give many examples of practical problems leading to the calculation of such a limit. We will discuss some of them later, but for the moment the examples already given will sufficiently indicate the importance of this idea. The limit (29) is called the definite integral of the function f(x) taken over the interval [a, b], and it is denoted by

The expression f(x)dx is called the integrand, a and b are the limits of integration; a is the lower limit, b is the upper limit.

The connection between differential and integral calculus. As an example of the direct calculation of a definite integral, we may take example 2, §1. We may now say that the problem considered there reduces to calculation of the definite integral

Another example was considered in §3, where we solved the problem of finding the area bounded by the parabola y = x². Here the problem reduces to calculation of the integral

We were able to calculate both these integrals directly, because we have simple formulas for the sum of the first n natural numbers and for the sum of their squares. But for an arbitrary function f(x), we are far from being able to add up the sum (28) (that is, to express the result in a simple formula) if the points ξi, and the increments Δxi are given to suit some particular problem. Moreover, even when such a summation is possible, there is no general method for carrying it out; various methods, each of a quite special character, must be used in the various cases.

So we are confronted by the problem of finding a general method for the calculation of defiqite integrals. Historically this question interested mathematicians for a long period of time, since there were many practical aspects involved in a general method for finding the area of curvilinear figures, the volume of bodies bounded by a curved surface, and so forth.

We have already noted that Archimedes was able to calculate the area of a segment and of certain other figures. The number of special problems that could be solved, involving areas, volumes, centers of gravity of solids, and so forth, gradually increased, but progress in finding a general method was at first extremely slow. The general method could not be discovered until sufficient theoretical and computational material had been accumulated through the demands of practical life. The work of gathering and generalizing this material proceeded very gradually until the end of the Middle Ages; and its subsequent energetic development was a direct consequence of the rapid growth in the productive powers of Europe resulting from the breakup of the former (feudal) methods of manufacturing and the creation of new ones (capitalistic).

The accumulation of facts connected with definite integrals proceeded alongside of the corresponding investigations of problems related to the derivative of a function. The reader already knows from §1 that this immense preparatory labor was crowned with success in the 17th century by the work of Newton and Leibnitz. It is in this sense that Newton and Leibnitz are the creators of the differential and integral calculus.

One of the fundamental contributions of Newton and Leibnitz consists of the fact that they finally cleared up the profound connection between differential and integral calculus, which provides us, in particular, with a general method of calculating definite integrals for an extremely wide class of functions.

To explain this connection, we turn tq an example from mechanics.

We suppose that a material point is moving along a straight line with velocity v = f(t), where t is the time. We already know that the distance a covered by our point in the time between t = t₁ and t = t₂ is given by the definite integral

Now let us assume that the law of motion of the point is known to us; that is, we know the function s = F(t) expressing the dependence on the time t of the distance s calculated from some initial point A on the straight line. The distance σ covered in the interval of time [t₁, t₂] is obviously equal to the difference

In this way we are led by physical considerations to the equality

which expresses the connection between the law of motion of our point and its velocity.

From a mathematical point of view the function F(t), as we already know from §5, may be defined as a function whose derivative for all values oft in the given interval is equal to f(t), that is

Such a function is called a primitive for f(t).

We must keep in mind that if the function f(t) has at least one primitive, then along with this one it will have an infinite number of others; for if F(t) is a primitive for f(t), then F(t) + C, where C is an arbitrary constant, is also a primitive. Moreover, in this way we exhaust the whole set of primitives for f(t), since if F₁(t) and F₂(t) are primitives for the same function f(t), then their difference ϕ(t) = F₁(t) − F₂(t) has a derivative ϕ(t) that is equal to zero at every point in a given interval so that ϕ(t) is a constant.*

From a physical point of view the various values of the constant C determine laws of motion which differ from one another only in the fact that they correspond to all possible choices for the initial point of the motion.

We are thus led to the result that for an extremely wide class of functions f(x), including all cases where the function f(x) may be considered as the velocity of a point at the time x, we have the following equality†

where F(x) is an arbitrary primitive for f(x).

This equality is the famous formula of Newton and Leibnitz, which reduces the problem of calculating the definite integral of a function to finding a primitive for the function and in this way forms a link between the differential and the integral calculus.

Many particular problems that were studied by the greatest mathematicians are automatically solved by this formula, stating that the definite integral of the function. f(x) on the interval [a, b] is equal to the difference between the values of any primitive at the left and right ends of the interval.‡ It is customary to write the difference (30) thus:

Example 1. The equality

shows that the function x³/3 is a primitive for the function x². Thus, by the formula of Newton and Leibnitz,

Example 2. Let c and c′ be two electric charges, on a straight line at distance r from each other. The attraction F between them is directed along this straight line and is equal to

(a = kcc′, where k is a constant). The work W done by this force, when the charge c remains fixed but c′ moves along the interval [R₁, R₂], may be calculated by dividing the interval [R₁, R₂] into parts Δri. On each of these parts we may consider the force to be approximately constant, so that the work done on each part is equal to . Making the parts smaller and smaller, we see that the work W is equal to the integral

The value of this integral can be calculated at once, if we recall that

so that

In particular, the work done by a force F as the charge c′, initially at a distance R₁, from c, moves out to infinity, is equal to

From the arguments given above for the formula of Newton and Leibnitz, it is clear that this formula gives mathematical expression to an actual tie existing in the objective world. It is a beautiful and important example of how mathematics gives expression to objective laws. We should remark that in his mathematical investigations, Newton always took a physical point of view. His work on the foundations of differential and integral calculus cannot be separated from his work on the foundations of mechanics.

The concepts of mathematical analysis, such as the derivative or the integral, as they presented themselves to Newton and his contemporaries, had not yet completely “broken away” from their physical and geometric origins, such as velocity and area. In fact, they were half mathematical in character and half physical. The conditions existing at that time were not yet suitable for producing a purely mathematical definition of these concepts. Consequently, the investigator could handle them correctly in complicated situations only if he remained in close contact with the practical aspects of his problem even during the intermediate (mathematical) stages of his argument.

From this point of view the creative work of Newton was different in character from that of Leibnitz.* Newton was guided at all stages by a physical way of looking at the problem. But the investigations of Leibnitz do not have such an immediate connection with physics, a fact that in the absence of clear-cut mathematical definitions sometimes led him to mistaken conclusions. On the other hand, the most characteristic feature of the creative activity of Leibnitz was his striving for generality, his efforts to find the most general methods for the problems of mathematical analysis.

The greatest merit of Leibnitz was his creation of a mathematical symbolism expressing the essence of the matter. The notations for such fundamental concepts of mathematical analysis as the differential dx, the second differential d₂x, the integral ∫y dx, and the derivative d/dx were proposed by Leibnitz. The fact that these notations are still used shows how well they were chosen.

One advantage of a well-chosen symbolism is that it makes our proofs and calculations shorter and easier; also, it sometimes protects us against mistaken conclusions. Leibnitz, who was well aware of this, paid especial attention in all his work to the choice of notation.

The evolution of the concepts of mathematical analysis (derivative, integral, and so forth) continued, of course, after Newton and Leibnitz and is still continuing in our day; but there is one stage in this evolution that should be mentioned especially. It took place at the beginning of the last century and is related particularly to the work of Cauchy.

Cauchy gave a clear-cut formal definition of the concept of a limit and used it as the basis for his definitions of continuity, derivative, differential, and integral. These definitions have been introduced at the corresponding places in the present chapter. They are widely used in present-day analysis.

The great importance of these achievements lies in the fact that it is now possible to operate in a purely formal way not only in arithmetic, algebra, and elementary geometry, but also in this new and very extensive branch of mathematics, in mathematical analysis, and to obtain correct results in so doing.

Regarding practical application of the results of mathematical analysis, it is now possible to say: If the original data are verified in the actual world, then the results of our mathematical arguments will also be verified there. If we are properly assured of the accuracy of the original data, then there is no need to make a practical check of the correctness of the mathematical results; it is sufficient to check only the correctness of the formal arguments.

