On the study and difficulties of mathematics [by A. De Morgan].


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Side 75 - XIII. •All parallelograms on the same or equal bases and between the same parallels...
Side 76 - 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 25 - 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 30 - Four persons purchased a farm in company for 4755 dollars ; of which B paid three times as much as A ; C paid as much as A and B ; and D paid as much as C and B. What did each pay 1 Prob. 32. It is required to divide the number...
Side 12 - A'H'C'D' contains ^ of G. 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...
Side 13 - J., and is found by multiplying the numerator of the first by the denominator of the second for the numerator of the result, and the denominator of the first by the numerator of the second for the denominator of the result. That this process does give the same result as ordinary division in all cases where ordinary division is applicable, we can easily shew from any two whole numbers, for example, 12 and 2, whose quotient is 6. Now 12 is...
Side 71 - ... what has just been observed; since in the comparison of two things with one and the same third thing, in order to ascertain their connexion or discrepancy, consists the whole of reasoning. Thus, the deduction without further process of the equation...
Side 90 - When it is said that the angle = — ^r- — , it is only meant that, on one particular supradius position, (namely, that the angle 1 is that angle whose arc is equal to the radius,) the number of these units in any other angle is found by dividing the number of linear units in its arc by the number of linear units in the radius. It only remains to give a formula for finding the number of degrees, minutes, and seconds in an angle, whose theoretical measure is given. It is proved in geometry that...
Side 25 - A fraction is not altered by multiplying or dividing both its numerator and denominator by the same quantity.
Side 3 - ... faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty.

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