I will suppose that every one, who attempts to divide a large astronomical instrument, will have it engraved first. Dividing is a most delicate operation, and every coarser one should precede it. Besides, its being numbered is particularly useful to distinguish one dot from another : thus, in the two annexed tables of errors, (see p. 831, 832.) the side columns give significant names to every dot, in terms of its value to the nearest tenth of a degree, and the mistaking of one for another is rendered nearly impossible.
The foregoing examination furnishes materials for the construction of the table of half differences, or apparent errors*. The first line of this table consists of two varie ties ; i. e. the micrometers were at 180° distance for ob taining the numbers which fill the columns of the first and third quadrant ; and at 90° for those of the second and fourth quadrant. The third variety makes one line, and was obtained with a distance of 45° : the fourth consists of two lines, with a distance of 22° 30': the fifth of four lines, with a distance of 11° 15' : the sixth of eight lines, with a distance of 5° 37' 30": the seventh of sixteen lines, with a distance of 2° 48' 45": and the eighth and last variety, being the remainder of the table, consists of thirty-two lines, and was obtained with a distance of 4° 13' 7".5.
The table of apparent errors, or half differences, just explained, furnishes data for computing the table of real errors. The rule is this : let a be the real error of the preceding dot, and b that of the following one, and c the apparent error, taken from the table of half dint ences, of a +6 the dot under investigation ; then is — c = its real 2 error. But, as this simple expression may not be so ge nerally understood by workmen as 1 wish, it may be neces sary to say the same thing less concisely. If the real errors of the preceding and following dots are both +, or both —, take half their sum, and prefix thereto the common sign ; but if one of them is + and the other —, take half their difference, prefixing the sign of the greater quantity : again, if the apparent error of the dot under in vestigation has the same sign of the quantity found above, give to their sum the common sign for the real error ; but if their signs are contrary, give to their difference the sign of the greater for the real error. I add a few examples.
found in like manner, and completes the first line. It is
convenient to put the error of the division 90° of each quadrant at the bottom of each column, although it is the same as the point 0° on the following quadrant. The line 45° is next filled up; the second example shows this ; but there is no occasion to dwell longer upon this explanation ; for every one, who is at all fit for such pursuits, will think what has already been said fully sufficient for his purpose However, I will just mention, that there can be no dan ger, in the formation of this table, of taking from a wrong line the real errors which are to be the criterion for find ing that of the one under trial, because they are in the next line to it, the others, which intervene in the full ta ble, not being yet inserted. The last course of all is, how ever, an exception ; for, as the examining microscopes could not be brought near enough to bisect the angle 2° 48' 45", recourse was had to that quantity and its half; on which account the examination is prosecuted by using errors at two lines distance, as is shown in the two last e xa m plc s.
When the table of real errors is constructed, the other table, although it is of no farther use, should not be thrown away ; for if any material mistake has been committed, it will be discovered as the operation of dividing is carried on, and in this case the table of apparent errors must be had recourse to ; indeed not a figure should be destroyed until the work is done.* Respecting the angular value of the numbers in these tables, it may be worth mentioning that it is not of the least importance, 100 of them being comprised in one revolu tion of the micrometer screw ; and, in the instance before me, 5.6 of them made no more than a second. It is not pretended that one of these parts was seen beyond a doubt, being scarcely 50 of an inch, much less the tenths, as exhibited in the tables ; but as they were visible upon the micrometer heads, it was judged best to take them into the account.
Having now completed the two first sections of my me thod of dividing ; namely, the first, which consists of mak ing 256 small round dots ; and the second, in finding the errors of these dots, and forming them into a table; I come now to the third and last part, which consists in using the erroneous dots in comparison with the tabulated errors, so as ultimately to make from them the true divisions.