Trigonometry and Double Algebra (Google eBook)

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Printed for Taylor, Walton and Maberly, 1849 - Trigonometry - 167 pages
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Page vi - Solutions of the Trigonometrical Problems proposed at St. John's College, Cambridge, from 1829 to 1846. By the Rev. T. GASKIN, MA 8vo. 9*.
Page vi - Foundation of Algebra,' No. III. (Read Nov. 27, 1843.) Cambridge, 1842 and 1845, 8vo. George Peacock. A Treatise on Algebra. Vol. I. Arithmetical Algebra. Vol. II. Symbolical Algebra and its applications to the geometry of position. London, 1843, 12mo. Martin Ohm [translated by Alexander John Ellis]. The Spirit of Mathematical Analysis, and its relation to a logical system.
Page 41 - the introduction of the unexplained symbol V — 1,"6 and goes on to say: "The use, which ought to have been called experimental, of the symbol V—l, under the name of an impossible quantity, shewed that; come how it might, the intelligible results (when such things occurred) of the experiment were always true, and otherwise demonstrable. I am now going to try some new experiments.
Page v - J.) on the objections against the geometrical representation of the square roots of negative quantities ii. 371 ; on the geometrical representation of the powers of quantities whose indices involve the square roots of negative quantities, ii.
Page 97 - M and N may be magnitudes, + the sign of addition of the second to the first. 2. M and N may be numbers, and + the sign of multiplying the first by the second. 3. M and N may be lines, and + a direction to make a rectangle with the antecedent for a base, and the consequent for an altitude. 4. M and N may be men, and + the assertion that the antecedent is the brother of the consequent. 5. M and N may be nations, and + the sign of the consequent having fought a battle with the antecedent: and so on.
Page v - Imaginaires. (Read June 20, 1805.) See also the review of this in vol. xii. of the Edinburgh Review, April— July, 1808 (written by Play fair).
Page 3 - In this method the right angle is divided into 90 equal parts, each of which is called a degree. Each degree is subdivided into 60 equal parts, each of which is called a minute.
Page 92 - As soon as the idea of acquiring symbols and laws of combination, without given meaning, has become familiar, the student has the notion of what I will call a symbolic calculus; which, with certain symbols and certain laws of combination, is symbolic algebra: an art, not a science; and an apparently useless art, except as it may afterwards furnish the grammar of a science.
Page 4 - ... able to construct. These results were, as we have pointed out, all derived by common arithmetical operations, based on the obvious truth that the circumference of a circle is greater than that of any inscribed, and less than that of any circumscribed polygon. They involve none of those more subtle ideas connected with Limits, Infinitesimals, or Differentials, which seem to render more recent results suspected by modern 'squarers.
Page 92 - No. 708) speaks of surfaces as being "ut est color in corpore." every student who has any mechanical associations connected with those symbols; that is, to every student who has previously used them in ordinary algebra. Geometrical reasons, and arithmetical process, have each its own office; to mix the two in elementary instruction, is injurious to the proper acquisition of...

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