A Treatise on Elementary Geometry, with Appendices Containing a Collection of Exercises for Students and an Introduction to Modern Geometry
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A Treatise on Elementary Geometry, with Appendices Containing a Collection of Exercises for Students and an Introduction to Modern Geometry

A Treatise on Elementary Geometry, with Appendices Containing a Collection of Exercises for Students and an Introduction to Modern Geometry


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About the Book

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. 1879 Excerpt: ...angles at S is less than the sum of the angles at O, that is, less than four right angles. PROPOSITION XXVII.--THEOREM. 71. Two triedral angles are either equal or symmetrical, when the three face angles of one are respectively equal to the three face angles of the other. In the trieifra-angles S and s, let ASB = asb, ASC = asc, and BSC--bsc; then, the diedral angle SA is equal to tLe diedral angle sa. On the edges of these angles take the six equal distances SA, SB, SO, sa, sb, sc, and draw AB, BC, AC, ab, be, ac. The isosceles triangles SAB and sab are equal, having an equal angle included by equal sides, hence AB = ab; and for the same reason, BC = be, AC = ac; therefore, the triangles ABC and abc are equal. At any point D in SA, draw DE in the face ASB and DF in the face ASC, perpendicular to SA; these lines meet AB and A C, respectively, for, the triangles ASB and ASC being isosceles, the angles SAB and SA C are acute; let E and F be the points of meeting, and join EF. Now on sa take sd = SD, and repeat the same construction in the triedral angle s. The triangles ADE and ade are equal, since AD = ad, and the angles at A and D are equal to the angles at a and d; hence, AE = ae and DE = de. In the same manner, we have AF = af and DF = df. Therefore, the triangles AEF and aef are equal (I. 76), and we have EF = ef. Finally, the triangles EDF and edf, being mutually equilateral, are equal; therefore, the angle EDF, which measures the diedral angle SA, is equal to the angle edf, which measures the diedral angle sa, and the diedral angles SA and sa are equal (41). In the same manner, it may be proved that the diedral angles SB and SC are equal to the diedral angles sb and sc, respectively. So far the demonstration applies to either of the two figures denoted...


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Product Details
  • ISBN-13: 9781130711875
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 94
  • Spine Width: 5 mm
  • Width: 189 mm
  • ISBN-10: 1130711870
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 181 gr

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A Treatise on Elementary Geometry, with Appendices Containing a Collection of Exercises for Students and an Introduction to Modern Geometry
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