This article is based on a lecture given by Peter Neumann (a son of Bernhard Neumann and Hanna Neumann) at a conference at the University of Sussex on 19 March 2001 to celebrate the 90th birthday of Walter Ledermann. The talk was entitled Introduction to the theory of finite groups the title of the famous text written by Walter Ledermann. This article is based on notes taken by EFR at that lecture.
The modern definition of a group is usually given in the following way.
Definition
A group G is a set with a binary operation G G G which assigns to every ordered pair of elements x, y of G a unique third element of G (usually called the product of x and y) denoted by xy such that the following four properties are satisfied:
Closure: if x, y are in G then xy is in G.
Associative law: if x, y, z are in G then x(yz) = (xy)z.
Identity element: there is an element e in G with ex = xe = x for all x in G.
Inverses: for every x in G there is an element u in G with xu = ux = e.
The first point to make is that 1. is debatable as an axiom since it is a consequence of the definition of a binary operation. However it is not our purpose to debate this here and it is convention that this axiom is included.
Where did this, now standard, definition come from? Particularly we wish to examine some moves towards this definition made in the 19th century. We are not, therefore, concerned here with the bulk of the work done in group theory in the 19th century which concerned the study of permutation groups required for Galois theory. It is important to realise that the abstract definition of a group was merely an esoteric sideline of group theory through the 19th century.
Let us first note that there were two meanings of the term "abstract group" during the first half of the 20th century - say from 1905 to 1955. One meaning was that of a group defined by the four axioms as above, while the second meaning was that of a group defined by generators and relations. For example Todd used the term in this second way when he talked about "the Mathieu groups as abstract groups". Here we are only interested in the first of these meanings.
The emergence of the abstract group concept was a remarkably slow process. The story begins with the prehistory which involves Galois and Cauchy. Galois defined a group in 1832 although it did not appear in print until Liouville published Galois' papers in 1846. The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Académie des Sciences in 1829. René Taton has found evidence in the archives of the Académie which suggest that Cauchy spoke with Galois and persuaded him to withdraw the paper and submit a new version of it for the Grand Prix of 1830. This is based on strong circumstantial evidence, but we do know that Galois submitted to Fourier a new version of his paper to be considered for the Grand Prix in March 1830.
Fourier died shortly after this and Galois' paper was lost. It was never considered for the prize which was awarded jointly to Abel (posthumously) and Jacobi in July 1830. Galois was invited by Poisson to submit a third version of his memoir on equations to the Académie and he did so on 17 January 1831. This version of the paper was refereed by Poisson who rejected it but wrote a very sympathetic report. Although Galois had proved the results in general, the paper only considered equations of prime degree. Poisson failed to understand the paper and suggested that the arguments were developed further. It was unclear to him how Galois' results classified which equations were soluble by radicals.
The night before he fought the duel which led to his death Galois made notes on his papers. One of these notes, made on 29 May 1832, was the following:-
... if in such a group one has the substitutions S and T then one has the substitution ST.
Although Galois had used groups extensively throughout his paper on equations, he had not given a definition. It is little wonder that Poisson found the paper hard to understand for it contains many explicit calculations in a group, yet the concept was not defined - poor Poisson!
Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition. He considered substitutions in n letters x, y, z, ... and defined derived substitutions to be all those which can be deduced by multiplying these substitutions together in any order. He then called the set of substitutions together with the derived substitutions, a "conjugate system of substitutions". For some time these two identical concepts, a "group" and a "conjugate system of substitutions", were both used. However, from 1863 when Jordan wrote a commentary on Galois' work in which he used "group", it became the standard term. This was reinforced when Jordan published his major group theory text Traité des substitutions et des équations algebraique in 1870. However, Cauchy's term "conjugate system of substitutions" continued to be used by some up to about 1880.
How much was Cauchy influenced by Galois? Although he had seen Galois' papers submitted to the Académie, there was no explicit definition of a group in them. On the other hand he must have at least been subliminally influenced. Both Galois and Cauchy, of course, define groups in terms of the closure property alone. The now familiar axioms of associativity, identity and inverses do not appear. Both were dealing with permutations which means that closure is all that is necessary to define a group, the other properties all following automatically. Cauchy went on to write 25 papers on this topic between September 1845 and January 1846.
[Note by EFR. Peter Neumann did not mention in his lecture any influence that Ruffini's ideas might have had on Cauchy. In 1821 Cauchy had written to Ruffini praising his work which he had clearly read. Although there is no explicit definition of a group in Ruffini's work, again the concept clearly appears and may have had as major an influence on Cauchy's thinking as the work of Galois.]
The first person to try to give an abstract definition of a group was Cayley. He wrote a paper on groups in 1854 which he published again in two separate journals in 1878. In the 1854 paper he attempted to give an abstract definition in terms of symbols which operate on a system (x, y, ... ) so that
(x, y, ... ) = (x', y', ... ) where x', y', ... are any functions of x, y, ... .
Cayley went on to define an identity symbol 1 which leaves the members of the system unaltered. He defines the element as the result of operating on the system first by then by . He notes that need not equal . Cayley also requires the associative law . = . to be satisfied. He then says that any set of such symbols with the property that the product of any two is in the set is called a group.
This is an important attempt at an abstract definition of a group but Cayley has overreached himself. What he has here is really a mess. Why does he require the associative law to hold if his symbols are operators? As for permutations the associative law follows automatically for operators. It is also not clear that (x', y', ... ) where x', y', ... are any functions of x, y, ... will be in the system. This definition is not entirely successful.
In 1878 Cayley wrote:-
A group is defined by the law of composition of its members.
This idea was picked up by Burnside, von Dyck and others. Burnside, in his book The Theory of Groups of Finite Order published in 1897 gave the following definition:-
Let A, B, C, ... represent a set of operations which can be performed on the same object or set of objects.
He then supposes that any two of the operations are distinct in the sense that no two produce the same effect. He follows Cayley in requiring closure, the associative law, and inverses. Again his definition is subject to the same criticism at Cayley's definition. If his elements are operations then why does he need to postulate the associative law? Rather strangely Burnside does not assume the existence of an identity, although one can infer it from the fact that A and A -1 are in the set and we have closure. Burnside repeats exactly the same definition in the second edition of his book which appeared in 1911.
It is worth noting that neither Cayley nor Burnside insisted that their groups were finite. The definition deliberately allowed for the possibility of infinite groups, and Burnside in particular was interested in studying infinite groups. What we have given here is part of a sequence of development which we might call the English school. There were other bits in the sequence between these major contributions which we have omitted. We now turn to another development of group theory which was going on at the same time which we might call the European school.
In 1870 Kronecker gave a definition of a group in a completely different context, namely the context of a class group in algebraic number theory. He takes a specifically finite set ', '', ''', ... such that from any two, a third can be derived by a specific method. He then assumes that the commutative and associative laws hold and also that ''' '''' if '' '''. This appears to be a separate development by Kronecker who does not tie it in with previous work on groups. However, Heinrich Weber in 1882 gave a very similar definition to that of Kronecker yet he did tie it in with previous work on groups.
Heinrich Weber defined a group of degree h, like Kronecker in the context of class groups, again to be a finite set. He required that from two elements of the system one can derive a third element of the system so that the following hold:-
(rs)t = r(st) = rst
r = s implies r = s.
A few comments here. The associative law is put in a slightly strange way. The expression rst has no meaning until the associative law is defined to hold, so in a modern treatment one might write:
(rs)t = r(st) so that either side may be denoted by rst.
One can still see that there is a little lack of clarity in the definition. What Heinrich Weber is in fact defining is a semigroup with cancellation and, given that it is finite, this is sufficient to ensure the existence of an identity and of inverses. That this fails for infinite systems was noted by Heinrich Weber in his famous text Lehrbuch der Algebra published in 1895. In this book he notes that the above definition of a group only works for finite groups and in the infinite case one needs to postulate inverses explicitly.
We should point out that there appears to be relatively little cross-fertilisation between the two lines of development. There was some, however. For instance Burnside did read Frobenius's papers, although he did so later than he should have done and had to apologise twice to Frobenius for not knowing what he had already published. Recently a letter from Burnside to Schur has been discovered and we know that Burnside corresponded with Hölder.
We do know where some of the 20th century influences came from - Emmy Noether, an important 20th century figure particularly influential through van der Waerden's Algebra book, was strongly influenced by Heinrich Weber. On the other hand Schur, another influential 20th century figure, was influenced by Frobenius.
Monday, October 15, 2007
The function concept
If today we try to answer the difficult question "What is mathematics?" we often respond with an answer such as "It is the study of relations on sets" or "It is the study of functions on sets" or "It is the study of dependencies among variable quantities". If these statements come anywhere close to the truth then it might be logical to suggest that the concept of a function must have arisen in the very earliest stages in the development of mathematics. Indeed if we look at Babylonian mathematics we find tables of squares of the natural numbers, cubes of the natural numbers, and reciprocals of the natural numbers. These tables certainly define functions from N to N. E T Bell wrote in 1945:-
It may not be too generous to credit the ancient Babylonians with the instinct for functionality; for a function has been successively defined as a table or a correspondence.
However this surely is the result of modern mathematicians seeing ancient mathematics through modern eyes. Although we can see that the Babylonians were dealing with functions, they would not have thought in these terms. We therefore have to reject the suggestion that the concept of a function was present in Babylonian mathematics even if we can see that they were studying particular functions.
If we move forward to Greek mathematics then we reach the work of Ptolemy. He computed chords of a circle which essentially means that he computed trigonometric functions. Surely, one might think, if he was computing trigonometric functions then Ptolemy must have understood the concept of a function. As O Petersen wrote in 1974 in [22]:-
But if we conceive a function, not as a formula, but as a more general relation associating the elements of one set of numbers with the elements of another set, it is obvious that functions in that sense abound throughout the Almagest.
Indeed Petersen is certainly correct to say that functions, in the modern sense, occur throughout the Almagest. Ptolemy dealt with functions, but it is very unlikely that he had any understanding of the concept of a function. As Thiele writes on the first page of [2]:-
From time to time, anachronistic comparisons like the one just given help us with the elucidation of documented facts, but not with the interpretation of their history.
Having suggested that the concept of a function is absent in these ancient pieces of mathematics, let us suggest, as Youschkevitch does in [32], that Oresme was getting closer in 1350 when he described the laws of nature as laws giving a dependence of one quantity on another. Youschkevitch writes [32]:-
The notion of a function first occurred in more general form in the 14th century in the schools of natural philosophy at Oxford and Paris.
Galileo was beginning to understand the concept even more clearly. His studies of motion contain the clear understanding of a relation between variables. Again another piece of his mathematics shows how he was beginning to grasp the concept of a mapping between sets. In 1638 he studied the problem of two concentric circles with centre O, the larger circle A with diameter twice that of the smaller one B. The familiar formula gives the circumference of A to be twice that of B. But taking any point P on the circle A, then PA cuts circle B in one point. So Galileo had constructed a function mapping each point of A to a point of B. Similarly if Q is a point on B then OQ produced cuts circle A in exactly one point. Again he has a function, this time from points of B to points of A. Although the circumference of A is twice the length of the circumference of B they have the same number of points. He also produced the standard one-to-one correspondence between the positive integers and their squares which (in modern terms) gave a bijection between N and a proper subset.
