MacTutor
GEORGE WILLIAM HILL was born at New York on 1838 March 3. His father had come from England in 1808, while his mother was descended from one of the original Huguenot colonists. Early in his life his parents removed to West Nyack, N.Y., and there, in a quiet country home, Hill spent his boyhood. At the age of seventeen he went to Rutgers College, New Brunswick, N.J., where, under Dr. Strong, he received his introduction to the works of Euler, Laplace, Lagrange, and the other great writers on dynamical astronomy. The knowledge he obtained there had the greatest influence on his subsequent work. From Rutgers, Hill proceeded in 1859 to Cambridge, Mass., to continue his studies. Soon he started his original work. For an essay extending Laplace's theory of the configuration of the Earth he was awarded a prize offered by the Mathematical Monthly. This essay served as an introduction to Runkle, the editor, who was superintendent of the American Nautical Almanac Office. Early in 1861 Hill was offered, and accepted, a post in the office, and there he laboured for thirty years to the great advantage of mathematical astronomy.
In his early years at the Nautical Almanac Office, Hill wrote a few papers dealing with problems of general mathematical interest, but he soon confined his attention to purely astronomical subjects. In one of several papers written for the purpose of facilitating astronontical computation, he gave a very full discussion of the derivation and reduction of star places. But more and more he turned towards dynamical astronomy. The determination of orbits was one of his favourite subjects. In particular we may mention a rigorous discussion of all the observations of the great comet of 1858, and a determination of the most probable orbit. The most important part of Hill's early work is in connection with the planet Venus. In view of the transit of 1874 he determined corrections to the elements of the orbit and discussed in great detail the phenomena of the transit.
But it was not merely the application of previously existing theories that engaged Hill's attention. Even from the beginning of his career he was eagerly inquiring after new methods for attacking the old problems and for simplifying the numerical calculations. The rate of discovery of new minor planets was increasing so rapidly that it appeared that the calculators would not be able to keep pace with the discoverers, and that many new planets would be discovered only to be lost. Hill was thus led to write several papers for the rapid calculation of approximate orbits. These orbits would be sufficiently accurate for following the body. till a rigorous orbit could be worked out. But Hill was not the man to be long content with approximate orbits, and he was soon at work on perturbations. This led him to the masses of the planets, a subject to which he devoted a great deal of attention throughout his life. The planet which produces the greatest effect on the minor planets is Jupiter, and Hill gave tables for calculating certain long inequalities produced by it on all minor planets with a daily mean motion between 550" and 650". Observations of these planets over a sufficient interval of time will yield a very accurate value of the mass of Jupiter.
In 1877 and 1878 there appeared the work by which Hill is best known, and which constitutes his most original contribution to our knowledge of celestial mechanics a new method of dealing with the Moon's motion. For some years the world failed to recognise the value of Hill's work – so novel was the method and so unknown the author. It is true that early in the history of the lunar problem Newton and Euler had used methods resembling that employed by Hill. But their work had never been employed in any of the serious attempts to solve the problem of the Moon's motion, and, latterly, astronomers had settled more or less into a groove as to the best line of attack. Successive attempts had led to more and more accurate results as the quantities neglected became smaller and smaller, but the formulæ were rapidly tending to become unmanageable on account of the enormous number of terms involved and the greater labour involved in calculating each new term. Astronomers were more inclined to look with amazement at what had already been accomplished (or, perhaps, to contemplate adding more terms to the already existing series) than to think of a new theory, when Hill's memoirs appeared. The entire treatment of the problem was altered. Instead of polar coordinates referred to fixed axes, rectangular coordinates referred to moving axes were introduced. By this means Hill obtained the differential equations in a simple algebraic form, very suitable for solution in infinite series. The use of a new parameter greatly increased the convergency of the series. The new method then led to the use of the "variational curve" as an orbit of reference. All previous lunar theorists had used an ellipse for this purpose. The characteristic of the "variational curve" which makes it so useful is that it contains a most important part of the solar perturbation, known as the "variation." One of the great difficulties which had been met in the older theories was due to the fact that the lunar perigee had a continuous motion, so that a fixed ellipse could not be found which approximated to the actual orbit of the Moon for any length of time. In the new theory this gave no difficulty. The value of Hill's method has been acknowledged throughout the world, but although Hill wrote sufficient to prove its usefulness, he never had the time to elaborate a lunar theory. Nevertheless, his ideas have been put to practical use. The great task of developing and applying the method has been successfully undertaken by Professor E. W. Brown. The new theory has been completed, and the necessary tables are being formed.
