Again, in Astronomy, the time is determined from an observed altitude of the Sun or of a Star, from their declination and the latitude of the place. It is not then a question of mere curiosity to determine in what position, or part of the heavens, the Sun or Star ought to be observed, in order that the instrumental error, supposed to be of a certain magnitude, may least vitiate the determination of the time. The determination of the least errors is only one branch of the general Problem, which assigns, in its solution, the relations between the corresponding errors in the data and results, that is, the given errors in one or more of the conditions of the Problem, and the consequent errors in the results. Thus, the right ascension and declination of the Sun are computed in the Nautical Almanack from the longitude furnished by the Solar Tables and from the obliquity of the ecliptic. Now, the determination of this latter condition is subject to some error. If we assign a value to that error, we may then investigate the corresponding errors in the right ascension and declination, and, in the result of such investigation, we should necessarily include the cases, in which the original error would least affect the values of the right ascension and declination. The errors, that hitherto have been spoken of, are small variations or increments in the angles and sides of rectilinear and spherical triangles. Hence, an investigation of their corresponding values will comprehend a great variety of Problems that occur in Astronomy. For instance, it would assign the effects of parallax, refraction, aberration, precession, &c. in declination, right ascension, &c. since the effects of these inequalities, always very small, may be represented by very small portions of the arcs or circles along which those effects originally take place. It is not here intended to extend this enquiry beyond triangles; but, there are a great variety of Problems belonging to other figures and other subjects of investigation that might have been included under the class of Errores in mixtâ Mathesi. This was the Title which Roger Cotes gave to his Tract * on * Æstimatio errorum in mixta Mathesi per Variationes partium Trianguli plani et Spherici. this subject; and Lacaille *, in treating of the same subject, properly describes the object of Cotes's Tract to be the determination of the limits of inevitable errors in the practice of Geometry and Astronomy. We purpose to treat this, as we have treated all the preceding subjects, analytically. Suppose the relation between an angle A and a side b, to be expressed by this equation, sin. A = m. tan. b; then, if A should be increased by AA, whilst b was increased by Ab (AA, Ab, representing the entire differences or increments of A and b), the equation belonging to the changed triangle would be sin. (A + AA) = m. tan. (b + b), and the corresponding errors of A and b, or AA, Ab, would be to be determined from this equation, which is the difference of the two former, namely, from sin. (A + A) - sin. A = m. { tan. (b + b) - tan. b}. If we expand † sin. (A + AA), the left-hand side of the equation will become Now, in most of the cases that come under this enquiry, AA, whether it represents the quantity of precession, or of parallax, or of aberration, &c. is always a very small quantity: so small, that without vitiating the result, we may reject all terms involving (AA)2; (AA)3, &c.; in which case, the preceding quantity would become * Le but de l'Auteur est de determiner les limites des erreurs inevitables dans la pratique de la Geometrie et de l'Astronomie.' Acad. des Sciences, 1741, p. 240. † See Principles of Analytical Calculation, pp. 72, 73. In like manner, if the right-hand side of the equation be evolved and the terms that involve (Ab), (b)3, &c. be rejected, it will be reduced to Hence AA, Ab, are to be determined by this equation, If AA, Ab, should not be very small, or if considerable accuracy were required, the terms involving (AA), (b) may be retained, in which case the equation will be d.sin. Ad.tan.b , and for deducing A A in terms of Ab, or Ab in terms of ∆Α, the solution of a quadratic would be requisite, (see Principles of Anal. Calc. p. 74,) are the differential coefficients of sin. A, tan. b, and are respectively equal to cos. A, sec.2 b. We have taken a particular form; but, if we assume a general one, the method will be the same and the formula of solution similar. For instance, let X denote any function of A, and Y of &, and let the equation be X =mY; then the equation for determining AA, Ab, will be And, in like manner, if I should be a function of C, and U of a, &c. and the finite equation of relation should be X + n. V + &c. = m Y + p U + &c. n, m, &c. being constant quantities, the equation of relation between AA, &a, &c. these quantities being very small, would be In order to facilitate the solutions of the following cases, we will prefix the values of the differential coefficients of sin. x, cos. x, &c. In a right-angled triangle, of which one side is h, the other a, and the angle opposite h, e, it is required to find the error or variation in h, from a given error in 0, (See Cotes's Est. Errorum in mixta Mathesi, p. 20.) Here, h=a.tan.0; ... Ah = α. Δθ. a.A0. sec.20 = h.0 d tan. 0 = 2h.40 consequently, if Ae be given, Ah will be least when sin. 20 is the greatest, that is, when 0 = 45°, and consequently, when a=h. Hence, if h represent the height of a tower, and Ae be the error of observation, (see p. 197, 1. 22,) it will be most advantageous to observe the angular height of the tower at a distance about equal to its height *. EXAMPLE 2. In a right-angled spherical triangle, where C is the right angle, and A is invariable, it is required to find the corresponding variations of the hypothenuse c and the side b. By Naper's first Rule, p. 146, making the complement of A the middle part, and the radius equal 1, Let now c be invariable, and let it be required to find the ratio between the variations of the sides a, and b. Make the complement of a the middle part, then, by Naper's second Rule, p. 146, * Commodissum erit ad eam distantiam (AC) observationem instituere ut angulus (ACB) sit graduum 45 quamproxime.' Cotes's Est. Errorum, p. 20. |