"The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered." (Augustus De Morgan, "Review of 'Théorie Analytique des Probabilites' par M. le Marquis de Laplace" 1820)
"Whenever I meet in Laplace with the words “Thus it plainly appears”, I am sure that hours and perhaps days, of hard study will alone enable me to discover how it plainly appears." (Nathaniel Bowditch, "Mécanique céleste", 1829-39)
"The influence of electricity in producing decompositions, although of inestimable value as an instrument of discovery in chemical inquiries, can hardly be said to have been applied to the practical purposes of life, until the same powerful genius [Davy] which detected the principle, applied it, by a singular felicity of reasoning, to arrest the corrosion of the copper-sheathing of vessels. […] this was regarded as by Laplace as the greatest of Sir Humphry's discoveries." (Charles Babbage, "Reflections on the Decline of Science in England", 1830)
"The name of Sir Isaac Newton has by general consent been placed at the head of those great men who have been the ornaments of their species. […] The philosopher [Laplace], indeed, to whom posterity will probably assign a place next to Newton, has characterized the Principia as pre-eminent above all the productions of human intellect." (David Brewster, "Life of Sir Isaac Newton", 1831)
"The persons who have been employed on these problems of applying the properties of matter and the laws of motion to the explanation of the phenomena of the world, and who have brought to them the high and admirable qualities which such an office requires, have justly excited in a very eminent degree the admiration which mankind feels for great intellectual powers. Their names occupy a distinguished place in literary history; and probably there are no scientific reputations of the last century higher, and none more merited, than those earned by great mathematicians who have laboured with such wonderful success in unfolding the mechanism of the heavens; such for instance as D ’Alembert, Clairaut, Euler, Lagrange, Laplace." (William Whewell, "Astronomy and General Physics", 1833)
"We pass with admiration along the great series of mathematicians, by whom the science of theoretical mechanics has been cultivated, from the time of Newton to our own. There is no group of men of science whose fame is higher or brighter. The great discoveries of Copernicus, Galileo, Newton, had fixed all eyes on those portions of human knowledge on which their successors employed their labors. The certainty belonging to this line of speculation seemed to elevate mathematicians above the students of other subjects; and the beauty of mathematical relations and the subtlety of intellect which may be shown in dealing with them, were fitted to win unbounded applause. The successors of Newton and the Bernoullis, as Euler, Clairaut, D’Alembert, Lagrange, Laplace, not to introduce living names, have been some of the most remarkable men of talent which the world has seen." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)
"Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced - the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity." (Charles Babbage, "The Ninth Bridgewater Treatise: A Fragment", 1838)
"In general I would be cautious against […] plays of fancy and would not make way for their reception into scientific astronomy, which must have quite a different character. Laplace’s cosmogenic hypotheses belong in that class. Indeed, I do not deny that I sometimes amuse myself in a similar manner, only I would never publish the stuff. My thoughts about the inhabitants of celestial bodies, for example, belong in that category. For my part, I am (contrary to the usual opinion) convinced […] that the larger the cosmic body, the smaller are the inhabitants and other products. For example, on the sun trees, which in the same ratio would be larger than ours, as the sun exceeds the earth in magnitude, would not be able to exist, for on account of the much greater weight on the surface of the sun, all branches would break themselves off, in so far as the materials are not of a sort entirely heterogeneous with those on earth." (Carl F Gauss, (Letter to Heinrich Schumacher] 1847)
"No one can read the history of astronomy without perceiving that Copernicus, Newton, Laplace, are not new men, or a new kind of men, but that Thales, Anaximenes, Hipparchus, Empodocles, Aristorchus, Pythagorus, Oenipodes, had anticipated them." (Ralph W Emerson, "The Conduct of Life", 1860)
"Laplace considers astronomy a science of observation, because we can only observe the movements of the planets; we cannot reach them, indeed, to alter their course and to experiment with them. 'On earth', said Laplace, 'we make phenomena vary by experiments; in the sky, we carefully define all the phenomena presented to us by celestial motion.' Certain physicians call medicine a science of observations, because they wrongly think that experimentation is inapplicable to it." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)
"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible." (W W Rouse Ball, "A Short Account of the History of Mathematics", 1888)
"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems." (Florian Cajori, "A History of Mathematics", 1893)
"From the infinitely great down to the infinitely small, all things are subject to [the laws of nature]. The sun and the planets follow the laws discovered by Newton and Laplace, just as the atoms in their combinations follow the laws of chemistry, as living creatures follow the laws of biology. It is only the imperfections of the human mind which multiply the divisions of the sciences, separating astronomy from physics or chemistry, the natural sciences from the social sciences. In essence, science is one. It is none other than the truth." (Vilfredo Pareto, "Cours d’Economie Politique", 1896-97)
"The first effect of the mind growing cultivated is that processes once multiple get to be performed in a single act. Lazarus has called this the progressive 'condensation' of thought. [...] Steps really sink from sight. An advanced thinker sees the relations of his topics is such masses and so instantaneously that when he comes to explain to younger minds it is often hard [...] Bowditch, who translated and annotated Laplace's Méchanique Céleste, said that whenever his author prefaced a proposition by the words 'it is evident', he knew that many hours of hard study lay before him." (William James, "The Principles of Psychology" Vol. II, 1918)
"The scientist who recognizes God knows only the God of Newton. To him the God imagined by Laplace and Comte is wholly inadequate. He feels that God is in nature, that the orderly ways in which nature works are themselves the manifestations of God's will and purpose. Its laws are his orderly way of working." (Arthur H Compton, "The Human Meaning of Science", 1940)
"As far as I see, such a theory [of the primeval atom] remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being. He may keep, for the bottom of space-time, the same attitude of mind he has been able to adopt for events occurring in non-singular places in space-time. For the believer, it removes any attempt to familiarity with God, as were Laplace’s chiquenaude or Jeans’ finger. It is consonant with the wording of Isaiah speaking of the 'Hidden God' hidden even in the beginning of the universe […] Science has not to surrender in face of the Universe and when Pascal tries to infer the existence of God from the supposed infinitude of Nature, we may think that he is looking in the wrong direction." (Monsignor G Lemaître, "The Primeval Atom Hypothesis and the Problem of Clusters of Galaxies", [in R. Stoops (ed.), "La Structure et l'Evolution de l'Univers"] 1958)
"Laplace's equation does not yield easily to straightforward treatment: fortunately in the development of conformal transformations there is no need to seek a formal solution of the equation. It is only necessary to note first that all fields and functions to be considered in this book are those that satisfy the inverse square equation when emanating from a point source and therefore they also satisfy Laplace's equation. The second point is that the equation is necessary for the development of other important equations that govern these particular fields. Such fields are called Laplacian. When the field is not Laplacian, more recondite methods are necessary for determining its distribution." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)
"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)
"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Why should a geometer, whose principal concern is in measurements of distance, desire to engage in analysis on a Riemannian manifold? For example, pondering the Laplacian, its eigenvalues and eigenfunctions? Here are some reasons, chosen from among many others. We note also here that the existence of a canonical elliptic differential operator on any Riemannian manifold, one which is moreover easy to define and manipulate, is one of the motivations to consider Riemannian geometry as a basic field of investigation. [...] Riemannian geometry is by its very essence differential, working on manifolds with a differentiable structure. This automatically leads to analysis. It is interesting to note here that, historically, many great contributions to the field of Riemannian geometry came from analysts." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
No comments:
Post a Comment