Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections: A Translation of Gauss's "Theoria Motus." With an AppendixLittle, Brown, 1857 - 326 pàgines |
Altres edicions - Mostra-ho tot
Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic ... Carl Friedrich Gauss Visualització completa - 1857 |
Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic ... Carl Friedrich Gauss Visualització completa - 1857 |
Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic ... Carl Friedrich Gauss Visualització completa - 1857 |
Frases i termes més freqüents
aberration according accuracy approximate value ascending node auxiliary angle calculation celestial sphere circle coefficients comets computed conic sections conveniently coördinates corrected value corresponding cos² cotan deduced denote derived determined difference differential distance earth eccentric anomaly eccentricity ecliptic elements ellipse equal equation error evident example expressed formulas fundamental plane geocentric place given heavenly body heliocentric motion heliocentric place Hence hyperbola hyperbolic motion inclination intersection intervals latter Log diff log Q log r log logarithm logr longitude and latitude longitude in orbit manner mean anomaly moreover negative nutation obtained parabola parallax perihelion perturbations plane preceding article precision problem radius vector reduced referred regarded right ascension right line roots sin¹ sin² sine solution substituted taken tan² third tion true anomaly unknown quantities whence Wherefore
Passatges populars
Pàgina 258 - Therefore, that wiM be the most probable system of values of the unknown quantities p, q, r, s, etc., in which the sum of the squares of the differences between the observed and computed values of the functions V, V, V", etc. is a minimum, if the same degree of accuracy is to be presumed in all the observations.
Pàgina 247 - ... all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unknown quantities. This problem can only be properly undertaken when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most...
Pàgina xv - Piazzif from the above date to the eleventh of February were published. Nowhere in the annals of astronomy do we meet with so great an opportunity, and a greater one could hardly be imagined, for showing most strikingly, the value of this problem, than in this crisis and urgent necessity, when all hope of discovering in the heavens this planetary atom, among innumerable small stars after the lapse of nearly a year, rested solely upon a sufficiently approximate knowledge of its orbit to be based upon...
Pàgina 247 - If the astronomical observations and other quantities, on which the computations of orbits is based, were absolutely correct, the elements also, whether deduced from three or four observations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of Kepler), and, therefore, if other observations were used, they might be confirmed but not corrected. But since all our measurements and observations are nothing more than approximations to the...
Pàgina 257 - The function just found cannot, it is true, express rigorously the probabilities of the errors : for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always to be zero; while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly . . . (ed., when the exponent)...
Pàgina 258 - ... a double error can be committed in the former system with the same facility as a single error in the latter, in which case, according to the common way of speaking, a double degree of precision is attributed to the latter observations.”” The fact of the matter is, however, that: “. . . different fields have particularly favorite ways of expressing precision. Most of these measures are multiples of the standard deviation; it is not always clear which multiple is meant. . “Some consider...
Pàgina 256 - ... base, the common principle, the excellence of which is generally acknowledged, depends. It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it. By putting, therefore, V = V = V
Pàgina 258 - The most probable value of the desired parameters will be that in which the sum of the squares of the differences between the actually observed and computed values multiplied by numbers that measure the degree of precision, is a minimum.
Pàgina 251 - The investigation of an orbit having, strictly speaking, the maximum probability, will depend upon a knowledge of the law according to which the probability of errors decreases as the errors increase in magnitude: but that depends upon so many vague and doubtful considerations — physiological included — which cannot be subjected to calculation, that it is scarcely, and indeed less than scarcely, possible to assign properly a law of this kind in any case of practical astronomy. Nevertheless, an...
Pàgina xiv - To determine the orbit of a heavenly body, without hypothetical assumption, from observations not embracing a great period of time, and not allowing a selection with a view to the application of special methods...