At almost the same time that Galileo was coming up with these ideas, Descartes was introducing algebra into geometry in La Géométrie. He says that a curve can be drawn by letting lines take successively an infinite number of different values. This again brings the concept of a function into the construction of a curve, for Descartes is thinking in terms of the magnitude of an algebraic expression taking an infinity of values as a magnitude from which the algebraic expression is composed takes an infinity of values.
Let us pause for a moment before reaching the first use of the word "function". It is important to understand that the concept developed over time, changing its meaning as well as being defined more precisely as decades went by. We have already suggested that a table of values, although defining a function, need not be thought of by the creator of the table as a function. Early uses of the word "function" did encapsulate ideas of the modern concept but in a much more restrictive way.
Like so many mathematical terms, the word function was first used with its usual non-mathematical meaning. Leibniz wrote in August 1673 of:-
... other kinds of lines which, in a given figure, perform some function.
Johann Bernoulli, in a letter to Leibniz written on 2 September 1694, described a function as:-
... a quantity somehow formed from indeterminate and constant quantities.
In a paper in 1698 on isoperimetric problems Johann Bernoulli writes of "functions of ordinates" (see [32]). Leibniz wrote to Bernoulli saying:-
... I am pleased that you use the term function in my sense.
It was a concept whose introduction was particularly well timed as far as Johann Bernoulli was concerned for he was looking at problems in the calculus of variations where functions occur as solutions. See [28] for more information about how the author considers the calculus of variations to be the mathematical theory which developed most intimately in connection with the concept of a function.
One can say that in 1748 the concept of a function leapt to prominence in mathematics. This was due to Euler who published Introductio in analysin infinitorum in that year in which he makes the function concept central to his presentation of analysis. Euler defined a function in the book as follows:-
A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
This is all very well but Euler gives no definition of "analytic expression" rather he assumes that the reader will understand it to mean expressions formed from the usual operations of addition, multiplication, powers, roots, etc. He divides his functions into different types such as algebraic and transcendental. The type depends on the nature of the analytic expression, for example transcendental functions are not algebraic such as:-
... exponentials, logarithms, and others which integral calculus supplies in abundance.
Euler allowed the algebraic operations in his analytic expressions to be used an infinite number of times, resulting in infinite series, infinite products, and infinite continued fractions. He later suggests that a transcendental function should be studied by expanding it in a power series. He does not claim that all transcendental functions can be expanded in this was but says that one should prove it in each specific case. However there was a difficulty in Euler's work which was to lead to confusion, for he failed to distinguish between a function and its representation. However Introductio in analysin infinitorum was to change the way that mathematicians thought about familiar concepts. Jahnke writes [2]:-
Until Euler the trigonometric quantities sine, cosine, tangent etc., were regarded as lines connected with the circle rather than functions. ... It was Euler who introduced the functional point of view.
The function concept had led Euler to make many important discoveries before he wrote Introductio in analysin infinitorum. For example it had led him to define the gamma function and to solve the problem which had defeated mathematicians for some considerable time, namely summing the series
1/12 + 1/22 + 1/32 + 1/42 + ...
He showed that the sum was π2/6, publishing the result in 1740.
Let us return to the contents of Introductio in analysin infinitorum. In it Euler introduced continuous, discontinuous and mixed functions but since the first two of these concepts have different modern meanings we will choose to call Euler's versions E-continuous and E-discontinuous to avoid confusion. An E-continuous function was one which was expressed by a single analytic expression, a mixed function was expressed in terms of two or more analytic expressions, and an E-discontinuous function included mixed functions but was a more general concept. Euler did not clearly indicate what he meant by an E-discontinuous function although it was clear that Euler thought of them as more general than mixed functions. He later defined them as those functions which had arbitrarily handdrawn curves as their graphs (rather confusingly essentailly what we call a continuous function today).
In 1746 d'Alembert published a solution to the problem of a vibrating stretched string. The solution, of course, depended on the initial form of the string and d'Alembert insisted in his solution that the function which described the initial velocities of the each point of the string had to be E-continuous, that is expressed by a single analytic expression. Euler published a paper in 1749 which objected to this restriction imposed by d'Alembert, claiming that for physical reasons more general expressions for the initial form of the string had to be allowed. Youschkevitch writes [32]:-
d'Alembert did not agree with Euler. Thus began the long controversy about the nature of functions to be allowed in the initial conditions and in the integrals of partial differential equations, which continued to appear in an ever increasing number in the theory of elasticity, hydrodynamics, aerodynamics, and differential geometry.
In 1755 Euler published another highly influential book, namely Institutiones calculi differentialis. In this book he defined a function in an entirely general way, giving what we might reasonably say was a truly modern definition of a function:-
If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by other. If, therefore, x denotes a variable quantity, then all quantities which depend upon x in any way, or are determined by it, are called functions of x.
This might have been a huge breakthrough but after giving this wide definition, Euler then devoted the book to the development of the differential calculus using only analytic functions. The first problems with Euler's definition of types of functions was pointed out in 1780 when it was shown that a mixed function, given by different formulas, could sometimes be given by a single formula. The clearest example of such a function was given by Cauchy in 1844 when he noted that the function
y = x for x 0, y = -x for x < 0
can be expressed by the single formula y = √(x2). Hence dividing functions into E-continuous or mixed was meaningless. However, a more serious objection came through the work of Fourier who stated in 1805 that Euler was wrong. Fourier showed that some discontinuous functions could be represented by what today we call a Fourier series. The distinction between E-continuous and E-discontinuous functions, therefore, did not exist. Fourier's work was not immediately accepted and leading mathematicians such as Lagrange did not accept his results at this stage. Luzin points out in [17] and [18] that confusion regarding functions had been due to a lack of understanding of the distinction between a "function" and its "representation", for example as a series of sines and cosines. Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
Other mathematicians gave their own versions of the definition of a function. Condorcet seems to have been the first to take up Euler's general definition of 1755, see [31] for details. In 1778 the first two parts of Condorcet intended five part work Traité du calcul integral was sent to the Paris Academy. It was never published but was seen by many leading French mathematicians. In this work Condorcet distinguished three types of functions: explicit functions, implicit functions given only by unsolved equations, and functions which are defined from physical considerations such as being the solution to a diffferential equation.
Lacroix, who had read Condorcet's unfinished work, wrote in 1797:-
Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operation it is necessary to use to arrive from the latter to the first.
Cauchy, in 1821, came up with a definition making the dependence between variables central to the function concept. He wrote in Cours d'anlyse:-
If variable quantities are so joined between themselves that, the value of one of these being given, one can conclude the values of all the others, one ordinarily conceives these diverse quantities expressed by means of the one of them, which then takes the name independent variable; and the other quantities expressed by means of the independent variable are those which one calls functions of this variable.
Notice that despite the generality of Cauchy's definition, which is designed to cover the case of explicit and implicit functions, he is still thinking of a function in terms of a formula. In fact he makes the distinction between explicit and implicit functions immediately after giving this definition. He also introduces concepts which indicate that he is still thinking in terms of analytic expressions.
Fourier, in Théorie analytique de la Chaleur in 1822, gave the following definition:-
In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given of the abscissa x, there are an equal number of ordinates f(x). All have actual numerical values, either positive or negative or nul. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as it were a single quantity.
It is clear that Fourier has given a definition which deliberately moves away from analytic expressions. However, despite this, when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he uses the fact that his arbitrary function is continuous in the modern sense!
Dirichlet, in 1837, accepted Fourier's definition of a function and immediately after giving this definition he defined a continuous function (using continuous in the modern sense). Dirichlet also gave an example of a function defined on the interval [ 0, 1] which is discontinuous at every point, namely f(x) which is defined to be 0 if x is rational and 1 if x is irrational.
In 1838 Lobachevsky gave a definition of a general function which still required it to be continuous:-
A function of x is a number which is given for each x and which changes gradually together with x. The value of the function could be given either by an analytic expression or by a condition which offers a means for testing all numbers and selecting one from them, or lastly the dependence may exist but remain unknown.
Certainly Dirichlet's everywhere discontinuous function will not be a function under Lobachevsky's definition. Hankel, in 1870, deplored the confusion which still reigned in the function concept:-
One person defines functions essentially in Euler's sense, the other requires that y must change with x according to a law, without giving an explanation of this obscure concept, the third defines it in Dirichlet's manner, the fourth does not define it at all. However, everybody deduces from his concept conclusions that are not contained in it.
Around this time many pathological functions were constructed. Cauchy gave an early example when he noted that f(x) = exp(-1/x2) for x 0, f(0) = 0, is a continuous function which has all its derivatives at 0 equal to 0. It therefore has a Taylor series which converges everywhere but only equals the function at 0. In 1876 Paul du Bois-Reymond made the distinction between a function and its representation even clearer when he constructed a continuous function whose Fourier series diverges at a point. This line was taken further in 1885 when Weierstrass showed that any continuous function is the limit of a uniformly convergent sequence of polynomials. Earlier, in 1872, Weierstrass had sent a paper to the Berlin Academy of Science giving an example of a continuous function which is nowhere differentiable. Lützen writes in [2]:-
Weierstrass's function contradicted an intuitive feeling held by most of his contemporaries to the effect that continuous functions were differentiable except at "special points". it created a sensation and, according to Hankel, disbelief when du Bois-Reymond published it in 1875.
Poincaré was unhappy with the direction that the definition of functions had taken. He wrote in 1899:-
For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. ... Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show that our ancestor's reasoning was at fault, and we shall never get anything more than that out of them. If logic were the teacher's only guide, he would have to begin with the most general, that is to say, the most weird functions.
Where have more modern definitions taken the concept? Goursat, in 1923, gave the definition which will appear in most textbooks today:-
One says that y is a function of x if to a value of x corresponds a value of y. One indicates this correspondence by the equation y = f(x).
Just in case this is not precise enough and involves undefined concepts such as 'value' and 'corresponds', look at the definition given by Patrick Suppes in 1960:-
Definition. A is a relation (x)(x A (y)(z)(x = (y, z)). We write y A z if (y, z) A.
Definition. f is a function f is a relation and (x)(y)(z)(x f y and x f z y = z).
What would Poincaré have thought of Suppes' definition?
It may not be too generous to credit the ancient Babylonians with the instinct for functionality; for a function has been successively defined as a table or a correspondence.
However this surely is the result of modern mathematicians seeing ancient mathematics through modern eyes. Although we can see that the Babylonians were dealing with functions, they would not have thought in these terms. We therefore have to reject the suggestion that the concept of a function was present in Babylonian mathematics even if we can see that they were studying particular functions.
If we move forward to Greek mathematics then we reach the work of Ptolemy. He computed chords of a circle which essentially means that he computed trigonometric functions. Surely, one might think, if he was computing trigonometric functions then Ptolemy must have understood the concept of a function. As O Petersen wrote in 1974 in [22]:-
But if we conceive a function, not as a formula, but as a more general relation associating the elements of one set of numbers with the elements of another set, it is obvious that functions in that sense abound throughout the Almagest.
Indeed Petersen is certainly correct to say that functions, in the modern sense, occur throughout the Almagest. Ptolemy dealt with functions, but it is very unlikely that he had any understanding of the concept of a function. As Thiele writes on the first page of [2]:-
From time to time, anachronistic comparisons like the one just given help us with the elucidation of documented facts, but not with the interpretation of their history.
