Numbers

“ You believe in Number as the foundation on which rests the edifice of what you call the exact sciences. Without number mathematics are impossible. Well, then, what impossible being, to whom life everlasting should be granted, could ever [Pg 99] finish counting—and in what sufficiently concise language could he utter—the numbers contained in the infinite number of which the existence is demonstrated by your reason. Ask the greatest human genius, and suppose him to sit for a thousand years leaning on a table, his head in his hands, what would he answer? ”

“ In judging how often one large number is contained in another, a first and rough guess may be made by striking off the same number of figures from both, and using the results instead of the numbers themselves. ”

“ In the Science of Numbers, the same rule must be applied; for it is not until a student has attained full comprehension of the introductory laws that govern names and numbers that he can hope to understand the more intricate vibrations that exert their influence upon the course of human events. ”

“ 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. ”

“ Therefore, in future, on this part of the subject I shall cease to talk of any particular units, such as pebbles or acres, and speak of numbers only. A number, considered without intending to allude to any particular things, is called an abstract number: and it then merely signifies repetitions of a unit, or the number of times a unit is repeated. ”

“ The clearness and distinctness of each mode of number from all others, even those that approach nearest, makes me apt to think that demonstrations in numbers, if they are not more evident and exact than in extension, yet they are more general in their use, and more determinate in their application. ”

“ The fundamental truths of the science of Number all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our sense all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. ”

“ And moreover, seeing the conditions and ratios of what pertains to harmony to consist in numbers, since other things seemed in their entire nature to be formed in the likeness of numbers, and in all nature numbers are the first, they supposed the elements of numbers to be the elements of all things. ”

“ When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers. ”

“ It is not easy to find a definition for the Science of Numbers that shall be both brief and comprehensive. We may say, however, that it is a method of enumeration by means of which the life of men and things may be fixed and rounded out to a state of approximate completion by being brought into harmony with the laws of Omnipotence. ”

“ In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic ”

“ To comprehend the apparently mysterious operation of the Science of Numbers, and its effect upon the health, happiness, and success of the individual, the student must first master the table that follows, for the very cornerstone of the Science rests upon the relationship which exists between the numbers, and the letters of the alphabet. ”

“ To a great extent proficiency in applying the the principles of the Science of Numbers is a question of practice, for, while the rules are exact, the phenomena of their manifestation varies so materially in each individual case that it would require a volume of huge proportions to describe all the operations of each rule, to say nothing of the many exceptions that affect this play of lights and shadows. ”

“ 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. ”

“ Number is everything, and is the essence of things, but the triad includes all number, since it contains the beginning, middle, and end. Everything is derived from the primitive one and from the principal number; and since this number in breathing its vital evolution into the void is divided into many units, everything is derived from the multiplicity of these units or numbers. ”

“ The merit of our actual system of numeration consists in including the expression of all numbers in a single idea, making the value of each figure ten times that to the right, and filling all intervals with zeros. ”

“ Numbers are, in the strictest propriety, names of objects. Two is surely a name of the things which are two, the two balls, the two fingers, &c. The process of adding one to one which forms two is connoted, not denoted, by the name two. Numerals, in short, are concrete, not abstract names: they denote the actual collections of things, and connote the mental process of counting them. It is not twoness and threeness that are fiveness: the twoness of my two hands and the threeness of the feet of the table cannot be added together to form another abstraction. ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ Again, let none be alarmed at vast numbers and fractions, for in calculation it is as easy to set down or to reflect upon a thou­sand as a unit, or the thousandth part of an integer as an integer itself. ”

“ Thus we speak of counting up to a certain number; and similarly mathematicians speak of high and ascending powers, while engineers speak of high pressure, high speed, high power, &c. This tendency is probably aided by the use of bricks or cubes in elementary number-teaching.
24. Primitive Ideas of Number.—The names of numbers give an idea of the way in which the idea of number has developed. Where civilization is at all advanced, there are usually certain names, the origin of which cannot be traced ”

“ In so small a volume, however, it would be folly to attempt so pretentious a task, especially in view of the fact that, for all present purposes, it Is sufficient that we should realize that the Science of Numbers has been known and credited since very early in the world’s history, and that, having been cherished with most tender solicitude through the turmoil and storms of the centuries, it has come to us with the signs of truth and authenticity indelibly fixed by the stamp of years. ”