This statement naturally requires the following limitation. In mathematical arguments the original data, which we take from the actual world, are true only up to a certain accuracy. This means that at every step of our mathematical argument the results obtained will contain certain errors, which may accumulate as the number of steps in the argument increases.*

Returning now to the definite integral, let us consider a question of fundamental importance. For what functions f(x), defined on the interval [a, b], is it possible to guarantee the existence of the definite integral , namely a number to which the sum , tends as limit as max Δxi, → O? It must be kept in view that this number is to be the same for all subdivisions of the interval [a, b] and all choices of the points ξi.

Functions for which the definite integral, namely the limit (29), exists are said to be integrable on the interval [a, b]. Investigations carried out in the last century show that all continuous functions are integrable.

But there are also discontinuous functions which are integrable. Among them, for example, are those functions which are bounded and either increasing or decreasing on the interval [a, b].

The function that is equal to zero at the rational points in [a, b] and equal to unity at the irrational points, may serve as an example of a nonintegrable function, since for an arbitrary subdivision the integral sum sn, will be equal to zero or unity, depending on whether we choose the points ξi, as rational numbers or irrational.

Let us note that in many cases the formula of Newton and Leibnitz provides an answer to the practical question of calculating a definite integral. But here arises the problem of finding a primitive for a given function; that is, of finding a function that has the given function for its derivative. We now proceed to discuss this problem. Let us note by the way that the problem of finding a primitive has great importance in other branches of mathematics also, particularly in the solution of differential equations.

§11. Indefinite Integrals; the Technique of Integration

An arbitrary primitive of a given function f(x) is usually called an indefinite integral of f(x) and is written in the form

In this way, if F(x) is a completely determined primitive of f(x), then the indefinite integral of f(x) is given by

where C is an arbitrary constant.

Let us also note that if the function f(x) is given on the interval [a, b] and, if F(x) is a primitive for f(x) and x is a point in the interval [a, b], then by the formula of Newton and Leibnitz we may write

Here the integral on the right side differs from the primitive F(x) only by the constant F(a). In such a case this integral, if we consider it as a function of its upper limit x (for variable x), is a completely determined primitive of f(x). Consequently, an indefinite integral of f(x) may also be written as follows:

where C is an arbitrary constant.

Let us set up a fundamental table of indefinite integrals, which can be obtained directly from the corresponding table of derivatives (see §6):

The general properties of indefinite integrals may also be deduced from the corresponding properties of derivatives. For example, from the rule for the differentiation of a sum we obtain the formula

and from the corresponding rule expressing the fact that a constant factor k may be taken outside the sign of differentiation we get

For example,

There are a number of methods for calculating indefinite integrals. Let us consider one of them, namely the method of substitution or change of variable, which is based on the following equality

where x = ϕ(t) is a differentiable function. The relation (33) is to be understood in the sense that if in the function

on the left side of equality (33), we set x = ϕ(t), we thereby obtain a function F[ϕ(t)] whose derivative with respect to t is equal to the expression under the sign of integration on the right side of equality (33). This fact follows immediately from the theorem on the derivative of a function of a function.

Let us give some examples of this method of substitution

(substituticn of = t, from which k dx = dt).

(substitution of = , from which .

(substitution of x = a sin u).

As can be seen from these examples, the method of substitution or change of variables greatly extends the class of elementary functions that we are able to integrate; that is, for which we can find primitives that are themselves elementary functions. But it must be noted that from the point of view of actually calculating the result, we are in a much worse position, generally speaking, with respect to integration than for differentiation.

From §6 we know that the derivative of an arbitrary elementary function is itself an elementary function, which we may effectively calculate by making use of the rules of differentiation. But the converse statement is in general untrue, since there exist elementary functions whose indefinite integrals are not elementary functions. Examples are e–x², 1/(ln x), (sin x)/x and so forth. To obtain integrals of these functions we must make use of approximative methods and also introduce new functions which can not be reduced to elementary ones. We can not spend more time here on this question but must simply note that even in elementary mathematics it is possible to find many examples in which a direct operation can be carried out on a certain class of numbers, while the inverse operation can not be carried out on the same class; thus, a square of an arbitrary rational number is again a rational number, but the square root of a rational number is by no means always rational. Analogously, differentiation of elementary functions produces a function that is again elementary, but integration may lead us outside the class of elementary functions.

Some of the integrals that cannot be expressed in terms of elementary functions have great importance in mathematics and its applications. An example is

which plays a very important role in the theory of probability (see Chapter XI). Other examples are the integrals

which are called elliptic integrals of the first and second kind respectively. We are led to the calculation of these integrals by a large number of problems in physics (see Chapter V, §1, example 3). Detailed tables of these integrals for various values of the arguments x and ϕ have been calculated by approximate methods but with great accuracy.

It must be emphasized that the proof of the very fact that a given elementary function cannot be integrated in terms of elementary functions is in each case quite difficult. Such questions occupied the attention of outstanding mathematicians in the last century and have played an important role in the development of analysis. Fundamental results were obtained here by Čebyšev, who gave a complete answer to the question of expressing in terms of elementary functions the integrals of the form

where m, s, and p are rational numbers. Up to his time three relations, obtained by Newton, were known for the exponents m, s, and p, which implied the integrability of this integral in terms of elementary functions. Čebyšev proved that in all other cases the integral cannot be expressed in terms of elementary functions.

We introduce here another method of integration, namely integration by parts. It is based on the formula we already know

for the derivative of the product of the functions u and v. This formula may also be written

Let us now integrate the left and right sides, keeping in mind that

We now finally obtain the equality

which is also called the formula of integration by parts. We have not written the constant C since we may consider that it is included in one of the indefinite integrals occurring in this equation.

Let us introduce some applications of this formula. Suppose we have to calculate ∫ xex dx. Here we will take u = x and v′ = ex, and thus u′ = 1, v = ex, and consequently

In the integral ∫ ln x dx it is convenient to take u = ln x, v′ = 1, so that u′ = 1/x, v = x and

In the following characteristic example it is necessary to integrate twice by parts and then to find the desired integral from the equations so obtained:

from which

We end this section here; from it the reader will have obtained only a superficial idea of the theory of integration. We have not given any attention to many different methods in this theory. In particular we have not touched here on the very interesting question of the integration of rational fractions, a theory in which an important contribution was made by the well-known mathematician and mechanician, Ostrogradskil.

§12.  Functions of Several Variables

Up to now we have spoken only of functions of one variable, but in practice it is often necessary to deal also with functions depending on two, three, or in general many variables. For example, the area of a rectangle is a function

of its base x and its height y. The volume of a rectangular parallelepiped is a function

of its three dimensions. The distance between two points A and B is a function

of the six coordinates of these points. The well-known formula

expresses the dependence of the volume v of a definite amount of gas on the pressure p and absolute temperature T.

Functions of several variables, like functions of one variable, are in many cases defined only on a certain region of values of the variables themselves. For example, the function

is defined only for values of x, y and z that satisfy the condition

(For other x, y, z its values are not real numbers.) The set of points of space whose coordinates satisfy the inequality (35) obviously fills up a sphere of unit radius with its center at the origin of coordinates. The points on the boundary are not included in this sphere; the surface of the sphere has been so to speak “peeled off.” Such a sphere is said to be open. The function (34) is defined only for such sets of three numbers (x, y, z) as are coordinates of points in the open sphere G. It is customary to state this fact concisely by saying that the function (34) is defined on the sphere G.

Let us give another example. The temperature of a nonuniformly heated body V is a function of the coordinates x, y, z of the points of the body. This function is not defined for all sets of three numbers x, y, z but only for such sets as are coordinates of points of the body V.

Finally, as a third example, let us consider the function

where ϕ is a function of one variable defined on the interval [0, 1]. Obvious-ly the function u is defined only for sets of three numbers (x, y, z) which are coordinates of points in the cube:

We now give a formal definition of a function of three variables. Suppose that we are given a set E of triples of numbers (x, y, z) (points of space). If to each of these triples of numbers (points) of E there corresponds a definite number u in accordance with some law, then u is said to be a function of x, y, z (of the point), defined on the set of triples of numbers (on the points) E, a fact which is written thus:

In place of F we may also write other letters: f, ϕ, ψ.