Having suggested that the concept of a function is absent in these ancient pieces of mathematics, let us suggest, as Youschkevitch does in [32], that Oresme was getting closer in 1350 when he described the laws of nature as laws giving a dependence of one quantity on another. Youschkevitch writes [32]:-
The notion of a function first occurred in more general form in the 14th century in the schools of natural philosophy at Oxford and Paris.
Galileo was beginning to understand the concept even more clearly. His studies of motion contain the clear understanding of a relation between variables. Again another piece of his mathematics shows how he was beginning to grasp the concept of a mapping between sets. In 1638 he studied the problem of two concentric circles with centre O, the larger circle A with diameter twice that of the smaller one B. The familiar formula gives the circumference of A to be twice that of B. But taking any point P on the circle A, then PA cuts circle B in one point. So Galileo had constructed a function mapping each point of A to a point of B. Similarly if Q is a point on B then OQ produced cuts circle A in exactly one point. Again he has a function, this time from points of B to points of A. Although the circumference of A is twice the length of the circumference of B they have the same number of points. He also produced the standard one-to-one correspondence between the positive integers and their squares which (in modern terms) gave a bijection between N and a proper subset.
At almost the same time that Galileo was coming up with these ideas, Descartes was introducing algebra into geometry in La Géométrie. He says that a curve can be drawn by letting lines take successively an infinite number of different values. This again brings the concept of a function into the construction of a curve, for Descartes is thinking in terms of the magnitude of an algebraic expression taking an infinity of values as a magnitude from which the algebraic expression is composed takes an infinity of values.
Let us pause for a moment before reaching the first use of the word "function". It is important to understand that the concept developed over time, changing its meaning as well as being defined more precisely as decades went by. We have already suggested that a table of values, although defining a function, need not be thought of by the creator of the table as a function. Early uses of the word "function" did encapsulate ideas of the modern concept but in a much more restrictive way.
Like so many mathematical terms, the word function was first used with its usual non-mathematical meaning. Leibniz wrote in August 1673 of:-
... other kinds of lines which, in a given figure, perform some function.
Johann Bernoulli, in a letter to Leibniz written on 2 September 1694, described a function as:-
... a quantity somehow formed from indeterminate and constant quantities.
In a paper in 1698 on isoperimetric problems Johann Bernoulli writes of "functions of ordinates" (see [32]). Leibniz wrote to Bernoulli saying:-
... I am pleased that you use the term function in my sense.
It was a concept whose introduction was particularly well timed as far as Johann Bernoulli was concerned for he was looking at problems in the calculus of variations where functions occur as solutions. See [28] for more information about how the author considers the calculus of variations to be the mathematical theory which developed most intimately in connection with the concept of a function.
One can say that in 1748 the concept of a function leapt to prominence in mathematics. This was due to Euler who published Introductio in analysin infinitorum in that year in which he makes the function concept central to his presentation of analysis. Euler defined a function in the book as follows:-
A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
This is all very well but Euler gives no definition of "analytic expression" rather he assumes that the reader will understand it to mean expressions formed from the usual operations of addition, multiplication, powers, roots, etc. He divides his functions into different types such as algebraic and transcendental. The type depends on the nature of the analytic expression, for example transcendental functions are not algebraic such as:-
... exponentials, logarithms, and others which integral calculus supplies in abundance.
Euler allowed the algebraic operations in his analytic expressions to be used an infinite number of times, resulting in infinite series, infinite products, and infinite continued fractions. He later suggests that a transcendental function should be studied by expanding it in a power series. He does not claim that all transcendental functions can be expanded in this was but says that one should prove it in each specific case. However there was a difficulty in Euler's work which was to lead to confusion, for he failed to distinguish between a function and its representation. However Introductio in analysin infinitorum was to change the way that mathematicians thought about familiar concepts. Jahnke writes [2]:-
Until Euler the trigonometric quantities sine, cosine, tangent etc., were regarded as lines connected with the circle rather than functions. ... It was Euler who introduced the functional point of view.
The function concept had led Euler to make many important discoveries before he wrote Introductio in analysin infinitorum. For example it had led him to define the gamma function and to solve the problem which had defeated mathematicians for some considerable time, namely summing the series
1/12 + 1/22 + 1/32 + 1/42 + ...
He showed that the sum was π2/6, publishing the result in 1740.
Let us return to the contents of Introductio in analysin infinitorum. In it Euler introduced continuous, discontinuous and mixed functions but since the first two of these concepts have different modern meanings we will choose to call Euler's versions E-continuous and E-discontinuous to avoid confusion. An E-continuous function was one which was expressed by a single analytic expression, a mixed function was expressed in terms of two or more analytic expressions, and an E-discontinuous function included mixed functions but was a more general concept. Euler did not clearly indicate what he meant by an E-discontinuous function although it was clear that Euler thought of them as more general than mixed functions. He later defined them as those functions which had arbitrarily handdrawn curves as their graphs (rather confusingly essentailly what we call a continuous function today).
In 1746 d'Alembert published a solution to the problem of a vibrating stretched string. The solution, of course, depended on the initial form of the string and d'Alembert insisted in his solution that the function which described the initial velocities of the each point of the string had to be E-continuous, that is expressed by a single analytic expression. Euler published a paper in 1749 which objected to this restriction imposed by d'Alembert, claiming that for physical reasons more general expressions for the initial form of the string had to be allowed. Youschkevitch writes [32]:-
d'Alembert did not agree with Euler. Thus began the long controversy about the nature of functions to be allowed in the initial conditions and in the integrals of partial differential equations, which continued to appear in an ever increasing number in the theory of elasticity, hydrodynamics, aerodynamics, and differential geometry.
In 1755 Euler published another highly influential book, namely Institutiones calculi differentialis. In this book he defined a function in an entirely general way, giving what we might reasonably say was a truly modern definition of a function:-
If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by other. If, therefore, x denotes a variable quantity, then all quantities which depend upon x in any way, or are determined by it, are called functions of x.
This might have been a huge breakthrough but after giving this wide definition, Euler then devoted the book to the development of the differential calculus using only analytic functions. The first problems with Euler's definition of types of functions was pointed out in 1780 when it was shown that a mixed function, given by different formulas, could sometimes be given by a single formula. The clearest example of such a function was given by Cauchy in 1844 when he noted that the function
y = x for x 0, y = -x for x < 0
can be expressed by the single formula y = √(x2). Hence dividing functions into E-continuous or mixed was meaningless. However, a more serious objection came through the work of Fourier who stated in 1805 that Euler was wrong. Fourier showed that some discontinuous functions could be represented by what today we call a Fourier series. The distinction between E-continuous and E-discontinuous functions, therefore, did not exist. Fourier's work was not immediately accepted and leading mathematicians such as Lagrange did not accept his results at this stage. Luzin points out in [17] and [18] that confusion regarding functions had been due to a lack of understanding of the distinction between a "function" and its "representation", for example as a series of sines and cosines. Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
Other mathematicians gave their own versions of the definition of a function. Condorcet seems to have been the first to take up Euler's general definition of 1755, see [31] for details. In 1778 the first two parts of Condorcet intended five part work Traité du calcul integral was sent to the Paris Academy. It was never published but was seen by many leading French mathematicians. In this work Condorcet distinguished three types of functions: explicit functions, implicit functions given only by unsolved equations, and functions which are defined from physical considerations such as being the solution to a diffferential equation.
Lacroix, who had read Condorcet's unfinished work, wrote in 1797:-
Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operation it is necessary to use to arrive from the latter to the first.
Cauchy, in 1821, came up with a definition making the dependence between variables central to the function concept. He wrote in Cours d'anlyse:-
If variable quantities are so joined between themselves that, the value of one of these being given, one can conclude the values of all the others, one ordinarily conceives these diverse quantities expressed by means of the one of them, which then takes the name independent variable; and the other quantities expressed by means of the independent variable are those which one calls functions of this variable.
Notice that despite the generality of Cauchy's definition, which is designed to cover the case of explicit and implicit functions, he is still thinking of a function in terms of a formula. In fact he makes the distinction between explicit and implicit functions immediately after giving this definition. He also introduces concepts which indicate that he is still thinking in terms of analytic expressions.
Fourier, in Théorie analytique de la Chaleur in 1822, gave the following definition:-
In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given of the abscissa x, there are an equal number of ordinates f(x). All have actual numerical values, either positive or negative or nul. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as it were a single quantity.
It is clear that Fourier has given a definition which deliberately moves away from analytic expressions. However, despite this, when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he uses the fact that his arbitrary function is continuous in the modern sense!
Dirichlet, in 1837, accepted Fourier's definition of a function and immediately after giving this definition he defined a continuous function (using continuous in the modern sense). Dirichlet also gave an example of a function defined on the interval [ 0, 1] which is discontinuous at every point, namely f(x) which is defined to be 0 if x is rational and 1 if x is irrational.
In 1838 Lobachevsky gave a definition of a general function which still required it to be continuous:-
A function of x is a number which is given for each x and which changes gradually together with x. The value of the function could be given either by an analytic expression or by a condition which offers a means for testing all numbers and selecting one from them, or lastly the dependence may exist but remain unknown.
Certainly Dirichlet's everywhere discontinuous function will not be a function under Lobachevsky's definition. Hankel, in 1870, deplored the confusion which still reigned in the function concept:-
One person defines functions essentially in Euler's sense, the other requires that y must change with x according to a law, without giving an explanation of this obscure concept, the third defines it in Dirichlet's manner, the fourth does not define it at all. However, everybody deduces from his concept conclusions that are not contained in it.
Around this time many pathological functions were constructed. Cauchy gave an early example when he noted that f(x) = exp(-1/x2) for x 0, f(0) = 0, is a continuous function which has all its derivatives at 0 equal to 0. It therefore has a Taylor series which converges everywhere but only equals the function at 0. In 1876 Paul du Bois-Reymond made the distinction between a function and its representation even clearer when he constructed a continuous function whose Fourier series diverges at a point. This line was taken further in 1885 when Weierstrass showed that any continuous function is the limit of a uniformly convergent sequence of polynomials. Earlier, in 1872, Weierstrass had sent a paper to the Berlin Academy of Science giving an example of a continuous function which is nowhere differentiable. Lützen writes in [2]:-
Weierstrass's function contradicted an intuitive feeling held by most of his contemporaries to the effect that continuous functions were differentiable except at "special points". it created a sensation and, according to Hankel, disbelief when du Bois-Reymond published it in 1875.
Poincaré was unhappy with the direction that the definition of functions had taken. He wrote in 1899:-
For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. ... Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show that our ancestor's reasoning was at fault, and we shall never get anything more than that out of them. If logic were the teacher's only guide, he would have to begin with the most general, that is to say, the most weird functions.
Where have more modern definitions taken the concept? Goursat, in 1923, gave the definition which will appear in most textbooks today:-
One says that y is a function of x if to a value of x corresponds a value of y. One indicates this correspondence by the equation y = f(x).
Just in case this is not precise enough and involves undefined concepts such as 'value' and 'corresponds', look at the definition given by Patrick Suppes in 1960:-
Definition. A is a relation (x)(x A (y)(z)(x = (y, z)). We write y A z if (y, z) A.
