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On the study and difficulties of mathematics [by A. De Morgan].
Augustus De Morgan
Uten tilgangsbegrensning - 1831
algebra alter appears apply arithmetic asserted becomes beginner called circle common considered contained continually decimal definition demonstration denominator derived difficulties direction divided division equal equation evident example exist expression fact factors figure four fraction geometry give given greater ideas inch increased instance length less letter logarithms magnitude manner mathematics meaning method move multiplied nature necessary negative never nevertheless notion observe opening operations particular positive possible present principles problem proceed proportion proposition proved quantity question quotient reasoning reduced remainder represented result right angles root rule shown sides simple solution square stand step student subtraction sufficient suppose symbol taken term theory thing third tion triangle true truth unit usually whole numbers write written
Side 225 - ... equal angles in each ; then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other.
Side 230 - Thus, that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, was an experimental discovery, or why did the discoverer sacrifice a hecatomb when he made out its proof ?
Side 78 - To divide a term of the second series by one which comes before it, subtract the exponent of the divisor from the exponent of the dividend, and make this difference the exponent of c.
Side 36 - Here then appears a connexion between the multiplication of whole numbers, and the formation of a fraction whose numerator is the product of two numerators, and its denominator the product of the corresponding denominators. These operations will always come together, that is whenever a question occurs in which, when whole numbers are given, those numbers are to be multiplied together ; when fractional numbers are given, it will be necessary, in the same case, to multiply the numerator by the numerator,...
Side 8 - So it is with our reasoning faculties: it is desirable that their powers should be exerted upon objects of such a nature, that we can tell by other means whether the results which we obtain are true or false, and this before it is safe to trust entirely to reason. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds: 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first...
Side 8 - When the conclusion is attained by -reasoning, its truth or falsehood can he ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if, as was said before, reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. Between the meanings of terms there is no distinction, except a total distinction, and all adjectives...