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VIII. New Series for the Quadrature of the Conic Sections, and the Computation of Logarithms

Published online by Cambridge University Press:  17 January 2013

William Wallace
William Wallace
Affiliation:
Professor of Mathematics in the Royal Military College at Great Marlow
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Extract

The Quadrature of the Conic Sections, and the Computation of Logarithms, are problems of considerable importance, not only in the elements of Mathematics, but also in the higher branches of that science. On this account, every successful attempt to simplify their resolution, as well as any new formulæ which may be found applicable to that purpose, must always be interesting, and must in some measure contribute to the improvement of mathematical knowledge.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 6 , 1812 , pp. 269 - 344
Copyright
Copyright © Royal Society of Edinburgh 1812

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References

page 273 note * We may here observe, that this formula may be considered as the analytic expression of a general theorem (which is not inelegant), relating to regular figures described about any arch of a circle; and others analogous to it will occur in the following investigations.

page 302 note * This formula, although very elegant as an analytical transformation, does not seem to admit of being applied with advantage to the rectification of an arch, on account of the great number of factors of the product which would be required to give a result tolerably correct.

page 310 note * The same series may also be put under another form, which it may not be improper to notice briefly, on account of the facility with which the terms may be deduced one from another by the help of the common trigonometrical tables. It is this,

The arches α′, α″, α‴, αiv, &c. are to be deduced one from another as follows.

Take α such that , &c. The symbols T(m), T(m + 1) and R, denote the same things as in the other form of the series.