In practice the set E will usually be a set of points, filling out some geometrical body or surface: sphere, cube, annulus, and so forth, and then we simply say that the function is defined on this body or surface. Functions of two, four, and so forth, variables are defined analogously.

Implicit definition of a function. Let us note that functions of two variables may serve, under certain circumstances, as a useful means for the definition of functions of one variable. Given a function F(x, y) of two variables let us set up the equation

In general, this equation will define a certain set of points (x, y) of the surface on which our function is equal to zero. Such sets of points usually represent curves that may be considered as the graphs of one or several one-valued functions y = ϕ(x) or x = ψ(y) of one variable. In such a case these one-valued functions are said to be defined implicitly by the equation (36). For example, the equation

gives an implicit definition of two functions of one variable

But it is necessary to keep in mind that an equation of the form (36) may fail to define any function at all. For example, the equation

obviously does not define any real function, since no pair of real numbers satisfies it.

Geometric representation. Functions of two variables may always be visualized as surfaces by means of a system of space coordinates. Thus the function

is represented in a three-dimensional rectangular coordinate system by a surface, which is the geometric locus of points M whose coordinates x, y, z satisfy equation (37) (figure 26).

There is another, extremely useful method, of representing the function (37), which has found wide application in practice. Let us choose a sequence of numbers z₁, z₂, ···, and then draw on one and the same plane Oxy the curves

which are the so-called level lines of the function f(x, y). From a set of level lines, if they correspond to values of z that are sufficiently close to one another, it is possible to form a very good image of the variation of the function f(x, y), just as from the level lines of a topographical map one may judge the variation in altitude of the locality.

Figure 27 shows a map of the level lines of the function z = x² + y², the diagram at the right indicating how the function is built up from its level lines. In Chapter III, figure 50, a similar map is drawn for the level lines of the function z = xy.

FIG. 26.

Partial derivatives and differential. Let us make some remarks about the differentiation of the functions of several variables. As an example we take the arbitrary function

FIG. 27.

of two variables. If we fix the value of y, that is if we consider it as not varying, then our function of two variables becomes a function of the one variable x. The derivative of this function with respect to x, if it exists, is called the partial derivative with respect to x and is denoted thus:

The last of these three notations indicates clearly that the partial derivative with respect to x is in general a function of x and y. The partial derivative with respect to y is defined similarly.

Geometrically the function f(x, y) represents a surface in a rectangular three-dimensional system of coordinates. The corresponding function of x for fixed y represents a plane curve (figure 28) obtained from the intersection of the surface with a plane parallel to the plane Oxz and at a distance y from it. The partial derivative ∂z/∂x is obviously equal to the trigonometric tangent of the angle between the tangent. to the curve at the point (x, y) and the positive direction of the x-axis.

FIG. 28.

More generally, if we consider a function z = f(x₁, x₂, . . ., xn) of the n variables x₁, x₂, . . ., xn, the partial derivative ∂z/∂x, is defined as the derivative of this function with respect to xi, calculated for fixed values of the other variables:

We may say that the partial derivative of a function with respect to the variable xi is the rate of change of this function in the direction of the change in xi. It would also be possible to define a derivative in an arbitrary assigned direction, not necessarily coinciding with any of the coordinate axis, but we will not take the time to do this.

Examples.

1.

2.

It is sometimes necessary to form the partial derivatives of these partial derivatives; that is; the so-called partial derivatives of second order. For functions of two variables there are four of them

However, if these derivatives are continuous, then it is not hard to prove that the second and third of these four (the so-called mixed derivatives) coincide:

For example, in the case of first function considered,

the two mixed derivatives are seen to coincide.

For functions of several variables, just as was done for functions of one variable, we may introduce the concept of a differential.

For definiteness let us consider a function

of two variables. If it has continuous partial derivatives, we can prove that its increment

corresponding to the increments Δx and Δy of its arguments, may be put in the form

where ∂f/∂x and ∂f/∂y are the partial derivatives of the function at the point (x, y) and the magnitude a depends on Δx and Δy in such a way that α → 0 as Δx → 0 and Δy → 0.

The sum of the first two components

is linearly dependent* on Δx and Δy and is called the differential of the function. The third summand, because of the presence of the factor α, tending to zero with Δx and Δy, is an infinitesimal of higher order than the magnitude

describing the change in x and y.

Let us give an application of the concept of differential. The period of oscillation of a pendulum is calculated from the formula

where l is its length and g is the acceleration of gravity. Let us suppose that l and g are known with errors respectively equal to Δl and Δg. Then the error in the calculation of T will be equal to the increment ΔT corresponding to the increments of the arguments Δl and Δg. Replacing ΔT approximately by dT, we will have

The signs of Δl and Δg are unknown, but we may obviously estimate ΔT by the inequality

from which after division by T we get

Thus we may consider in practice that the relative error for T is equal to the sum of the relative errors for l and g.

For symmetry of notation, the increments of the independent variables Δx and Δy are usually denoted by the symbols dx and dy and are also called differentials. With this notation the differential of the function u = f(x, y, z) may be written thus:

Partial derivatives play a large role whenever we have to do with functions of several variables, as happens in many of the applications of analysis to technology and physics. We shall be dealing in Chapter VI with the problem of reconstructing a function from the properties of its partial derivatives.

In the following paragraphs, we give some simple examples of applications of partial derivatives in analysis.

Differentiation of implicit functions. Suppose we wish to find the derivative of y, where y is a function of x defined implicitly by the relation

between these variables. If x and y satisfy the relation (38) and we give x the increment Δx, then y will receive an increment Δy such that x + Δx and y + Δy again satisfy (38). Consequently*

Thus, provided ∂F/∂y ≠ 0, it follows that

In this way we have obtained a method for finding the derivative of an implicit function y without first solving the equation (38) for y.

Maximum and minimum problems. If a function, let us say of two variables z = f(x, y), attains its maximum at the point (x₀, y₀), that is if for all points (x, y) close to (x₀, y₀), then this point must also be the point of maximum altitude for any line formed by the intersection of the surface z = f(x, y) with a plane parallel to Oxz or Oyz. So at such a point we must have

The same equations must also hold for a point of local minimum. Consequently, the greatest or least values of the function are to be sought first of all at points where the conditions (39) are satisfied, but we must also not forget about points on the boundary of the domain of definition of the function and points where the function fails to have a derivative, if such points exist.

To establish whether a point (x, y) satisfying (39) is actually a maximum or minimum point, use is frequently made of various indirect arguments. For example, if for any reason it is clear that the function is differentiable and attains its minimum inside the region and that there is only one point where the conditions (39) are fulfilled, then obviously the minimum must be attained at this point.

For example, let it be required to make a rectangular tin box (without lid) with assigned volume V, using the smallest possible amount of material. If the sides of the base of this box are denoted by x and y, then its height h will be equal to V/xy, and consequently the surface S will be given by the function

of x and y. Since x and y by the terms of the problem must be positive, the question has been reduced to finding the minimum of the function S(x, y) for all possible points (x, y) in the first quadrant of the plane (x, y), which we will denote by the letter G.

If the minimum is attained at some point of the region G, then the partial derivatives must be equal to zero

that is yx² = 2V, xy² = 2V, from which we find as the dimensions of the box:

We have solved the problem but have not altogether proved that our solution is correct. A rigorous mathematician will say to us: “You have supposed from the very beginning that under the given conditions the box with minimum surface actually exists and, proceeding from this assumption, you have found its dimensions. So you have really obtained only the following result: If there exists a point (x, y) in G for which the function S attains its minimum, then the coordinates of this point must necessarily be determined by the equation (41). But now you must show that the minimum of S does exist for some point in G and then I will admit the correctness of your result.” This remark is a very reasonable one, since, for example, our function S, as we shall soon see, does not possess any maximum in the region G. But let us show how it is possible to convince ourselves that in the given case the function actually does attain its minimum at a certain point (x, y) of the region G.

