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Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse
Published online by Cambridge University Press: 08 April 2017
Extract
It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 5 , Issue 2 , 1805 , pp. 271 - 293
- Copyright
- Copyright © Royal Society of Edinburgh 1805
References
page 272 note * This discovery was made by Mr Landen, who published it first in the Philosophical Transactions for 1775, and afterwards in his Mathematical Memoirs.
page 280 note * The properties of the ellipse here alluded to have been explained by Euler, and some of them have also been observed by Landen.
page 284 note * See Fig. page 279.
page 286 note * Hence, by the way, it appears, that instead of the semi-perimeters of two ellipses which we have used in the preceding paper, for expressing the coefficients of the development of (a 1 + b 2 — 2ab cos φ)n, we may substitute any two of an indefinite number of elliptic arches, and certain algebraic functions of the axes of these ellipses; therefore, the different infinite series, which may be used to express the coefficients A and B, are really innumerable.