The memoirs in which Hill announced his new method, besides being of the greatest importance as having led to a new lunar theory, deserve detailed attention on other accounts. The "variational curve," which is fundamental to the theory, is a rigorous solution of the problem presented by the differential equations of motion when the eccentricities, the inclination, and the parallax, vanish. The actual motion of the Moon is obtained as the result of "free" and "forced" oscillations about the "variational curve." The "variational curve" is the first example we have of a periodic curve, which is a rigorous solution of the problem of three bodies. It is the starting-point of a general theory of periodic orbits which has been developed by Poincaré, Darwin, Brown, Moulton, and others. The importance of periodic orbits is due to the fact that they are valid for all time. No known orbit is periodic, but our hopes of obtaining lunar and planetary theories which will be accurate for very long intervals of time are intimately connected with periodic orbits.
In regions of pure mathematics, also, Hill's memoirs on the lunar theory have given rise to important advances. In discussing the eccentricity of the Moon's orbit, Hill by a series of skilful transformations made his results depend on the solution of a differential equation of the form
The invention of this new method is not Hill's only contribution to lunar theory. At various times throughout his long life he wrote important papers dealing with definite branches of the theory, and, particularly, extensions of Delaunay's theory. As is well known, Delaunay did not quite complete his theory: at the time of his death there were several sources of perturbation still to be considered. Hill always expressed the greatest admiration for Delaunay's work and spent a great deal of time in calculating the outstanding perturbations. In one of his papers he calculated all the perturbations due to the Earth's ellipticity. Hill shows his usual thoroughness in this paper by determining for himself the value of the constant which enters. This entailed an elaborate discussion of numerous pendulum experiments. In other papers dealing with the Moon, Hill dealt with perturbations produced by planets, and inequalities due to the motion of the ecliptic.
Turning again to Hill's work on planetary theory we find a memoir "On Gauss's Method of Computing Secular Perturbations, with an Application to the Action of Venus on Mercury." This is a rediscussion of a celebrated investigation of Gauss's, in which the formulæ are obtained in a very simple manner. The object of this paper was to emphasise the importance of Gauss's original memoir and to reduce the formulæ to the form best adapted for practical requirements.
Hill had at various times been engaged in work connected with Jupiter and Saturn. The appointment of Newcomb to the directorship of the Nautical Almanac Office in 1877 led to his definitely taking up the problem presented by the two great planets. Newcomb had conceived the idea of revising the tables of all the planets on a uniform basis. The most difficult part was connected with Jupiter and Saturn, and Newcomb left it entirely to Hill. These two planets had always given trouble. The cause of that trouble had been found by Laplace to lie in the fact that five periods of Jupiter are very nearly equal to two of Saturn. Laplace, Pontécoulant, and Plana had not obtained the required degree of accuracy, as they had not an adequate method for obtaining the perturbations of the second order. Such a method was subsequently invented by Hansen, but he had left his work on Jupiter and Saturn unfinished in order to apply his method to the Moon. Hill concluded that Hansen's method, suitably developed to give perturbations of still higher orders, would suffice for his purpose. This was, to use his own words, "to form theories of Jupiter and Saturn which would be practically serviceable for a space of three hundred years on each side of the central epoch taken near the centre of gravity of all the times of observation theories whose errors in this interval would simply result, not from neglected terms in the developments, but from the unavoidable imperfections in the values of the arbitrary constants and the masses, adopted from the indications of observation." At this work Hill laboured for nearly fifteen years. All the original calculations he made himself, and although Newcomb pressed him to accept assistance, he only consented in so far as duplication was concerned. The resulting tables, which are as accurate as any we possess, have been used in the construction of all the great national ephemerides (except the Conn. des Temps) since 1900.