FIG. 29.

The fundamental theorem on which we shall base our argument is one that is proved in analysis with complete rigor; it amounts to the following. If the function f of one or several variables is everywhere continuous in a certain finite region H which is bounded and includes its boundary, then there always exists in H at least one point at which the function attains its minimum (maximum). With this theorem we can easily complete our analysis of the problem.

Let us consider an arbitrary point (x₀, y₀) of the region G; at this point let S(x₀, y₀) = N. Let us also choose a number R satisfying the two inequalities R > N, 2VR > N and construct a square ΩR, with side R², as in figure 29, where AB = CD = 1/R.

We now give a lower bound for the values of our function S(x, y) at points of the region G lying outside the square ΩR. If the point of the region G has abscissa x < 1/R, then

Analogously, if the point of the region G has its ordinate y < 1/R, then also S > N. Also, if the point of the region G has its abscissa x > 1/R and if it lies above the straight line AF or has its ordinate y > 1/R and lies to the right of the straight line CE, then

Thus, for all points (x, y) of the region G lying outside the square ΩR, the inequality S(x, y) > N holds, and since S(x₀, y₀) = N, the point (x₀, y₀) must belong to the square and consequently the minimum of our function on G is equal to its minimum on the square.

But the function S(x, y) is continuous in this square and on its boundary, so that by the theorem stated earlier there exists in the square a point (x, y) where our function assumes its minimum for points in the square and consequently for the entire region G. Thus the existence of a minimum has been proved.

This argument may serve as an example of the way that it is possible to discuss the existence of a maximum or a minimum for a function defined on an unbounded domain.

The Taylor formula. Like functions of one variable, functions of several variables may be represented by a Taylor formula. For example, an expansion of the function

in the neighborhood of the point (x₀, y₀) has the following form, if we confine ourselves to the first and second powers of x − x₀ and y − y₀:

If the function f(x, y) has continuous partial derivatives of the second order, the remainder term here will approach zero faster than

that is, faster than the square of the distance between the points (x, y) and (x₀, y₀), as r → 0. The Taylor formula provides a widely used method of defining and approximately calculating the values of various functions.

Let us note that with the help of this formula we can also answer the question asked earlier, whether a given function actually has a maximum or minimum at a point where ∂f/∂y = ∂f/∂y = 0. In fact, if these conditions are satisfied at the point (x₀, y₀), then for points (x, y) close to (x₀, y₀), the value of the function will, by the Taylor formula, differ from f(x₀, y₀) by the amount

where A, B, and C denote respectively the second partial derivatives , at the point (x₀, y₀).

If it turns out that the function

is positive for arbitrary values of (x − x₀) and (y − y₀) not both equal to zero, then the right side of equation (42) will also be positive for small values of (x − x₀) and (y − y₀), since for sufficiently small (x − x₀) and (y − y₀) the quantity R₃, is known to be less in absolute value than . Thus it will follow that at the point (x₀, y₀) the function f attains its minimum. On the other hand, if the function Φ(x, y) is negative for arbitrary (x − x₀) and (y − y₀) the right side of (42) will be negative for (x − x₀) and (y − y₀), so that at the point (x₀, y₀) the function will have a maximum. In more complicated cases it is necessary to consider the succeeding terms in the Taylor formula.

Problems concerning the maximum or the minimum of functions of three or more variables may be treated in a completely analogous fashion. As an exercise the reader may prove that if given masses

are arranged in space at given points

the moment (of inertia) M of this system of masses about the point P(x, y, z), defined as the sum of the products of the masses and the squares of their distances from the point P,

will be a minimum if the point P is at the so-called center of gravity of the system, with the coordinates

Maxima and minima with subsidiary conditions. For functions of several variables we may set up various problems concerning maximum and minimum. Let us illustrate with a simple example. Suppose that among all rectangles inscribed in a circle of radius R, we wish to find the one with greatest area. The area of a rectangle is equal to the product xy of its sides, where x and y are positive numbers connected in this case by the relation x² + y² = (2R)², as is clear from figure 30. Thus we are required to find the maximum of the function f(x, y) = xy for all x and y satisfying the relation x² + y² = 4R².

Problems of this sort, where it is necessary to find the maximum (or minimum) of a function f(x, y) for those values only of x and y that satisfy a certain relation that ϕ(x, y) are very common in practice.

FIG. 30.

Of course, it would be possible to solve the FIG.30. equation ϕ(x, y) for y, to substitute the solution into the function f(x, y) and in this way to seek the ordinary maximum for a function of one variable x. But this method is usually complicated and sometimes impossible.

For the solution of such problems in analysis, a much more convenient procedure called the method of Lagrange multipliers, has been worked out. The idea behind it is extremely simple. Let us consider the function

where λ is an arbitrary positive number. Obviously, for x, y satisfying the condition ϕ(x, y) = 0, the values of F(x, y) coincide with those of f(x, y).

For function F(x, y) let us seek a maximum without conditions of any kind on x and y. At the maximum point the conditions ∂F/∂x = ∂F/∂y = 0* must hold; in other words

The values of x and y at the maximum point for F(x, y), being a solution of the system (43) and (44), depend on the coefficient λ in these equations. Let us now suppose that we have succeeded in choosing the number λ in such a way that the coordinates of the maximum point satisfy the condition

Then this point will be an exact local maximum for the original problem.

FIG. 31.

In fact, we may consider the problem geometrically as follows. The function f(x, y) is defined on a certain region G (figure 31). The condition ϕ(x, y) = 0 will ordinarily be satisfied by the points of some curve Γ. We are required to find the greatest value of x and y on points of the line Γ. If F(x, y) attains its maximum on the curve Γ, then F(x, y) does not increase for small shifts in an arbitrary direction from this point, and in particular for shifts along the curve Γ. But for shifts along Γ, the values of F(x, y), coincide with those of f(x, y) which means that for small shifts along the curve the function f(x, y) does not increase, or in other words it has the local maximum at the point.

These arguments indicate a simple method of solving the problem. We solve equations (43), (44), (45) for the unknowns x, y, and λ, obtaining one or more solutions

To the points (x₁, y₁), (x₂, y₂), ··· so determined we adjoin those points of the boundary of G where the curve Γ leaves the region G. Then from all these points we choose that one at which f(x, y) takes on its greatest (or smallest) value.

Of course, the arguments here are far from proving the correctness of the method. In fact, we have not yet even proved that the points of local maximum for f(x, y) on the curve Γ can be obtained as maximum points for the function F(x, y) for some value of λ. However, it is possible to prove, as is done in the textbooks in analysis, that every point (x₀, y₀) where f(x, y) has a local maximum on the curve will be obtained by the method indicated, provided only that at this point the partial derivatives ϕ′x(x₀, y₀) and ϕ′y(x₀, y₀) are not both equal to zero.*

Let us use the method of Lagrange to solve the problem at the beginning of the present section. In this case f(x, y) = xy; ϕ(x, y) = x² + y² – 4R². We set up the equations (43), (44), (45)

for which, taking into account that x and y are positive, we find the unique solution

For these values of x and y, which are equal to one another so that the inscribed rectangle is a square, the area is in fact a maximum.

The method of Lagrange may be extended to deal with functions of three or more variables. There may be any number of subsidiary conditions (smaller than the number of variables) of the type of condition (45), and we will introduce the corresponding number of auxiliary multipliers.

Let us give some examples of problems involving maxima or minima with subsidiary conditions.

Example 1. For what height h and radius r will an open cylindrical tank of given volume V require the least amount of sheet metal for its manufacture; that is, the area of its sides and circular base will be a minimum ?

The problem obviously reduces to finding the minimum of the function of the variables r and h

under the condition πr²h = V, which may be written in the form

Example 2. A moving point is required to pass from A to B (figure 32). On the path AM it moves with the velocity of v₁, and on MB with the velocity v₂. Where should the point M be placed on the line DD′ so that the entire path from A to B may be covered as quickly as possible?

