Numerical superiority

“ When numbers, however, are taken in a concrete sense, and are applied to express peculiar modes of quantity,—such as angular, linear, superficial, and solid magnitudes,—a new set of numerical relations arise, and a large number of computations are required.
To express angular magnitude, and the various relations of linear magnitude with which it is connected, involves the consideration of a vast variety of Geometrical and Trigonometrical tables; such as tables of the natural sines, co-sines, tangents, secants, co-tangents, &c. &c.; tables of arcs and angles in terms of the radius ”

“ Its place in the organization of society as well as in the regulation of other aspects of human life has been taken by a numerical system that still dominates our entire culture—the decimal system. Even prior to the age of Columbus, the decimal system made its appearance in certain more civilized parts of the Western world where the duodecimal system was never known. ”

“ Superiority of numbers The superiority in numbers being a material condition, it was chosen from amongst all the factors required to produce victory, because it could be brought under mathematical laws through combinations of time and space. ”

“ The comparison between the numerical results obtained at one time and place and another, and between those presented in different countries, is becoming a political method increasing in prevalence and repute. In many quarters, indeed, the value of purely numerical estimates has been much exaggerated, and its peculiar liability to error, when made a basis of political reasoning, has been too much ignored. ”

“ To have a really correct apprehension of the principle of decimal notation, for instance, to understand that it is purely arbitrary, and that we might in the same way take any other number than ten as the base of a numerical scale,—that we might increase for instance by fives, or eights, or nines, or twelves, just as well as by tens—all this requires considerable maturity of intellect, and some subtlety of reasoning. ”

“ And what is the nature of the mental process by which numbers are found capable of having their relations expressed algebraically? It is just the same. It is the formation of certain abstract conceptions of numerical functions which are the same 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. And transcendental analysis stands to algebra in the same position that algebra stands in to arithmetic. ”

“ Before any absolutely true numerical results can be reached, it is requisite that the units be absolutely equal. The only way in which we can establish a numerical relationship between things that do not yield us like impressions, is to divide them into parts that do yield us like impressions. Two unlike magnitudes of extension, force, time, weight, or what not, can have their relative amounts estimated, only by means of some small unit that is contained many times in both ”

“ Considering that almost all the quantities which we treat in physical and social science are approximate only, it seems desirable that attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements. ”

“ With great propriety he has rejected many statements and rules which in the progress of nautical science have fallen into disuse, and in his selection of methods of computation, has, in general, kept in view those modern improvements in this branch of practical mathematics in which simplicity and accuracy are most happily combined. Much attention to numerical correctness seems to pervade the work. ”

“ Notation of Numerical Quantities.—Over a large part of the civilized world the introduction of the metric system (§ 118) has caused the notation of all numerical quantities to be in the denary scale. In Great Britain and her colonies, however, and in the United States, other systems of notation still survive, though there is none which is consistently in one scale, other than the denary. The method is to form quantities into groups, and these again into larger groups; but the number of groups making one of the next largest groups varies as we proceed along the scale. ”

“ Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. ”

“ The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ The problem first raised by the discovery of incommensurables proved, as time went on, to be one of the most severe and at the same time most far-reaching problems that have confronted the human intellect in its endeavour to understand the world. It showed at once that numerical measurement of lengths, if it was to be made accurate, must require an arithmetic more advanced and more difficult than any that the ancients possessed. ”

“ In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit of the same kind, we have only to remember that every expression for the quantity consists of two factors, the unit and the numerical part which expresses how often the unit is to be taken. Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indicated by the dimensions of the derived unit. ”

“ 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. ”

“ 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? ”

“ The transformation of functions belongs strictly to algebra. Teachers who love arithmetic, and who have themselves success in working out the so-called numerical conundrums, defend with much earnestness the current practice which uses so much time for arithmetic. They see in it a valuable training for ingenuity and logical analysis, and believe that the industry which discovers arithmetical ways of transforming the functions given in such problems into plain numerical operations of adding, subtracting, multiplying, or dividing is well bestowed. ”

