Mathematical instrument

“ If we take the word measurement in its direct and general mathematical acceptation, which signifies simply the determination of the value of the ratios between any homogeneous magnitudes, we must consider, in geometry, that the measurement of surfaces and of volumes, unlike that of lines, is never conceived, even in the most simple and the most favourable cases, as being effected directly. ”

“ Leaving aside these technical difficulties, the fact remains that in a number of cases mathematics has been found applicable to nature at least in an approximate way, and as a result mathematical laws have been discovered. We shall now proceed to show how these theories of mathematical physics have evolved. To begin with, a large number of empirical facts are first discovered by the experimenter. ”

“ The difficulties which we have indicated in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c. It is this fact which makes necessary the formation of mathematical science, as we are going to see; for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus that it has been led to the creation of mathematics. ”

“ I.—At the beginning of the eighteenth century, mathematical instruments are carried to such perfection as to warrant the belief that all physical phenomena may be analyzed, light, electricity, sound, crystallization, heat, elasticity, cohesion and other effects of molecular forces.—See "Whewell's History of the Inductive Sciences. ”

“ And when such processes as Newton thus deduced from the conception of a Limit, are represented by means of general algebraical symbols instead of geometrical diagrams, we have then before us the Method of Fluxions, or the Differential Calculus; a mode of treating mathematical problems justly considered as the principal weapon by which the splendid triumphs of modern mathematics have been achieved. ”

“ Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. ”

“ Not confined to constant differences, it is available in every case of differences that follow a definite law, reducible therefore to an equation. An engine adjusted to the purpose being set to work, will produce any distant term, or succession of terms, required—thus presenting the numerical solution of a problem, even though the analytical solution be yet undetermined.' That the future path of some important branches of mathematical enquiry must now in some measure be directed by the dictates of mechanism, is sufficiently evident ”

“ It appears, therefore, that what we want is a knowledge of the forms of equipotential surfaces and lines of induction in as many different cases as we can collect together and remember. In certain classes of cases, such as those relating to spheres, we may proceed by mathematical methods. In other cases we cannot afford to despise the humbler method of actually drawing tentative figures on paper, and selecting that which appears least unlike the figure we require. ”

“ It was far otherwise with mathematical physics. The more you study this subject, the more you will find yourself astonished by the almost incredible triumphs of intellect which it exhibits. The great mathematical physicists of the eighteenth and first few years of the nineteenth century, most of them French, are a case in point: Maupertuis, Clairaut, D’Alembert, Lagrange, Laplace, Fourier, form a series of names, such that each recalls to mind some achievement of the first rank. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ We have insisted particularly on this generalising tendency of mathematics, as it is the one which will play a prominent part when we consider the methodology of mathematical physics as applied to the world of reality. ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ The commonly distinctive feature of a modern mathematical treatise, in any branch of physics, is that the investigation of any problem is initially conducted on the widest and most comprehensive basis, equations being first obtained in their most general form, the simpler and more obvious cases being allowed to follow naturally, the greater including the less. ”

“ The mathematician deals with numbers and their relations to one another. Progressing from the 2-plus-2 stage into higher mathematics, this science is essential to all the others—from the simple task of counting test tubes in a cabinet to an incredibly complex mathematical idea.
The mathematician speaks the language of all sciences using his special tool. Without him modern technology would not exist because mathematics interprets and explains all other sciences. ”

“ The older methods are effective with a portion of the student body which has the greater mathematical ability and training, but they discourage a large majority of the pupils who are not gifted or prepared for severe mathematical analysis. For this reason, many of the more difficult mathematical demonstrations often given in physics texts are omitted. Most of the problems involve only the units employed in practical every-day measurements. ”

“ Geometry has always been a favorite mode of exercising this propensity: and that science, along with Trigonometry, Plane and Spherical, to which the early problems of astronomy gave rise, have, up to the present day, been a constant field for the exercise of mathematical ingenuity; a few simple astronomical truths being assumed as the basis of the reasoning. ”

“ In this, and in all other instances, as was explained above, the particular numerical data and the numerical results are determined by means and by portions of the mechanism which act quite independently of those that regulate the operations. In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated. ”