FIG. 32.

Let us take as unknowns the angles α and β marked in figure 32. The lengths a and b of the perpendiculars from the points A and B to the straight line DD′ and the distance c between them are known. The time required for covering the entire path is represented, as can easily be seen, by the formula

It is required to find the minimum of this expression, taking into account the fact that α and β are connected by the relation

The reader may solve these examples by the Lagrange method. In the second example he will find that the best position for M is given by the condition

This is the well-known law for the refraction of light. Consequently, a ray of light will be refracted in its passage from one medium to another in such a way that the time for its passage from a point in one medium to a point in the other is a minimum. Conclusions of this sort are interesting not only for computational purposes but also from a general philosophical point of view; they have inspired researchers in the exact sciences to penetrate further and further into the profound and general laws of nature.

Finally let us note that the multipliers λ, introduced in the solution of problems by the method of Lagrange, are not merely auxiliary numbers. In each case they are closely connected with the essential nature of the particular problem and have a concrete interpretation.

§13.   Generalizations of the Concept of Integral

In §10 we defined the definite integral of the function f(x) on the interval [a, b] as the limit of the sum

when the length of the greatest segment Δxi in the subdivision of [a, b] approaches zero. In spite of the fact that the class of functions f(x) for which this limit actually exists (the class of integrable functions) is a very wide one, and in particular includes all continuous and even many discontinuous functions, this class of functions has a serious shortcoming. If we add, subtract, or multiply, or under certain conditions divide the values of two integrable functions f(x) and ϕ(x), we obtain functions which, as may easily be proved, are again integrable. For f(x)/ϕ(x) this will be true in all cases in which 1/ϕ(x) remains bounded on [a, b]. But if a function is obtained as a result of a limiting process from a sequence of approximating integrable functions f₁(x), f₂(x), f₃(x), ··· such that for all values of x in the interval [a, b]

then the limit function f(x) is not necessarily integrable.

In many cases this and other circumstances give rise to considerable complication, since the process of passing to a limit is widely used.

A way out of the difficulty was discovered by making further generalizations of the concept of an integral. The most important of these is the integral of Lebesgue, with which the reader will become acquainted in Chapter XV on the theory of functions of a real variable. But here we will confine ourselves to generalizations of the integral in other directions, which are also of the greatest importance in practice.

Multiple integrals. We have already studied the process of integration for functions of one variable defined on a one-dimensional region, namely an interval. But the analogous process may be extended to functions of two, three, or more variables, defined on corresponding regions.

For example, let us consider a surface

defined in a rectangular system of coordinates, and on the plane Oxy let there be given a region G bounded by a closed curve Γ. It is required to find the volume bounded by the surface, by the plane Oxy and by the cylindrical surface passing through the curve Γ with generators parallel to the Oz axis (figure 33). To solve this problem we divide the plane region G into subregions by a network of straight lines parallel to the axes Ox and Oy and denote by

FIG. 33.

those subregions which consist of complete rectangles. If the net is sufficiently fine, then practically the whole of the region G will be covered by the enumerated rectangles. In each of them we choose at will a point

and, assuming for simplicity that Gi denotes not only the rectangle but also its area, we set up the sum

It is clear that, if the surface is continuous and the net is sufficiently fine, this sum may be brought as near as we like to the desired volume V. We will obtain the desired volume exactly if we take the limit of the sum (47) for finer and finer subdivisions (that is, for subdivisions such that the greatest of the diagonals of our rectangles approaches zero)

From the point of view of analysis it is therefore necessary, in order to determine the volume V, to carry out a certain mathematical operation on the function f(x, y) and its domain of definition G, an operation indicated by the left side of equality (48). This operation is called the integration of the function f over the region G, and its result is the integral of f over G. It is customary to denote this result in the following way:

Similarly, we may define the integral of a function of three variables over a three-dimensional region G, representing a certain body in space. Again we divide the region G into parts, this time by planes parallel to the coordinate planes. Among these parts we choose the ones which represent complete parallelepipeds and enumerate them

In each of these we choose an arbitrary point

and set up the sum

where Gi denotes the volume of the parallelepiped Gi. Finally we define the integral of f(x, y, z) over the region G as the limit

to which the sum (50) tends when the greatest diagonal d(Gi) approaches zero.

Let us consider an example. We imagine the region G is filled with a nonhomogeneous mass whose density at each point in G is given by a known function ρ(x, y, z). The density ρ(x, y, z) of the mass at the point (x, y, z) is defined as the limit approached by the ratio of the mass of an arbitrary small region containing the point (x, y, z) to the volume of this region as its diameter approaches zero.* To determine the mass of the body G it is natural to proceed as follows. We divide the region G into parts by planes parallel to the coordinate planes and enumerate the complete parallelepipeds formed in this way

Assuming that the dividing planes are sufficiently close to one another, we will make only a small error if we neglect the irregular regions of the body and define the mass of each of the regular regions Gi (the complete parallelepipeds) as the product

where (ξi, ηi, ζi) is an arbitrary point Gi. As a result the approximate value of the mass M will be expressed by the sum

and its exact value will clearly be the limit of this sum as the greatest diagonal Gi approaches zero; that is

The integrals (49) and (51) are called double and triple integrals respectively.

FIG. 34.

Let us examine a problem which leads to a double integral. We imagine that water is flowing over a plane surface. Also, on this surface the underground water is seeping through (or soaking back into the ground) with an intensity f(x, y) which is different at different points. We consider a region G bounded by a closed contour (figure 34) and assume that at every point of G we know the intensity f(x, y), namely the amount of underground water seeping through per minute per cm² of surface; we will have f(x, y) > 0 where the water is seeping through and f(x, y) < 0 where it is soaking into the ground. How much water will accumulate on the surface G per minute ?

If we divide G into small parts, consider the rate of seepage as approximately constant in each part and then pass to the limit for finer and finer subdivisions, we will obtain an expression for the whole amount of accumulated water in the form of an integral

Double (two-fold) integrals were first introduced by Euler. Multiple integrals form an instrument which is used everyday in calculations and investigations of the most varied kind.

It would also be possible to show, though we will not do it here, that calculation of multiple integrals may be reduced, as a rule, to iterated calculation of ordinary one-dimensional integrals.

Contour and surface integrals. Finally, we must mention that still other generalizations of the integral are possible. For example, the problem of defining the work done by a variable force applied to a material point, as the latter moves along a given curve, naturally leads to a so-called curvilinear integral, and the problem of finding the general charge on a surface on which electricity is continuously distributed with a given surface density leads to another new concept, an integral over a curved surface.

FIG. 35.

For example, suppose that a liquid is flowing through space (figure 35) and that the velocity of a particle of the liquid at the point (x, y)is given by a function P(x, y), not depending on z. If we wish to determine the amount of liquid flowing per minute through the contour Γ,*we may reason in the following way. Let us divide Γ up into segments Δsi. The amount of water flowing through one segment Δsi is approximately equal to the column of liquid shaded in figure 35; this column may be considered as the amount of liquid forcing its way per minute through that segment of the contour. But the area of the shaded parallelogram is equal to

where αi is the angle between the direction of the x-axis and the outward normal of the surface bounded by the contour Γ; this normal is the perpendicular to the tangent, which we may consider as defining the direction of the segment Δsi. By summing up the areas of such parallelograms and passing to the limit for finer and finer subdivisions of the contour Γ, we determine the amount of water flowing per minute through the contour Γ; it is denoted thus:

and is called a curvilinear integral. If the flow is not everywhere parallel, then its velocity at each point (x, y) will have a component P(x, y) along the x-axis and a component Q(x, y) along the y-axis. In this case we can show by an analogous argument that the quantity of water flowing through the contour will be equal to

When we speak of an integral over a curved surface G for a function f(M) of its points M(x, y, z), we mean the limit of sums of the form

for finer and finer subdivisions of the region G into segments whose areas are equal to Δσi.

General methods exist for transforming multiple, curvilinear, and surface integrals into other forms and for calculating their values, either exactly or approximately.

