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JULES HENRI POINCARÉ was born at Nancy on 1854 April 29.
His father, who came of a Lorraine family, was an eminent physician; the President of France, M. Raymond Poincaré, was his cousin; and another cousin, M. Lucien Poincaré, is a distinguished physicist. As a boy he suffered from poor health, and doubtless it was partly owing to this that he acquired those habits of preoccupation and inward concentration of thought, which were characteristic of him afterwards. He was educated successively at school at Nancy, at the École Polytechnique, and at the École Supérieure des Mines. At the age of twenty-five he received the degree of Doctor in the University of Paris, and about the same time was appointed Professor of Analysis at Caen. The wide variety of his scientific investigations is reflected in the diverse professorships which at one time and another he held. At Paris University he filled successively the chairs of Mechanical Physics, Mathematical Physics, Calculus of Probability, and finally, for the last sixteen years, Celestial Mechanics; and at the Ecole Polytechnique he was lecturer in Analysis and afterwards Professor of Astronomy. Yet one best-known branch of his activity is not represented in these. M. Darboux wrote to him: "Why does not the Faculty possess a chair of Scientific Philosophy? I should have been able to ask you to occupy it also." To the general public in his country he first became celebrated by winning the international prize offered by King Oscar II. of Sweden in 1887 for a memoir on the Problem of Three Bodies; the award was received in France with great enthusiasm, and the prize essay, which was the foundation of Poincaré's Les Méthodes Nouvelles de la Mécanique Céleste, remains one of his greatest achievements. It is impossible to enumerate the honours bestowed on him in his own country and abroad; thirty-two academies received him as an associate. We may mention, however, that he received the Sylvester medal of the Royal Society, and the gold medal of this Society was awarded to him in 1900. He was received into the French Academy in 1909.
Whilst continually engaged in the most profound researches, and given to fits of abstraction and absent-mindedness, he nevertheless took an active part in all those administrative and consultative duties which are pressed upon an eminent man of science. As a professor he was a conscientious teacher. Of extraneous offices, he was President of the Conseil des Observatoires and of the Société Astronomique de France; in 1906 he was President of the Académie des Sciences; he was also a member of the Bureau des Longitudes. When presiding at official meetings he could throw aside his preoccupied habits and conduct the business with alert attention. With all the pressure of professorial and other work, the output of his scientific papers averaged one a week for the thirty-three years of his active career.
It is perhaps pardonable to claim M. Poincaré as pre-eminently an astronomer, though few men have the encyclopædic knowledge necessary to compare the merits of his contributions to pure mathematics, to mathematical physics, to astronomy, and to philosophy. Each of these studies owes a great debt to his genius. His was one of those rare minds that recognise no limits between the sciences; and we can dimly perceive how much of his success may have been due to the ease with which he could apply the weapons of one branch to the problems of another. His marvellous faculty of generalisation demanded an unrestricted field, and it was backed by an activity of brain which enabled him to enter deeply into so many diverse lines of knowledge. The problems of astronomy, of the evolution of the solar system and of the universe, appear, however, to have been his first interest; and in the last fifteen years of his life, by the professorial chairs he occupied, and by his presidency of astronomical societies and committees, he seems to have recognised it as the chief claim. His contributions to astronomy are so numerous that it is impossible to attempt a general survey of them. His last published work, a critical examination of the various cosmogonic hypotheses, represents one line of his investigations. An important paper on the problem of Saturn's ring may be mentioned. The modern theory of tides owes much to Poincaré. There are, however, two special investigations, which by common consent are regarded as his chief title to fame, and these we may here consider especially.
In Lés Methodes Nouvelles de la Mécanique Céleste, a great work in three volumes published between 1892 and 1899, the brilliancy and originality of his powers are most highly manifested. Sir George Darwin describes it as "the mine from which, for more than half a century to come, humbler investigators will excavate their materials." Professor Moulton, speaking of the mathematical processes employed, says: "In power and elegance they are as much beyond those of Laplace, as his were beyond the geometry of Newton." One of Poincaré's theorems showed that the series used in celestial mechanics are ultimately divergent for large values of the time. The various periodic terms in the motions of the planets, which suffice for ordinary discussions, hold approximately true; but no conclusion as to the ultimate stability of the solar system can be drawn, for the applications of the series are not valid beyond certain limits of time. This possible instability (for the conclusion is merely negative, and does not decide between stability and instability) must not be confused with that arising from the dissipative forces of the tides, to which Poincaré also devoted researches. The result we are now considering arises from the historic Problem of Three Bodies in its simple form.