Definition. f is a function f is a relation and (x)(y)(z)(x f y and x f z y = z).
What would Poincaré have thought of Suppes' definition?
An overview of the history of mathematics
Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.
In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra.
Geometric problems relating to similar figures, area and volume were also studied and values obtained for π.
The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18th Century.
However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of Newton. Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.
The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.
Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.
The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.
Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.
The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century.
Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.
Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers
Analysis was driven by the requirements of mathematical physics and astronomy. Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.
The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm's work led to Hilbert and the development of functional analysis.
Notation and communication
There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.
For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.
What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton's notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.
Let us think for a moment how dependent we all are on mathematical notation and convention. Ask any mathematician to solve ax = b and you will be given the answer x = b/a. I would be very surprised if you were given the answer a = b/x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities.
It was not always like this: Harriot used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.
Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign "=" was introduced by Recorde in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written!
Brilliant discoveries?
It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period.
For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered. Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.
Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at "the right time". This too would be completely unfair (although it does go some why to explain why two or more people often discovered something independently around the same time). There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly.
How we view history
We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.
There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.
Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.
A challenge
If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs.
Then the world changed. The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever.
Here is the challenge. What will replace the calculator? You might say that this is an unfair question. However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us.
We can think of faster calculators, smaller calculators, better calculators but I'm asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, .... .
In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra.
Geometric problems relating to similar figures, area and volume were also studied and values obtained for π.
The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18th Century.
However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of Newton. Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.
The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.
Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.
The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.
Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.
The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century.
Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.
Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers
Analysis was driven by the requirements of mathematical physics and astronomy. Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.
The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm's work led to Hilbert and the development of functional analysis.
Notation and communication
There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.
For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.
What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz's notation lead more easily to extending the ideas of the calculus, while Newton's notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton's notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.
Let us think for a moment how dependent we all are on mathematical notation and convention. Ask any mathematician to solve ax = b and you will be given the answer x = b/a. I would be very surprised if you were given the answer a = b/x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities.
It was not always like this: Harriot used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.
Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign "=" was introduced by Recorde in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written!
Brilliant discoveries?
It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period.
For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered. Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.
Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at "the right time". This too would be completely unfair (although it does go some why to explain why two or more people often discovered something independently around the same time). There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly.
How we view history
We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.
There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.
Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.
A challenge
If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs.
Then the world changed. The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever.
Here is the challenge. What will replace the calculator? You might say that this is an unfair question. However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us.
We can think of faster calculators, smaller calculators, better calculators but I'm asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, .... .
Orbits and gravitation
Although the motions of the planets were discussed by the Greeks they believed that the planets revolved round the Earth so are of little interest to us in this article although the method of epicycles is an early application of Fourier series.
The first to propose a system of planetary paths which would set the scene for major advances was Copernicus who in De revolutionibus orbium coelestium (1543), argued that the planets and the Earth moved round the Sun. Although a major breakthrough, Copernicus proposed circular paths for the planets and accurate astronomical observations soon began to show that his proposal was not strictly accurate.
You can see a diagram from De revolutionibus orbium coelestium showing Copernicus's solar system.
In 1600 Kepler became assistant to Tycho Brahe who was making accurate observations of the planets. After Brahe died in 1601 Kepler continued the work, calculating planetary paths to unprecedented accuracy.
Kepler showed that a planet moves round the Sun in an elliptical path which has the Sun in one of its two foci. He also showed that a line joining the planet to the Sun sweeps out equal areas in equal times as the planet describes its path. Both these laws were first formulated for the planet Mars, and published in Astronomia Nova (1609).
You can see a diagram from Astronomia Nova showing Kepler's elliptical path for Mars.
However scientists certainly did not accept Kepler's first two laws with enthusiasm. The first was given a cool reception and was certainly thought to require further work to confirm it. The second of Kepler's laws suffered an even worse fate in being essentially ignored by scientists for around 80 years.
Kepler's third law, that the squares of the periods of planets are proportional to the cubes of the mean radii of their paths, appeared in Harmonice mundi (1619) and, perhaps surprisingly in view of the above comments, was widely accepted right from the time of its publication.
In 1679 Hooke wrote a letter to Newton. In the letter he explained how he considered planetary motion to be the result of a central force continuously diverting the planet from its path in a straight line. Newton did not answer this directly but explained his own idea that the rotation of the Earth could be proved from the fact that an object dropped from the top of a tower should have a greater tangential velocity than one dropped near the foot of the tower.
Newton provided a sketch of the path that the particle would follow, quite incorrectly showing it spiral towards the centre of the Earth. Hooke replied that his theory of planetary motion would lead to the path of the particle being an ellipse so that the particle, were it not for the fact that the Earth was in the way, would return to its original position after traversing the ellipse.
Newton, not one to like being corrected, had to admit that his original sketch was incorrect but he "corrected" Hooke's sketch on the assumption that gravity was constant. Hooke replied to Newton that his own theory involved an inverse square law for gravitational attraction. Many years later Hooke was to claim priority for proposing the inverse square law of gravitation and used this letter to Newton to support his claim.
It is worth emphasising that there is a major step to be made from an inverse square law of force to explain planetary motion and a universal law of gravitation. Certainly the motion of the Moon round the Earth was not seen to necessarily be part of the same laws which govern the motion of the planets round the Sun.
Fifty years after these events Newton was to record his own recollections of these events which, although interesting, do not really agree with the known historical facts! [I preserve Newton's old English.]
In the same year I began to think of gravity extending to ye orb of the Moon and (having found out how to estimate the force with wch globe revolving within a sphere presses the surface of a sphere) from Kepler's rule of the periodical times of the Planets being in sesquialternate proportion to their distances from the centres of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must reciprocally as the squares of their distances from the centres about wch they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly. All this was in the two plague years of 1665-1666...
In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance from the Sun. Halley wrote that
Mr Hook said that he had it, but that he would conceale it for some time so that others, triing and failing might know how to value it, when he should make it publick.
Later in the same year in August, Halley visited Newton in Cambridge and asked him what orbit a body would follow under an inverse square law of force
Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy and amasement asked him how he knew it, why, said he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it him.
Despite the claims by Newton in the above quote, he had in fact proved this result in 1680 as a direct result of the letters from Hooke. Newton indeed reworked his proof and sent a nine page paper De motu corporum in gyrum (On the motion of bodies in an orbit) to Halley. It did not state the law of universal gravitation nor Newton's three laws of motion. All this was to develop over the next couple of years to become the basis for the Principia.
Halley was largely responsible for ensuring that the Principia was published. He received Newton's complete manuscript by April of 1687 but there were many problems not the least being that Newton tried to prevent the publication of the Book III when Hooke claimed priority with the inverse square law of force.
In the Principia the problem of two attracting bodies with an inverse square law of force is completely solved (in Propositions 1-17, 57-60 in Book I). Newton argues that an inverse square law must give produce elliptical, parabolic or hyperbolic orbits.
A bright comet had appeared on 14 November 1680. It remained visible until 5 December 1680 when it moved too close to the Sun to be observed. It reappeared two weeks later moving away from the Sun along almost the same path along which it had approached. Newton found good agreement between its orbit and a parabola. He uses the orbit of this comet, and comets in general, to support his inverse square law of gravitation in the Principia.
You can see a diagram of the orbit of the comet of 1680 from the Principia.
In the Principia Newton also deduced Kepler's third law. He looked briefly (in Propositions 65 and 66) at the problem of three bodies. However Newton later said that an exact solution for three bodies
exceeds, if I am not mistaken, the force of any human mind.
It is important at this stage to examine the problems which now arose. Newton had completely solved the theoretical problem of the motion of two point masses under an inverse square law of attraction. For more than two point masses only approximations to the motion of the bodies could be found and this line of research led to a large effort by mathematicians to develop methods to attack this three body problem. However, the problem of the actual motion of the planets and moons in the solar system was highly complicated by other considerations.
Even if the Earth - Moon system were considered as a two body problem, theoretically solved in the Principia, the orbits would not be simple ellipses. Neither the Earth nor the Moon is a perfect sphere so does not behave as a point mass. This was to lead to the development of mechanics of rigid bodies, but even this would not give a completely accurate picture of the two body problem since tidal forces mean that neither the Earth nor Moon is rigid.
The observational data used by Newton in the Principia was provided by the Royal Greenwich Observatory. However modern scholars such as Richard Westfall claim that Newton sometimes adjusted his calculations to fit his theories. Certainly the observational evidence could not be used to prove the inverse square law of gravitation. Many problems relating observation to theory existed at the time of the Principia and more would arise.
Halley used Newton's method and found almost parabolic orbits for a number of comets. When he computed the orbits for three comets which had appeared in 1537, 1607 and one Halley observed himself in 1682, he found that the characteristics of the orbits were almost identical. Halley deduced they were the same comet and later was able to identify it with one which had appeared in 1456 and 1378. He computed an elliptical orbit for the comet and he noticed that Jupiter and Saturn were perturbing the orbit slightly between each return of the comet. Taking the perturbations into account Halley predicted the comet would return and reach perihelion (the point nearest the Sun) on 13 April 1759. He gave an error of one month on either side of this date. The comet was actually first seen again in December 1758 reaching perihelion on 12 March 1759.
In 1713 a second edition of the Principia, edited by Roger Cotes, appeared. Cotes wrote a preface defending the theory of gravitation given in the Principia. Cotes was himself to provide the next mathematical steps by finding the derivatives of the trigonometric functions, results published after his death.
Euler developed methods of integrating linear differential equations in 1739 and made known Cotes' work on trigonometric functions. He drew up lunar tables in 1744, clearly already studying gravitational attraction in the Earth, Moon, Sun system. Clairaut and d'Alembert were also studying perturbations of the Moon and, in 1747, Clairaut proposed adding a 1/r4 term to the gravitational law to explain the observed motion of the perihelion, the point in the orbit of the Moon where it is closest to the Earth.
However by the end of 1748 Clairaut had discovered that a more accurate application of the inverse square law came close to explaining the orbit. He published his version in 1752 and, two years later, d'Alembert published his calculations going to more terms in his approximation than Clairaut. In fact this work was of importance in having Newton's inverse square law of force accepted in Continental Europe.
The Earth's axis of rotation precesses, that is the direction of the axis of rotation itself rotates in a circle with a period of about 26000 years. Precession is caused by the gravitational attraction of the Sun on the equatorial bulge of the Earth, the bulge being predicted by Newton. Cassini made a measurement of an arc of longitude in 1712 but obtained a result which wrongly suggested that the Earth was elongated at the poles. In 1736 Maupertuis obtained the correct result verifying Newton's predictions. However, this illustrates the problems encountered by mathematicians at this time with basic data about bodies in the solar system, even the Earth, being highly inaccurate.
There is a small periodic effect called nutation superimposed on precession caused by the motion of the perihelion of the Moon. This superimposed effect has a period of 18.6 years and was first observed by Bradley in 1730 but not announced until 18 years later when he had observed the full cycle. D'Alembert quickly showed that Bradley's observed period was deducible from the inverse square law and Euler further clarified this with further work on the mechanics of rigid bodies during the 1750's.