Immediately after he had completed this work, Hill retired from the Nautical Almanac, to quote the words of Newcomb, "doubtless feeling, as well he might, that he had done his whole duty to science and the government." It was only from his official duties, however, that he retired, for he continued to work at many problems connected with the theory of perturbations. For a short time, indeed, he undertook to lecture on celestial dynamics at Columbia University, but he preferred to live quietly at his old country home. There he worked continuously to the very end of his life, writing many papers of importance. These papers cover a very wide field, and it is impossible to do justice to them all. All questions involving the fundamental constants of the solar system received his attention. He extended his researches in lunar theory; he calculated the masses of several planets; and he wrote new papers on the theories of Jupiter and Saturn. A comparison of his new tables for these planets with the most recent observations showed that his theory was sensibly correct. In his later years, however, his mind tended towards those methods of calculating general perturbations which seemed hopeful of giving results valid for longer intervals of time than the ordinary theories afforded. We thus find memoirs on periodic orbits, intermediary orbits, and periphlegmatic orbits. The researches of Poincaré and Gylden therefore received a close examination by him. He expressed high admiration for Poincaré's work on periodic orbits, and believed that it would lead to results still more valuable than those already obtained, Gyldén attempted by quite different methods from Poincaré to obtain theories suitable for indefinitely long intervals of time. He undertook his researches from a much more practical point of view than Poincaré, and on that account his work was specially attractive to Hill. Unfortunately Gyldén had set himself the tremendous task of dealing with all the eight great planets at once, and this together with his early death prevented him from getting anything completed. Hill believed that by taking more restricted problems results of value could be obtained in a comparatively short time. In particular he derived results for Jupiter and Saturn when the inclination of their orbits is neglected and various constants are assumed known. The difference between the actual and ideal cases is calculable. In other papers he dealt with other aspects of Gyldén's work. His last paper was published in 1914 January in the Astronomical Journal, of which he was an associate editor.
It is difficult to find such another man as Hill in the whole history of science. Although holding only a humble post in the Nautical Almanac Office, he lived happily in the liberty it afforded him, and never sought to improve his material position. All he asked was peace and freedom to prosecute his work. Newcomb did his best to secure official recognition for his illustrious assistant, but Hill seemed to take little interest in the matter. The post he occupied, happily, did not prevent his getting due scientific recognition. Newcomb's words put the situation very plainly: "Here was perhaps the greatest living master in the highest and most difficult branch of astronomy, winning world-wide recognition for his country in the science, and receiving the salary of a department clerk." Europe, as well as his own country, bestowed on him the highest scientific honours. He was awarded our Gold Medal in 1887. The Royal Society elected him a foreign member in 1902, and awarded him the Copley Medal in 1909. When he visited this country in 1892, Cambridge gave him an honorary degree. France elected him a corresponding member of its Institute. Mathematicians throughout the world have declared that they found inspiration in his work. He was President of the American Mathematical Society, 1894-96. Four large quarto volumes of his collected works were published by the Carnegie Institution of Washington, 1905-07. His later works could form a fifth volume.
Hill stands in the foremost ranks of dynamical astronomers. Few have the time or inclination to read his works. A glance at the volumes, particularly that on Jupiter and Saturn, will show their magnitude. The fact that his tables have been generally accepted as the best is proof of their accuracy. Poincaré's praise and the labour devoted by Brown and others to the development of his lunar theory are the best proofs of its value. But only those who actually read his works can form a true estimate of his greatness. Even his earliest papers show that he had a firm grasp of the whole field of celestial dynamics as it had been presented by the great masters of the past. Almost everyone who studies carefully any of the great works will be fascinated by them. Celestial dynamics seemed to enter into the soul of Hill. Yet his knowledge of what had already been done did not tie him down to the old methods. His mind seems to have been so singularly clear that he could comprehend a whole theory as a single thought. It was the clearness of his intellect together with the highest mathematical genius which enabled him to do so much. To him mathematics was the handmaid of astronomy. Although capable of the highest mathematical research, he did not aim so much at the formation of elegant formulæ as at the determination of the actual quantities of which dynamical astronomy stands in need. His new method for forming a lunar theory will probably remain the greatest monument to his fame, but the most marked characteristic of his work is its eminently practical character.
He died at West Nyack 1914 April 16.
He was elected an Associate 1878 November 8.