From the discovery of the divergence or non-periodicity of planetary orbits, Poincaré was led to the consideration of theoretical orbits which are strictly periodic and therefore valid for all time. The subject of Periodic Orbits, which has now an extensive literature, is one with which the practical astronomer finds it difficult to feel much sympathy; he is inclined to condemn the whole matter as misguided ingenuity, or as pure mathematics in a thin disguise. But this attitude is not a just one. The subject originally arose in the study of the lunar theory, and G. W. Hill's fertile method depends on the "variational curve," which is the, first example of a periodic orbit. To some investigators (and notably to E. W. Brown) the interest of these orbits lies in their application to the stability and possible distribution of the outer satellites of the planets. To Sir George Darwin it was a conviction that their study would ultimately afford a clue to the empirical law of Bode as to the distances of planets from the Sun. To Poincaré it was the one method by which it seemed possible to penetrate into the far-distant future of the solar system; or, to use his metaphor, "they furnish the only breach by which we may hope to penetrate the fortress of a problem hitherto deemed impregnable." Just as, although the Moon does not move along the variational curve, yet its deviation from this periodic orbit is always comparatively small, Poincaré believed that a similar use of periodic orbits might be made in determining expressions for the motions of planets valid for all time, for he showed that a periodic solution can always be found which shall differ by as small a quantity as we please from any given motion of the perturbed body.
Perhaps the researches which have aroused the widest attention are those in which Poincaré considered the possible forms of equilibrium of rotating masses of liquid, resulting in his discovery of the so-called pear-shaped figure of equilibrium. The possibility that in this way may be traced the first stages in the separation of a satellite from its primary, or the evolution of a double star, renders this work of great interest to all those who have speculated on the origin of celestial systems. But the real beauty of the investigation, and the far-reaching general principles which it contains, are often lost sight of through attention being focussed on the immediate practical result. As in the study of the Problem of Three Bodies, it was a question of stability which guided Poincaré's researches, and his results in the first instance apply with perfect generality to any mechanical system. He found that, where there is more than one form of equilibrium possible, if these forms are arranged in families, an interchange of stability and instability takes place between two families wherever they cross. If a stable and an unstable family converge, so that one limiting form may be considered a member common to both families, then on tracing the forms beyond this point, what was formerly the stable family becomes unstable, and vice versa. In the application to rotating fluid masses, it was known that when the ordinary oblate spheroid becomes unstable, the Jacobi ellipsoid with three unequal axes supersedes it as the stable configuration. Poincaré found that a point was reached when this also became unstable, and he argued that this must be a point where a new series of figures of equilibrium branches off. He investigated these new figures and showed their pear-shaped character.
In pure mathematics one of his most fertile generalisations was the invention of automorphic or, as he named them, Fuchsian functions. These functions are arrived at by an extension of the ideas of the theory of elliptic functions. The essential character of the elliptic function is its double periodicity, so that it is unchanged by a certain doubly infinite series of substitutions, just as is unchanged by the singly infinite series of substitutions for . The Fuchsian functions have the property of being unchanged by a still more general series of substitutions, namely, of the form , subject to the condition . Poincaré was able to form zeta- and theta-Fuchsian functions corresponding to the zeta and theta-elliptic functions. From this he was led to the discovery that the new functions sufficed for the integration of any linear differential equation with algebraic coefficients. They further enable the allied problem of expressing the coordinates of an algebraic curve in terms of a parameter to be solved in all cases.