The problem of the orbits of Jupiter and Saturn had troubled astronomers and mathematicians from Kepler's first theory of elliptical orbits. The Paris Académie des Sciences offered Prizes for work on this topic in 1748, 1750 and 1752. In 1748 Euler's studies of the perturbation of Saturn's orbit won him the Prize. His work for the 1752 Prize, however, contains many mathematical errors and was not published until 17 years later. It did contain significant ideas, however, which were independently discovered since Euler's work was not known.
Lagrange won the Académie des Sciences Prize in 1764 for a work on the libration of the Moon. This is a periodic movement in the axis the Moon pointing towards the Earth which allows, over a period of time, more than 50% of the surface of the Moon to be seen. He also won the Académie des Sciences of 1766 for work on the orbits of the moons of Jupiter where he gave a mathematical analysis to explain an observed inequality in the sequence of eclipses of the moons.
Euler, from 1760 onwards, seems to be the first to study the general problem of three bodies under mutual gravitation (rather than looking at bodies in the solar system) although at first he only considered the restricted three body problem when one of the bodies has negligible mass. When one body has negligible mass it is assumed that the motions of the other two can be solved as a two body problem, the body of negligible mass having no effect on the other two. Then the problem is to determine the motion of the third body attracted to the other two bodies which orbit each other. Even in this form the problem does not lead to exact solutions. Euler, however, found a particular solution with all three bodies in a straight line.
The first comet to have an elliptical orbit calculated which was far from a parabola was observed by Messier in 1769. The elliptical orbit was computed by Lexell who correctly realised that the small elliptical orbit had been produced by perturbations by Jupiter. The comet made no reappearance and again Lexell correctly deduced that Jupiter had changed the orbit so much that it was thrown far away from the Sun.
The Académie des Sciences Prize of 1772 for work on the orbit of the Moon was jointly won by Lagrange and Euler. Lagrange submitted Essai sur le problème des trois corps in which he showed that Euler's restricted three body solution held for the general three body problem. He also found another solution where the three bodies were at the vertices of an equilateral triangle. Lagrange considers his solutions do not apply to the solar system but we now know the both the Earth and Jupiter have asteroids sharing their orbits in the equilateral triangle solution configuration discovered by Lagrange. For Jupiter these bodies are called Trojan planets, the first to be discovered being Achilles in 1908. The Trojan planets move 60 in front and 60 behind Jupiter at what are now called the Lagrangian points.
You can see the known asteroids. This picture shows the positions of nearly 6000 asteroids whose orbits are now known. The effect of the Lagrangian points is readily seen. Jupiter is the outermost planet depicted in the picture.
However all this work on the orbits of bodies in the solar system failed to keep pace with observations which always seemed one step ahead, giving further and yet further problems for the theorists to explain. Laplace, from 1774 onwards, became an important contributor to the attempt of the theoreticians to explain the observations of the observers.
Lagrange introduced the method of variation of the arbitrary constants in a paper in 1776 stating that the method was of interest in celestial mechanics and, in special cases, had been already been used by Euler, Laplace and himself. Lagrange published further major papers in 1783 and 1784 on the theory of perturbations of orbits using methods of variations of the arbitrary constants and, in 1785, applied his theory to the orbits of Jupiter and Saturn.
An important development occurred on 13 March 1781 when the astronomer William Herschel (father of John Herschel) observing in his private observatory in Bath, England found
... a curious either nebulous star or perhaps a comet.
Almost immediately it was realised that it was a planet and within a year of its discovery it was shown to have an almost circular orbit. The name Uranus was eventually adopted although William Herschel himself proposed Georgium Sidus (perhaps in the hope of more funds from King George!) while in France it was known as Herschel until the middle of the following century.
Laplace read a memoir to the Académie des Sciences on 23 November 1785 in which he gave a theoretical explanation of all the remaining major discrepancies between theory and observation of all the planets and their moons excluding Uranus. He also addressed the question of the stability of the solar system for the first time. This work was to culminate in the publication of Mécanique céleste (1799) in which, among many other important results, he claimed to prove the stability of the solar system.
The remaining observations not explained by theory at the end of the 18 Century concerned the motion of the Moon. Laplace's work of 1787, that of Adams of 1854 and later Delaunay's work described below eventually provided solutions. Observations of Uranus in the early years of the 19th Century showed there were problems with its orbit and by 1830 Uranus had departed by 15" from the best fitting ellipse.
The next body to be discovered in the solar system was the minor planet Ceres, discovered in 1801. In 1766 J D Titus and in 1772 J E Bode had noted that
(1+4)/10, (3+4)/10, (6+4)/10, (12+4)/10, (24+4)/10, (48+4)/10, (96+4)/10
gave the distances of the 6 known planets from the Sun (taking the Earth's distance to be 1) except there was no planet at distance 2.8. The discovery of Uranus at distance 19.2 was close to the next term of the sequence 19.6.
A search was made for a planet at distance 2.8 and on 1 January 1801 G Piazzi discovered such a body. On 11 February Piazzi fell ill and ended his observations. The new planet, unobserved by other astronomers, passed behind the Sun and was lost. However Gauss in a brilliant piece of work was able to compute an orbit from the small number of observations. In fact Gauss' s method requires only 3 observations and is still essentially that used today in calculating orbits. Ceres, so named by Piazzi, was found to be where Gauss predicted by Olbers. Its distance from the Sun fitted exactly the 2.8 prediction of the Titus-Bode law.
Johann Encke, a student of Gauss, computed (using Gauss's method) an elliptical orbit for the comet of 1818. It had the shortest known period of 3.3 years. The period showed a periodic decrease which Encke could not explain by perturbations by other planets.
Work on the general three body problem during the 19th Century had begun to take two distinct lines. One was the developing of highly complicated methods of approximating the motions of the bodies. The other line was to produce a sophisticated theory to transform and integrate the equations of motion. The first of these lines was celestial mechanics while the second was rational or analytic mechanics. Both the theory of perturbations and the theory of variations of the arbitrary constants were of major mathematical significance as well as contributing greatly to the understanding of planetary orbits.
Papers published by Hamilton in 1834 and 1835 made major contributions to the mechanics of orbiting bodies. as did the significant paper published by Jacobi in 1843 where he reduced the problem of two actual planets orbiting a sun to the motion of two theoretical point masses. As a first approximation the theoretical point masses orbited the centre of gravity of the original system in ellipses. He then used a method, first discovered by Lagrange, to compute the perturbations. Bertrand extended Jacobi's work in 1852.
In 1836 Liouville studied planetary theory, the three body problem and the motion of the minor planets Ceres and Vesta. Many mathematicians around this period devoted much of their time to these problems. Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
By around 1840 irregularities in the orbit of Uranus prompted many scientists to seek reasons them. Alexis Bouvard (a collector of planetary data) proposed that a planet might explain the irregularities and he wrote to the English Astronomer Royal Airy proposing this idea. Bessel also proposed this solution to the problem but died before completing his calculations. Delaunay, famed for his work on the orbit of the Moon, investigated the perturbations in a paper of 1842. Arago urged Le Verrier to work on the problem and on 1 June 1846 Le Verrier showed that the irregularities could be explained by an unknown planet and he determined the coordinates at which the planet would be found. The astronomer Galle in Berlin found the new planet on 26 September remarkably close to the position predicted by Le Verrier. The observations were confirmed on 29 September 1846 at the Paris observatory.
This was a remarkable achievement for Newton's theory of gravitation and of celestial mechanics. Le Verrier's personal triumph however was somewhat diminished when, on 15 October, a letter was published from the English astronomer Challis claiming that John Couch Adams of Cambridge University had made similar calculations to those of Le Verrier which he had completed in September 1845. His predicted position for the new planet had been almost as accurate as Le Verrier's but the English astronomers had been much less industrious in their search. John Herschel and Airy also supported Adams' claim. In fact Challis had, after a long delay, begun to search for the new planet on 29 July 1846. He observed it on 4 August but did not compare his observations with those of the previous night so only realised he had observed the planet after its discovery in Berlin about 7 weeks later. Arago was unimpressed by Adams' priority claims
Mr Adams does not have the right to appear in the history of the discovery of the planet Le Verrier either with a detailed citation or even with the faintest allusion. In the eyes of all impartial men, this discovery will remain one of the most magnificent triumphs of theoretical astronomy, one of the glories of the Académie and one of the most beautiful distinctions of our country.
The success of the mathematical analysis of both Le Verrier and Adams was somewhat fortunate. The orbits which they predicted were different and both not particularly good except around the 1840's. An argument over the naming of the new planet was, however, unfortunate. Arago was given the task of selecting a name by Le Verrier and Le Verrier made his wishes known in an unsubtle way by writing a paper on Herschel's planet, insisting that Uranus should be named after its discoverer. Encke, Gauss's student referred to above, suggested Neptune as a name. However Arago said
I commit myself never to call the new planet by any other name than Le Verrier. In this way, I think I will give an impeachable token of my love for science and follow the inspiration of a legitimate national sentiment.
The argument over a name led to Le Verrier resigning from the Bureau des Longitude and eventually Arago lost his battle over the name which became accepted as Neptune.
Delaunay, mentioned above for his work on the perturbations of Uranus, worked for 20 years on lunar theory. He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon. The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.
Delaunay detected discrepancies between the observed motion of the Moon and his predictions. Le Verrier claimed that Delaunay's methods were in error but Delaunay claimed that the discrepancies were due to unknown factors. In 1865 Delaunay suggested that the discrepancies arose from a slowing of the Earth's rotation due to tidal friction, an explanation which is today believed to be correct.
Le Verrier had published an account of his theory of Mercury in 1859. He pointed out that there was a discrepancy of 38" per century between the predicted motion of the perihelion (the point of closest approach of the planet to the Sun) which was 527" per century and the observed value of 565" per century. In fact the actual discrepancy was 43" per century and this was pointed out by later by Simon Newcomb. Le Verrier was convinced that a planet or ring of material lay inside the orbit of Mercury but being close to the Sun had not been observed.
Le Verrier's search proved in vain and by 1896 Tisserand had concluded that no such perturbing body existed. Newcomb explained the discrepancy in the motion of the perihelion by assuming a minute departure from an inverse square law of gravitation. This was the first time that Newton's theory had been questioned for a long time. In fact this discrepancy in the motion of the perihelion of Mercury was to provide the proof that Newtonian theory had to give way to Einstein's theory of relativity. More details relating to the advance of Mercury's perihelion are contained in the article on general relativity.
G W Hill published an account of his lunar theory in 1878. Earlier approaches started with an elliptic orbit of the Moon round the Earth, assuming the Sun had no effect, then perturbing the orbit to take account of the gravitation of the Sun. Hill, on the other hand, started with circular orbits for the Sun and Moon about the Earth and went on to examine the perturbations caused by assuming elliptic orbits.
The final major step forward in the study of the three body problem which we shall consider was that of Poincaré. Bruns proved in 1887 that apart from the 10 classical integrals, 6 for the centre of gravity, 3 for angular momentum and one for energy, no others could exist. In 1889 Poincaré proved that for the restricted three body problem no integrals exist apart from the Jacobian. In 1890 Poincaré proved his famous recurrence theorem, namely that in any small region of phase space trajectories exist which pass through the region infinitely often. Poincaré published 3 volumes of Les méthods nouvelle de la mécanique celeste between 1892 and 1899. He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent. The stability proofs of Lagrange and Laplace became inconclusive after this result.
