Captain Kater would first divide the circle into five equal parts, then trisect each of these, and then would per form another trisection. This is not the course that an artist would take: his instruments arc to be read off at opposite divisions, and he would not toil through the most difficult part of the work without a trial. De Chaulnes, in the first instance, bisected the circle, and whatever course he afterwards followed, every division would have its opposite; but Captain Kater's scheme admits of no op posites until his arcs of 8° are bisected. The Duke, in every subdivision, left waxed pieces at every step of an in terval, which he could examine again and again, before he cut the divisions. But Captain Kater, by using but two adjustable pieces, deprives his method of that advantage ; for, in order to obtain a proper opening between the mi croscopes, the two pieces are set alternately in advance of each other, and a blank space is left behind.
In all operations of this kind, there is a certain space that one may either see or not see, and which Mr Lax, in his circle of one foot radius, which is the same that Cap tain Kater proposes to divide, states to be three quarters of a second. Now if, at the time that the opening of 72° for quinquisecting the circle was adjusted, a succession of five + errors to that amount were committed, the circle would appear to be equally quinquisectcd, by an opening of ".75 — its proper measure. And with this error of opening in the similar operation for cutting the divisions, let it be supposed, that at each step of shifting the adjus table pieces, a— error to the same amount were commit ted, then it is evident, that the division 288° would err by a quantity 8 x " 75= 6". This is Professor Lax's way of reckoning the amount of error, which is indeed too severe ; but, in the case which we have exemplified, there is no thing improbable in the supposition, that the error might amount to more than half that quantity, .end yet neither the apparatus nor artist be at all to blame. Captain Kilter ob serves, that errors, if any, would be seen: True ; but di•, visions have been cut, and what is to be clone with them ? They are of course to be rubbed out, and a trial again made, in hopes of better luck.
Besides this, the method before us is still farther very objectionable. A division, when cut, whatever its error may be, communicates that et ror to others made front it, through every course of subdivision; and thus the sins of the fathers are visited upon the children down to the seventh generation.
There is another thing that is not adverted to by Captain Kater: no division can be viewed by the microscopes, to any useful degree of accuracy, until the bur is polished off; and the frequent recourse that must be m -de to this operation, which could not be done without carrying the circle round, to get it out of the way of the apparatus, would prove a source, not indeed of inaccuracy, hut of very much embarrassment and delay. It may be observed,
that the method of coaxing is fully as much liable to this objection, and that Troughton's is the only one that is per fectly free from it.
Captain Kater's apparatus is the most complex, and, if made, would be the most expensive, that has been either used or proposed for graduating; and we should remark, that the method would turn out to be extremely tedious. The Captain expresses himself upon both these matters as being of quite a different opinion ; we are not surprised at this, considering, as we do, how much faster the nimble ideas of a speculator get on, than the fumbling fingers of an artist.
The gentlemen graduators have one and all rejected the maxim of Graham : they know that it is not geometrically true, and to what extent it is practically so are perhaps in capable of judging. The bisection of an arc, as Graham meant it to be understood, consisted in setting off the chord of half that arc from left to right, and then from right to left. It was a condition, that the points of the beam com pass should be so sharp, short, and strong, that they would not be deflected towards the arc struck in the first position, when that in the second was made, even were the distance between the points somewhat erroneous ; for, in this case, the line of division would only be intersected by a broader stroke, which. would still occupy the mean distance. Gra ham would have said, that this kind of bisection continual ly halved the error; while that of the gentlemen, which he would have denominated stepping with the chord of half the arc, was continually extending it. As division by three, five, &c. cannot be performed without stepping, Graham pronounced them to be practically impossible.
It is our opinion, that the art of graduation is now in a state not to be improved by any one who is not a practi tioner. Those who excel in the highest branch, are men capable of thinking for themselves, and of course, for im provement, will naturally look inwards. On the other hand, the effusions of gentleman can do no harm, for their schemes will ever be inconsistent with practice. Should they, which is possible, suggest an useful hint to the prac tical man, it ought to be accepted with gratitude; and, at any rate, their well-meant endeavours should be kindly re ceived. With this view before us, we think, that when the real graduators of instruments shall be incorporated into a society, Professor Lax should be chosen an efficient member ; that Captain Kater ought to be admitted an ho norary one; and that we ourselves, after all this, may look up to the latter distinction.