Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/38

 11.

The inverse problem, celebrated under the title of Kepler’s problem, that of finding the true anomaly and the radius vector from the mean anomaly, is much more frequently used. Astronomers are in the habit of putting the equation of the centre in the form of an infinite series proceeding according to the sines of the angles $M,$ $2 M,$  $3 M,$  etc., each one of the coefficients of these sines being a series extending to infinity according to the powers of the eccentricity. We have considered it the less necessary to dwell upon this formula for the equation of the centre, which several authors have developed, because, in our opinion, it is by no means so well suited to practical use, especially should the eccentricity not be very small, as the indirect method, which, therefore, we will explain somewhat more at length in that form which appears to us most convenient.

Equation XII., $E=M+e \sin E,$ which is to be referred to the class of transcendental equations, and admits of no solution by means of direct and complete methods, must be solved by trial, beginning with any approximate value of $E,$  which is corrected by suitable methods repeated often enough to satisfy the preceding equation, that is, either with all the accuracy the tables of sines admit, or at least with sufficient accuracy for the end in view. If now, these corrections are introduced, not at random, but according to a safe and established rule, there is scarcely any essential distinction between such an indirect method and the solution by series, except that in the former the first value of the unknown quantity is in a measure arbitrary, which is rather to be considered an advantage since a value suitably chosen allows the corrections to be made with remarkable rapidity. Let us suppose $\varepsilon$ to be an approximate value of $E,$  and $x$  expressed in seconds the correction to be added to it, of such a value as will satisfy our equation $E=\varepsilon+x.$  Let $e \sin \varepsilon,$  in seconds, be computed by logarithms, and when this is done, let the change of the $\log \sin \varepsilon$  for the change of $1^{\prime \prime}$  in $\varepsilon$  itself be taken from the tables; and also the variation of $\log e \sin \varepsilon$  for the change of a unit in the number $e \sin \varepsilon;$  let these changes, without regard to signs, be respectively $\lambda,$  $\mu,$  in which it is hardly necessary to remark that both logarithms are presumed to contain an equal number of decimals. Now, if $\varepsilon$ approaches so near the correct value of $E$