Number and Its Algebra: Syllabus of Lectures on the Theory of Number and Its Algebra Introductory to a Collegiate Course in Algebra

D.C. Heath & Company, 1896 - 230 σελίδες

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Σελίδα 79 - Thou canst not wave thy staff in air, Or dip thy paddle in the lake, But it carves the bow of beauty there, And the ripples in rhymes the oar forsake. The wood is wiser far than thou; The wood and wave each other know Not unrelated, unaffied, But to each thought and thing allied, Is perfect Nature's every part, Rooted in the mighty Heart.
Σελίδα 86 - ... let him, for the present, reject the example in which it occurs, and defer the consideration of such equations until he has read the explanation of them to which we shall soon come. Above all, he must reject the definition still sometimes given of the quantity - a, that it is less than nothing. It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing ; above all, that the notion should have outlived the belief in judicial...
Σελίδα 36 - ... adopt the first method we shall often have difficulty in interpreting terms which make their appearance during the calculations. We shall therefore consider all the written symbols as mere numerical quantities, and therefore subject to all the operations of arithmetic during the process of calculation. But in the original equations and the final equations, in which every term has to be interpreted in a physical sense, we must convert every numerical expression into a concrete quantity by multiplying...
Σελίδα 34 - If, in the equation 1/1 = 1X1, 1 be taken as the unit of length, then the members of the equation have evidently not the same meaning, 1/1 being merely a numerical quantity, while 1 X 1 is a unit of area, it being a fundamental geometrical conception that the product of a length by a length is an area, that of a length by an area a volume, while the ratio of two quantities of the same order as that of a length to a length is a mere number of the order zero. In quaternions, however, we have the remarkable...
Σελίδα 145 - Every mathematical book that is worth reading must be read "backwards and forwards," if I may use the expression. I would modify Lagrange's advice a little and say, " Go on, but often return to strengthen your faith." When you come on a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.
Σελίδα 190 - Or we may enunciate the laws thus : the coefficient of the second term with its sign changed is equal to the sum of the roots ; the coefficient of the third term is equal to the sum of the products of...
Σελίδα 33 - Two things may be observed on this comparison, First, how very much the shorthand expression gains in clearness from its brevity. Secondly, that it is only shorthand for something which is just straightforward common sense and nothing else. We may always depend upon it that algebra, which cannot be translated into good English and sound common sense is bad algebra.
Σελίδα 222 - ... means to prevent or to cure the latter. It may therefore have, like Medicine, the three departments of Physiology, Pathology, and Therapeutics. §2. Since Pedagogics is capable of no such exact definitions of its principle and no such logical deduction as other sciences, the treatises written upon it abound more in shallowness than any other literature. Short-sightedness and arrogance find in it a most congenial atmosphere, and criticism and declamatory bombast flourish in perfection as nowhere...
Σελίδα 197 - If the number of independent equations be greater than the number of variables, there is in general no solution, and the system of equations is said to be inconsistent.
Σελίδα 136 - ... lay before you a considerable number of mathematical papers, which give collectively a fairly complete account of contemporaneous mathematical activity in Germany. Reserving for the mathematical section a detailed summary of these papers, I mention here only certain points of more general interest. When we contemplate the development of mathematics in this nineteenth century, we find something similar to what has taken place in other sciences. The famous investigators of the preceding period,...

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