Chapter XIII: The Nineteenth Century (5)

[124] 69 miles is 364,320 feet, so that the two northern degrees were a little more and the Peruvian are a little less than 69 miles.

[125] The remaining 8,000 stars were not “reduced” by Lacaille. The whole number were first published in the “reduced” form by the British Association in 1845.

[126] A mural quadrant.

[127] The ordinary approximate theory of the _collimation error_, _level error_, and _deviation error_ of a transit, as given in textbooks of spherical and practical astronomy, is substantially his.

[128] The title-page is dated 1767; but it is known not to have been actually published till three years later.

[129] For a more detailed discussion of the transit of Venus, see Airy’s _Popular Astronomy_ and Newcomb’s _Popular Astronomy_.

[130] _Some_ other influences are known—_e.g._ the sun’s heat causes various motions of our air and water, and has a certain minute effect on the earth’s rate of rotation, and presumably produces similar effects on other bodies.

[131] The arithmetical processes of working out, figure by figure, a non-terminating decimal or a square root are simple cases of successive approximation.

[132] “C’est que je viens d’un pays où, quand on parle, on est pendu.”

[133] Longevity has been a remarkable characteristic of the great mathematical astronomers: Newton died in his 85th year; Euler, Lagrange, and Laplace lived to be more than 75, and D’Alembert was almost 66 at his death.

[134] This body, which is primarily literary, has to be distinguished from the much less famous Paris Academy of Sciences, constantly referred to (often simply as the Academy) in this chapter and the preceding.

[135] E.g. _Mélanges de Philosophie, de l’Histoire, et de Littérature_; _Éléments de Philosophie_; _Sur la Destruction des Jésuites_.

[136] _I.e._ he assumed a law of attraction represented by μ∕_r^2_ + ν∕_r^3_.

[137] This appendix is memorable as giving for the first time the method of _variation of parameters_ which Lagrange afterwards developed and used with such success.

[138] That of the distinguished American astronomer Dr. G. W. Hill (chapter XIII., § 286).

[139] They give about ·78 for the mass of Venus compared to that of the earth.

[140] The orbit might be a parabola or hyperbola, though this does not occur in the case of any known planet.

[141] On the _Calculus of Variations_.

[142] The establishment of the general equations of motion by a combination of _virtual velocities_ and _D’Alembert’s principle_.

[143] _Théorie des Fonctions Analytiques_ (1797); _Resolution des Équations Numériques_ (1798); _Leçons sur le Calcul des Fonctions_ (1805).

[144] _Théorie Analytique des Probabilités._

[145] The fact that the post was then given by Napoleon to his brother Lucien suggests some doubts as to the unprejudiced character of the verdict of incompetence pronounced by Napoleon against Laplace.

[146] _Outlines of Astronomy_, § 656.

[147] Laplace, _Système du Monde_.

[148] If _n_, _n′_ are the mean motions of the two planets, the expression for the disturbing force contains terms of the type = _sin_(_np_ ± _n′p′_) _t_, _cos_ where _p_, _p′_ are integers, and the coefficient is of the order _p_⁓_p′_ in the eccentricities and inclinations. If now _p_ and _p′_ are such that _np_⁓_n′p′_ is small, the corresponding inequality has a period 2π∕(_np_⁓_n′p′_), and though its coefficient is of order _p_⁓_p′_, it has the small factor _np_⁓_np′_ (or its square) in the denominator and may therefore be considerable. In the case of Jupiter and Saturn, for example, _n_ = 109,257 in seconds of arc per annum, _n′_ = 43,996; 5_n′_ - 2_n_ = 1,466; there is therefore an inequality of the _third_ order, with a period (in years) = 360°∕1,466″ = 900.

[149] This statement requires some qualification when perturbations are taken into account. But the point is not very important, and is too technical to be discussed.

