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

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### Common terms and phrases

abstract according addition affirmed Algebra analysis Analytical angles applicable argument arithmetic axioms basis branches called combinations common compared comparison conclusion connection considered consists contains deductive defined definition denominate distinct divided division elementary elements employed equal equation exact example exist explain expressed facts figures four fractions Geometry give given Hence idea important increase indicated Induction inferred kind knowledge known language laws length Logic marks mathematical means measure merely method mind multiplied nature necessary objects observed once operations particular plane practical predicate premises presented principles properties proportion proposition proved quantity question ratio reasoning reference regarded relation rules scale sense sides simple solid space square straight surface syllogism symbols taken term things third tion triangle true truths unit weight whole yard

### Popular passages

Page 305 - In the mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it ; if too wandering, they fix it; if too inherent in the sense, they abstract it.

Page 235 - 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 333 - Rock, which stands on one side of the harbour'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 317 - 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 50 - 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 333 - Horn. Arrived within a week's sail of Rio, he set seriously about determining, by lunar observations, the precise line of the ship's course, and its situation in it at a determinate moment, and having ascertained this within from five to ten miles, ran the rest of the way by those more ready and compendious methods, known to navigators, which can be safely employed for short trips between one known point and another, but which cannot be trusted in long voyages, where the moon is their only guide.

Page 73 - ... whatever is predicated (ie affirmed or denied) universally, of any class of things, may be predicated in like manner, (viz. affirmed or denied,) of any thing comprehended in that class.

Page 243 - 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 27 - All definitions are of names, and of names only; but, in some definitions, it is clearly apparent, that nothing is intended except to explain the meaning of the word; while in others, besides explaining the meaning of the word, it is intended to be implied that there exists a thing, corresponding to the word.

Page 70 - ... comprehended in it : now it is evident, that whatever is said of the whole of a class, may be said of any thing comprehended in that class ; so that we are thus authorized to say of the world, that " it had an intelligent author.