Loquacity

“ If we now determine a line , such that this integral is equal to
,
where and are the areas of the two sections, the length of will be the same whatever unit of length we adopt, and whatever system of logarithms we use. If we suppose the sections divided into elements of equal size, then the logarithm of , multiplied by the number of pairs of elements, will be equal to the sum of the logarithms of the distances of all the pairs of elements. Here may be considered as the geometrical mean of all the distances between pairs of elements. ”

“ As I am well aware that this challenge will never be accepted, I may conclude that no experiment will ever be in contradiction with Euclid's postulate; but, on the other hand, no experiment will ever be in contradiction with Lobatschewsky's postulate.
5. But it is not sufficient that the Euclidean (or non-Euclidean) geometry can ever be directly contradicted by experiment. Nor could it happen that it can only agree with experiment by a violation of the principle of sufficient reason, and of that of the relativity of space. Let me explain myself. Consider any material system whatever. ”

“ Hence the reality of mathematical knowledge. I doubt not but it will be easily granted, that the knowledge we have of mathematical truths is not only certain, but real knowledge ; and not the bare empty vision of vain, insignificant chimeras of the brain : and yet, if we will consider, we shall find that it is only of our own ideas. The mathematician considers the truth and properties belonging to a rectangle or circle only as they are in idea in his own mind. ”

“ To assume, for example, that objects which we see continue, after we have turned away our eyes, to be more or less analogous to what they were while we were looking, is a less violent assumption than to assume that such objects are composed of an infinite number of mathematical points. Hence in the physical study of the geometry of physical space, points must not be assumed ab initio as they are in the logical treatment of geometry, but must be constructed as systems composed of data and hypothetical analogues of data. ”

“ We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. [Pg 143] It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example. ”

“ You have then (kind Reader) in this little book most amply unfolded the theory of the construction of logarithms, there called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and various relations to their natural numbers, are clearly demonstrated.
It seemed desirable also to add to the theory an Appendix as to the construction of another and better kind of logarithms (mentioned by the Author in the preface to his Rabdologiæ) in which the logarithm of unity is 0. ”

“ We determine the length of the great arc on the surface of the earth by triangulation, which depends upon a base measured with exactitude. But whatever precision may be brought to the measure of the angles, the inevitable errors can, by accumulating, cause the value of the arc concluded from a great number of triangles to deviate appreciably from the truth. We recognize this value, then, only imperfectly unless the probability that its error is comprised within given limits can be assigned. The error of a geodetic result is a function of the errors of the angles of each triangle. ”

“ The tables contain seven-figure logarithms to 108,000, and to 1200 to 20 places, some antilogarithms to 20 places, hyperbolic logarithms from 1 to 10 at intervals of ·01 and to 1200 at intervals of unity to 7 places, logistic logarithms, log sines and tangents to every second of the first two degrees, and natural and log sines, tangents, secants, and versed sines for every minute of the quadrant to 7 places. The natural functions occupy the left-hand pages and the logarithmic the right-hand. ”

“ And indeed the necessary motion of one particle of matter into the place from whence another particle of matter is removed, is but a consequence from the supposition of plenitude ; which will therefore need some better proof than a supposed matter of fact, which experiment can never make out ; – our own clear and distinct ideas plainly satisfying us, that there is no necessary connexion between space and solidity, since we can conceive the one without the other. And those who dispute for or against a vacuum, do thereby confess they have distinct ideas of vacuum and plenum, i.e. ”

“ Not all physical quantities, in the present state of our knowledge, can be assigned a definite dimensional formula, and this indicates that not all of physics has as yet been reduced to a clearly established mechanical basis. The dimensional formula thus affords a valuable criterion of the extent and boundaries of our strictly definite knowledge of physics. Within these boundaries we are on safe and easy ground and are dealing, independent of all speculation, with the relations between precisely defined quantities. ”

“ Geometrical Co-existence, when it is not a matter of definition (as 'a square is a rectangle with four equal sides') , is deduced from the definitions and axioms: as when it is shown that in triangles the greater side is opposite the greater angle. The deductions of theorems or secondary laws, in Geometry is a type of what is desirable in the Physical Sciences: the demonstration, namely, that all the connections of phenomena, whether successive or co-existent, are consequences of the redistribution of matter and energy according to the principle of Causation. ”

“ Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparison and measurement. But it was found practicable, [Pg 155] and being practicable, was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which are obvious to the most careless observer, and which compose the first principles and ultimate premises of the science. ”

“ I must therefore resign myself, in the enunciation of the law of relativity, to including velocities of every kind among the data which define the state of the bodies. However that may be, the difficulty is the same for both Euclid's geometry and for Lobatschewsky's. I need not therefore trouble about it further, and I have only mentioned it incidentally. To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.
6. Experiments only teach us the relations of bodies to one another. ”