Poincaré introduced further topological methods in 1912 for the theory of stability of orbits in the three body problem. It fact Poincaré essentially invented topology in his attempt to answer stability questions in the three body problem. He conjectured that there are infinitely many periodic solutions of the restricted problem, the conjecture being later proved by Birkhoff. The stability of the orbits in the three body problem was also investigated by Levi-Civita, Birkhoff and others.
The first to propose a system of planetary paths which would set the scene for major advances was Copernicus who in De revolutionibus orbium coelestium (1543), argued that the planets and the Earth moved round the Sun. Although a major breakthrough, Copernicus proposed circular paths for the planets and accurate astronomical observations soon began to show that his proposal was not strictly accurate.
You can see a diagram from De revolutionibus orbium coelestium showing Copernicus's solar system.
In 1600 Kepler became assistant to Tycho Brahe who was making accurate observations of the planets. After Brahe died in 1601 Kepler continued the work, calculating planetary paths to unprecedented accuracy.
Kepler showed that a planet moves round the Sun in an elliptical path which has the Sun in one of its two foci. He also showed that a line joining the planet to the Sun sweeps out equal areas in equal times as the planet describes its path. Both these laws were first formulated for the planet Mars, and published in Astronomia Nova (1609).
You can see a diagram from Astronomia Nova showing Kepler's elliptical path for Mars.
However scientists certainly did not accept Kepler's first two laws with enthusiasm. The first was given a cool reception and was certainly thought to require further work to confirm it. The second of Kepler's laws suffered an even worse fate in being essentially ignored by scientists for around 80 years.
Kepler's third law, that the squares of the periods of planets are proportional to the cubes of the mean radii of their paths, appeared in Harmonice mundi (1619) and, perhaps surprisingly in view of the above comments, was widely accepted right from the time of its publication.
In 1679 Hooke wrote a letter to Newton. In the letter he explained how he considered planetary motion to be the result of a central force continuously diverting the planet from its path in a straight line. Newton did not answer this directly but explained his own idea that the rotation of the Earth could be proved from the fact that an object dropped from the top of a tower should have a greater tangential velocity than one dropped near the foot of the tower.
Newton provided a sketch of the path that the particle would follow, quite incorrectly showing it spiral towards the centre of the Earth. Hooke replied that his theory of planetary motion would lead to the path of the particle being an ellipse so that the particle, were it not for the fact that the Earth was in the way, would return to its original position after traversing the ellipse.
Newton, not one to like being corrected, had to admit that his original sketch was incorrect but he "corrected" Hooke's sketch on the assumption that gravity was constant. Hooke replied to Newton that his own theory involved an inverse square law for gravitational attraction. Many years later Hooke was to claim priority for proposing the inverse square law of gravitation and used this letter to Newton to support his claim.
It is worth emphasising that there is a major step to be made from an inverse square law of force to explain planetary motion and a universal law of gravitation. Certainly the motion of the Moon round the Earth was not seen to necessarily be part of the same laws which govern the motion of the planets round the Sun.
Fifty years after these events Newton was to record his own recollections of these events which, although interesting, do not really agree with the known historical facts! [I preserve Newton's old English.]
In the same year I began to think of gravity extending to ye orb of the Moon and (having found out how to estimate the force with wch globe revolving within a sphere presses the surface of a sphere) from Kepler's rule of the periodical times of the Planets being in sesquialternate proportion to their distances from the centres of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must reciprocally as the squares of their distances from the centres about wch they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly. All this was in the two plague years of 1665-1666...
In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance from the Sun. Halley wrote that
Mr Hook said that he had it, but that he would conceale it for some time so that others, triing and failing might know how to value it, when he should make it publick.
Later in the same year in August, Halley visited Newton in Cambridge and asked him what orbit a body would follow under an inverse square law of force
Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy and amasement asked him how he knew it, why, said he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it him.
Despite the claims by Newton in the above quote, he had in fact proved this result in 1680 as a direct result of the letters from Hooke. Newton indeed reworked his proof and sent a nine page paper De motu corporum in gyrum (On the motion of bodies in an orbit) to Halley. It did not state the law of universal gravitation nor Newton's three laws of motion. All this was to develop over the next couple of years to become the basis for the Principia.
Halley was largely responsible for ensuring that the Principia was published. He received Newton's complete manuscript by April of 1687 but there were many problems not the least being that Newton tried to prevent the publication of the Book III when Hooke claimed priority with the inverse square law of force.
In the Principia the problem of two attracting bodies with an inverse square law of force is completely solved (in Propositions 1-17, 57-60 in Book I). Newton argues that an inverse square law must give produce elliptical, parabolic or hyperbolic orbits.
A bright comet had appeared on 14 November 1680. It remained visible until 5 December 1680 when it moved too close to the Sun to be observed. It reappeared two weeks later moving away from the Sun along almost the same path along which it had approached. Newton found good agreement between its orbit and a parabola. He uses the orbit of this comet, and comets in general, to support his inverse square law of gravitation in the Principia.
You can see a diagram of the orbit of the comet of 1680 from the Principia.
In the Principia Newton also deduced Kepler's third law. He looked briefly (in Propositions 65 and 66) at the problem of three bodies. However Newton later said that an exact solution for three bodies
exceeds, if I am not mistaken, the force of any human mind.
It is important at this stage to examine the problems which now arose. Newton had completely solved the theoretical problem of the motion of two point masses under an inverse square law of attraction. For more than two point masses only approximations to the motion of the bodies could be found and this line of research led to a large effort by mathematicians to develop methods to attack this three body problem. However, the problem of the actual motion of the planets and moons in the solar system was highly complicated by other considerations.
Even if the Earth - Moon system were considered as a two body problem, theoretically solved in the Principia, the orbits would not be simple ellipses. Neither the Earth nor the Moon is a perfect sphere so does not behave as a point mass. This was to lead to the development of mechanics of rigid bodies, but even this would not give a completely accurate picture of the two body problem since tidal forces mean that neither the Earth nor Moon is rigid.
The observational data used by Newton in the Principia was provided by the Royal Greenwich Observatory. However modern scholars such as Richard Westfall claim that Newton sometimes adjusted his calculations to fit his theories. Certainly the observational evidence could not be used to prove the inverse square law of gravitation. Many problems relating observation to theory existed at the time of the Principia and more would arise.
Halley used Newton's method and found almost parabolic orbits for a number of comets. When he computed the orbits for three comets which had appeared in 1537, 1607 and one Halley observed himself in 1682, he found that the characteristics of the orbits were almost identical. Halley deduced they were the same comet and later was able to identify it with one which had appeared in 1456 and 1378. He computed an elliptical orbit for the comet and he noticed that Jupiter and Saturn were perturbing the orbit slightly between each return of the comet. Taking the perturbations into account Halley predicted the comet would return and reach perihelion (the point nearest the Sun) on 13 April 1759. He gave an error of one month on either side of this date. The comet was actually first seen again in December 1758 reaching perihelion on 12 March 1759.
In 1713 a second edition of the Principia, edited by Roger Cotes, appeared. Cotes wrote a preface defending the theory of gravitation given in the Principia. Cotes was himself to provide the next mathematical steps by finding the derivatives of the trigonometric functions, results published after his death.
Euler developed methods of integrating linear differential equations in 1739 and made known Cotes' work on trigonometric functions. He drew up lunar tables in 1744, clearly already studying gravitational attraction in the Earth, Moon, Sun system. Clairaut and d'Alembert were also studying perturbations of the Moon and, in 1747, Clairaut proposed adding a 1/r4 term to the gravitational law to explain the observed motion of the perihelion, the point in the orbit of the Moon where it is closest to the Earth.
However by the end of 1748 Clairaut had discovered that a more accurate application of the inverse square law came close to explaining the orbit. He published his version in 1752 and, two years later, d'Alembert published his calculations going to more terms in his approximation than Clairaut. In fact this work was of importance in having Newton's inverse square law of force accepted in Continental Europe.
The Earth's axis of rotation precesses, that is the direction of the axis of rotation itself rotates in a circle with a period of about 26000 years. Precession is caused by the gravitational attraction of the Sun on the equatorial bulge of the Earth, the bulge being predicted by Newton. Cassini made a measurement of an arc of longitude in 1712 but obtained a result which wrongly suggested that the Earth was elongated at the poles. In 1736 Maupertuis obtained the correct result verifying Newton's predictions. However, this illustrates the problems encountered by mathematicians at this time with basic data about bodies in the solar system, even the Earth, being highly inaccurate.
There is a small periodic effect called nutation superimposed on precession caused by the motion of the perihelion of the Moon. This superimposed effect has a period of 18.6 years and was first observed by Bradley in 1730 but not announced until 18 years later when he had observed the full cycle. D'Alembert quickly showed that Bradley's observed period was deducible from the inverse square law and Euler further clarified this with further work on the mechanics of rigid bodies during the 1750's.
The problem of the orbits of Jupiter and Saturn had troubled astronomers and mathematicians from Kepler's first theory of elliptical orbits. The Paris Académie des Sciences offered Prizes for work on this topic in 1748, 1750 and 1752. In 1748 Euler's studies of the perturbation of Saturn's orbit won him the Prize. His work for the 1752 Prize, however, contains many mathematical errors and was not published until 17 years later. It did contain significant ideas, however, which were independently discovered since Euler's work was not known.
Lagrange won the Académie des Sciences Prize in 1764 for a work on the libration of the Moon. This is a periodic movement in the axis the Moon pointing towards the Earth which allows, over a period of time, more than 50% of the surface of the Moon to be seen. He also won the Académie des Sciences of 1766 for work on the orbits of the moons of Jupiter where he gave a mathematical analysis to explain an observed inequality in the sequence of eclipses of the moons.
Euler, from 1760 onwards, seems to be the first to study the general problem of three bodies under mutual gravitation (rather than looking at bodies in the solar system) although at first he only considered the restricted three body problem when one of the bodies has negligible mass. When one body has negligible mass it is assumed that the motions of the other two can be solved as a two body problem, the body of negligible mass having no effect on the other two. Then the problem is to determine the motion of the third body attracted to the other two bodies which orbit each other. Even in this form the problem does not lead to exact solutions. Euler, however, found a particular solution with all three bodies in a straight line.
The first comet to have an elliptical orbit calculated which was far from a parabola was observed by Messier in 1769. The elliptical orbit was computed by Lexell who correctly realised that the small elliptical orbit had been produced by perturbations by Jupiter. The comet made no reappearance and again Lexell correctly deduced that Jupiter had changed the orbit so much that it was thrown far away from the Sun.
The Académie des Sciences Prize of 1772 for work on the orbit of the Moon was jointly won by Lagrange and Euler. Lagrange submitted Essai sur le problème des trois corps in which he showed that Euler's restricted three body solution held for the general three body problem. He also found another solution where the three bodies were at the vertices of an equilateral triangle. Lagrange considers his solutions do not apply to the solar system but we now know the both the Earth and Jupiter have asteroids sharing their orbits in the equilateral triangle solution configuration discovered by Lagrange. For Jupiter these bodies are called Trojan planets, the first to be discovered being Achilles in 1908. The Trojan planets move 60 in front and 60 behind Jupiter at what are now called the Lagrangian points.