J.J.
In his early years at the Nautical Almanac Office, Hill wrote a few papers dealing with problems of general mathematical interest, but he soon confined his attention to purely astronomical subjects. In one of several papers written for the purpose of facilitating astronontical computation, he gave a very full discussion of the derivation and reduction of star places. But more and more he turned towards dynamical astronomy. The determination of orbits was one of his favourite subjects. In particular we may mention a rigorous discussion of all the observations of the great comet of 1858, and a determination of the most probable orbit. The most important part of Hill's early work is in connection with the planet Venus. In view of the transit of 1874 he determined corrections to the elements of the orbit and discussed in great detail the phenomena of the transit.
But it was not merely the application of previously existing theories that engaged Hill's attention. Even from the beginning of his career he was eagerly inquiring after new methods for attacking the old problems and for simplifying the numerical calculations. The rate of discovery of new minor planets was increasing so rapidly that it appeared that the calculators would not be able to keep pace with the discoverers, and that many new planets would be discovered only to be lost. Hill was thus led to write several papers for the rapid calculation of approximate orbits. These orbits would be sufficiently accurate for following the body. till a rigorous orbit could be worked out. But Hill was not the man to be long content with approximate orbits, and he was soon at work on perturbations. This led him to the masses of the planets, a subject to which he devoted a great deal of attention throughout his life. The planet which produces the greatest effect on the minor planets is Jupiter, and Hill gave tables for calculating certain long inequalities produced by it on all minor planets with a daily mean motion between 550" and 650". Observations of these planets over a sufficient interval of time will yield a very accurate value of the mass of Jupiter.
In 1877 and 1878 there appeared the work by which Hill is best known, and which constitutes his most original contribution to our knowledge of celestial mechanics a new method of dealing with the Moon's motion. For some years the world failed to recognise the value of Hill's work – so novel was the method and so unknown the author. It is true that early in the history of the lunar problem Newton and Euler had used methods resembling that employed by Hill. But their work had never been employed in any of the serious attempts to solve the problem of the Moon's motion, and, latterly, astronomers had settled more or less into a groove as to the best line of attack. Successive attempts had led to more and more accurate results as the quantities neglected became smaller and smaller, but the formulæ were rapidly tending to become unmanageable on account of the enormous number of terms involved and the greater labour involved in calculating each new term. Astronomers were more inclined to look with amazement at what had already been accomplished (or, perhaps, to contemplate adding more terms to the already existing series) than to think of a new theory, when Hill's memoirs appeared. The entire treatment of the problem was altered. Instead of polar coordinates referred to fixed axes, rectangular coordinates referred to moving axes were introduced. By this means Hill obtained the differential equations in a simple algebraic form, very suitable for solution in infinite series. The use of a new parameter greatly increased the convergency of the series. The new method then led to the use of the "variational curve" as an orbit of reference. All previous lunar theorists had used an ellipse for this purpose. The characteristic of the "variational curve" which makes it so useful is that it contains a most important part of the solar perturbation, known as the "variation." One of the great difficulties which had been met in the older theories was due to the fact that the lunar perigee had a continuous motion, so that a fixed ellipse could not be found which approximated to the actual orbit of the Moon for any length of time. In the new theory this gave no difficulty. The value of Hill's method has been acknowledged throughout the world, but although Hill wrote sufficient to prove its usefulness, he never had the time to elaborate a lunar theory. Nevertheless, his ideas have been put to practical use. The great task of developing and applying the method has been successfully undertaken by Professor E. W. Brown. The new theory has been completed, and the necessary tables are being formed.
The memoirs in which Hill announced his new method, besides being of the greatest importance as having led to a new lunar theory, deserve detailed attention on other accounts. The "variational curve," which is fundamental to the theory, is a rigorous solution of the problem presented by the differential equations of motion when the eccentricities, the inclination, and the parallax, vanish. The actual motion of the Moon is obtained as the result of "free" and "forced" oscillations about the "variational curve." The "variational curve" is the first example we have of a periodic curve, which is a rigorous solution of the problem of three bodies. It is the starting-point of a general theory of periodic orbits which has been developed by Poincaré, Darwin, Brown, Moulton, and others. The importance of periodic orbits is due to the fact that they are valid for all time. No known orbit is periodic, but our hopes of obtaining lunar and planetary theories which will be accurate for very long intervals of time are intimately connected with periodic orbits.