“ That the squares of the periodic times of the planets are proportional to the cubes of their distances from [Pg 145] the sun, is an uniformity derived from the laws of the causes (or forces) which produce the planetary motions; but that the square of any number is four times the square of half the number, is true independently of any cause. The only laws of resemblance, therefore, which we are called upon to consider independently of causation, belong to the province of mathematics. ”

“ The analogies that exist between relativity theory and Lobachevskian geometry are in any case interesting. The formulas of recent mechanics for are reduced to the formulas of Newtonian mechanics. Similarly also the Lobachevskian geometry, if we take the so called radius of curvature as infinite, goes over into the euclidean geometry. ”

“ When the radii of these spheres are plotted down on a diagram as ordinates, and a series of integers as abscissas, the general form of the logarithmic curve becomes apparent; and close scrutiny has shown that either the logarithmic curve, or some curve lying very close to it, expresses the real law of nature. ”

“ 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. ”

“ But if their quantities of matter are unequal, then, at an average of all the chances, each will intercept a number of particles proportional to its quantity of matter, and so the forces with which the bodies are impelled toward one another, will be as the quantity of matter directly, and the square of the distance inversely. ”

“ It is intended merely to show the kind of way in which, given a world with the kind of properties that psychologists find in the world of sense, it may be possible, by means of purely logical constructions, to make it amenable to mathematical treatment by defining series or classes of sense-data which can be called respectively particles, points, and instants. If such constructions are possible, then mathematical physics is applicable to the real world, in spite of the fact that its particles, points, and instants are not to be found among actually existing entities. ”

“ For as the logarithm, or the squaring of the circle, approaches infinitely near to correctness through infinite fractions, so, through manifold experience, the induction, i.e., the knowledge of the cause from the effects, approaches, not infinitely indeed, but yet so near mathematical evidence, i.e., knowledge of the effects from the cause, that the possibility of mistake is small enough to be neglected, but yet the possibility exists ”

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ Thus, that the three angles of a triangle are quite equal to two right ones is a truth as certain as anything can be, and I think more evident than many of those propositions that go for principles; and yet there are millions, however expert in other things, who know not this at all, because they never set their thoughts on work about such angles. And he that certainly knows this proposition may yet be utterly ignorant of the truth of other propositions, in mathematics itself, which are as clear and evident as this ”

“ So in the space of our world-lines, the space may be deformed in any imaginable manner without introducing any new intersections or changing the sequence of the existing intersections. It is this sequence which gives us the mathematical expression of our so-called experimental laws; a deformation of our space is equivalent mathematically to a transformation of axes, consequently we see why it is that the form of our laws must be the same when referred to any and all sets of axes, that is, must remain unaltered by any mathematical transformation. ”

“ Mathematics is the model of exact inference; and in physics we have endeavoured to replace all cruder inference by this rigorous type. Where we cannot complete the mathematical chain we confess that we are wandering in the dark and are unable to assert real knowledge. Small wonder then that physical science should have evolved a conception of the world consisting of entities rigorously bound to one another by mathematical equations forming a deterministic scheme. ”

“ There can never be complete absence of friction, nor perfectly ideal suspension of the weight or application of the power. And the necessary atmospheric disturbances, even in a "vacuum," to say nothing of the difficulties of absolute measurements, are not less fatal. Only as method, therefore, which really means as procedure according to standards of strictest accuracy and of highest logical consistency, or as closest, most constant loyalty to a spirit of truth, not as doctrine, can mathematics be said to be freely applicable. ”

“ Difficulties are indeed involved in the application of the calculus of probabilities to the exact sciences. Why are the decimals of a table of logarithms, why are those of the number π distributed in accordance with the laws of chance? Elsewhere I have already studied the question in so far as it concerns [Pg 412] logarithms, and there it is easy. It is clear that a slight difference of argument will give a slight difference of logarithm, but a great difference in the sixth decimal of the logarithm. Always we find again the same criterion. ”

“ The extent of his mathematical knowledge, however, is suggested by the fact that he computed in great detail the number of grains of sand that would be required to cover the sphere of the sun's orbit, making certain hypothetical assumptions as to the size of the earth and the distance of the sun for the purposes of argument. Mathematicians find his computation peculiarly interesting because it evidences a crude conception of the idea of logarithms. ”

“ With his standard rigid bodies defined in this way by physical objects, the physicist can perform measurements in the space of his frame and in consequence obtains the numerical results of three-dimensional Euclidean geometry. Inasmuch as the more careful he is to guard against such contingent influences as variations of temperature, the more accurately will his numerical results approximate to those of pure Euclidean geometry, he feels justified in stating that rigid objects behave like rigid Euclidean bodies and that the space of our experience is rigorously Euclidean. ”