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

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abstract affirmed Algebra algebraic quantity analysis Analytical Analytical Geometry antece applicable apprehended argument Aristotle arithmetic axioms basis bers branch called common compared conclusion connection copula deductive deductive reasoning definition denominate numbers denote divided division elementary elements employed ence Enthymeme equal equation eral example explain expressed facts figures fractional units genus Geometry gism Hence ical idea implies induction inferred integer integer units INTEGRAL CALCULUS kind knowledge known language laws learner length Logic marks of design mathematical science means method metic middle term mind multiplied Natural Philosophy nature objects operations philosophy plane practical predicate premises principles properties proportion proposition quantities quotient ratio reasoning regarded relation right angles rules sides signification signs simple solid species spelling square straight line subtraction surface syllogism symbols thing tion triangle Troy Weight true truths unit of measure unity whole words yards

### Popular passages

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 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 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 60 - 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 62 - ... logically ; which is, in fact, as great a blunder as if any one were to mistake grammar for a peculiar language, and to suppose it possible to speak correctly without speaking grammatically. — They have, in short, considered logic as an art of reasoning ; whereas (so far as it is an art) it is the art of reasoning ; the logician's object being, not to lay down principles by which one may reason, but by which all must reason, even though they are not distinctly aware of them : — to lay down...

Page 243 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.

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 88 - ... syllogism can be faulty which does not violate these canons ; none correct which does : hence on these two canons are built the rules or cautions which are to be observed with respect to syllogisms, for the purpose of ascertaining whether those canons have been strictly observed or not. 1st. Every syllogism has three, and only three terms : viz. the middle term, and the two terms (or extremes, as they are commonly called) of the Conclusion or Question. Of these...

Page 88 - We will speak first of pure categorical syllogisms ; and the axioms or canons by which their validity is to be explained : viz. first, if two terms agree with one and the same third, they agree with each other: secondly, if one term agrees and another disagrees with one and the same third, these two disagree with each other.