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XXII.—Investigation of Analogous Properties of Co-ordinates of Elliptic and Hyperbolic Sectors

Published online by Cambridge University Press:  17 January 2013

William Wallace
William Wallace
Affiliation:
Cambridge P.S. &c.Emeritus Professor of Mathematics in the University of Edinburgh.
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Extract

The analogy between certain properties of the co-ordinates of elliptic and hyperbolic sectors, which forms the subject of this memoir, was observed by the early writers on the fluxional or differential calculus, and employed by them in its improvement. But, independently of this important application, the propositions in question are some of the most elegant theorems in geometry, and highly interesting as abstract truths.

Maclaurin, in the third chapter of the second book of his Treatise of Fluxions, has proved the truth of this theorem: “Supposing n to be any number, let E and n × E be elliptic sectors, which stand on arcs that begin at a vertex of either axis, and H and n × H hyperbolic sectors, which stand on arcs that begin at the extremity of the transverse axis: the algebraic equation which expresses the relation between x and z, ordinates drawn to the other axis from the extremities of the arcs, must be the very same in the two curves.”

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 14 , Issue 2 , 1840 , pp. 431 - 447
Copyright
Copyright © Royal Society of Edinburgh 1840

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References

page 437 note * Maclaurin's Fluxions, Article 757.

page 439 note * Vieta, FrancisciOpera Mathematica. Leyden, 1646 (pp. 295, 297)Google Scholar.

page 442 note * These important analytical expressions were found by De Moivre in 1707, and inserted in the Philosophical Transactions of that year; and again in the Transactions for 1722. They are also in his Miscellanea Analytica, printed at London 1730.

page 443 note * Bernouilli, Joannis, Opera, vol. i. pp. 387 and 511Google Scholar.

page 446 note * Gregorii, a S. Vincentio Vera Quadratura Circuli et Hyperbolœ. Antwerp, 1647Google Scholar.

Page 446 note † Mercator, . Logarithmotechma, &c. London, 1668Google Scholar.

page 446 note ‡ Philosophical Transactions (No. 27), vol. i., Lowthorpe's Abridgment.

page 447 note * Miscellanea Berolinensia, tome vii.; and Introductio in Analysin Infinitum, t. i.

page 447 note † Lagrange, . Leçons sur le Calcul des Fonctions, p. 114Google Scholar.