In regions of pure mathematics, also, Hill's memoirs on the lunar theory have given rise to important advances. In discussing the eccentricity of the Moon's orbit, Hill by a series of skilful transformations made his results depend on the solution of a differential equation of the form
,
where are a series of decreasing small quantities. This equation is of frequent occurrence in physical problems, and in connection with these has been treated by Lord Rayleigh and others. Even before Hill's work, the equation had been obtained by Adams in connection with the motion of the Moon's node, in which problem it arises naturally. Adams had actually solved the equation with considerable accuracy in much the same way as Hill. But Adams' work was unpublished till after Hill's appeared. Hill's discussion shows the extraordinary originality of his mathematical genius. The solution involves the use of an infinite determinant – an entirely new mathematical idea. In all problems involving an infinite number of terms the utmost care has to be taken if gross errors are to be avoided. In the case of the infinite determinant the danger is particularly great. But Hill successfully overcame all the difficulties and obtained a result of great analytical beauty which enabled him to calculate the motion of the lunar perigee with a degree of accuracy far surpassing anything previously obtained.
The invention of this new method is not Hill's only contribution to lunar theory. At various times throughout his long life he wrote important papers dealing with definite branches of the theory, and, particularly, extensions of Delaunay's theory. As is well known, Delaunay did not quite complete his theory: at the time of his death there were several sources of perturbation still to be considered. Hill always expressed the greatest admiration for Delaunay's work and spent a great deal of time in calculating the outstanding perturbations. In one of his papers he calculated all the perturbations due to the Earth's ellipticity. Hill shows his usual thoroughness in this paper by determining for himself the value of the constant which enters. This entailed an elaborate discussion of numerous pendulum experiments. In other papers dealing with the Moon, Hill dealt with perturbations produced by planets, and inequalities due to the motion of the ecliptic.
Turning again to Hill's work on planetary theory we find a memoir "On Gauss's Method of Computing Secular Perturbations, with an Application to the Action of Venus on Mercury." This is a rediscussion of a celebrated investigation of Gauss's, in which the formulæ are obtained in a very simple manner. The object of this paper was to emphasise the importance of Gauss's original memoir and to reduce the formulæ to the form best adapted for practical requirements.
Hill had at various times been engaged in work connected with Jupiter and Saturn. The appointment of Newcomb to the directorship of the Nautical Almanac Office in 1877 led to his definitely taking up the problem presented by the two great planets. Newcomb had conceived the idea of revising the tables of all the planets on a uniform basis. The most difficult part was connected with Jupiter and Saturn, and Newcomb left it entirely to Hill. These two planets had always given trouble. The cause of that trouble had been found by Laplace to lie in the fact that five periods of Jupiter are very nearly equal to two of Saturn. Laplace, Pontécoulant, and Plana had not obtained the required degree of accuracy, as they had not an adequate method for obtaining the perturbations of the second order. Such a method was subsequently invented by Hansen, but he had left his work on Jupiter and Saturn unfinished in order to apply his method to the Moon. Hill concluded that Hansen's method, suitably developed to give perturbations of still higher orders, would suffice for his purpose. This was, to use his own words, "to form theories of Jupiter and Saturn which would be practically serviceable for a space of three hundred years on each side of the central epoch taken near the centre of gravity of all the times of observation theories whose errors in this interval would simply result, not from neglected terms in the developments, but from the unavoidable imperfections in the values of the arbitrary constants and the masses, adopted from the indications of observation." At this work Hill laboured for nearly fifteen years. All the original calculations he made himself, and although Newcomb pressed him to accept assistance, he only consented in so far as duplication was concerned. The resulting tables, which are as accurate as any we possess, have been used in the construction of all the great national ephemerides (except the Conn. des Temps) since 1900.
