Mathematical table

“ Notwithstanding these and other tables of the same kind, the Board of Longitude considered that still more extended tables might be useful to science, and employed the late Dr. Hutton to form a multiplication table of all numbers from 1 to 1000, multiplied by all numbers less than 100. These were printed by their directions; and it is to be presumed that no expense was spared to render them accurate: yet in one page only of those tables (page 20) no less than forty errors occur, not one of which is noticed in the printed list of the errata. ”

“ Tables relating to ordinary arithmetical operations are first described, and afterwards an account is given of the most useful and least technical of the more strictly mathematical tables, such as factorials, gamma functions, integrals, Bessel's functions, &c. It is difficult to classify the tables described in a perfectly satisfactory manner without prolixity, as many collections contain valuable sets belonging to a variety of classes. ”

“ See "The Black Star Passes", Ace Books, F-346.
I
Three men sat around a table which was littered with graphs, sketches of mathematical functions, and books of tensor formulae. Beside the table stood a Munson-Bradley integraph calculator which one of the men was using to check some of the equations he had already derived. The results they were getting seemed to indicate something well above and beyond what they had expected. ”

“ W. Hall, Four-figure Tables and Constants (Cambridge, 1905) , 60 pp., chiefly for nautical computation; A. du P. Denning, Five-figure Mathematical Tables for School and Laboratory Purposes (12 pp. of tables, large octavo) ; A. R. Hinks, Cambridge Four-figure Mathematical Tables (12 pp.) . C. M. Willich, Popular Tables (London, 1853) , is a useful book for an amateur; it gives Briggian and hyperbolic logarithms to 1200 to 7 places, squares, &c., to 343, &c. ”

“ Mathematical Tables (W. & R. Chambers, Edinburgh) , containing logarithms of numbers to 100,000, and a canon to every minute of log sines, tangents, and secants and of natural sines to 7 places, besides proportional logarithms and other small tables, is cheap and suitable for schools, though not to be compared as regards matter or typography to the best tables described above.
Of six-figure tables C. Bremiker’s Logarithmorum VI. decimalium nova tabula Berolinensis (Berlin 1852) is probably one of the best. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ That nearly all tables of numbers which follow any law, however complicated, may be formed, to a greater or less extent, solely by the proper arrangement of the successive addition and subtraction of numbers befitting each table, is a general principle which can be demonstrated to those only who are well acquainted with mathematics; but the mind, even of the reader who is but very slightly acquainted with that science, will readily conceive that it is not impossible, by attending to the following example. ”

“ For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving xy as tabular result for x and y as arguments. The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more Valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. ”

“ The notion of order is one which has enormous importance in mathematics. Not only the integers, but also rational fractions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The order of points on a line is essential to geometry; so is the slightly more complicated order of lines through a point in a plane, or of planes through a line. Dimensions, in geometry, are a development of order. The conception of a limit, which underlies all higher mathematics, is a serial conception. ”

“ The statement that mathematics is the science of number would be untrue in two different ways. On the one hand, there are recognised branches of mathematics which have nothing to do with number—all geometry that does not use co-ordinates or measurement, for example: projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater and less. ”

“ The mathematical problems and tables are in simple arithmetical terms, and involve no algebra or higher mathematics. The method of treatment is strictly progressive, following the successive steps necessary to becoming an intelligent and skilled mechanic. ”

“ Geometry, on the other hand, gave great assistance to algebra, by affording concrete representations of relations which would otherwise be too abstract for easy comprehension. A curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression. As soon as we know, too, that every regular geometrical curve represents some algebraic633 equation, we are presented by observation of mechanical movements with abundant suggestions towards the discovery of mathematical problems. ”

“ A given point on a line may be the limit of a set of points on the line, without its being necessary to bring in co-ordinates or measurement or anything quantitative. The cardinal number is the limit (in the order of magnitude) of the cardinal numbers 1, 2, 3, ... , ..., although the numerical difference between and a finite cardinal is constant and infinite: from a quantitative point of view, finite numbers get no nearer to as they grow larger. ”

