Quotient

“ How can number 45 be divided into four such parts that, if to the first part you add 2, from the second part you subtract 2, the third part you multiply by 2, and the fourth part you divide by 2, the sum of the addition, the remainder of the subtraction, the product of the multiplication, and the quotient of the division, be all equal? ”

“ The reason for the recurrence of the figures of the quotient in the same order is as follows: If 1000, &c. be divided by the number 247, the remainder at each step of the division is less than 247, being either 0, or one of the first 246 numbers. If, then, the remainder never become nothing, by carrying the division far enough, one remainder will occur a second time. ”

“ Theorem III. The limit of the quotient of two variables is equal to the quotient of the limits of the separate variables, provided the limit of the denominator is not zero.
Before proving these theorems it is necessary to establish the following properties of infinitesimals.
(1) The sum of a finite number of infinitesimals is an infinitesimal. ”

“ The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un = anun-1 + bnun-2. When we can solve this equation we have an expression for the nth convergent to the fraction, generally in the form of the quotient of two series, each of n terms. ”

“ Multiply thing and two dirhems by the quotient, that is a half. The then the computation is this: Multiply thing and two dirhems by the quotient, that is a half. The product is half a thing and one dirhem, equal to thing. Remove now half a dirhem on account of the half dirhem on the other side. The remainder is one dirhem, equal to half a thing. Double it: then you have thing, equal to two dirhems. This is one of the two numbers [40] , and the other is four.
Instance: “You divide one dirhem amongst a certain number of men, which number is thing. ”

“ The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal. ”

“ Or whether any one will say, that if the Equation 2 × o = 5 × o, be divided by o, the Quotients on both Sides are equal? Whether therefore a Case may not be general with respect to all Quantities, and yet not extend to Nothings, or include the Case of Nothing? And whether the bringing Nothing under the Notion of Quantity may not have betrayed Men into false Reasoning?
Qu. 41. Whether in the most general Reasonings about Equalities and Proportions, Men may not demonstrate as well as in Geometry? Whether in such Demonstrations, they are not obliged to the same strict Reasoning as in Geometry? ”

“ Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to one another and to the whole; so that it may be increased by addition of like parts, and diminished by subtraction, may be multiplied and divided, and, in a word, may bear any proportion to another quantity of the same kind, that one line or number can bear to another. That this is essential to all mathematical quantity, is evident from the first elements of algebra, which treats of quantity in general, or of those relations and properties which are common to all kinds of quantity. ”

“ Mathematics contain properly the doctrine of measure; and the object of this science is commonly said to be quantity; therefore quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which, I apprehend, has led some persons to apply mathematical reasoning to subjects that do not admit of it. ”

“ Therefore, one has much to say for the thesis that the whole logical value of the quantitative conceptions in science is due, not to any peculiar advantage of the concept of quantity, as such, but to the exactness of the forms of serial order which are discoverable in case of any quantitative realm of facts. ”

“ Strictly speaking, only the things which I have mentioned belong to the category of quantity: everything else that is called quantitative is a quantity in a secondary sense. It is because we have in mind some one of these quantities, properly so called, that we apply quantitative terms to other things. We speak of what is white as large, because the surface over which the white extends is large; we speak of an action or a process as lengthy, because the time covered is long ”

“ We can not, therefore, reason respecting causation, without introducing considerations of quantity and extension at every step; and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy, the laws of quantity become the grand instrument for calculating forward to an effect, or backward to a cause. That in all other sciences, as well as in geometry, questions of quality are scarcely ever independent of questions of quantity, may be seen from the most familiar phenomena. ”

“ From the nature of quantity we may see what it is that gives mathematics such advantage over other sciences in clearness and certainty; namely, that quantity admits of a much greater variety of relations than any other subject of human reasoning; and, at the same time, every relation or proportion of quantities may, by the help of lines and numbers, be so distinctly defined as to be easily distinguished from all others, without any danger of mistake. ”

“ Though proportion be measurable in its own nature, and therefore hath proper quantity, yet as things cannot have proportion which have not quantity of some other kind, it follows that whatever has quantity must have it in one or other of these three kinds, extension, duration, or number. ”

“ But if it comes to be discovered that variations of quality in any class of phenomena, correspond regularly to variations of quantity either in those same or in some other phenomena; [Pg 249] every formula of mathematics applicable to quantities which vary in that particular manner, becomes a mark of a corresponding general truth respecting the variations in quality which accompany them: and the science of quantity being (as far as any science can be) altogether deductive, the theory of that particular kind of qualities becomes, to this extent, deductive likewise. ”

“ The determination of values consists so completely of a simple transformation, that when the primitive expression of the number is found to be already conformed to the regular system of numeration, there is no longer any determination of value, properly speaking, or, rather, the question is answered by the question itself. ”

“ It is not easy to say how many kinds of improper quantity may in time be introduced into mathematics, or to what new subjects measures may be applied; but this, I think, we may conclude, that there is no foundation in nature for, nor can any valuable end be served by, applying measure to anything but what has these two properties. ”

