At the time alluded to, my brother, in the art of dividing, was justly considered the rival of Ramsdell ; but he was then almost unknown beyond the narrow circle of the ma thematical and optical instrument-makers, for whom lie was chiefly occupied in the division, by hand, of small astro nomical quadrants, and Hadley's sextants of large radius. Notwithstanding my own employment at that time was of a much inferior nature, yet I closely inspected his work, and tried, at leisure hours, on waste materials, to imitate it. With as steady a hand, and as good an eye as young men generally have, 1 was much disappointed at finding, that, after having made two points, neat and small, to my liking, I could not bisect the distance between them, with out enlarging, displacing, or deforming them with the points of the compasses. This circumstance gave me an early dis like to the tools then in use ; and occasioned me the more uneasiness, as I foresaw, that it was an evil which no prac tice, care, or habit, could entirely cure ;—beam compasses, spring dividers, and a scale of equal parts, in short, appear ed to me little better than so many sources of mischief.
had already acquired a good share of dexterity as a general workman. Of the different branches of our art, that of turning alone seemed to MC to border on perfection. This juvenile conceit, fallacious as I afterwards found it, furnished the first train of thoughts which led to the me thod about to be described ; for it occurred to me, that if I could, by any means, apply the principle of turning to the art of dividing instruments, the tools liable to objection might be dispensed with. The means of doing this were first suggested by seeing the action of the perambulator, or measuring wheel ; the surface of the Earth presenting itself as the edge of the instrument to be divided, and the wheel of the perambulator as a narrow roller acting on that edge ; and hence at ose an idea, that some easy contrivance might be devised, for marking off the revolutions and parts of the roller upon the instrument. Since the year above-mentioned, several persons have proposed to me, as new, diving by the roller, and I have been told, that it also occurred long ago to Hook, Sisson, and others ; but, as Ilatton on watch-making says, " I do not consider the man an inventor, who merely thinks of a thing. To be an in ventor, in my opinion, he must act successfully upon the thought, so as to make it useful." I had no occasion, how ever, to have made an apology for acting upon a thought, which, unknown to me, had been previously conceived by others; for it will be seen in the sequel, how little the roll er has to do in the result, and with what extreme caution it is found necessary to employ it.
When a roller is properly proportioned to the radius of the circle to be divided, and, with its edge, made a small matter conical, t--) that one side may be too great, and the other side too little, it may be adjusted so exactly, that it may be carried several times around the circle, without the error of a single second; and it acts with so much steadiness, that it may not unaptly be considered as a wheel and pinion of indefinitely high numbers. Yet, such is the imperfection of the edges of the circle and roller, that, when worked with the greatest care, the intermediate parts, on a radius of two feet, will sometimes he unequal to the value of half a minute or more. After having found the terminating point of a quadrant or circle so permanent, although I was not prepared to expect pet feet equality throughout, yet I was much mortified to find the errors so great, at least ten times as much as I expected ; which fact indicated, beyond a doubt, that if the roller is to be trusted at all, it must only be trusted through a very short arc. Ilad there been any thing slippery in the action,
which would have been indicated by measuring- the same part, at different times, differently, there would have been an end of it at once; but this not being the case in any sensible degree, the roller becomes a useful auxiliary to fill up short intervals, the limits of which have been cor rected by more certain means.• Bird, who enjoyed the undisputed reputation of being the most accurate divider of the age in which he lived, was the first who contrived the means how to render the usual divisions of the quadrant bisectional ; which proper ty, except his being unusually careful in avoiding the effects of unequal expansion from change of temperature, chiefly distinguished his method from others who divided by hand. This desirable object he accomplished by the use which he made of a finely divided scale of equal parts. The thing aimed at was, to obtain a point upon the arc at the highest bisectional number of divisions from 0, which in his eight feet quadrants was 1024= 85° 20'. The extent of the beam compasses, with which he traced the arc upon the limb of the instrument to be divided, being set off upon that arc, gave the points V' and 6u° ; which being bisected, gave 30° more to complete the total arc. A second order of bisections gave points at 15° distance from each other : but that which denoted 75° was most useful. Now, from the known length of the radius, as measured upon the scale, the length of the chord of 10° 20' was computed, taken off from the scale, and protracted from 75° forwards ; and the chord of 4° 4(1, being ascertained in the same man ner, was set off from 90° backwards, meeting the chord of VP 20' in the continually bisectional arc of 20'. This point being found, the work was carried on by bisections, and the chords, as they became small enough, were set off beyond this point, to supply the remainder of the quad rantal arc. My brother, whom I mentioned before, from mere want of a scale of equal parts upon which he could rely, contrived the means of dividing bisectionally without one. His method I will briefly state as follows, in the manner in which it would apply to dividing a mural qua drant. The arcs of 60° and 30° give the total arc as be fore ; and let the last arc of 30° be bisected, also the last arc of 15°, and again the last arc of 7° SO': the two marks next 90° will now be 82° 30' and 86° 15', consequently the point sought lies between them. Bisections will serve us no longer ; but if we divide this space equally into three parts, the most forward of the two intermediate marks will give us and if we divide the portion of the arc be tween this mark and t5' also into three, the most back ward of the two marks will denote 85° 25'. Lastly, if we divide any one of these last spaces into five, and set off one of these fifth parts backwards from 85° 25', we shall have the desired point at 1024 divisions upon the arc from 0°. All the rest of the divisions which have been made in this operation, which I have called marks, because they should be made as Lint as possible, must be erased ; for my brother would not suffer a mark to remain upon the arc, to interfere with his future bisections.