“ Then came the use of numbers—a thing necessary to human life, and at the same time immutable and eternal; a science which first urged us to raise our views to heaven, and not gaze without an object on the motions of the stars, and the distribution of days and nights. ”

“ but still each of these sciences has a subject which is different from the science. I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other. Is not that true?
Yes, he said.
And the odd and even numbers are not the same with the art of computation?
They are not.
The art of weighing, again, has to do with lighter and heavier; but the art of weighing is one thing, and the heavy and the light another. Do you admit that?
Yes.
Now, I want to know, what is that which is not wisdom, and of which wisdom is the science? ”

“ All numbers must be numbers of something: there are no such things as numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of any thing. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience. All things possess quantity ”

“ In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry. Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number. A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number ”

“ The reason why numbers signify things, or rather that they are as a kind of adjectives to substantives, denoting some quality in things, is that number in itself is natural; for natural things are determined by nubers, but spiritual by things and their states. ”

“ The difference, therefore, of our conduct in preferring the greater number depends not upon our passions, but upon custom, and general rules. We have found in a multitude of instances, that the augmenting the numbers of any sum augments the passion, where the numbers are precise and the difference sensible. ”

“ If this work shall show more clearly the value of our number system, and shall make the study of mathematics seem more real to the teacher and student, and shall offer material for interesting some pupil more fully in his work with numbers, the authors will feel that the considerable labor involved in its preparation has not been in vain. ”

“ It is the invention of general symbols serving to express the numerical relations of entities, whatever be their special characters. And what is the nature of the mental process by which numbers are found capable of having their relations expressed algebraically? It is the same. {21} It is the formation of certain abstract conceptions of numerical functions which are constant whatever be the magnitudes of the numbers. It is the invention of general symbols serving to express the relations between numbers, as numbers express the relations between things. ”

“ For we often do, and might always, verify the simplest numerical propositions, by reckoning up the numbers. The operations of addition, subtraction, multiplication, division, and extraction of roots, with all the most complex ones relating to algebraic quantities, considered as the exponents of numbers, are no more than methods of producing this coincidence of words, founded upon and rising above one another. ”

“ Although a poem be not made by counting of syllables upon the fingers, yet "numbers" is the most poetical synonym we have for verse, and "measure" the most significant equivalent for beauty, for goodness, and perhaps even for truth. Those early and profound philosophers, the followers of Pythagoras, saw the essence of all things in number, and it was by weight, measure, and number, as we read in the Bible, that the Creator first brought Nature out of the void. ”

“ For, the several simple modes of numbers being in our minds but so many combinations of units, which have no variety, nor are capable of any other difference but more or less, names or marks for each distinct combination seem more necessary than in any other sort of ideas. For, without such names or marks, we can hardly well make use of numbers in reckoning, especially where the combination is made up of any great multitude of units ”

“ In illustration and as an example it may be noticed that the eyes of the majority of persons convey the consciousness of numbers precisely, up to a certain point, without any operation of counting. Most people can say at a glance, of any small group of objects, that there are two, three, four, or even seven. With almost all individuals, counting, and counting from the beginning, becomes necessary when there are eight or more objects together. ”

“ When we consider the importance of numbers to man, as an intellectual being, and compare the advancement he has made by this knowledge, beyond the [Pg 52] animals who have wanted the means of acquiring such information, the importance of investigating this curious subject will be fully acknowledged. Without numbers, by which the divisions of time, space, and value are characterised, man could have possessed no knowledge of the order and succession of events ”

“ Spell out, in ordinary reading matter, all round numbers and numbers of less than three digits, unless of a statistical or technical character, and unless occurring in groups of six or more, following one another in close succession. Treat all numbers in connected groups alike: if the largest contains three or more digits, use figures for all. ”

“ Aim. Readiness in the four mathematical operations in whole numbers, decimal numbers, and simple fractions, written and oral. Ability to independently solve various examples in the forms in which they generally appear in practical life, also examples in proportion and examples in percentage, together with problems of planes and solids. Course and mode of its execution as given in a text—Instruction in Mathematics (Regneundervisningen) —followed in all essentials. ”