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. 1833 Excerpt: ... From the point C as a centre with the radius CB, describe the semicircle FBE. Then, the angle B being right, AB is a tangent to the circumference; consequently (Prop. XXVI. Cor. 1.) AE: AB:: AB: AF; and therefore (Prop. XIX. B. V.), AE, AB are incommensurable, and consequently, AC, AB are also incommensurable; for if ...
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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. 1833 Excerpt: ... From the point C as a centre with the radius CB, describe the semicircle FBE. Then, the angle B being right, AB is a tangent to the circumference; consequently (Prop. XXVI. Cor. 1.) AE: AB:: AB: AF; and therefore (Prop. XIX. B. V.), AE, AB are incommensurable, and consequently, AC, AB are also incommensurable; for if these had a common measure, the same also would measure their sum AE (Prop. XVII. B. V.), which has been proved to be incommensurable with AB: hence the diagonal of a square is incommensurable with its side. Scholium. It appears from the above demonstration that it would be in vain to attempt to express accurately by numbers the side and diagonal of a square; a fact which might, indeed, have been inferred from the third corollary to Proposition X. Book II. For, representing the side of a square by unity, double the square of the side will be 2; and, consequently, the diagonal will be expressed by the square root of 2. Now V2 is a surd expression, that is to say, its numerical value can never be accurately found, although it may be approximated to sufficiently near for every practical purpose. This circumstance affords a striking instance of the insufficiency of numbers to answer rigorously all the purposes of geometry. We cannot, for instance, take upon ourselves to say that any two lines that may be promiscuously proposed, shall be susceptible of accurate numerical representation, without first inquiring whether these lines are commensurable or not: since, for aught we know to the contrary, one of the proposed lines may be equal to the side, and the other to the diagonal of the same square, or else they may be similarly related to each other. The reasonings of geometry, however, are quite independent of any proviso, of this kind. That triangl..
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Add this copy of Elements of Geometry With Notes to cart. $65.41, good condition, Sold by Bonita rated 4.0 out of 5 stars, ships from Santa Clarita, CA, UNITED STATES, published 2016 by Palala Press.
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