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addition algebraic angle applied argument arithmetical base becomes begin calculation called clear common complete considered constant corresponding course curve definite described determined differential directed directed numbers distance division equal equation examples exercises expression fact figure follows former formula fraction function give given graduation graph graphic height idea identical illustrated important increase indices instance integral interest kind latter length less limit logarithms mathematical means measure method movement moving multiplied natural negative obtained original parabola positive possible practical present principle problem pupil question ratio reached reason rectangle regarded relation represented result roots rule scale shows side simple sine square statement steps strip student suggest suppose symbols taken teacher term tion unit variables whole
5. lappuse - The ideal of mathematics," writes Dr. Whitehead,1 "should be to erect a calculus to facilitate reasoning in connexion with every province of thought, or of external experience, in which the succession of thoughts or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.
27. lappuse - ... and formularies should employ consistently the same symbols, and that these should readily suggest the verbal units for which they stand.1 1 The Institute of Actuaries have published an official set of symbols which are used, without explanation, in all the papers and discussions of their members. When the verbal statement of the rule contains a numerical constant the practice must be taught of placing it before the literal symbols. Thus the rule that the volume of a pyramidal solid is obtained...
79. lappuse - It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
412. lappuse - To wit, when we seek to subject them to numeration ... we find that they flee away perpetually, so that not one of them can be apprehended precisely in itself. . . . Now that cannot be called a true number which is of such a nature that it lacks precision. . . . Therefore, just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity.
454. lappuse - A = tan a/sin b . . (4) and the derived formula : sin a sin b _ sin c sin A sin B sin C ' ' The " supplemental formulas " corresponding to these are given (as a cheap luxury) in the supplementary exercise, but "Napier's analogies...
384. lappuse - The essentials of the ideas connoted by the terms " indeterminate value " and " singular points " find their place here. In Ex. LXXVI an inquiry into the properties of a few functions of two variables is made the occasion for extending the method of rectangular coordinates to the analysis and description of curved surfaces. The investigation is undertaken in the spirit of ch. iv., § 9 ; that is, the surfaces are treated as tri-dimensional graphs to be studied not so much for their own sake as for...
17. lappuse - The view that they represent wholly distinct forms of intellectual activity is partial, unhistorical, and unphilosophical. A more serious charge against it is that it has produced an infinite amount of harm in the teaching of mathematics. Our purpose in teaching mathematics in school should be to enable the pupil to realize, at least in an elementary way, this two-fold significance of mathematical progress. A person, to be really " educated," should have been taught the importance of mathematics...
46. lappuse - ... is less compact and less easily reproduced. Its message is frequently inarticulate and obscure. For these and similar reasons it should be regarded as a subsidiary algebraic instrument which fulfils its best office when it either leads up to a formula by which it may itself be superseded, or serves to unfold more fully the implications of a formula whose properties have been only partially explored.