Formula of Ostrogradskiĭ. Several important and very general formulas relating an integral over a volume to an integral over its surface (and also an integral over a surface, curved or plane, to an integral around its boundary) were discovered in the middle of the past century by Ostrogradskiĭ.

We shall not try to give here a proof of the general formula of Ostrogradskiĭ, which has very wide application, but will merely illustrate it by an example of its simplest particular case.

Let us imagine, as we did before, that over a plane surface there is a horizontal flow of water that is also soaking into the ground or seeping out again from it. We mark off a region G, bounded by a curve Γ, and assume that for each point of the region we know the components P(x, y) and Q(x, y) of the velocity of the water in the direction of the x-axis and of the y-axis respectively.

Let us calculate the rate at which the water is seeping from the ground at a point with coordinates (x, y). For this purpose we consider a small rectangle with sides Δx and Δy situated at the point (x, y).

As a result of the velocity P(x, y) through the left vertical edge of this rectangle, there will flow approximately P(x, y)Δy units of water per minute into the rectangle, and through the right side in the same time will flow out approximately P(x + Δx, y)Δy units. In general, the net amount of water leaving a square unit of surface as a result of the flow through its left and right vertical sides will be approximately

If we let Δx approach zero, we obtain in the limit

Correspondingly, the net rate of flow of water per unit area in the direction of the y-axis will be given by

This means that the intensity of the seepage of ground water at the point with coordinates (x, y) will be equal to

But in general, as we saw earlier, the quantity of water coming out from the ground will be given by the double integral of the function expressing the intensity of the seepage of ground water at each point, namely

But, since the water is incompressible, this entire quantity must flow out during the same time through the boundaries of the contour Γ. The quantity of water flowing out through the contour Γ is expressed, as we saw earlier, by the curvilinear integral over Γ

The equality of the magnitudes (52) and (53) expresses the formula of Ostrogradskiĭ in its simplest two-dimensional case

We have merely explained the meaning of this formula by a physical example, but it can be proved mathematically.

In this way the mathematical theorem of Ostrogradskiĭ reflects a widespread phenomenon in the external world, which in our example we interpreted in a readily visualized way as preservation of the volume of an incompressible fluid.

Ostrogradskiĭ established a considerably more general formula expressing the connection between an integral over a multidimensional volume and an integral over its surface. In particular, for a three-dimensional body G, bounded by the surface Γ, his formula is

where dσ is the element of surface.

It is interesting to note that the fundamental formula of the integral calculus

may be considered as a one-dimensional case of the formula of Ostrogradskiĭ. The equation (54) connects the integral over an interval with the “integral” over its “null-dimensional” boundary, consisting of the two end points.

Formula (54) may be illustrated by the following analogy. Let us imagine that in a straight pipe with constant cross section s = 1 water is flowing with velocity F(x), which is different for different cross sections (figure 36). Through the porous walls of the pipe, water is seeping into it (or out of it) at a rate which is also different for different cross sections.

FIG. 36.

If we consider a segment of the pipe from x to x + Δx, the quantity of water seeping into it in unit time must be compensated by the difference F(x + Δx) – F(x) between the quantity flowing out of this segment and the quantity flowing into it along the pipe. So the quantity seeping into the segment is equal to the difference F(x + Δx) – F(x), and consequently the rate of seepage per unit length of pipe (the ratio of the seepage over an infinitesimal segment to the length of the segment) will be equal to

More generally, the quantity of water seeping into the pipe over the whole section [a, b] must be equal to the amount lost by flow through the ends of the pipe. But the amount seeping through the walls is equal to and the amount lost by flow through the ends is F(b) – F(a). The equality of these two magnitudes produces formula (54).

§14.  Series

Concept of a series. A series in mathematics is an expression of the form

The numbers uk are called the terms of the series. There is an infinite number of them, and they are arranged in a definite order, so that to each natural number k = 0, 1, 2, ··· there corresponds a definite value uk.

The reader must keep in mind that we have not said whether it is possible to calculate a value for such expressions or, in case it is possible, how to do it. The presence of a plus sign between the terms uk in our expression seems to indicate that in some way all the terms should be added. But there are infinitely many of them and addition of numbers is defined only for a finite number of terms.

Let us denote by Sn the sum of the first n terms of the series; we will call it the nth partial sum. As a result we obtain a sequence of numbers

and we may speak of a variable quantity Sn, where n = 1,2, ···.

The series is said to be convergent if, as n → ∞, the variable Sn approaches a definite finite limit

This limit is called the sum of the series, and in this case we write

But if, as n → ∞, the limit Sn does not exist, then the series is said to be divergent and in this case there is no sense in speaking of its sum.* But if all the un have the same sign, then it is customary to say that the sum of the series is equal to infinity with the corresponding sign.

As an example, let us consider the series

whose terms form a geometric progression with common ratio x.

The sum of the first n terms is equal to

if | x | < 1 this sum has a limit

and so for | x | < 1 we may write

If | x | > 1, then obviously

and the series diverges. The same situation holds also for x = 1, as may be seen immediately without use of formula (55), which for x = 1 has no meaning. Finally, if x = –1 the partial sums take the values +1 and 0 alternately, so that this series also is divergent.

To each series there corresponds a definite sequence of values of its partial sums S₁, S₂, S₃, ··· such that the convergence of the series depends on the fact that the sums approach a limit. Conversely, to an arbitrary sequence of numbers S₁, S₂, S₃, ··· corresponds a series

the partial sums of which will be the numbers of the sequence. Thus the theory of variables ranging over a sequence may be reduced to the theory of the corresponding series, and conversely. Yet each of these theories has independent significance. In some cases it is more convenient to study the variable directly and in others to consider the equivalent series.

Let us note that series have long served as an important method of representing various entities (above all, functions) and of calculating their value. Of course, the views of mathematicians concerning series have changed with the passage of time, corresponding to the changes in their ideas about infinitesimals. The above clear-cut definition of convergence and divergence of a series was formulated at the beginning of the last century at the same time as the closely associated concept of a limit.

If the series converges, then its general term approaches zero with increasing n, since

From examples given in the following paragraphs, it will be clear that the converse statement is in general false. But the criterion is still a useful one, since it provides a necessary condition for the convergence of a series. For example, the divergence of a geometric progression with common ratio x > 1 follows immediately from the fact its general term does not approach zero.

If the series consists of positive terms, then its partial sum Sn increases with increasing n and only two cases can exist: Either the variable Sn becomes and remains greater than any preassigned number A for sufficiently large n, in which case limn→∞ Sn = ∞, so that the series diverges; or else there exists a number A such that for all n the value of Sn does not exceed A; but then the variable Sn necessarily approaches a definite finite limit not greater than A and the series is convergent.

Convergence of a series. The question whether a given series converges or diverges may often be settled by comparing it with another series. Here it is customary to make use of the following criterion.

If we are given two series

with positive terms such that for all values of n, beginning with a certain one, we have the inequality

then the convergence of the second series implies the convergence of the first, and the divergence of the first implies the divergence of the second.

For example, let us consider the so-called harmonic series

Its terms are correspondingly not less than the terms of the series

in which the sum of the underlined terms in each case is equal to .

It is clear that the sum Sn of the second series approaches infinity with increasing n, and consequently that the harmonic series diverges.

The series

where α is a positive number less than unity, also obviously diverges, since for arbitrary n

On the other hand, it is possible to prove that series (56) for α > 1 is convergent. We will prove this here only for the case α 2; for this purpose we consider the series

with positive terms. It converges to unity as its sum, since its partial sums Sn, are equal to

On the other hand, the general term of this series satisfies the inequality

from which it follows that the series

converges. All the more then will the series (56) converge with α > 2.

Let us give here without proof another useful criterion for convergence and divergence of series with positive terms, the so-called criterion of d’Alembert.

Let us suppose that, as n approaches infinity, the ratio (un + 1)/un has a limit q. Then for q < 1 the sequence will certainly converge, while for q > 1 it will diverge. But for q = 1 the question of its convergence remains open.

