FIG. 259. means of this projection you can determine immediately what rumb you are to sail upon. For let FIG. 259. represent a partial map of this construction; AB representing 4° of longitude, and AC perpendicular to it, 4° of latitude; the former degrees are equal, and the latter are increasing (1227), and are to be laid down by a Table of meridional parts (1228). Suppose a ship wants to go from a in longitude 7° and latitude 31°, to b in longitude 10° and latitude 33°; and it is required to find the rumb it must sail upon. Join ab and that is the rumb (1229). Now to determine what rumb this is, there is always in these maps, one or more points from which are drawn thirty-two straight lines representing the thirty-two points of the compass, and you may easily discover to which of these lines, or nearest to which, ab is parallel, and thus you get the point of the compass you are to sail upon. For this purpose, a parallel ruler may be very useful, laying one edge to coincide with ab, and bringing the other edge over the point from which the lines of the compass are drawn. ON THE USE OF INTERPOLATIONS IN ASTRONOMY, Art. 1241. IF any quantity vary, and its value at certain intervals of time be known, it is the business of interpolation to find its value, accurately, or nearly so, at any other point of time. In Astronomy, the quantities between which we want to interpolate are of such a nature, that if they be taken at equal intervals of time, and you take their differences, and the differences of those differences, and so on, the last differences will become accurately, or nearly equal to nothing. Hence, if a represent the interval from the first time at which the value of the quantity was taken, and y be the value of the quantity corresponding, then may y be represented by A + B × x + С × x × x − 1 + D × × × x − 1 × x −2+ &c. for if we take r terms, and for a write any equidistant set of numbers, as 0, 1, 2, 3, 4, &c. the 7th differences of the results, or of the values of y, will become equal to nothing. For take two terms, A + Bæ; and for a write 0, 1, 2, 3, 4, &c. and we have these results, A A+ 2B The first differences of which are B, and the second differences=0. Take three terms, A+ Bx+C × xxx-1; and for a write 0, 1, 2, 3, 4, &c. and we have these results, A A+2B+2C A+ 3B+ 6C A+4B +12C The first differences of which are, B B+2C B+4C B+6C The second differences are 2C, and the third differences = 0. Thus it appears, that the last differences will always become=0. 1242. In general therefore, let the successive values of y be a, b, c, d, &c. and let it be required to find the coefficients A, B, C, &c. First, take the successive differences thus, d−3c+3b-a, &c. &c. &c. nd put Pba, Q = c — 2b + a, R= d -3c+3b-a, &c. Now for a write 0, 1, 2, 3, 4, &c. and the successive corresponding values of y being a, b, c, d, &c. we have the following equations, a A, b= A + B, C = A + 2B + 2C, d=A+3B+6C + 6D, &c. Hence, Aza; B=b-a=P; C—C—A−2B = 2 = ; &c. therefore y 2.3 + &c. where the law of con tinuation is manifest. Hence, if a, b, c, d, &c. be the values of a variable quantity taken at any successive equal intervals of time, beginning at any instant, and if such be their law that their last differences always become =0, we shall get at any intermediate time the accurate value of that quantity, because then all its intermediate values follow the same law as the values of y from the equation; but if the differences do not at last become accurately =0, we shall then get only an approximate value, because then the intermediate values do not follow accurately the same law, whereas the values of y found from our equation must always follow the same law, and therefore the value of y will be only an approximation to the value of the quantity at any intermediate time between those at which y was assumed accurately equal to it; the approximation however will be sufficiently accurate for all practical purposes, provided the differences become at last very small, which is the case in the application of this rule to interpolations in Astronomy. The use of interpolations is therefore to determine the place of a body, or the value of a quantity at any time, from knowing the place or value at three or four times near to the given time., 1243. But besides the use of the above equation to find the value of any term of a series from its position being given, the converse is often required, that is, having given any term to find its position or distance from the first term. In this case, we have the value of y given to find x, which will be determined from the solution of the equation, which will rise in its dimensions as it may be necessary to increase the number of terms, and this depends upon how many orders of differences you must take before they become equal to nothing. EXAMPLE I. On March, 1783, the sun's declination at noon at Greenwich by the Nautical Ephemeris, was as follows: on the 19th N. 28′. 41′′=1721′′=a; on the 20th N. 5′ = 300′′=b; on the 21st S.-18'. 41′′=—1121′′=c; to find the time of the equinox. The value of c is here written negative, because the declination has passed through 0. Hence, we proceed thus, Here, a=1721, P=-1421; hence, y=1721-1421 x ; now when the sun comes to the equator, y, the declination, becomes = 0, therefore 1721-1421 x x=0, and x = Hit=1d. 5h. 3′. 53′′ the time from the 19th; hence, 20d. 5h. 3'. 53" is the time required. If at any place we observe the sun's declination for three or four days at the equinox by the astronomical quadrant, we may thus determine the time at that place when the sun comes to the equator, without the Ephemeris. EXAMPLE II. To find the time, from the Nautical Almanac, when the sun entered the tropic in June, 1783. The sun enters the tropic when its longitude is three signs. Here y represents the longitude; let us therefore take three longitudes the nearest to the time, which in this case will be a sufficient number. Now on the 20th day the longitude is 2o. 28°. 55'. 7′′, on the 21st it is 2, 29°. 52′. 21" and on the 22d it is 3o. 0°. 49′. 34"; but it will render the operation a little more simple, that is, the numbers will be smaller, if we take from each two signs, in which case it is manifest that y begins at the beginning of the third sign, and consequently when y becomes equal to one sign the sun enters the tropic. Therefore 28°. 55. 7" 104107", 29°. 52. 21" 107541"b, 1. 0°. 49. 34"-110974" c; hence, 104107, 107541, 110974, 3434, 3433, 1. Here P-3434, and Q=1, which being so very small compared with P, we may omit it; consequently y=104107 +3434r; but at the tropic, y = 1 sign 108000"; hence, 108000=104107 +3434r, therefore x-3=1d. 3h. 12. 28" the time from the 20th day, and therefore the sun enters the tropic the 21d. 3h. 12. 28". EXAMPLE III. Given five places of a Comet as follows; on November 5, at 9h. 17' in Cancer 2°. 30′ 150a; on the 6th at 8h. 17′ in 4°. 7′ = 247′ = b; on the 7th at 8h. 17′ in 6°. 20' 380′-c; on the 8th at 8h. 17′ in 9o. 10′=550′= d; on the 9th at 8h. 17' in 12°. 40′=760'=e; to find its place on the 7th at 14h. 17'. First, subtract 5d. 8h. 17' from 7d. 14h. 17' and there remains 2d. 6h.=2,25 for the interval of time between the first observation and the given time at which the place is required; this therefore is the value of x to which we want to find the corresponding value of y; hence, Here P=97, Q=36, R=1, S=2; hence, y=150+97 × 2,25 + 36 × 2,25 × 1,25 + 1 × 2,25 × 1,25 × ,25 +3.4 × 2,25 × 1,25 × ,25 × -,75418′,96 = 6o, 58'. 57" the place required. 2.3 EXAMPLE IV. In October, 1788, the Moon's declination at noon at Greenwich appears, from the Nautical Almanac, to have been as follows; on the 9th S. 12°. 42'=762'-a; on the 10th S. 8°. 44-524b; on the 11th S. 4°. 24' 264' c; on the 12th N. -0°. 10'-d; on the 13th N.-4°. 49'-289' = e; to find the declination on the 10th at 6h. 80'. |