In the modern developments of physics Poincaré was in the front rank of workers. A monumental work on the diffraction of electric waves is remarkable alike for the novelty of the analytical methods and the practical importance of the results. In his later years the principle of relativity and the allied problems of the dynamics of an electron claimed a large share of his attention. His work on this subject was perhaps mainly critical, rather than novel; but he has done much to reduce the various investigations into systematic relation; his broad outlook has helped to clear a subject of bewildering confusion. To the new developments of thermodynamics and Planck's theory of quanta, which appear destined to effect the greatest revolution that physical ideas have yet undergone, he devoted a searching examination. His conclusion, apparently reached with extreme reluctance, was that there can be no escape from at least some of the consequences of these new ideas. Newtonian mechanics is only applicable to matter in bulk, and the ultimate minute constituents demand some other basis of treatment. One result of Planck's quantum hypothesis is that it ceases to be possible to represent ultimate dynamics by differential equations. This is a staggering blow at what seemed to be the last article of faith of the mathematical physicist, who, bravely abandoning mechanical models, prepared to throw over the æther and even the electron if need be, yet appeared to have arrived at some ultimate facts of nature in the differential equations which remained.
In no subject has the influence of Poincaré been more widely felt than in his philosophy of science and mathematics. Much of his more technical work is hardly yet appreciated, and it will be many years before the new methods he introduced will reap their full harvest; but his philosophical books, La Science et l'Hypothèse, La Valeur de la Science, Science et Méthode, written in his delight. fully clear style, have met immediately with a multitude of readers. They have come to a world which was eager for them. Many scientists are repelled from the ordinary philosophic works, feeling that a hopeless difference of standpoint divides them from the author from the outset. To such the writings of Poincaré come as a complete revelation. His way of thought seems to fill exactly the need of which they are conscious. There must be many of the younger generation of physicists and mathematicians whose philosophical ideas are based almost wholly on his teaching. The view which is particularly associated with the name of Poincaré is his attitude towards space and time and geometry; to him space is neither Euclidean nor non-Euclidean; the postulates of geometry are not to be derived from experiment. No system of geometry is forced on us; we may adopt whichever we will. But a certain geometry may be more convenient than any other, because it enables the facts of nature to be expressed with greater simplicity. Euclidean geometry is convenient because, when it is adopted, we find that light travels in straight lines, and that ordinary bodies are more or less rigid. If, as the principle of relativity tells us, it is found that bodies are not rigid, but systematically change their shape when they are differently oriented, we need not change our geometry though it may be more convenient to do so. "Geometry is a convention, a sort of cloak badly fitted between our love of simplicity and our desire not to depart too far from what our instruments teach us." Poincaré applied this principle to other physical ideas. "What shall be our position, faced with these new conceptions? Are we going to be forced to modify our former conclusions? Not at all: we adopted a convention because it seemed convenient, and we said that nothing could compel us to abandon it. Today certain physicists want to adopt a new convention. They are not forced to do so; they consider the new convention more useful, that is all; and those who are not of the same opinion can rightly retain the old if they do not care to disturb their old habits of thought. Between ourselves, I think they will do so for a long time to come."
We have had to examine his work under the different divisions adopted by custom; but it is interesting to trace the unity throughout it all. To quote M. Bosler: "The Fuchsian functions connect themselves with non-Euclidean geometry, which leads to the relativity of space, and that leads straight to the idealism of Berkeley." Poincaré believed that science can only advance "by unlooked-for drawing together of its diverse branches." Perhaps no other man in modern days has broken down so completely the barrier between pure and applied mathematics; he has shown by his example that the physicist may be helped by even the most advanced and specialised methods of pure mathematics. But he was a believer in mathematics for its own sake, apart from its utility and application. "Mathematical science is bound to reflect on itself, and that is well, for to reflect on itself is to reflect on the human mind which created it."
His death took place on 1912 July 17, at a time when in England an unprecedented concourse of scientific men had met together to celebrate the 250th anniversary of the foundation of the Royal Society. The news, which was quite unexpected, cast a gloom over the final days of the meetings. A few months earlier, Poincaré had come over to England to deliver a course of four lectures on mathematical philosophy in the University of London. In August his loss was especially felt at the International Congress of Mathematicians at Cambridge, over which his great fellow-worker Sir George Darwin presided. It is a strange stroke of fate by which in the same year France has lost Poincaré and England Darwin; in many respects their methods of research were in striking contrast, but their names are ever closely linked together by their contributions to astronomy, and it is pleasant to reflect on the mutual appreciation and cordial relations between these two foremost investigators of celestial mechanics.
Poincaré leaves a widow and four children.
He was elected an Associate of the Society 1894 November 9.