The sum of a finite number of summands does not change if we permute the summands. But in general this is no longer true for infinite series. There exist convergent series for which it is possible to permute the terms in such a way as to change their sum and even to turn them into divergent series. Series with unstable sums of this sort fail to possess one of the fundamental properties of ordinary sums, permutability of the summands. So it is important to distinguish those series which preserve this property. It turns out that they are the so-called absolutely convergent series. The series

is said to be absolutely convergent if the series

of absolute values of its terms is also convergent. It is possible to prove that an absolutely convergent series is always convergent; in other words, that its partial sums Sn approach a finite limit. It is obvious that every convergent series with terms of one sign is absolutely convergent.

The series

is an example of an absolutely convergent series, since the terms of the series

are not greater than the corresponding terms of the convergent series

An example of a series which is convergent, but not absolutely convergent, is the following

as the reader may prove for himself.

Series of functions; uniformly convergent series. In analysis we often have to do with series whose terms are functions of x. In the preceding paragraphs we have already had examples of this sort, for instance, the series 1 + x + x² + x³ + ···. For some values of x this series converges, but for others it diverges. Particularly important in applications are series of functions convergent for all values of x belonging to a certain interval, which may in particular be the whole of the real axis or the positive half of it and so forth. Then the necessity arises for differentiating such series term by term, integrating them, deciding whether their sum is continuous, and so forth. For the familiar case of the sum of a finite number of terms, there are simple general rules. We know that the derivative of a sum of differentiable functions is equal to the sum of their derivatives, the integral of a sum of continuous functions is the sum of their integrals, and a sum of continuous functions is itself a continuous function: All this holds for the sum of a finite number of terms.

But for infinite series these simple rules are in general no longer true. We could give many examples of convergent series of functions for which the rules of termwise integration and differentiation are false. In the same way a series of continuous functions may turn out to have a discontinuous sum. On the other hand many infinite series behave like finite sums with respect to these rules.

Profound investigations of this question have shown that these rules may still be applied if the infinite series in question are not only convergent at each separate point of the interval of definition (the domain over which x, varies) but if they are uniformly convergent over the whole interval. In this way there was crystallized in mathematical analysis, in the middle of the 19th century, the important concept of the uniform convergence of a series.

Let us consider the series

whose terms are functions defined on the interval [a, b]. We suppose that for each separate value of x in the interval this series converges to a certain sum S(x). The sum of the first n terms of the series

is also a certain function of x, defined on [a, b].

We now introduce a magnitude ηn, which is equal to the least upper bound of the values* | S(x) – Sn(x) |, as x varies on the interval [a, b]. This magnitude is written as follows†

In case the quantity S(x) – Sn(x) attains its maximum value, which will certainly occur for example, when S(x) and Sn(x) are continuous, then ηn is simply the maximum of | S(x) – Sn(x) | on [a, b].

From the assumed convergence of our series, we have for every individual value of x in the interval [a, b]

But the magnitude ηn, may approach zero or it may not. If ηn, approaches zero as n → ∞, then the series is said to be uniformly convergent, and in the opposite case nonuniformly convergent. In the same sense it is possible to speak of the uniform or nonuniform convergence of a sequence of functions Sn(x) without necessarily interpreting them as partial sums of a series.

Example 1. The series of functions

which we take to be defined only for nonnegative values of x, namely on the half line [0, ∞), may be written in the form

from which we see that its partial sums are equal to

and

Thus the series is convergent for all nonnegative x and has the sum S(x) = 0. Furthermore,

so that the series is uniformly convergent to zero on the half axis [0, ∞). Figure 37 shows the graphs of some of the partial sums Sn(x).

Example 2. The series

may be written in the form

from which

and therefore

Thus the sum of the series is discontinuous on the interval [0, 1] with a discontinuity at the point x = 1. The quantity | Sn(x) – S(x) | is less than unity for every x in [0, 1] but for x close to x = 1 it is arbitrarily close to unity. So,

for all n = 1, 2, ···. Thus the series is nonuniformly convergent on the interval [0, 1]. Figure 38 shows some of the graphs of the function Sn(x). The graph of the sum of the series consists of the segment of the x-axis omitting the right end point and of the point (1,1).

FIG. 37.

FIG. 38.

This example shows that the sum of a nonuniformly convergent series of continuous functions may in fact be a discontinuous function.

On the other hand, if we consider the series on the interval with q < 1, then

so that on this interval the series converges uniformly and its sum is continuous. The fact that the sum of a uniformly convergent series of continuous functions is itself a continuous function is a general rule, as was pointed out earlier, which can be rigorously proved.

Example 3. The sum of the first n terms of the series Sn(x) has the graph represented by the heavy broken line in figure 39. Obviously, for all n we have Sn(0) = 0, but if 0 < x 1, then for all n 1/x, we will have Sn(x) = 0, and consequently for arbitrary x in the interval [0, 1],

On the other hand,

So the quantity ηn does not approach zero but even approaches infinity. We now note that the series corresponding to this sequence Sn(x) cannot be integrated term by term on the interval [0, 1], since

so that the series

reduces to the divergent series

Let us state without proof the fundamental properties of uniformly convergent series:

FIG. 39.

1. The sum of a series of continuous functions which is uniformly convergent on the interval [a, b] is a continuous function on this interval.

2. If the series of continuous functions

converges uniformly on the interval [a, b], then it may be integrated term by term on this interval; that is, for all x₁, x₂ in [a, b] we have the equality

3. If on the interval [a, b] the series (57) converges and the functions uk(x) have continuous derivatives, then the equality

obtained by termwise differentiation of (57) will be valid on the interval [a, b] if the series on the right in (58) converges uniformly.

Power series. In §9, a function f(x) defined on an interval [a, b] was called analytic, if on this interval it has derivatives of arbitrary order and if in a sufficiently small neighborhood of any point x₀ of the interval [a, b] it may be expanded in a convergent Taylor series

If we introduce the notation

this series may be written in the following form

A series of this sort, where the numbers a₁, a₂, ··· are constants independent of x, iswalled a power series.

As an example let us consider the power series

whose terms form a geometric progression.

We know that for all values of x in the interval –1 < x < 1 this series converges and its sum is equal to

For other values of x the series diverges.

It is also easy to see that the difference between the sum of the series and the sum of its first n terms is given by the formula

and if where q is a positive number less than unity, then

From this it is clear that ηn approaches zero with increasing n so that the series is uniformly convergent on the interval for all positive values of q < 1.

It is easy to verify that the function

has a derivative of nth order, which is equal to

from which

and the sum of the first n terms of the Taylor series for the function S(x) exactly coincides for x₀ = 0 with the sum of the first n terms of the series (59). Moreover, we know that the remainder term of the formula, given by the equality (62), approaches zero with increasing n, for all x on the interval –1 < x < 1. Thus we have shown that the series (61) is the Taylor series of its sum S(x).

Let us note one further fact. From the interval of convergence –1 < x < 1 of our series, let us choose an arbitrary point x₀. It is easy to see that for all x sufficiently close to x₀, namely for all x satisfying the inequality

we have the equality

The reader may prove without difficulty that

Consequently, series (63) is the Taylor series of its sum S(x) and converges to it in a sufficiently small neighborhood of any point x₀ belonging to the interval of convergence of (61). Since the point x₀ is arbitrary, this means that the function S(x) is analytic on the interval.

All these facts that we have observed for the particular power series (61) hold for arbitrary power series.* Namely, for every power series of the form (60) where the constants ak are chosen by any given law, there exists a certain nonnegative number R (which may also be infinite), called the radius of convergence of the series (60), with the following properties:

1. For all values of x from the interval x₀ – R < x < x₀ + R, which is called its interval of convergence, the series converges and its sum S(x) is an analytic function of x in its interval. Here the convergence is uniform for every interval [a, b] lying completely within the interval of convergence. The series itself is the Taylor series of its sum.

