Ration

“ Perhaps it is within the ‘Senior Wrangler’s’ knowledge of physical exploration, that when the physicist finds a result not bearing that ratio to its assigned cause which the two were ascertained in other cases to have, he immediately assumes the presence of some perturbing cause or causes, which modify the ratio. ”

“ It is mathematically demonstrable that any arrest, alteration, or extra occurrence of a physical process necessarily implies increase of Energy, and ultimately of mass of Matter in the world.... Any conceivable alteration in the prior order of atoms, centres of force, or elemental activities, reckoned in any terms of Motion, must, in fact, have the effect of increasing the sum total. ”

“ Socrates: Yes, that is usually what people mean by ratios. The learned people call numbers made from the ratios, rational. Does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers, whether they be odd or even whole numbers?
Boy: You want me to call the numbers made from ratios of whole numbers something called rational? A ratio makes a rational number?
Socrates: Yes boy, can you do that?
Boy: Certainly, Socrates.
Socrates: Do you agree with the way I told him this, Meno? ”

“ The following is a simple particular case of this refined problem: Let integral rational functions of one variable with integral rational coefficients, and a prime number be given. Consider the system of those integral rational functions of which can be expressed in the form
where is a rational integral function of the arguments and is any power of the prime number . ”

“ We shall suppose that there are no other equations of condition, and that the independently variable components are the same throughout the whole mass; and we shall at first limit ourselves to the consideration of the conditions of equilibrium with respect to the changes which may be expressed by infinitesimal variations of the quantities which define the initial state of the mass, without regarding the possibility of the formation at any place of infinitesimal masses entirely different from any initially existing in the same vicinity. ”

“ For demonstrations are more contracted by the method of indivisibles: but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's ”

“ Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance, is the class of proper fractions. [Pg 72]
In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, the truth is that it is the limit of the corresponding set of rational real numbers in the series of segments ordered by whole and part. For example, is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. ”

“ If we now recur to the consideration of the principal axes of strain and the principal ratios of elongation and denote by and the principal axes of strain in the strained and unstrained element respectively, it is evident that the sign of for example, depends upon the direction in which we regard as corresponding to a given direction in . ”

“ We desire to form a rational function φ (η) , or say φ (A (y) ) , of η1, ... η, in which the η2 constants Aij shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y1, ... yn were connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions y1, ... ”

“ for this is the custom—and properly so—in those sciences where mathematical demonstrations are applied to natural phenomena, as is seen in the case of perspective, astronomy, mechanics, music, and others where the principles, once established by well-chosen experiments, become the foundations of the entire superstructure. ”

“ To sum up, the aim of mathematical physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations. It is besides, it is above all, to reveal to him the hidden harmony of things in making him see them in a new way. ”

“ What is mass? According to Newton, it is the product of the volume by the density. According to Thomson and Tait, it would be better to say that density is the quotient of the mass by the volume. What is force? It is, replies Lagrange, that which moves or tends to move a body. It is, Kirchhoff will say, the product of the mass by the acceleration. But then, why not say the mass is the quotient of the force by the acceleration?
These difficulties are inextricable.
When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. ”

“ We may lay down as an incontesible axiom, that, in all the operations of art and nature, nothing is created; an equal quantity of matter exists both before and after the experiment; the quality and quantity of the elements remain precisely the s; and nothing takes place beyond the changes and modifications in the combination of these elements. Upon this principle the whole art of performing chemical experiments experiments depends: We must always suppose an exact quantity between the elements of the body examined and those products of its analysis. ”

“ INCE the ancients (as we are told by Pappus) made great account of the science of mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavored to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated Mathematics so far as it regards Philosophy. The ancients considered Mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical Mechanics all the manual arts belong, from which Mechanics took its name. ”

“ The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers; ”

“ Sure when I see such dreadful Consequences rising from a Principle, I would be as fully convinced of the Truth of it, as of a Mathematical Demonstration, before I would venture to act upon it, or make it a part of my Religion. ”