You can see the known asteroids. This picture shows the positions of nearly 6000 asteroids whose orbits are now known. The effect of the Lagrangian points is readily seen. Jupiter is the outermost planet depicted in the picture.
However all this work on the orbits of bodies in the solar system failed to keep pace with observations which always seemed one step ahead, giving further and yet further problems for the theorists to explain. Laplace, from 1774 onwards, became an important contributor to the attempt of the theoreticians to explain the observations of the observers.
Lagrange introduced the method of variation of the arbitrary constants in a paper in 1776 stating that the method was of interest in celestial mechanics and, in special cases, had been already been used by Euler, Laplace and himself. Lagrange published further major papers in 1783 and 1784 on the theory of perturbations of orbits using methods of variations of the arbitrary constants and, in 1785, applied his theory to the orbits of Jupiter and Saturn.
An important development occurred on 13 March 1781 when the astronomer William Herschel (father of John Herschel) observing in his private observatory in Bath, England found
... a curious either nebulous star or perhaps a comet.
Almost immediately it was realised that it was a planet and within a year of its discovery it was shown to have an almost circular orbit. The name Uranus was eventually adopted although William Herschel himself proposed Georgium Sidus (perhaps in the hope of more funds from King George!) while in France it was known as Herschel until the middle of the following century.
Laplace read a memoir to the Académie des Sciences on 23 November 1785 in which he gave a theoretical explanation of all the remaining major discrepancies between theory and observation of all the planets and their moons excluding Uranus. He also addressed the question of the stability of the solar system for the first time. This work was to culminate in the publication of Mécanique céleste (1799) in which, among many other important results, he claimed to prove the stability of the solar system.
The remaining observations not explained by theory at the end of the 18 Century concerned the motion of the Moon. Laplace's work of 1787, that of Adams of 1854 and later Delaunay's work described below eventually provided solutions. Observations of Uranus in the early years of the 19th Century showed there were problems with its orbit and by 1830 Uranus had departed by 15" from the best fitting ellipse.
The next body to be discovered in the solar system was the minor planet Ceres, discovered in 1801. In 1766 J D Titus and in 1772 J E Bode had noted that
(1+4)/10, (3+4)/10, (6+4)/10, (12+4)/10, (24+4)/10, (48+4)/10, (96+4)/10
gave the distances of the 6 known planets from the Sun (taking the Earth's distance to be 1) except there was no planet at distance 2.8. The discovery of Uranus at distance 19.2 was close to the next term of the sequence 19.6.
A search was made for a planet at distance 2.8 and on 1 January 1801 G Piazzi discovered such a body. On 11 February Piazzi fell ill and ended his observations. The new planet, unobserved by other astronomers, passed behind the Sun and was lost. However Gauss in a brilliant piece of work was able to compute an orbit from the small number of observations. In fact Gauss' s method requires only 3 observations and is still essentially that used today in calculating orbits. Ceres, so named by Piazzi, was found to be where Gauss predicted by Olbers. Its distance from the Sun fitted exactly the 2.8 prediction of the Titus-Bode law.
Johann Encke, a student of Gauss, computed (using Gauss's method) an elliptical orbit for the comet of 1818. It had the shortest known period of 3.3 years. The period showed a periodic decrease which Encke could not explain by perturbations by other planets.
Work on the general three body problem during the 19th Century had begun to take two distinct lines. One was the developing of highly complicated methods of approximating the motions of the bodies. The other line was to produce a sophisticated theory to transform and integrate the equations of motion. The first of these lines was celestial mechanics while the second was rational or analytic mechanics. Both the theory of perturbations and the theory of variations of the arbitrary constants were of major mathematical significance as well as contributing greatly to the understanding of planetary orbits.
Papers published by Hamilton in 1834 and 1835 made major contributions to the mechanics of orbiting bodies. as did the significant paper published by Jacobi in 1843 where he reduced the problem of two actual planets orbiting a sun to the motion of two theoretical point masses. As a first approximation the theoretical point masses orbited the centre of gravity of the original system in ellipses. He then used a method, first discovered by Lagrange, to compute the perturbations. Bertrand extended Jacobi's work in 1852.
In 1836 Liouville studied planetary theory, the three body problem and the motion of the minor planets Ceres and Vesta. Many mathematicians around this period devoted much of their time to these problems. Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
By around 1840 irregularities in the orbit of Uranus prompted many scientists to seek reasons them. Alexis Bouvard (a collector of planetary data) proposed that a planet might explain the irregularities and he wrote to the English Astronomer Royal Airy proposing this idea. Bessel also proposed this solution to the problem but died before completing his calculations. Delaunay, famed for his work on the orbit of the Moon, investigated the perturbations in a paper of 1842. Arago urged Le Verrier to work on the problem and on 1 June 1846 Le Verrier showed that the irregularities could be explained by an unknown planet and he determined the coordinates at which the planet would be found. The astronomer Galle in Berlin found the new planet on 26 September remarkably close to the position predicted by Le Verrier. The observations were confirmed on 29 September 1846 at the Paris observatory.
This was a remarkable achievement for Newton's theory of gravitation and of celestial mechanics. Le Verrier's personal triumph however was somewhat diminished when, on 15 October, a letter was published from the English astronomer Challis claiming that John Couch Adams of Cambridge University had made similar calculations to those of Le Verrier which he had completed in September 1845. His predicted position for the new planet had been almost as accurate as Le Verrier's but the English astronomers had been much less industrious in their search. John Herschel and Airy also supported Adams' claim. In fact Challis had, after a long delay, begun to search for the new planet on 29 July 1846. He observed it on 4 August but did not compare his observations with those of the previous night so only realised he had observed the planet after its discovery in Berlin about 7 weeks later. Arago was unimpressed by Adams' priority claims
Mr Adams does not have the right to appear in the history of the discovery of the planet Le Verrier either with a detailed citation or even with the faintest allusion. In the eyes of all impartial men, this discovery will remain one of the most magnificent triumphs of theoretical astronomy, one of the glories of the Académie and one of the most beautiful distinctions of our country.
The success of the mathematical analysis of both Le Verrier and Adams was somewhat fortunate. The orbits which they predicted were different and both not particularly good except around the 1840's. An argument over the naming of the new planet was, however, unfortunate. Arago was given the task of selecting a name by Le Verrier and Le Verrier made his wishes known in an unsubtle way by writing a paper on Herschel's planet, insisting that Uranus should be named after its discoverer. Encke, Gauss's student referred to above, suggested Neptune as a name. However Arago said
I commit myself never to call the new planet by any other name than Le Verrier. In this way, I think I will give an impeachable token of my love for science and follow the inspiration of a legitimate national sentiment.
The argument over a name led to Le Verrier resigning from the Bureau des Longitude and eventually Arago lost his battle over the name which became accepted as Neptune.
Delaunay, mentioned above for his work on the perturbations of Uranus, worked for 20 years on lunar theory. He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon. The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.
Delaunay detected discrepancies between the observed motion of the Moon and his predictions. Le Verrier claimed that Delaunay's methods were in error but Delaunay claimed that the discrepancies were due to unknown factors. In 1865 Delaunay suggested that the discrepancies arose from a slowing of the Earth's rotation due to tidal friction, an explanation which is today believed to be correct.
Le Verrier had published an account of his theory of Mercury in 1859. He pointed out that there was a discrepancy of 38" per century between the predicted motion of the perihelion (the point of closest approach of the planet to the Sun) which was 527" per century and the observed value of 565" per century. In fact the actual discrepancy was 43" per century and this was pointed out by later by Simon Newcomb. Le Verrier was convinced that a planet or ring of material lay inside the orbit of Mercury but being close to the Sun had not been observed.
Le Verrier's search proved in vain and by 1896 Tisserand had concluded that no such perturbing body existed. Newcomb explained the discrepancy in the motion of the perihelion by assuming a minute departure from an inverse square law of gravitation. This was the first time that Newton's theory had been questioned for a long time. In fact this discrepancy in the motion of the perihelion of Mercury was to provide the proof that Newtonian theory had to give way to Einstein's theory of relativity. More details relating to the advance of Mercury's perihelion are contained in the article on general relativity.
G W Hill published an account of his lunar theory in 1878. Earlier approaches started with an elliptic orbit of the Moon round the Earth, assuming the Sun had no effect, then perturbing the orbit to take account of the gravitation of the Sun. Hill, on the other hand, started with circular orbits for the Sun and Moon about the Earth and went on to examine the perturbations caused by assuming elliptic orbits.
The final major step forward in the study of the three body problem which we shall consider was that of Poincaré. Bruns proved in 1887 that apart from the 10 classical integrals, 6 for the centre of gravity, 3 for angular momentum and one for energy, no others could exist. In 1889 Poincaré proved that for the restricted three body problem no integrals exist apart from the Jacobian. In 1890 Poincaré proved his famous recurrence theorem, namely that in any small region of phase space trajectories exist which pass through the region infinitely often. Poincaré published 3 volumes of Les méthods nouvelle de la mécanique celeste between 1892 and 1899. He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent. The stability proofs of Lagrange and Laplace became inconclusive after this result.
Poincaré introduced further topological methods in 1912 for the theory of stability of orbits in the three body problem. It fact Poincaré essentially invented topology in his attempt to answer stability questions in the three body problem. He conjectured that there are infinitely many periodic solutions of the restricted problem, the conjecture being later proved by Birkhoff. The stability of the orbits in the three body problem was also investigated by Levi-Civita, Birkhoff and others.
A history of Topology
Topological ideas are present in almost all areas of today's mathematics. The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common. We shall trace the rise of topological concepts in a number of different situations.
Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
The paper not only shows that the problem of crossing the seven bridges in a single journey is impossible, but generalises the problem to show that, in today's notation,
A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.
The next step in freeing mathematics from being a subject about measurement was also due to Euler. In 1750 he wrote a letter to Christian Goldbach which, as well as commenting on a dispute Goldbach was having with a bookseller, gives Euler's famous formula for a polyhedron
v - e + f = 2
where v is the number of vertices of the polyhedron, e is the number of edges and f is the number of faces. It is interesting to realise that this, really rather simple, formula seems to have been missed by Archimedes and Descartes although both wrote extensively on polyhedra. Again the reason must be that to everyone before Euler, it had been impossible to think of geometrical properties without measurement being involved.
Euler published details of his formula in 1752 in two papers, the first admits that Euler cannot prove the result but the second gives a proof based dissecting solids into tetrahedral slices. Euler overlooks some problems with his remarkably clever proof. In particular he assumed that the solids were convex, that is a straight line joining any two points always lies entirely within the solid.
The route started by Euler with his polyhedral formula was followed by a little known mathematician Antoine-Jean Lhuilier (1750 -1840) who worked for most of his life on problems relating to Euler's formula. In 1813 Lhuilier published an important work. He noticed that Euler's formula was wrong for solids with holes in them. If a solid has g holes the Lhuilier showed that
v - e + f = 2 - 2g.
This was the first known result on a topological invariant.
Möbius published a description of a Möbius band in 1865. He tried to describe the 'one-sided' property of the Möbius band in terms of non-orientability. He thought of the surface being covered by oriented triangles. He found that the Möbius band could not be filled with compatibly oriented triangles.