“ In plane geometry we have a regressive product of two Liniengrössen, which gives the point of intersection of the lines with a certain numerical coefficient.
The fundamental operations relating to the pointy line, and plane are thus translated into analysis by multiplications. The immense flexibility and power of such an analysis will be appreciated by anyone who considers what generalized multiplication in connection with additive relations has done in other fields, as in quaternions, or in the theory of matrices, or in the algebra of logic. ”

“ Arithmetical numbers appealed to the Middle Ages just as they do to us, because they are difficult to deal with, and because the greatest masters of numbers, the Newtons and Leibnitzes, rank among the greatest men. ”

“ It is incontrovertible that in dealing with the division of material substances, the binary scale comes into mental and practical play as easy and convenient. Not all the specious pleas of those who advocate the so-called reform can turn us from it. Its advantages over the decimal system in all the operations of every day life are apparent. ”

“ Even natural units need translation, and the artificial units we devise might be content to get on a footing with natural ones. How small a purpose, indeed, would be served if the names of the measures were the same, but of the numbers not the same, nor of the things measured! Such are some, by no means all, of the incurable faults and defects of the metric nomenclature.
The obstacles to metric reform have been chiefly artificial. Like little David in Saul's armor, the system has been weighed down with superfluities. ”

“ I have an immense respect for a man of talents plus “the mathematics.” But the calculating power alone should seem to be the least human of qualities, and to have the smallest amount of reason in it; since a machine can be made to do the work of three or four calculators, and better than any one of them. Sometimes I have been troubled that I had not a deeper intuitive apprehension of the relations of numbers. But the triumph of the ciphering hand-organ has consoled me. I always fancy I can hear the wheels clicking in a calculator’s brain. ”

“ 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. ”

“ They merely determine the operations [1] to be effected, which operations may of course be performed on an infinite variety of particular numerical values, and do not bring out any definite numerical results unless the numerical data of the problem have been impressed on the requisite portions of the train of mechanism. In the above example, the first essential step towards an arithmetical result, would be the substitution of specific numbers for , and for the other primitive quantities which enter into the function. ”

“ 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. ”

“ For it is the peculiarity of linear extension that it alone allows its magnitudes to be placed in absolute juxtaposition, or, rather, in coincident position; it alone can test the equality of two magnitudes by observing whether they will coalesce, as two equal mathematical lines do, when placed between the same points; it alone can test equality by trying whether it will become identity. Hence, then, the fact, that all exact science is reducible, by an ultimate analysis, to results measured in equal units of linear extension. ”

“ The late Professor Newcomb made the matter plain even to the non-mathematical mind, when he said that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." ”

“ Yet in fact, how is a hundred millions of miles a great distance? how is a hundred millions of times a great ratio? Not in itself: this greatness is no quality of the numbers which can be proved like their mathematical properties; on the contrary, all that absolutely belongs to number, space, and ratio, must, we know demonstrably, be equally true of the largest and the smallest. ”

“ But mere counting does not imply either the possession or the use of anything that can be really called the mathematical faculty, the exercise of which in any broad sense has only been possible since the introduction of the decimal notation. The Greeks, the Romans, the Egyptians, the Jews, and the Chinese had all such cumbrous systems, that anything like a science of arithmetic, beyond very simple operations, was impossible ”

“ Experience has shown where the dream of universal mathematics leads us. Number is driven to the heart of phenomena and nature dissected with this delicate scalpel. Speaking in more general terms, we adopt spatial relation as the perfect example of intelligible relation. I do not wish to deny the use of such a method now and again, the services it may render, or the beauty of construction peculiar to the systems it inspires. But we must see what price we pay for these advantages. Do we choose geometry for an informing and regulating science? ”

“ Let z equal resolution and y equal speed and we have z plus y which equals resolution and speed, that is victory.”
“I hope it will work out that way,” said Dick, “but war isn't altogether mathematics.”
“Not altogether, but that beautiful study plays a great part in every campaign. People are apt to abuse mathematics, when they don't know what they're talking about. The science of mathematics is the very basis of music, divine melody, heaven's harmony.” ”