And from these equations the values of x, y, and z might be found by the usual algebraic process.

It is manifest that, in the first of the three equations, the coefficients of x, y, and z must consist of the sums of the products of the several coefficients of x, y, and z in the original equations of condition when, in each separately, every term is multiplied by the coefficient of x in that equation, and that the coefficient of x must be the sum of the squares of those coefficients: in the second, the coefficients of x, y, and z must consist of the sums of the products of the several coefficients of x, y, and z in the original equations when, in each separately, every term is multiplied by the coefficient of y; and, in the third, the coefficients of x, y, and z must consist of the sums of the products of the several coefficients when, in each equation separately, every term is multiplied by the coefficient of z. Therefore, in applying the method of least squares, the given equations are to be so multiplied: and three equations of the form

A + B x + Cy+Dz=0

will be obtained by taking the sum of all those which were multiplied by the coefficients of x, the sum of all those which were multiplied by the coefficients of y, and the sum of all those which were multiplied by the coefficients of z. From the three equations the values of x, y, and z are to be found; and these will be the most probable values of the elements.

Instead of the ordinary process for determining x, y, and z, from these last equations, the following method has been proposed (Gauss, Theoria combinationis &c. Gottingen, 1823. Galloway, Treatise on Probability, 1839, art. 159.). Let the second members of the preceding equations (the differentials of the sum of the squares of the presumed errors) be represented by R1, R2, and R, respectively; and from the equations, by the usual rules of algebra, obtain three equations of the forms

the most probable values of x, y, z respectively, and will coincide with those which would have been obtained from the M1 M2 M3 will G1' H2' Ks

three equations above mentioned: also

G1

be the weights due to x, y, and z; and the constant number

0.476936 divided by the square roots of the several weights will give the relative values of the probable errors.

In the method of least squares the values of x, y, and are determined from equations in which the coefficients of those quantities are the sums of the squares of the proper coefficients of the same quantities in the original equations; and as these squares are necessarily positive, their sum, in each equation, is greater than the coefficient in either of the other terms of the same equation, the latter coefficients being made up of the sums of quantities, some of which are positive and some negative: thus the unknown quantities, being determined by a division in which the divisors are the greatest possible consistently with the conditions of the subject, must contain the least possible amount of error.

It should be observed that, in forming the equations of condition, every error which may be considered as constant, such as those which depend upon a false position of the instrument, must be previously corrected, or must have an allowance made for it; so that, in each equation, equal deviations from truth in excess and defect may be equally probable, or that all the equations which are employed may have equal weights. Should the weights be unequal, the several equations must be reduced to the same standard in this respect (since the degree of precision is proportional to the square root of the weight), by being multiplied into the square root of its proper weight. The inaccuracy which frequently attends the estimate formed of the relative weights of the several equations is the chief cause of imperfection in every method of determining the most probable values of elements from the equations of condition.

CHAP. XVI.

NAUTICAL ASTRONOMY.

PROBLEMS FOR DETERMINING THE GEOGRAPHICAL POSITION OF A SHIP OR STATION, THE LOCAL TIME, AND THE DECLINATION OF THE MAGNETIC NEEDLE.

331. THE determination of the place of a ship at sea by the distance sailed and by the angle which the ship's course makes with the meridian is, from the nature of the means used to ascertain those elements, as well as from the deviations produced by currents, by the set of the waters, and other causes, too uncertain to be relied on when the ship has been long out of the sight of land. Even the magnetic needle and the machine for measuring time may fail in giving correctly the indications for which they were provided; but celestial observations skilfully made will always enable the mariner to find his place on the ocean with the requisite precision, and hence it becomes his duty to make such observations as often as circumstances will permit.

The nature of the observations, and the manner of applying to the purposes of navigation the formulæ deduced from the investigations of general astronomy, or the tabulated results of such formulæ, constitute the subjects of that branch of the science which from the circumstances of its application is called nautical. The like observations and reductions are, however, equally requisite for the scientific traveller on land, in order that he may be enabled to fix the geographical positions of the remarkable stations at which he may arrive; and hence the several problems which follow, though particularly subservient to navigation, must be considered as belonging to a general course of practical astronomy.