Johann Benedict Listing (1802-1882) was the first to use the word topology. Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology. Listing wrote a paper in 1847 called Vorstudien zur Topologie although he had already used the word for ten years in correspondence. The 1847 paper is not very important, although he also introduces the idea of a complex, since it is extremely elementary. In 1861 Listing published a much more important paper in which he described the Möbius band (4 years before Möbius) and studied components of surfaces and connectivity.
Listing was not the first to examine connectivity of surfaces. Riemann had studied the concept in 1851 and again in 1857 when he introduced the Riemann surfaces. The problem arose from studying a polynomial equation f(w, z) = 0 and considering how the roots vary as w and z vary. Riemann introduced Riemann surfaces, determined by the function f(w, z), so that the function w(z) defined by the equation f(w, z) = 0 is single valued on the surfaces.
Jordan introduced another method for examining the connectivity of a surface. He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point. If a general circuit c can be transformed into a system of irreducible circuits a1, a2, ...., an so that c describes ai mi times then he wrote
c = m1a1 + m2a2 + ....+ mnan .
The circuit c is reducible if
m1a1 + m2a2 + ....+ mnan = 0. (*)
A system of irreducible circuits a1, a2, ...., an is called independent if they satisfy no relation of the form (*) and complete if any circuit can be expressed in terms of them. Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.
Listing had examined connectivity in three dimensional Euclidean space but Betti extended his ideas to n dimensions. This is not as straightforward as it might appear since even in three dimensions it is possible to have a surface that cannot be reduced to a point yet closed curves on the surface can be reduced to a point. Betti's definition of connectivity left something to be desired and criticisms were made by Heegaard.
The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler's convex polyhedra formula had been generalised to not necessarily convex polyhedra by Jonquières in 1890 and now Poincaré put it into a completely general setting of a p-dimensional variety V.
Also while dealing with connectivity Poincaré introduced the fundamental group of a variety and the concept of homotopy was introduced in the same 1895 papers.
A second way in which topology developed was through the generalisation of the ideas of convergence. This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set. He also defined closed subsets of the real line as subsets containing their first derived set. Cantor also introduced the idea of an open set another fundamental concept in point set topology.
Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano-Weierstrass theorem which states
A bounded infinite subset S of the real numbers possesses at least one point of accumulation p, i.e. p satisfies the property that given any > 0 there is an infinite sequence (pn) of points of S with |p - pn | < .
Hence the concept of neighbourhood of a point was introduced.
Hilbert used the concept of a neighbourhood in 1902 when he answered in the affirmative one of his own questions, namely
Is a continuous transformation group differentiable?
In 1906 Fréchet called a space compact if any infinite bounded subset contains a point of accumulation. However Fréchet was able to extend the concept of convergence from Euclidean space by defining metric spaces. He also showed that Cantor's ideas of open and closed subsets extended naturally to metric spaces.
Riesz, in a paper to the International Congress of Mathematics in Rome (1909), disposed of the metric completely and proposed a new axiomatic approach to topology. The definition was based on an set definition of limit points, with no concept of distance. A few years later in 1914 Hausdorff defined neighbourhoods by four axioms so again there were no metric considerations. This work of Riesz and Hausdorff really allows the definition of abstract topological spaces.
There is a third way in which topological concepts entered mathematics, namely via functional analysis. This was a topic which arose from mathematical physics and astronomy, brought about because the methods of classical analysis were somewhat inadequate in tackling certain types of problems. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
Hadamard introduced the word 'functional' in 1903 when he studied linear functionals F of the form
F(f) = lim f(x) gn(x) dx
where the limit is taken as n ∞ and the integral is from a to b. Fréchet continued the development of functional by defining the derivative of a functional in 1904.
Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series. Distance was defined via an inner product. Schmidt's work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Fréchet.
A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces. Banach took Fréchet's linear functionals and showed that they had a natural setting in normed spaces.
Poincaré developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. His study of autonomous systems
dx/dt = f(x, y) , dy/dt = g(x, y)
involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier. The collection of methods developed by Poincaré was built into a complete topological theory by Brouwer in 1912.
Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
The paper not only shows that the problem of crossing the seven bridges in a single journey is impossible, but generalises the problem to show that, in today's notation,
A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.
The next step in freeing mathematics from being a subject about measurement was also due to Euler. In 1750 he wrote a letter to Christian Goldbach which, as well as commenting on a dispute Goldbach was having with a bookseller, gives Euler's famous formula for a polyhedron
v - e + f = 2
where v is the number of vertices of the polyhedron, e is the number of edges and f is the number of faces. It is interesting to realise that this, really rather simple, formula seems to have been missed by Archimedes and Descartes although both wrote extensively on polyhedra. Again the reason must be that to everyone before Euler, it had been impossible to think of geometrical properties without measurement being involved.
Euler published details of his formula in 1752 in two papers, the first admits that Euler cannot prove the result but the second gives a proof based dissecting solids into tetrahedral slices. Euler overlooks some problems with his remarkably clever proof. In particular he assumed that the solids were convex, that is a straight line joining any two points always lies entirely within the solid.
The route started by Euler with his polyhedral formula was followed by a little known mathematician Antoine-Jean Lhuilier (1750 -1840) who worked for most of his life on problems relating to Euler's formula. In 1813 Lhuilier published an important work. He noticed that Euler's formula was wrong for solids with holes in them. If a solid has g holes the Lhuilier showed that
v - e + f = 2 - 2g.
This was the first known result on a topological invariant.
Möbius published a description of a Möbius band in 1865. He tried to describe the 'one-sided' property of the Möbius band in terms of non-orientability. He thought of the surface being covered by oriented triangles. He found that the Möbius band could not be filled with compatibly oriented triangles.
Johann Benedict Listing (1802-1882) was the first to use the word topology. Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology. Listing wrote a paper in 1847 called Vorstudien zur Topologie although he had already used the word for ten years in correspondence. The 1847 paper is not very important, although he also introduces the idea of a complex, since it is extremely elementary. In 1861 Listing published a much more important paper in which he described the Möbius band (4 years before Möbius) and studied components of surfaces and connectivity.
Listing was not the first to examine connectivity of surfaces. Riemann had studied the concept in 1851 and again in 1857 when he introduced the Riemann surfaces. The problem arose from studying a polynomial equation f(w, z) = 0 and considering how the roots vary as w and z vary. Riemann introduced Riemann surfaces, determined by the function f(w, z), so that the function w(z) defined by the equation f(w, z) = 0 is single valued on the surfaces.
Jordan introduced another method for examining the connectivity of a surface. He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point. If a general circuit c can be transformed into a system of irreducible circuits a1, a2, ...., an so that c describes ai mi times then he wrote
c = m1a1 + m2a2 + ....+ mnan .
The circuit c is reducible if
m1a1 + m2a2 + ....+ mnan = 0. (*)
A system of irreducible circuits a1, a2, ...., an is called independent if they satisfy no relation of the form (*) and complete if any circuit can be expressed in terms of them. Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.
Listing had examined connectivity in three dimensional Euclidean space but Betti extended his ideas to n dimensions. This is not as straightforward as it might appear since even in three dimensions it is possible to have a surface that cannot be reduced to a point yet closed curves on the surface can be reduced to a point. Betti's definition of connectivity left something to be desired and criticisms were made by Heegaard.
The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler's convex polyhedra formula had been generalised to not necessarily convex polyhedra by Jonquières in 1890 and now Poincaré put it into a completely general setting of a p-dimensional variety V.
Also while dealing with connectivity Poincaré introduced the fundamental group of a variety and the concept of homotopy was introduced in the same 1895 papers.
A second way in which topology developed was through the generalisation of the ideas of convergence. This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set. He also defined closed subsets of the real line as subsets containing their first derived set. Cantor also introduced the idea of an open set another fundamental concept in point set topology.
Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano-Weierstrass theorem which states
A bounded infinite subset S of the real numbers possesses at least one point of accumulation p, i.e. p satisfies the property that given any > 0 there is an infinite sequence (pn) of points of S with |p - pn | < .
Hence the concept of neighbourhood of a point was introduced.
Hilbert used the concept of a neighbourhood in 1902 when he answered in the affirmative one of his own questions, namely
Is a continuous transformation group differentiable?
In 1906 Fréchet called a space compact if any infinite bounded subset contains a point of accumulation. However Fréchet was able to extend the concept of convergence from Euclidean space by defining metric spaces. He also showed that Cantor's ideas of open and closed subsets extended naturally to metric spaces.
Riesz, in a paper to the International Congress of Mathematics in Rome (1909), disposed of the metric completely and proposed a new axiomatic approach to topology. The definition was based on an set definition of limit points, with no concept of distance. A few years later in 1914 Hausdorff defined neighbourhoods by four axioms so again there were no metric considerations. This work of Riesz and Hausdorff really allows the definition of abstract topological spaces.
There is a third way in which topological concepts entered mathematics, namely via functional analysis. This was a topic which arose from mathematical physics and astronomy, brought about because the methods of classical analysis were somewhat inadequate in tackling certain types of problems. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
Hadamard introduced the word 'functional' in 1903 when he studied linear functionals F of the form
F(f) = lim f(x) gn(x) dx
where the limit is taken as n ∞ and the integral is from a to b. Fréchet continued the development of functional by defining the derivative of a functional in 1904.
Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series. Distance was defined via an inner product. Schmidt's work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Fréchet.
A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces. Banach took Fréchet's linear functionals and showed that they had a natural setting in normed spaces.
Poincaré developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. His study of autonomous systems
dx/dt = f(x, y) , dy/dt = g(x, y)
involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier. The collection of methods developed by Poincaré was built into a complete topological theory by Brouwer in 1912.
Tuesday, September 25, 2007
From the Ashes of the First Stars
Above, an artist's impression shows a primordial quasar as it might have been, surrounded by sheets of gas, dust, stars and early star clusters. Exacting observations of three distant quasars now indicate emission of very specific colors of the element iron. These Hubble Space Telescope observations, which bolster recent results from the WMAP mission, indicate that a whole complete cycle of stars was born, created this iron, and died within the first few hundred million years of the universe.
Young Stars Hatching in Orion
The latest image released from the Spitzer Space Telescope shows infant stars “hatching” in the head of Orion. Astronomers think that a supernova 3 million years ago sent shockwaves through the region, collapsing clouds of gas and dust, and beginning a new generation of star formation.
The region imaged by Spitzer is called Barnard 30, located about 1,300 light-years from Earth in the constellation of Orion. More specifically, it’s located right beside the star considered to be Orion’s head, Lambda Orionis.
Since the region is shrouded in dark clouds of gas and dust that obscure visible light images, this was an ideal target for Spitzer, which can peer right through them in the infrared spectrum. The tints of orange-red glow are dust particles warmed by the newly forming stars. The reddish-pink dots are the young stars themselves, embedded in the clouds of gas and dust.
The region imaged by Spitzer is called Barnard 30, located about 1,300 light-years from Earth in the constellation of Orion. More specifically, it’s located right beside the star considered to be Orion’s head, Lambda Orionis.
Since the region is shrouded in dark clouds of gas and dust that obscure visible light images, this was an ideal target for Spitzer, which can peer right through them in the infrared spectrum. The tints of orange-red glow are dust particles warmed by the newly forming stars. The reddish-pink dots are the young stars themselves, embedded in the clouds of gas and dust.
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