A. S. E.
His father, who came of a Lorraine family, was an eminent physician; the President of France, M. Raymond Poincaré, was his cousin; and another cousin, M. Lucien Poincaré, is a distinguished physicist. As a boy he suffered from poor health, and doubtless it was partly owing to this that he acquired those habits of preoccupation and inward concentration of thought, which were characteristic of him afterwards. He was educated successively at school at Nancy, at the École Polytechnique, and at the École Supérieure des Mines. At the age of twenty-five he received the degree of Doctor in the University of Paris, and about the same time was appointed Professor of Analysis at Caen. The wide variety of his scientific investigations is reflected in the diverse professorships which at one time and another he held. At Paris University he filled successively the chairs of Mechanical Physics, Mathematical Physics, Calculus of Probability, and finally, for the last sixteen years, Celestial Mechanics; and at the Ecole Polytechnique he was lecturer in Analysis and afterwards Professor of Astronomy. Yet one best-known branch of his activity is not represented in these. M. Darboux wrote to him: "Why does not the Faculty possess a chair of Scientific Philosophy? I should have been able to ask you to occupy it also." To the general public in his country he first became celebrated by winning the international prize offered by King Oscar II. of Sweden in 1887 for a memoir on the Problem of Three Bodies; the award was received in France with great enthusiasm, and the prize essay, which was the foundation of Poincaré's Les Méthodes Nouvelles de la Mécanique Céleste, remains one of his greatest achievements. It is impossible to enumerate the honours bestowed on him in his own country and abroad; thirty-two academies received him as an associate. We may mention, however, that he received the Sylvester medal of the Royal Society, and the gold medal of this Society was awarded to him in 1900. He was received into the French Academy in 1909.
Whilst continually engaged in the most profound researches, and given to fits of abstraction and absent-mindedness, he nevertheless took an active part in all those administrative and consultative duties which are pressed upon an eminent man of science. As a professor he was a conscientious teacher. Of extraneous offices, he was President of the Conseil des Observatoires and of the Société Astronomique de France; in 1906 he was President of the Académie des Sciences; he was also a member of the Bureau des Longitudes. When presiding at official meetings he could throw aside his preoccupied habits and conduct the business with alert attention. With all the pressure of professorial and other work, the output of his scientific papers averaged one a week for the thirty-three years of his active career.
It is perhaps pardonable to claim M. Poincaré as pre-eminently an astronomer, though few men have the encyclopædic knowledge necessary to compare the merits of his contributions to pure mathematics, to mathematical physics, to astronomy, and to philosophy. Each of these studies owes a great debt to his genius. His was one of those rare minds that recognise no limits between the sciences; and we can dimly perceive how much of his success may have been due to the ease with which he could apply the weapons of one branch to the problems of another. His marvellous faculty of generalisation demanded an unrestricted field, and it was backed by an activity of brain which enabled him to enter deeply into so many diverse lines of knowledge. The problems of astronomy, of the evolution of the solar system and of the universe, appear, however, to have been his first interest; and in the last fifteen years of his life, by the professorial chairs he occupied, and by his presidency of astronomical societies and committees, he seems to have recognised it as the chief claim. His contributions to astronomy are so numerous that it is impossible to attempt a general survey of them. His last published work, a critical examination of the various cosmogonic hypotheses, represents one line of his investigations. An important paper on the problem of Saturn's ring may be mentioned. The modern theory of tides owes much to Poincaré. There are, however, two special investigations, which by common consent are regarded as his chief title to fame, and these we may here consider especially.
In Lés Methodes Nouvelles de la Mécanique Céleste, a great work in three volumes published between 1892 and 1899, the brilliancy and originality of his powers are most highly manifested. Sir George Darwin describes it as "the mine from which, for more than half a century to come, humbler investigators will excavate their materials." Professor Moulton, speaking of the mathematical processes employed, says: "In power and elegance they are as much beyond those of Laplace, as his were beyond the geometry of Newton." One of Poincaré's theorems showed that the series used in celestial mechanics are ultimately divergent for large values of the time. The various periodic terms in the motions of the planets, which suffice for ordinary discussions, hold approximately true; but no conclusion as to the ultimate stability of the solar system can be drawn, for the applications of the series are not valid beyond certain limits of time. This possible instability (for the conclusion is merely negative, and does not decide between stability and instability) must not be confused with that arising from the dissipative forces of the tides, to which Poincaré also devoted researches. The result we are now considering arises from the historic Problem of Three Bodies in its simple form.
