It must already have been anticipated, that dividing by the eye is equally applicable to straight lines as it is to cir cles. Au apparatus for this purpose should consist of a bar or brass, three quarters or an inch thick, and not less than three inches broad ; six feet may do very well for the length ; it may be laid upon a deal plank strengthened by another plank screwed edge-wise on its lower surface. The bar should be planed on both its edges and on its surface, with the greatest exactness; and it will be better if it has a narrow slip of silver, inlaid through its whole length, for receiving the dots. An apparatus nearly similar to the other should slide along its surface, carrying a roller, the circumference of which is 12.8 inches, and turned a little conical for the sake of adjustment. The roller may be di vided into 32 parts, each of which, when transferred to the bar, will give intervals of 0 4 of an inch each : the angle of the subdividing sector should of course be II' 15' and sub divided into four parts, which will divide the inch into tenths : the surface may also receive other lines, with sub divisions suited to the different purposes for which it may be wanted. The revolutions of the roller and its A- parts must be dotted upon the bar ; taking care, by sizing the roller, to come as near the true standard measure as possi ble : when this is done, compare the extent of the greater bisectional number that is contained in the length, i. e. 128 intervals or 51.2 inches, with the standard measure ; noting the difference as indicated by the micrometer heads : the examination and construction of the table of errors may then be conducted just as was done for the circle.
Being now ready for the performance of its work, the scale to be divided must be laid alongside of the bar, and the true divisions must be cut upon it by an appeal, as be fore, to the erroneous dots on the bar, corrected by a cor responding table of errors. The apparatus, remaining en tire in the possession of the workman, with its primitive dots, the table of errors, &c. is ready for dividing another standard, which will be precisely similar to others that have been, or may be divided from it. It may be considered, in deed, as a kind of engine ;• and, as it is not vitiated by the coarse operation of racking with a screw, but performed by only looking at the work, the method will command about three times the accuracy that can be derived from the usual straight-line dividing engine. Should it be asked, if an engine thus appointed would succeed for dividing circles ? I answer, Yes but I would not recommend it ; because, be yond a certain extent of radius, it is not necessary ; for the errors, which would be introduced into the work by the violence of racking a large wheel, are sufficiently reduced by the comparative shortness of the radius of such instru ments as we divide by that method : and, what is still more to the purpose, the dividing engine is lbw. times more ex peditious, and bears rough usage better. I cannot quit the subject of dividing straight lines without observing, that I never had my apparatus complete. The standard which I made for Sir George Shuckburgh Evelyn in 1796, was done by a mere make-shift contrivance, upon the principle of di viding by the eye ; how I succeeded, may be seen in Sir George's papers on Weights and Measures, in the Phil. Trans. For 1798. I made a second, sonic years after, for Professor Pictet of Geneva, which became the subject of comparison with the new measure of France, before the National Institute ; and their report, drawn Up by Mr Pictet, has been ably re-stated and corrected by Dr Young, as published in the Journals of the Royal Institution. I made a third for the magistrates of Aberdeen. I notice the two latter, principally to give myself an opportunity of saying, that•if those three scales were to be compared to gether, notwithstanding they were divided at distant periods of time, and at different seasons of the year, they would he found to agree with each other as nearly as the different pints of the same scale agree.
I hope I may here be allowed to allude to an inadvertency which has been committed in the paper ; and which Sir George intended to have cot rected, had he lived to conclude his useful endeavours to harmonize the discordant weights and measures of this country. The in struments which he has brought into comparison are, his own five feet standard measure and equatorial ; General Roy's forty-two inch scale ; the standard of Mr Aubert ; and that of the Royal Society. The inadvertency is this ; in his equatorial, and the standard of the Royal Society, he has charged- the error of the most erroneous extent, when compared with the mean extent, alike to both divisions ; i. e. he has supposed one of the divisions, which bound the erroneous extent, to be too much to the right, and the other too much to the left. and that by equal quantities. This is certainly a good-natured way of stating the errors of work ; and perhaps not unjustly so, where the worst part has been selected ; but in the other three instances, namely, in Gene ral Roy's, Mr Aubert's, and his own standards, he has charged the whole error of the most erroneous extent to one of the bounding lines.
I was well confirmed in my high opinion of the general accuracy of Bird's dividing, when, last winter,* I measured the chords of many arcs of the Greenwich quadrant. instrument has indeed suffered, both from a change in its figure, and from the wearing of its centre ; but the gradua tion, considering the time when it was done, I found to be very good. Sir George, in his paper upon the equatorial, (Phil. Trans. for 1793,) after some compliments paid to the divider of his instrument, says, " the late Mr John Bird seems to have admitted a probable discrepancy in the divi sions of his eight feet quadrant, amounting to 3" ;" and he refers to Bird on the construction of the Greenwich quad rant. This quantity being three times as great as any er rors that I met with, I was lately induced to inquire how the matter stood. Bird, in the paper referred to, says, " in dividing this instrument, I never met with an equality that exceeded one second. I will suppose, that in the 90 arch this error lay towards the left hand, and in the 96 arch that it lay towards the right, it will cause a difference between the two arches of two seconds ; and, if an o•or of one second be allowed to the observer in reading off his obser vation, the whole amount is no more than three seconds, which is agreeable to what I have heard, &c." Sir George's examination of his own equatorial furnishes me with the means of a direct comparison : in his account of the de clination circle, we find an error +2".35, and another to these add an error of half a second in each, for reading off, which Sir George also admits, we shall then have a discrepancy of 4".85 ; but, as the errors of reading off are not errors of division, let them be discharged from both, and the errors will then stand for the quadrant 2", and for the circle 3".85. As the radius of the former, however, is four times greater than that of the latter, it will appear, by this mode of trial, that the equatorial is rather more than twice as accurately divided as the quadrant. In doing justice to Bird in this instance, I have only done as I would be done by : for, should any future writer set me back a century on the chronological scale of progressive improve ment, I hope some one will be found to restore me to my proper niche. I now subjoin a restatement of the greatest error of each of the instruments that arc brought into com parison by Sir George, after having reduced them all by one rule, viz. allowing each of the two points which bound the most erroneous extent, to divide the apparent error equally between them. They are expressed in parts of an inch, and follow each other in the order of their accuracy.