“ They are all calculated in the same manner; and we take for a single example the series of the cubes of whole numbers, which form an arithmetical progression of the third order (Fig. 8) . We get by direct calculation the first four terms, 1, 8, 27, 64; then, by subtraction, the first three terms of the column D1, the first two of the column D2, and the first term, 6, of D3, The table is then completed by the law given above. ”

“ The more reason that we have here to consider geometry as being at the present day essentially analytical, the more necessary was it to guard against this abusive exaggeration of mathematical analysis, according to which all geometrical observation would be dispensed with, in establishing upon pure algebraical abstractions the very foundations of this natural science. ”

“ A higher development of mathematical thought shows how, by the addition of certain conceptions, one can pass from the series of conceived objects and relations of objects that projective geometry finds in space, to the series of the metrical geometer, and vice versa. ”

“ If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-and-so many places, exactly fulfils the above conditions. We can equally well form a descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to later terms of the series, and differing, sooner or later, by less than any assigned amount. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. ”

“ Our present knowledge of the theory of quadratic number fields [25] puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients. ”

“ Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members.
On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. ”

“ In some tables the labour required in the construction is the same as if all the tabular results had been calculated separately; but in the majority of instances a table can be formed by expeditious methods which are inapplicable to the calculation of an individual result. This is the case with tables of a continuous quantity, which may frequently be constructed by differences. The most striking instance perhaps is afforded by a factor table or a table of primes ”

“ The diagram was present in my mind from early childhood; I remember that I learnt the multiplication table by reference to it at the age of seven or eight. I need hardly say that the impression is not that of perfectly straight lines, I have therefore used no ruler in drawing it."
J.S. "The figures are about a quarter of an inch in length, and in ordinary type. They are black on a white ground. The numeral 200 generally takes the place of 100 and obliterates it. There is no light or shade, and the picture is invariable." ”

“ Although the algebra of today guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen. ”

“ In multiplications or divisions of any length it is generally convenient to begin by forming a table of the first nine multiples of the multiplicand or divisor, and Napier’s bones at best merely provide such a table, and in an incomplete form, for the additions of the two figures in the same parallelogram have to be performed each time the rods are used. ”

“ But to form a table of prime numbers the process is theoretically simple and rapid, for we have only to range all the numbers in a line and strike out every second number beginning from 2, every third beginning from 3, and so on, those that remain being primes. Even when the tabular results are constructed separately, the method of differences or other methods connecting together different tabular results may afford valuable verifications. By having recourse to tables not only does the computer save time and labour, but he also obtains the certainty of accuracy. ”

“ The task is not without difficulty; it is not enough to open a book at random and to analyse any proof we may come across. First of all, geometry must be excluded, or the question becomes complicated by difficult problems relating to the rôle of the postulates, the nature and the origin of the idea of space. For analogous reasons we cannot avail ourselves of the infinitesimal calculus. We must seek mathematical thought where it has remained pure—i.e., in Arithmetic. ”

“ A list of the most convenient factors and their logarithms is given in a separate table. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is one that is frequently employed in the calculation of logarithms, but the peculiarity of the present work is that it forms part of the process of using the table. ”

“ If you calculate your combinations by working upwards from a basic triangle lying on the table, you will get half the correct number of ways, because you overlook the fact that an equal number of pyramids may be built on that triangle downwards, so to speak, through the table. ”

“ Multiplication Tables.—A multiplication table is usually of double entry, the two arguments being the two factors; when so arranged it is frequently called a Pythagorean table. The largest and most useful work is A. L. Crelle’s Rechentafeln (Bremiker’s edition, 1857, stereotyped; many subsequent editions with German, French, and English title-pages) , which gives in one volume all the products up to 1000 ⨯ 1000, so arranged that all the multiples of any one number appear on the same page. The original edition was published in 1820 and consisted of two thick octavo volumes. ”