From the discovery of the divergence or non-periodicity of planetary orbits, Poincaré was led to the consideration of theoretical orbits which are strictly periodic and therefore valid for all time. The subject of Periodic Orbits, which has now an extensive literature, is one with which the practical astronomer finds it difficult to feel much sympathy; he is inclined to condemn the whole matter as misguided ingenuity, or as pure mathematics in a thin disguise. But this attitude is not a just one. The subject originally arose in the study of the lunar theory, and G. W. Hill's fertile method depends on the "variational curve," which is the, first example of a periodic orbit. To some investigators (and notably to E. W. Brown) the interest of these orbits lies in their application to the stability and possible distribution of the outer satellites of the planets. To Sir George Darwin it was a conviction that their study would ultimately afford a clue to the empirical law of Bode as to the distances of planets from the Sun. To Poincaré it was the one method by which it seemed possible to penetrate into the far-distant future of the solar system; or, to use his metaphor, "they furnish the only breach by which we may hope to penetrate the fortress of a problem hitherto deemed impregnable." Just as, although the Moon does not move along the variational curve, yet its deviation from this periodic orbit is always comparatively small, Poincaré believed that a similar use of periodic orbits might be made in determining expressions for the motions of planets valid for all time, for he showed that a periodic solution can always be found which shall differ by as small a quantity as we please from any given motion of the perturbed body.
Perhaps the researches which have aroused the widest attention are those in which Poincaré considered the possible forms of equilibrium of rotating masses of liquid, resulting in his discovery of the so-called pear-shaped figure of equilibrium. The possibility that in this way may be traced the first stages in the separation of a satellite from its primary, or the evolution of a double star, renders this work of great interest to all those who have speculated on the origin of celestial systems. But the real beauty of the investigation, and the far-reaching general principles which it contains, are often lost sight of through attention being focussed on the immediate practical result. As in the study of the Problem of Three Bodies, it was a question of stability which guided Poincaré's researches, and his results in the first instance apply with perfect generality to any mechanical system. He found that, where there is more than one form of equilibrium possible, if these forms are arranged in families, an interchange of stability and instability takes place between two families wherever they cross. If a stable and an unstable family converge, so that one limiting form may be considered a member common to both families, then on tracing the forms beyond this point, what was formerly the stable family becomes unstable, and vice versa. In the application to rotating fluid masses, it was known that when the ordinary oblate spheroid becomes unstable, the Jacobi ellipsoid with three unequal axes supersedes it as the stable configuration. Poincaré found that a point was reached when this also became unstable, and he argued that this must be a point where a new series of figures of equilibrium branches off. He investigated these new figures and showed their pear-shaped character.
In pure mathematics one of his most fertile generalisations was the invention of automorphic or, as he named them, Fuchsian functions. These functions are arrived at by an extension of the ideas of the theory of elliptic functions. The essential character of the elliptic function is its double periodicity, so that it is unchanged by a certain doubly infinite series of substitutions, just as is unchanged by the singly infinite series of substitutions for . The Fuchsian functions have the property of being unchanged by a still more general series of substitutions, namely, of the form , subject to the condition . Poincaré was able to form zeta- and theta-Fuchsian functions corresponding to the zeta and theta-elliptic functions. From this he was led to the discovery that the new functions sufficed for the integration of any linear differential equation with algebraic coefficients. They further enable the allied problem of expressing the coordinates of an algebraic curve in terms of a parameter to be solved in all cases.
In the modern developments of physics Poincaré was in the front rank of workers. A monumental work on the diffraction of electric waves is remarkable alike for the novelty of the analytical methods and the practical importance of the results. In his later years the principle of relativity and the allied problems of the dynamics of an electron claimed a large share of his attention. His work on this subject was perhaps mainly critical, rather than novel; but he has done much to reduce the various investigations into systematic relation; his broad outlook has helped to clear a subject of bewildering confusion. To the new developments of thermodynamics and Planck's theory of quanta, which appear destined to effect the greatest revolution that physical ideas have yet undergone, he devoted a searching examination. His conclusion, apparently reached with extreme reluctance, was that there can be no escape from at least some of the consequences of these new ideas. Newtonian mechanics is only applicable to matter in bulk, and the ultimate minute constituents demand some other basis of treatment. One result of Planck's quantum hypothesis is that it ceases to be possible to represent ultimate dynamics by differential equations. This is a staggering blow at what seemed to be the last article of faith of the mathematical physicist, who, bravely abandoning mechanical models, prepared to throw over the æther and even the electron if need be, yet appeared to have arrived at some ultimate facts of nature in the differential equations which remained.
