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Page 295 - The four fundamental operations for rational algebraic expressions. Factoring, determination of highest common factor and lowest common multiple by factoring; fractions, including complex fractions, and ratio and proportion; linear equations, both numerical and literal, containing one or more unknown quantities; problems depending on linear equations; radicals, including the extraction of the square root of polynomials and of numbers; exponents, including the fractional and the negative.
Page 77 - And therefore, while in all things that we see or do, we are to desire perfection, and strive for it, we are nevertheless not to set the meaner thing, in its narrow accomplishment, above the nobler thing, in its mighty progress; not to esteem smooth minuteness above shattered majesty ; not to prefer mean victory to honourable defeat; not to lower the level of our aim, that we may the more surely enjoy the complacency of success.
Page 60 - Soc. And how much are three times three feet ? Boy. Nine. Soc. And how much is the double of four ? Boy. Eight. Soc. Then the figure of eight is not made out of a line of three ? Boy. No.
Page 106 - General formulas which men have devised to express groups of details, and which have severally simplified their conceptions by uniting many facts into one fact, they have supposed must simplify the conceptions of a child also.
Page 295 - ... and geometric progressions, with applications. It is assumed that pupils will be required throughout the course to solve numerous problems which involve putting questions into equations. Some of these problems should be chosen from mensuration, from physics...
Page 20 - Give a boy accurate perceptions. Teach him the difference between the similar and the same. Make him call things by their right names. Pardon in him no blunder. Then he will give you solid satisfaction as long as he lives.
Page 127 - With the algebraists, however, who are Pagans themselves, the 'Pagan fables' are believed, and the inferences are made, not so much through lapse of memory as through an unaccountable addling of the brains. In short, I never yet encountered the mere mathematician who...
Page 17 - ... mathematical study furnishes the first scientific key to the existence of bodies and their various motions. Mathematics in its pure form, as arithmetic, algebra, geometry, and the application of the analytical method, as well as mathematics applied to matter and force or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this, its quantitative aspect. Mathematics furnishes the instrument, the tool of thought, which gives...
Page 295 - Quadratic equations, both numerical and literal. Simple cases of equations with one or more unknown quantities, that can be solved by the methods of linear or quadratic equations. Problems depending on quadratic equations. The binomial theorem for positive integral exponents. The formulas for the nth term and the sum of the terms of arithmetical and geometric progressions, with applications.
Page 60 - ... that is to say of eight feet ; and I want to know whether you still say that a double square comes from a double line? BOY. Yes. Soc. But does not this line become doubled if we add another such line here? BOY. Certainly. Soc. And four such lines will make a space containing eight feet? BOY. Yes. Soc. Let us describe such a figure: is not that what you would say is the figure of eight feet?