Immediately after he had completed this work, Hill retired from the Nautical Almanac, to quote the words of Newcomb, "doubtless feeling, as well he might, that he had done his whole duty to science and the government." It was only from his official duties, however, that he retired, for he continued to work at many problems connected with the theory of perturbations. For a short time, indeed, he undertook to lecture on celestial dynamics at Columbia University, but he preferred to live quietly at his old country home. There he worked continuously to the very end of his life, writing many papers of importance. These papers cover a very wide field, and it is impossible to do justice to them all. All questions involving the fundamental constants of the solar system received his attention. He extended his researches in lunar theory; he calculated the masses of several planets; and he wrote new papers on the theories of Jupiter and Saturn. A comparison of his new tables for these planets with the most recent observations showed that his theory was sensibly correct. In his later years, however, his mind tended towards those methods of calculating general perturbations which seemed hopeful of giving results valid for longer intervals of time than the ordinary theories afforded. We thus find memoirs on periodic orbits, intermediary orbits, and periphlegmatic orbits. The researches of Poincaré and Gylden therefore received a close examination by him. He expressed high admiration for Poincaré's work on periodic orbits, and believed that it would lead to results still more valuable than those already obtained, Gyldén attempted by quite different methods from Poincaré to obtain theories suitable for indefinitely long intervals of time. He undertook his researches from a much more practical point of view than Poincaré, and on that account his work was specially attractive to Hill. Unfortunately Gyldén had set himself the tremendous task of dealing with all the eight great planets at once, and this together with his early death prevented him from getting anything completed. Hill believed that by taking more restricted problems results of value could be obtained in a comparatively short time. In particular he derived results for Jupiter and Saturn when the inclination of their orbits is neglected and various constants are assumed known. The difference between the actual and ideal cases is calculable. In other papers he dealt with other aspects of Gyldén's work. His last paper was published in 1914 January in the Astronomical Journal, of which he was an associate editor.
It is difficult to find such another man as Hill in the whole history of science. Although holding only a humble post in the Nautical Almanac Office, he lived happily in the liberty it afforded him, and never sought to improve his material position. All he asked was peace and freedom to prosecute his work. Newcomb did his best to secure official recognition for his illustrious assistant, but Hill seemed to take little interest in the matter. The post he occupied, happily, did not prevent his getting due scientific recognition. Newcomb's words put the situation very plainly: "Here was perhaps the greatest living master in the highest and most difficult branch of astronomy, winning world-wide recognition for his country in the science, and receiving the salary of a department clerk." Europe, as well as his own country, bestowed on him the highest scientific honours. He was awarded our Gold Medal in 1887. The Royal Society elected him a foreign member in 1902, and awarded him the Copley Medal in 1909. When he visited this country in 1892, Cambridge gave him an honorary degree. France elected him a corresponding member of its Institute. Mathematicians throughout the world have declared that they found inspiration in his work. He was President of the American Mathematical Society, 1894-96. Four large quarto volumes of his collected works were published by the Carnegie Institution of Washington, 1905-07. His later works could form a fifth volume.
Hill stands in the foremost ranks of dynamical astronomers. Few have the time or inclination to read his works. A glance at the volumes, particularly that on Jupiter and Saturn, will show their magnitude. The fact that his tables have been generally accepted as the best is proof of their accuracy. Poincaré's praise and the labour devoted by Brown and others to the development of his lunar theory are the best proofs of its value. But only those who actually read his works can form a true estimate of his greatness. Even his earliest papers show that he had a firm grasp of the whole field of celestial dynamics as it had been presented by the great masters of the past. Almost everyone who studies carefully any of the great works will be fascinated by them. Celestial dynamics seemed to enter into the soul of Hill. Yet his knowledge of what had already been done did not tie him down to the old methods. His mind seems to have been so singularly clear that he could comprehend a whole theory as a single thought. It was the clearness of his intellect together with the highest mathematical genius which enabled him to do so much. To him mathematics was the handmaid of astronomy. Although capable of the highest mathematical research, he did not aim so much at the formation of elegant formulæ as at the determination of the actual quantities of which dynamical astronomy stands in need. His new method for forming a lunar theory will probably remain the greatest monument to his fame, but the most marked characteristic of his work is its eminently practical character.
He died at West Nyack 1914 April 16.
He was elected an Associate 1878 November 8.
J.J.
George William Hill's obituary appeared in Journal of the Royal Astronomical Society 75:4 (1915), 258-264.