In no subject has the influence of Poincaré been more widely felt than in his philosophy of science and mathematics. Much of his more technical work is hardly yet appreciated, and it will be many years before the new methods he introduced will reap their full harvest; but his philosophical books, La Science et l'Hypothèse, La Valeur de la Science, Science et Méthode, written in his delight. fully clear style, have met immediately with a multitude of readers. They have come to a world which was eager for them. Many scientists are repelled from the ordinary philosophic works, feeling that a hopeless difference of standpoint divides them from the author from the outset. To such the writings of Poincaré come as a complete revelation. His way of thought seems to fill exactly the need of which they are conscious. There must be many of the younger generation of physicists and mathematicians whose philosophical ideas are based almost wholly on his teaching. The view which is particularly associated with the name of Poincaré is his attitude towards space and time and geometry; to him space is neither Euclidean nor non-Euclidean; the postulates of geometry are not to be derived from experiment. No system of geometry is forced on us; we may adopt whichever we will. But a certain geometry may be more convenient than any other, because it enables the facts of nature to be expressed with greater simplicity. Euclidean geometry is convenient because, when it is adopted, we find that light travels in straight lines, and that ordinary bodies are more or less rigid. If, as the principle of relativity tells us, it is found that bodies are not rigid, but systematically change their shape when they are differently oriented, we need not change our geometry though it may be more convenient to do so. "Geometry is a convention, a sort of cloak badly fitted between our love of simplicity and our desire not to depart too far from what our instruments teach us." Poincaré applied this principle to other physical ideas. "What shall be our position, faced with these new conceptions? Are we going to be forced to modify our former conclusions? Not at all: we adopted a convention because it seemed convenient, and we said that nothing could compel us to abandon it. Today certain physicists want to adopt a new convention. They are not forced to do so; they consider the new convention more useful, that is all; and those who are not of the same opinion can rightly retain the old if they do not care to disturb their old habits of thought. Between ourselves, I think they will do so for a long time to come."
We have had to examine his work under the different divisions adopted by custom; but it is interesting to trace the unity throughout it all. To quote M. Bosler: "The Fuchsian functions connect themselves with non-Euclidean geometry, which leads to the relativity of space, and that leads straight to the idealism of Berkeley." Poincaré believed that science can only advance "by unlooked-for drawing together of its diverse branches." Perhaps no other man in modern days has broken down so completely the barrier between pure and applied mathematics; he has shown by his example that the physicist may be helped by even the most advanced and specialised methods of pure mathematics. But he was a believer in mathematics for its own sake, apart from its utility and application. "Mathematical science is bound to reflect on itself, and that is well, for to reflect on itself is to reflect on the human mind which created it."
His death took place on 1912 July 17, at a time when in England an unprecedented concourse of scientific men had met together to celebrate the 250th anniversary of the foundation of the Royal Society. The news, which was quite unexpected, cast a gloom over the final days of the meetings. A few months earlier, Poincaré had come over to England to deliver a course of four lectures on mathematical philosophy in the University of London. In August his loss was especially felt at the International Congress of Mathematicians at Cambridge, over which his great fellow-worker Sir George Darwin presided. It is a strange stroke of fate by which in the same year France has lost Poincaré and England Darwin; in many respects their methods of research were in striking contrast, but their names are ever closely linked together by their contributions to astronomy, and it is pleasant to reflect on the mutual appreciation and cordial relations between these two foremost investigators of celestial mechanics.
Poincaré leaves a widow and four children.
He was elected an Associate of the Society 1894 November 9.
A. S. E.
Jules Henri Poincaré's obituary appeared in Journal of the Royal Astronomical Society 73:4 (1913), 223-228.