[150] ∑_e^2m_√_a_ = _c_, ∑_tan^2im_√_a_ = _c′_, where _m_ is the mass of any planet, _a_, _e_, _i_ are the semi-major axis, eccentricity, and inclination of the orbit. The equation is true as far as squares of small quantities, and therefore it is indifferent whether or not _tan i_ is replaced as in the text by _i_.

[151] Nearly the whole of the “eccentricity fund” and of the “inclination fund” of the solar system is shared between Jupiter and Saturn. If Jupiter were to absorb the whole of each fund, the eccentricity of its orbit would only be increased by about 25 per cent., and the inclination to the ecliptic would not be doubled.

[152] Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those of _von Zach_ (1804) and _Delambre_ (1806); and the chief planetary ones were those of _Lalande_ (1771), of _Lindenau_ for Venus, Mars, and Mercury (1810-13), and of _Bouvard_ for Jupiter, Saturn, and Uranus (1808 and 1821).

[153] The motion of the satellites of Uranus (chapter XII., § 253, 255) is in the opposite direction. When Laplace first published his theory their motion was doubtful, and he does not appear to have thought it worth while to notice the exception in later editions of his book.

[154] This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapter XIII., § 294), many of which have orbits that are far more eccentric than those of the other planets and are inclined to the ecliptic at considerable angles.

[155] _Système du Monde_, Book V., chapter VI.

[156] In his paper of 1817 Herschel gives the number as 863, but a reference to the original paper of 1785 shews that this must be a printer’s error.

[157] The motion of Castor has become slower since Herschel’s time, and the present estimate of the period is about 1,000 years, but it is by no means certain.

[158] More precisely, counting motions in right ascension and in declination separately, he had 27 observed motions to deal with (one of the stars having no motion in declination); 22 agreed in sign with those which would result from the assumed motion of the sun.

[159] The method was published by Legendre in 1806 and by Gauss in 1809, but it was invented and used by the latter more than 20 years earlier.

[160] The figure has to be enormously exaggerated, the angle SσE as shewn there being about 10°, and therefore about 100,000 times too great.

[161] Sir R. S. Ball and the late Professor Pritchard (§ 279) have obtained respectively ·47″ and ·43″; the mean of these, ·45″, may be provisionally accepted as not very far from the truth.

[162] An average star of the 14th magnitude is 10,000 times fainter than one of the 4th magnitude, which again is about 150 times less bright than Sirius. See § 316.

[163] Newcomb’s velocity of light and Nyrén’s constant of aberration (20″·4921) give 8″·794; Struve’s constant of aberration (20″·445), Loewy’s (20″·447), and Hall’s (20″·454) each give 8″·81.

[164] _Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat._

[165] _Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen._

[166] _E.g._ in Grant’s _History of Physical Astronomy_, Herschel’s _Outlines of Astronomy_, Miss Clerke’s _History of Astronomy in the Nineteenth Century_, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’s _Collected Papers_.

[167] This had been suggested as a possibility by several earlier writers.

[168] The discovery of a terrestrial substance with this line in its spectrum has been announced while this book has been passing through the press.

[169] Observations made on Mont Blanc under the direction of M. Janssen in 1897 indicate a slightly larger number than Dr. Langley’s.

[170] _Catalogus novus stellarum duplicium_, _Stellarum duplicium et multiplicium mensurae micrometricae_, and _Stellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830_.

[171] _I.e._ 2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)^{5∕2} = 10.

[172] If L be the ratio of the light received from a star to that received from a standard first magnitude star, such as Aldebaran or Altair, then its magnitude _m_ is given by the formula

L = (1∕2·512)^{m - 1} = (1∕100)^{(m - 1)∕5}, whence m - 1 = -5∕2log L.

A star brighter than Aldebaran has a magnitude less than 1, while the magnitude of Sirius, which is about nine times as bright as Aldebaran, is a _negative_ quantity,-1·4, according to the Harvard photometry.

End of Project Gutenberg's A Short History of Astronomy, by Arthur Berry