The Logic and Utility of Mathematics: With the Best Methods of Instruction Explained and Illustrated

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A.S. Barnes & Company, 1850 - Logic - 375 pages
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Page 257 - The square described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides.
Page 305 - Admission to its sanctuary, and to the privileges and feelings of a- votary, is only to be gained by one means — sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range.
Page 46 - INDUCTION, then, is that operation of the mind, by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, Induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times.
Page 243 - The area of a rectangle is equal to the product of its base and altitude.
Page 239 - AD c, have two sides, and the included angle of the one equal to two sides and the included angle of the other, each to each, and are equal in all their parts...
Page 56 - IN every instance in which we reason, in the strict sense of the word, ie make use of arguments, whether for the sake of refuting an adversary, or of conveying instruction, or of satisfying our own minds on any point, whatever may be the subject we are engaged on, a certain process takes place in the mind, which is one and the same in all cases, provided it be correctly conducted.
Page 321 - Rock, which stands on one side of the harbor's mouth, so nearly right ahead that we had not to alter our course above a point in order to hit the entrance of Rio. This was the first land we had seen for three months, after crossing so many seas, and being set backwards and forwards by innumerable currents and foul winds.
Page 56 - This indeed is, and cannot but be, the case with every other process respecting which any system has been formed ; the practice not only may exist independently of the theory, but must have preceded the theory. There must have been Language before a system of grammar could be devised ; and musical compositions previous to the science of Music. This, by the way, will serve to expose the futility of the popular objection against Logic, that men may reason very well who know nothing of it.
Page 321 - I hove-to at four in the morning till the day should break, and then bore up ; for although it was very hazy, we could see before us a couple of miles or so. About eight o'clock it became so foggy that I did not like to stand in farther, and was just bringing the ship to the wind again before sending the people to breakfast, when it suddenly cleared off, and I had the satisfaction of seeing the great...
Page 88 - i_ •LJ unsound mode of arguing, which appears to demand our conviction, and to be decisive of the question in hand, when in fairness it is not.

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