“ The great inventions of the seventeenth century—Analytical Geometry and the Infinitesimal Calculus—were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. Philosophers, who should have taken up the task, had too little mathematical ability to invent the new branches of mathematics which have now been found necessary for any adequate discussion. ”

“ And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics. ”

“ Every branch of physics gives rise to an application of mathematics. A prophecy may be hazarded that in the future these applications will unify themselves into a mathematical theory of a hypothetical substructure of the universe, uniform under all the diverse phenomena. ”

“ My object is to shew how it comes about that we have inherited a mathematical scheme which in its final form embraces subjects physically so different as Heat-Conduction, Hydrodynamics, Elasticity, Magnetism, Electricity, and Light, and can be made to include any one of these by assigning proper names to the symbols. ”

“ The eminent scientists who designed that system were able to solve the most difficult problems in higher mathematics, but failed to comprehend what system of weights and measures was best suited for the carder, spinner, weaver and finisher of wool, cotton, linen and silk. The glamor of their fame failed to make the centimetre suitable for counting ‘picks.’ Their system had to stand or fall on its merits, and falling has proved that the highest of mathematical abilities is not inconsistent with a dense ignorance of the practical affairs of every-day life. ”

“ The mathematics of history is a human reality. We are facing an undeniable fact, the logical organization of the events of history, without a pure mathematics that is not derived from one that is integral or from a differential equation but, within the mathematical concept of applied sciences where pure results are not achieved with the “Standard Meter”, the model unit of measure that is in Paris that, although it is kept hermetically sealed, is always under the influence of that which is material. ”

“ A higher development of mathematical thought shows how, by the addition of certain conceptions, one can pass from the series of conceived objects and relations of objects that projective geometry finds in space, to the series of the metrical geometer, and vice versa. ”

“ The mathematical problems and tables are in simple arithmetical terms, and involve no algebra or higher mathematics. The method of treatment is strictly progressive, following the successive steps necessary to becoming an intelligent and skilled mechanic. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ Just so, pure mathematical science has no logical dependence upon physics. Yet we have all heard how largely physical science has influenced the lines of investigation followed by the modern mathematicians. Within the mathematical realm itself, pure Algebra, when once abstractly defined, is not logically dependent upon Geometry for its principles or for its theories, yet some theories of modern Algebra have actually developed largely under the spell, as it were, of ideas of an unquestionable geometrical origin. ”

“ In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. ”

“ Origin of Mathematical Physics.—Let us penetrate further, and study more closely the conditions that have permitted the development of mathematical physics. We observe at once that the efforts of scientists have always aimed to resolve the complex phenomenon directly given by experiment into a very large number of elementary phenomena. ”

“ The principle selected by Mr Babbage as the basis of that part of the machinery which calculates, is the Method of Differences; and he has in fact literally thrown this mathematical principle into wheel-work. In order to form a notion of the nature of the machinery, it will be necessary, first to convey to the reader some idea of the mathematical principle just alluded to.
A numerical table, of whatever kind, is a series of numbers which possess some common character, and which proceed increasing or decreasing according to some general law. ”

“ Tools and instruments used by mechanicians are very often made of the material most used in their profession; for instance, a blacksmith’s tools are generally of iron, a carpenter’s tools largely of wood, and a glass-blower uses many things made of glass, and so on. Mathematicians are no exception to this general rule, and seem to imagine that everything can be accomplished by pure mathematical formulæ. ”

“ Without this, indeed, it is scarcely possible that any theory should ever be fairly brought to the test by a comparison with facts. Accordingly, the mathematical theory of electrical equilibrium, and the laws of the distribution of the electric fluids over the surfaces of bodies in which they are accumulated, have been made the subject of elaborate geometrical investigation by the most expert mathematicians, and have attained a degree of extent and elegance which places this branch of science in a very high rank in the scale of mathematico-physical enquiry. ”

“ The task is not without difficulty; it is not enough to open a book at random and to analyse any proof we may come across. First of all, geometry must be excluded, or the question becomes complicated by difficult problems relating to the rôle of the postulates, the nature and the origin of the idea of space. For analogous reasons we cannot avail ourselves of the infinitesimal calculus. We must seek mathematical thought where it has remained pure—i.e., in Arithmetic. ”