“ Such is the character of the judgments which the calculus enables us to form with respect to the occurrence of a certain difference between the real value of any quantity under measurement and the value assigned to it by the measurement. The confidence that the constants which we have determined are accurate within certain limits is a subjective feeling which cannot be dislodged from an important part of probabilities. ”

“ And so the balance sways and the story runs—here a gain, there a loss, and Nature’s great average struck across the two, whereof the sum total forms one of the factors in that mighty equation in which the result will equal the unknown quantity of her purpose. ”

“ Quantity, on the contrary, is the character external to being, and does not affect the being at all. Thus e.g. a house remains what it is, whether it be greater or smaller; and red remains red, whether it be brighter or darker. Measure, the third grade of being, which is the unity of the first two, is a qualitative quantity. All things have their measure: i.e. the quantitative terms of their existence, their being so or so great, does not matter within certain limits ”

“ In quantity (a translation of the Greek ‘posotes’—‘how-muchness’—and seemingly formed by Aristotle on the analogy of Plato’s ‘poiotēs’, from which we have quality) we can perhaps see the beginning of that interest in the calculable aspect of the objects of the visible world from which the exact sciences have arisen. The human mind had now begun to weigh and measure, to examine and compare; and that weighing and measuring has gone on—with intervals—for twenty-three centuries. ”

“ The process of measure, instead of being only the wrong infinite of an endless progression, in the shape of an ever-recurrent recoil from quality to quantity, and from quantity to quality, is also the true infinity of coincidence with self in another. In measure, quality and quantity originally confront each other, like some and other. But quality is implicitly quantity, and conversely quantity is implicitly quality. ”

“ Quantity in all its forms is but a development of number. That which is mathematical is not the less logical; if anything it is more logical, in the sense that it presents logical results in a higher degree of complexity and variety.
Before proceeding then from Perfect to Imperfect Induction I must devote a portion of this work to treating the logical conditions of number. I shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves, and the probability or improbability of their occurrence under definite circumstances. ”

“ Continued Fractions.—The theory of continued fractions (q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the form where b, c, d, . . . are integers, and a is an integer or zero. ”

“ Such is the general acceptation in which we should conceive the Calculus of Variations, in order suitably to appreciate the importance of this admirable logical [Pg 163] instrument, the most powerful that the human mind has as yet constructed.
The method of variations being only an immense extension of the general transcendental analysis, I have no need of proving specially that it is susceptible of being considered under the different fundamental points of view which the calculus of indirect functions, considered as a whole, admits of. ”

“ The general spirit of the transcendental analysis, from this point of view, consists in introducing as auxiliaries, in the place of the primitive quantities, or concurrently with them, in order to facilitate the establishment of equations, the limits of the ratios of the simultaneous increments of these quantities ”

“ For this purpose it is indispensable to perceive, first of all, that, in the purely logical point of view, this science is by itself necessarily and rigorously universal; for there is no question whatever which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations, and consequently as admitting of reduction, in final analysis, to a simple question of numbers. In all our researches, indeed, on whatever subject, our object is to arrive at numbers, at quantities, though often in a very imperfect manner and by very uncertain methods. ”

“ As science has been developed, the domain of quantity has everywhere encroached on that of quality, till the process of scientific inquiry seems to have become simply the measurement and registration of quantities, combined with a mathematical discussion of the numbers thus obtained. It is this scientific method of directing our attention to those features of phenomena which may be regarded as quantities which brings physical research under the influence of mathematical reasoning. ”

“ In the same manner as an exponent expresses one operation on quantity, namely, the multiplication of it by itself a certain number of times, or raising it to the power expressed by that exponent, so the term Function, which has been introduced into modern analysis, is intended to express generally the results of all the various operations that can be performed upon quantity. ”

“ Let us now proceed to consider the modes in which from numerical results we can establish the actual relation between the quantity of the cause and that of the effect. What we want is a rational formula or function, which will exhibit the reason or exact nature and origin of the law in question. There is no word more frequently used by mathematicians than the word function, and yet it is difficult to define its meaning with perfect accuracy. ”

“ In the first place, the possibility of drawing them at all, manifestly supposes that we are acquainted not only with the variations, but with the absolute quantities both of A and a. If we do not know the total quantities, we can not, of course, determine the real numerical relation according to which those quantities vary. ”

“ Let me procure a larger sheet, as the measureless sea of infinite and nameless numbers is flowing fast upon me. Next reckon the chances in the case of two persons, and they swell to the vast sum of nine millions, seven hundred and sixty-five thousand, six hundred and twenty-five; while the chances for four men will be the square of that number, and so on for ever. ”

“ Similarly, in chemistry, we try to ascertain the quantity of the matters brought into combination, in order to find out the measures or proportions conditioning such combinations, that is to say, those quantities which give rise to definite qualities. In statistics, too, the numbers with which the study is engaged are important only from the qualitative results conditioned by them. Mere collection of numerical facts, prosecuted without regard to the ends here noted, is justly called an exercise of idle curiosity, of neither theoretical nor practical interest. ”