Elements of Geometry and Plane Trigonometry; With an Appendix, and Copious Notes and Illustrations

Elements of Geometry and Plane Trigonometry; With an Appendix, and Copious Notes and Illustrations

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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. 1817 edition. Excerpt: ...stand on equal arcs AE and EB; consequently AE: AB:: CO: CE. But, ACBE being a quadrilateral figure contained in a circle, CE.AB = AE.CB]EB.CA = AE (CA + CB), or AE: AB:: CE: CA + CB; wherefore CO: CE:: CE: CA+CB, or CE = CA(CA + CB) Cor. Hence, in small arcs, the ratio of the sine to the arc approaches that of equality. For, let the semiarcs AE and EB be again bisected in the points F and G; and, continuing their subdivision indefinitely, let the successive intermediate chords be drawn. The ratio of the sine BD to the arc AB may be viewed as compounded of the ratio of BD to the chord AB, of that of AB to the two chords AE and EB, of that of AE and EB to the four chords AF, FE, EG, and GB, and so forth. But these ratios, it has been shown, are the same respectively as those of the supplemental chords CB, CE, CF, &c. to the diameter CA. And since each of the ratios CB: CA, CE: CA, CF: CA, &c. approaches to equality, it is evident that their compounded ratio, or that of the sine to its corresponding arc, must also approach to equality. Scholium. Hence the ratio of the sine BD to the arc AB is expressed numerically, by the ratio of the continued product of the series of supplemental chords CB, CE, CF, &c. to the relative continued power of the diameter CA. The ratio may, therefore, be determined to any degree o exactness, by the repeated application of the proposition in computing those derivative chords. But a very convenient approximation is more readily assigned. Make CD to GI as CB to CA, CI to CK as CE to CA, CK to CL as CF to CA, and so forth, tending always towards the limit Z; then the ratio of CD to CZ, being compounded of these ratios, must express the ratio of the sine BD to its corresponding arc AB. Now CD: CB:: CB: CA;show more

Product details

  • Paperback | 90 pages
  • 189 x 246 x 5mm | 177g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236615298
  • 9781236615299