“ ⁠By these considerations, however, Kepler was led to enunciate his third law, that the squares of the periodic times of planets are proportional to the cubes of their mean distances from the sun.—Kepler, Prodromus Dissertationum Mathematicarum continens Mysterium Cosmographicum, etc. ”

“ Iteration, like friction, is likely to generate heat instead of progress, and Mr Tulliver’s heat was certainly more and more palpable. ”

“ We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them.
But what is the nature of these rules? No general rule, no rigorous rule; a multitude of little rules applicable to each particular case.
These rules are not imposed upon us and we might amuse ourselves in inventing others; but they could not be cast aside without greatly complicating the enunciation of the laws of physics, mechanics and astronomy.
We therefore choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: ”

“ This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n − 1, it is true of n, and thence conclude that it is true for all the whole numbers. [Pg 37]
We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ The same is true for any other elements, C, D, E, F, &c., as related to A. But by applying the general law to the ternary compound ABC the same conclusion may be drawn as to the ratio A : C in all compounds containing A and C, or B : C in the corresponding compounds. By reasoning further in the same way, we come to the conclusion that only such compounds are possible which contain elements according to certain ratio-numbers, i.e. their combining weight. Any other ratio would violate the law of the integral reaction of compounds. ”

“ The potential for any substance in any homogeneous mass is equal to the work required to bring a unit of the substance by a reversible process from a state in which and the temperature is the same as that of the given mass into combination with this mass, which at the close of the process must have the same volume and temperature as at first, and which is supposed so large as not to be sensibly altered in any part. ”

“ The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. ”

“ To see them, all fire and flame and tenderness, at a trembling distance, lavishing caresses of eye and voice with every action, through every silence—their love driving them toward each other, and they withholding like fluttering moths, each to the other a candle-flame, and revolving each about the other in the mad gyrations of an amazing orbit-flight! ”

“ For determining the state of a given mass when in equilibrium and having a given volume and given energy or entropy, the condition of equilibrium affords an additional equation corresponding to each of the independent relations between the n component substances. But the equations which express our knowledge of the matter in the given mass will be correspondingly diminished, being in number, like the equations of condition relating to the quantities of the component substances, which may be derived from the former by differentiation. ”

“ The accumulation of matter, however, and its relative quantity, are rendered sensible by weight; for weight is proportionate to the quantity of matter, as regards the parts of a tangible substance, but spirit and its quantity of matter are not to be computed by weight, which spirit rather diminishes than augments. ”

“ Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another.
First, let A and B have the same ratio to C: A shall be equal to B.
For, if A is not equal to B, one of them must be greater than the other; let A be the greater. ”

“ Quantities, and the ratio's of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. ”

“ It is obvious, that if elementary matter consists of indestructible and indivisible particles, each of which constantly preserves the same weight relatively to all the others, compounds formed by the aggregation of two, three, four, or more such particles must exemplify the rule of combination in definite proportions deduced from observation. ”

“ If in comparing indetermined quantities of different sorts with another, any one is said to be as any other directly or inversly: the meaning is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. and if any one is said to be as any other two or more directly or inversly: the meaning is, that the first is augmented or diminished in the ratio's in which the others, or the reciprocals of the others, are augmented or diminshed. ”

“ In fact the atomic weight, the atomic volume, and the atomic heat of an element, are quantities so closely connected that the determination of one will lead to that of the others. The chemist has to solve a complicated problem in deciding in the case of each of 60 or 70 elements which mode of determination is most accurate. Modern chemistry presents us with an almost infinitely extensive web of numerical ratios developed out of a few fundamental ratios. ”

“ When men began to use the balance in chemical analysis, they did not prove by trial, but took for granted, as self-evident, that the weight of the whole must be found in the aggregate weight of the elements.”89 True, it is assumed; but, I apprehend, no otherwise than as all experimental inquiry assumes provisionally some theory or hypothesis, which is to be finally held true or not, according as the experiments decide. ”