assisted by the best optical Means of viewing and measuring minute Quantities. By Edward Troughton.-By speculative men, this paper will be regarded as very curious and interesting, and to the artist it is highly important. We have read it with great interest; and we regret that the want of plates prevents us from giving so full an account of it as it deserves : but still we hope that we shall be able to render intelligible to our readers the nature of its most important contents. The merit of Mr. Troughton's method will be better appreciated, if we previously attend to the former modes of dividing large astronomical instruments. Bird, who was esteemed the most accurate di. vider of his time, invented a method of dividing a quadrant by continually biseeting a part of it. A quadrant of 900 contains 90x60 minutes, or 90 X12, or 1080 portions of 5 minutes each : now 240 x 5' = 1024.X 5'; therefore an arc equal to 1024X5 (85° 20') is the are nearest to 90°, which can be repeatedly bisected, and of which the arc determined by the tenth bisection is 5: The means of obtaining this point, corresponding to 85° 20', were thus effected: • The extent of the beam-compasses, with which Bird traced the arc upon the limb of the instrument to be divided, being set off upon that arc, gave the points go and 69°; which, being bisected, gave 30° more to complete the total arc. A second order of bisections gave points at 150 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 759 for. wards ; and the chord of 4° 40', being ascertained in the same manner, was set off from gou backwards, meeting the chord of 10° 20' in the continually bisectional arc of 87° 23This 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 re. mainder of the quadrantal arc. My brother, whom I mentioned be. fore, from mere wait of a scale of equal parls upon which he could rely, contrived the means of dividing bisectionally without one. method I will briefly state as follows, in the manner which it would apply to dividing a mural quadrant. The arcs of 60° and 3co give the total arc as before ; and let the last arc of 30° be bisected, also the last arc of 15°, and again the last arc of 70 30': the two marks oest 90u will now be 87° 30 and 86° 15, conecquently 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 85", and if we divide the portion of the arc between this mark and 86°15' also into three, the most back: ward of the two marks will denote 8530'. Lasily, if we divide any one of these last spaces into five, and set off one of these fifth parts backwards from 850 30', we shall have the desired point ac 1024 divisions upon the arc from on. All the rest of the divisions which bave been made in this operation, which I have called marks because because they should be made as faint as possible, must be erased; for my brother would not suffer a mark to remain upon the arc to interfere with his future bisections.' Ramsden's method is applicable only to small instruments; for which, however, Mr. Troughton thinks, it is admirably calculated. • The only method (says Mr.T.) of dividing large instruments now practised in London, that I know of, besides my own, has not yet, I believe, been made public. It consists in dividing by hand with beam compasses and spring dividers, in the usual way; with the addition of examining the work by microscopes, and correcting it, as it proceeds, by pressing forwards or backwards by hand, with a fine conical point, those dots which appear erroneous; and thus adjusting them to their proper places. The method admits of consi. derable accuracy, provided the operator has a steady hand and good cye; but his work will ever be irregular and inelegant. He must have a circular line passing through the middle of his dots, to enable him to make and keep them at an equal distance from the centre. The bisectional arcs, also, which cut them across, deform them much; and, what is worse, the dots which require correction (about two thirds perhaps of the whole) will become larger than the rest, and unequally so in proportion to the number of attempts which have been found necessary to adjust them. In the course of which operation, some of them grow insulferably too large, and it becomes necessary to reduce them to an equality with their neighbours. This is done with the burnisher, and causes a hollow in the surface, which has a very disagreeable appearance.' Mr. T. subsequently adds; I will dismiss this method of dividing, with observing that it is tedious in the extreme; and did I not know the contrary beyond a doubt, I should have supposed it to have surpassed the utmost limit of human patience.' We now proceed to describe. Mr. Troughton's method. Suppose the radius of a large circle to be 40 inches, and that of a small circle or cylinder to be 4 inches; then, if this latter rolled equally along the circumference of the former, it would perform the circuit after having made ten rotations. Suppose next the circumference of the cylinder or roller to be divided into ten equal parts: then such divisions might be transferred to the circumference of the greater circle, which would thus be divided into 100 parts. If the roller in ten rotations exactly made the circuit, if its divisions were precisely equal, and if it revolved along the circumference in every part alike, then would the circumference of the circle be thus exactly divided into a hundred equal parts. This, however, is not in reality the author's method of equally dividing the circle. He finds the means, indeed, of obtaining the first condition : the second, if acquired, would be rendered useless by the impracticability of attaining attaining the third ; and the ultimate equal division of the circle is, in fact, entirely independent of equality in the primary division. If the roller were a perfect cylinder, it would be practically extremely difficult to turn it with such nicety that it should exactly measure the circle : but this difficulty is dexterously obviated by forming the roller slightly conical : so that an easy adjustment makes a section of it (that which is to revolve on the convex border of the circle) exactly measure the circle. A circumstance relating to the circuit of the roller is particularly worthy of notice. In the different parts of its journey round' the circle, the roller measures the latter very differently; that is, a division of it transferred to the circle in one point would be unequal to the same division transferred at any other point, 60o distant, for instance; notwithstanding which, the roller, having reached the point on the