332. On the hypothesis that the earth is at rest in the centre of the imaginary sphere of the fixed stars, the planes of the meridian and horizon of an observer, while he remains in one place, may be indicated by two fixed circles in the heavens ; and the diurnal rotation of the earth on its axis may be represented by a general rotation of the heavens from east to west. The sun, moon, and planets may, also, be considered as attached to the surfaces of spheres whose radii are their distances from the earth, and which, like the sphere of the

fixed stars, have movements of rotation. Now, let it be imagined that the observer is on the surface of a small sphere or spheroid representing the earth in the centre of the sphere of the stars: then, if the celestial body which is the subject of observation be not in the plane of the meridian, a spherical triangle can be imagined to be formed by the colatitude of the observer's station, the polar distance of the body and its zenith distance, or the complement of its observed altitude above the horizon: the sides and angles of one such triangle, or of two or more combined, will constitute the data and the things required; and these last may consequently be found by the rules of spherical trigonometry. Several corrections must however be applied to the observed altitude of a celestial body in order to reduce it to its true value, and these being alike, whatever be the nature of the problem of which the altitude is one of the data, it will be convenient to describe them before the particular problems are enunciated.

333. The octant, sextant, or circle may have an index error arising from the mirrors not being parallel to one another when the index of the vernier is at the zero of the graduations. At sea, the eye of the observer is above the plane of the sensible horizon by a quantity which depends upon the height of the ship; the refraction of light in the atmosphere causes the celestial body to appear too high, and finally, the effect of parallax is to make it appear too low. The index error of the instrument having been previously found (art. 136.) must be applied to the observed altitude as a correction, by addition or subtraction, as the case may be: the dip, or angular depression of the horizon, if the observation be made at sea (art. 165.), must be taken from the tables and subtracted from the observed altitude; or if an artificial horizon be employed, half the altitude, after correcting the index error, must be taken. The refraction may be had from the proper tables (art. 145.); and the horizontal parallax may be taken from the Nautical Almanac, but this term must be multiplied by the cosine of the observed altitude in order to reduce it to the value of the parallax for that altitude (art. 154.). It may be stated, however, that the parallaxes of the sun or of a planet are seldom used in reducing ordinary observations, and a fixed star has no sensible parallax. The refraction is greater than the parallax, for the sun or a planet, but the parallax of the moon exceeds her refraction: hence the difference between the corrections for parallax and refraction must be subtracted from the apparent altitude of the sun or of a planet, and added to that of the moon; and as it is proper to observe the altitude of the upper or lower limb of

the sun or moon, the correction thus applied gives the true altitude of the observed limb. The angular semidiameter of the celestial body must then be added to, or subtracted from the altitude of the limb, according as the lower or upper has been observed, and this may be found in the Nautical Almanac; but if the celestial body be the moon, the augmentation of her semidiameter on account of her altitude (art. 163.) must be added to that value of the semidiameter which is taken from the Almanac; and thus the correct altitude of the centre of the luminary is obtained.

The following examples will illustrate the process of correcting the observed altitudes of the sun and moon preparatory to the employment of such altitudes or the zenith distances in any geographical or nautical problem.

Ex. 1. Sept. 30. 1842, at Sandhurst, there was observed, by reflexion from mercury, the double altitude of the sun's upper limb, which was found to be

60° 40′ 40′′ 8 40

2) 60 32 0

30 16 0

diff.

}}

1 32

Ex. 2. Nov. 29. 1843, at Sandhurst, at 6 h. 39 min. P.M. (Greenwich mean time) by reflexion from mercury, the double altitude of the moon's lower limb when on the meridian was found to be Index error of the sextant (subtractive)

75° 33' 30' 42

Determination of moon's augmented semidiam. (Art. 163.)

Moon's semidiam. (Naut. Alm.)=14′ 57′′. Therefore the altitude of the moon's centre is (nearly) 38°.

Log. sin. alt. moon's centre (nearly)

Augmentation in the zenith

9.7893

= 16′′., log. = 1.2041