Induction Volume 2

Induction Volume 2

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1870 edition. Excerpt: ...and adaptation of the fundamental and indefinable notion of equality. The ninth axiom is 'The whole is greater than its part.' This also violates the first test; it is not a real proposition; the predicate is not different from the subject. It is a property implicated in the common fundamental notion that gives a meaning to addition, subtraction, whole, part. The concrete experience implied by all these words is one and the same experience, and in it is implicated the fact that what we call a sum is greater than any one of the amounts summed up; or what we call a whole is greater than any of the parts. We could not possess the notion of whole and part without possessing the fact that the whole is a larger magnitude than the part. If, therefore, there be any necessity for distinctly announcing this peculiar aspect of the great fundamental notion of addition, it should be given as one of the forms of expressing the notion of Addition, when that notion is first introduced at the threshold of Arithmetic. The tenth axiom, 'All right angles are equal' is implicated in the definition of a right angle; and should be stated as an appendage to that definition. The eleventh axiom, in Euclid's text, is a difficult theorem preparatory to the propositions respecting parallel lines. It is usually given in a modified and simpler form. Thus (by De Morgan)--' If a straight line be taken, and a point exterior to it; of all the straight lines that can be drawn through the point, one only will be parallel to the first-mentioned straight line.' In whatever form given, it is not an axiom, but a proposition deducible from the definition of parallel lines; in fact, it ought to appear among the Theorems of the first book, unless, indeed, it be so nearly identified with...show more

Product details

  • Paperback | 178 pages
  • 189 x 246 x 10mm | 327g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236653394
  • 9781236653390