circle whence it set out, performs a second, third, &c. course of revolutions, without any sensible deviations from its former track. This is, however, satisfactorily explained by Mr. T. The brass in one part may be more porous than in another, and, with regard to the travelling of the roller, may be related as a ploughed field and a gravel walk are. Again, the convex edge of the circle is indented in the first circuit of the roller; and in succeeding circuits it follows its first track. In our illustration, we made the diameters of the circle and the roller as ten to one : but in Mr. Troughton's construction they are as sixteen to one. The roller is divided into sixteen parts; and consequently the circle will be separated by the above mentioned method of transferring the divisions into 10x 16 or 256 parts : but the operation, advanced as far as this division, is incomplete for two reasons: first, the divisions are unequal; and, secondly, if equal, their value would be 1° 24' 22",5 each, whereas it is desirable to obtain divisions of five minutes each. The 256 divisions are not in fact made equal, but their respective inequalities or errors are ascertained and estimated by the following most ingenious process, which is a main part of the author's method : • The apparatus (that by which the 256 dots are made) must now be taken off, and the circle mounted in the same manner, that it will be in the Observatory. The two microscopes, which have divided heads, must also be firmly fixed to the support of the instrument, on opposite sides, and their wires brought to bisect the first dot, and the one which should be 1800 distant. Noir, the microscopes remaining fixed, turn the circle half round, or until the first microscope coincides with the opposite dot; and, if the other microscope be exactly at the other dot, it is obvious that these dots are 18. apart, or in the true diameter of the circle; and and if they disagree, it is obvious that half the quantity by which they disagree, as measured by the divisions of the micrometer head, is the error of the opposite division ; for the quantity measured is that by which the greater portion of the circle exceeds the less. It is convenient to note these errors + or-, as the dots are found too forward or too backward, according to the numbering of the degrees ; and for the purpose of distinguishing the + and - errors, the heads, as mentioned before, are numbered backwards and for. wards to fifty. One of the microscopes remaining as before, remove the other to a position at right angles ; and, considering for the present both the former dots to be true, examine the others by them; i e. as before, try by the micrometer how many divisions of the head the greater half of the semi-circle exceeds the less, and note half the quantity + or -, as before, and do the same for the other semicircle. One of the micrometers must now be set at an angle of 45° with the other, and the half differences of the two parts of each of the four quadrants registered with their respective signs. When the circle is a vertical one, as in the present instance, it is much the best to proceed so far in the examination with it in that position, for fear of any general bending or spring of the figure ; but, for the examipation of smaller arcs than 45°, it will be perfectly safe, and more convenient, to have it horizontal ; because the dividing apparatus will then carry the micrometers, several perforations being made in the plate B for the limb to be seen through at proper intervals. The. micrometers must now be placed at a distance of 22° 30', and the half differences of the parts of all the arcs of 459 measured and noted as before ; thus descending by bisections to 110 15',5° 37' 30", and 2° 48' 45". Half this last quantity is too small io allow the micrometers to be brought near enough ; but it will have the desired ef. fect, if they are placed at that quantity and its half, i.e. 4° 13' 7"',5; in which case the examination, instead of being made at the next, will take place at the next division but one, to that which is the subject of trial. During the whole of the time that the examination is made, all the dots, except the one under examination, are for the present supposed to be in their true places ; and the only thing in this most important part of the business, from first to last, is to ascertain with the utmost care, in divisions of the micrometer head, liow much one of the parts of the interval under examination exceeds the other, and carefully to tabulate the half of their difference.' Having, by this process of examination, ascertained the errors of the respective dots, the author arranges these errors in a table, and, from what he calls the apparent errors, deduces the real. If A be an arc, and it be required to ascertain the error of the intermediate dot, (which ought, if every thing were exactly performed, to be in the middle,] suppose the lesser А arc to be x, and the greater +x + x, then 2 2 (1-x) - 2 x. Now the micrometer shews this quantity 2 2 This () however is only 2 2 2 2 d d 2 x. Suppose it =d, then x = is the apparent ertor, and the arc is divided by the dot under examination into d d + and an apparent error, since the two dots which are the extremities of the arc A are erroneously placed. Suppose the arc A to be within the two true points, and let the intervals between the true points and the erroneous extremities be respectively a and b; then the true arc is Ata+b; and the true half arc, which determines the real position of the middle point, is b + + -: but the distance of the dot under examination, 2 2 from the true upper extremity of the arc, is and hence the real error 6 + , 4- +a; + --ta + 2 2 + ) 2 2 2 Gr --which is in fact Mr. Troughton's rule, since, according to his directions, a is to be put + a, b—b, and d-d. ( answers to his c.) He observes; • The rule is this ; let a be the real error of the preceding dot, anul b that of the following one, and c the apparent error, taken from the table of half differences, of the dot under investigation ; then ato is +c= its real error. But, as this simple expression may not be so generally understood by workmen as I wish, it may be necessary 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 investigation has the sanie sign of the quantity found above, give to their sum the common sign, for the real error ; but if their sigus are contrary, give to their difference the sign of the greater for the real error. I add a few examples. Example 1. 0,0 Real error of the first point of the third quadrant - 6,9 Half |