2. At the end points of the interval of convergence, the series may converge or diverge, depending on its individual character. But it will certainly diverge outside the closed interval

We suggest to the reader that he consider the power series

and convince himself that their radii of convergence are respectively infinity, zero, and unity.

By the definition given earlier every analytic function may be expanded, in a sufficiently small neighborhood of an arbitrary point where it is defined, into a power series which converges to the function. Conversely, from what has been said it follows that the sum of every power series whose radius of convergence is not zero is an analytic function in its interval of convergence.

So we see that power series are organically connected with analytic functions. We could even say that on their interval of convergence power series are the natural means of representing analytic functions, and consequently they are also the natural means of approximating analytic functions by algebraic polynomials.*

For example, from the fact that the function 1/(1 – x) may be expanded in the power series

which is convergent on the interval –1 < x < 1, it follows that the power series is uniformly convergent on an arbitrary interval with a < 1, and this implies the possibility of approximating the function on the whole interval [–a, a] by the partial sums of the series with any preassigned degree of accuracy.

Let us suppose that we are required to approximate the function 1/(1 – x) by polynomials on the interval with an accuracy of 0.01. We note that for all x in this interval we have the inequality

and since 2⁶ = 64, and 2⁷ = 128, the desired polynomial, approximating the function on the whole interval with an accuracy of 0.01, will have the form

Let us note one further extremely valuable property of power series: They may be differentiated termwise everywhere in the interval of convergence. This property finds extremely wide application in the solution of various problems in mathematics.

For example, let it be required to find the solution of the differential equation y′ = y under the auxiliary condition y(0) = 1. We will seek the solution in the form of a power series,

Because of the auxiliary condition, we must set a₀ = 1. Assuming that this series converges, we may differentiate it termwise; as a result we obtain

If we substitute these two series into the differential equation and equate coefficients for each of the powers of x, we obtain

and the desired solution has the form

It is well known that this series converges for all values of x and that its sum is equal to y = ex.

In this case we have obtained a series whose sum is a well-known elementary function. But this does not always happen; it may turn out that a convergent power series so obtained has a sum that is not an elementary function. An example is the series

obtained as a solution of Bessel’s differential equation, which is of great importance in applications. In this way power series may serve to define functions.

Suggested Reading

R. Courant, Differential and integral calculus, 2 vols., Interscience, New York, 1938.

A. Dresden, Introduction to the calculus, Henry Holt, New York, 1940.

F. Klein, Elementary mathematics from an advanced standpoint. Arithmetic–algebra–analysis, Dover, New York, 1953.

K. Knopp, Infinite sequences and series, Dover, New York, 1956.

*This section is based on the beautiful essay of Academician S. I. Vavilov “Galileo” (Great Soviet Encyclopedia, Volume 10, 1952).

† Nowadays formula (1) is deduced from the general laws of mechanics, but historically it was just this formula which, after being established experimentally by Galileo, served as a part of the accumulation of experience that was subsequently generalized by those laws.

*The number x is the abscissa and y is the ordinate of the point M.

*This does not mean that such an expression is impossible. In Chapter XII we will show how to obtain a single formula.

*Of course, the above statement will be completely clear to the reader only after we have given a precise definition of exactly what is meant in mathematics by the term “formula” and “analytic expression.”

*If in the obvious equalities (k + 1)³ – k³ = 3k² + 3k + 1, for the different values k = 1, 2, ···, n – 1, we add the left and right sides separately, we obtain the equation

where σn = 1² + 2² + . . . + (n – 1)². Solving this equation for σn, we get

† For example, if = 0.001, we may take N = 500. In fact, since

for positive integral n, therefore

for arbitrary n > 500. In the same way it would be possible to assign arbitrarily small values , for example:

and for each of them to choose, as above, appropriate values N = N₁, N₂, ···.

*In this connection we should remark that addition, subtraction, multiplication, and division (excluding division by zero) of rational numbers, that is numbers of the form p/q where p and q are integers, leads to rational numbers. But this is not necessarily the case for the operation of taking a limit. The limit of a sequence of rational numbers may be irrational number.

† It is not important that the polygon should be regular. The only essential feature is that the greatest side of the variable inscribed polygon should approach zero.

*It is sufficient to set the function equal to unity at the irrational points.

*It is understood that we are speaking here of the case where the limit in question actually exists. If this limit does not exist, then we say that at the point x the function does not have a derivative.

*We always assume here that h ≠ 0.

*Here Ax is never equal to zero.

†The final summand here approaches zero for Δx + 0, since Δv/Δx approaches a finite number, equal to the derivative v′, which was assumed from the beginning to exist, and Δu + 0, since the function u, assumed to have a derivative, is continuous.

*In deducing this formula we have tacitly assumed that, as Δx approaches zero, Δu is never equal to zero. But the formula remains true even when this assumption does not hold.

*Strictly defined, the “convexity upward” is that property of the curve that consists of its lying above (more precisely “not below”) the chord joining any two of its points; analogously, for “convexity dowriward” (which is also simply called “concavity”), the curve does not lie above its chords.

*In more complicated cases, where the second derivative itself changes sign, the problem of determining the character of the stationary point is solved by means of the Taylor formula (§9).

*This formula gives all the solutions.

*Of course these arguments only give a geometric interpretation of the theorem and by no means form a rigorous proof.

† Formula (23) can be derived by a simple application of the mean-value theorem to the function

‡ By the symbols [a, b] and (a, b) we denote the sets of values of x satisfying the inequalities and a < x < b respectively.

*The same rule is valid for finding the limit of a fractional expression in which the numerator and the denominator both approach infinity. This method, which is very convenient for finding such limits (or, as we say, for the removal of indeterminacies), will be used, for example, in §3 of Chapter XII.

*This is only one of the possible forms for the remainder term Rn + 1(x).

*By the mean value theorem

when ν lies between t and t₀. Thus ϕ(t) = ϕ(t₀) = const for all t.

† It is possible to prove mathematically, without recourse to examples from mechanics, that if the function f(x) is continuous (and even if it is discontinuous but Lebesguesummable; see Chapter XV) on the interval [a, b], then there exists a primitive F(x) satisfying equality (30).

‡This formula has been generalized in various ways (see for example §13, the formula of Ostrogradskiĭ).

*The discoveries of Newton and Leibnitz were made independently.

*For example, it follows formally from a = b and b = c that a = c. But in practice this relation appears as follows: From the facts that a = b is known with accuracy up to and b = c is known with the same accuracy, it follows that a = c is known with accuracy up to 2.

* For x > 0. (In | x |)′ = (In x)′ = l/x; for x < 0, (In | x |)′ = [In(–x)]′ = 1/– x(–1) = l/x.

* In general a function Ax + By + C, where A, B, C are constants, is called a linear function of x and y. If C = 0, it is called a homogeneous linear function. Here we omit the word “homogeneous.”

* We assume that F(x, y) has continuous derivatives with respect to x and y.

* We are speaking here, of course, of a maximum attained in the domain of definition of the function F(x, y). The functions f(x, y) and ϕ(x, y) are assumed to be differentiable.

* In the course in higher mathematics of V. I. Smirnov, the reader will find a simple example where this particular feature of the situation would lead to the loss of a solution if we apply the method of Lagrange mechanically and do not consider, in addition to the points mentioned above, a point where not only (45) holds but also:

* The diameter of a region is defined as the least upper bound of the distance between two points of the region.

* More precisely, through a cylindrical surface with the contour for its base and with height equal to unity.

* Since for small displacements along the curve the differential of the coordinate y is equal to cos(, ) ds and the differential dx is equal to –cos(, ) ds, this latter integral is often written in the form

* Let us note that it is also possible to give generalized definitions of the sum of a series, by virtue of which it is possible to assign to certain divergent series a more or less natural concept of “generalized sum.” Such series are said to be summable. Operations with generalized sums of divergent series are sometimes useful.

* See Chapter XV.

† sup is an abbreviation for the Latin word supremum (highest).

* For more detailed information on this point see Chapter IX.

* Approximations going beyond the. limits of the interval of convergence of a power series require other methods. (See Chapter XII.)

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