sive some error might arise from quinquisection, and trisection, in order to examine the accuracy of the divisions, he described another circle inch within the former, by continual bisections, but found no sensible difference between the two sets of divisions. It appears also, that Mr. Bird, notwithstanding all his objections to, and declamations against the prac tice of stepping, sometimes used it himself.*

* The late skilful divider, Mr. John Troughton, from the difficulty of constructing a sufficiently accurate scale of equal parts upon which he could rely, contrived the means of dividing bisectionally without one, which differs from Bird's method, only in the means he adopted to reach the point which terminates the great bisectional arc, and is thus described, as a preferable method for a mural quadrant, by his ingenious and more skilful brother, Mr. Edward Troughton. "The arcs of 60° and 80° give the total arc, as before stated, and let the last arc of 80° be bisected, also the last arc of 15°, and again the last arc of 7° 30′. 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 85°; and if we divide the portion of the arc between this mark and 86° 15′ also into three, the most backward of the two marks will denote 85 30'; lastly, if we divide any one of these last spaces into five, and set off one of these fifth parts backwards from 85° 30′, we shall have the desired point at 1024 divisions upon the arc from 0o. All the rest of the divisions which have been made in this operation, which I have called marks, 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."

The use of mural quadrants in our observatories is now superseded by the more complete, more accurate and manageable instrument called the Astronomical Circle. To the fertile inventive genius and transcendent mechanical skill of the late Mr. Jesse Ramsden is the science of astronomy indebted for the various constructions of this circle, with its microscopes and other appendages for reading off their divisions with extraordinary accuracy, and also for improved constructions of the large Equatorial, &c. The following is one of his known methods of dividing large instruments. He divided in the usual way by beam compasses and spring dividers, accompanied with microscopes for minute examination, correcting in the progress by pressing backwards or forwards, by hand with a fine conical point, those dots apparently erroneous, and so adjusting them to their true positions. Mr. Troughton al

OF THE NONIUS DIVISIONS.

It will be necessary to give the young practitioner some account of the nature and use of that admirable contrivance commonly called a nonius, by which the divisions on the limbs of instruments are subdivided.

The nonius depends on this simple circumstance,

lows this method to admit of considerable accuracy under a steady hand and good eye, but judges the divider's work will ever be irregular and inelegant. He must have a circular line passing through the middle of his dots, to preserve them at an equal distance from the centre: the bisectional arcs also which cut them across deform them; and, what is worse, the dots requiring correction will become larger than the rest, and unequally so in proportion to their requisite adjustment. To reduce them to an equality with neighbouring ones, a burnisher is sometimes used, which may cause hollows in the surface, and dots so burnished up are generally of a bad figure or ill-defined. Mr. Troughton thinks it would be an improvement to divide the whole by hand at once, and afterwards to revise it, which will prevent the corrected dots, as in the above method, from being injured or moved by the subsequent application of the compasses.

Mr. Ramsden had other methods, besides the above, to divide his large Circles; and I am told by a workman from his workshop, now in our employ, that he had a contrivance by fixed double microscopes, with micrometer wires, for dividing the circle in a less tedious and inaccurate manner, by successively bringing the surface of the circle under the wires, punctuating the dots, and subsequently examining and correcting them. Before this time, Mr. Troughton states, that he "used a frame which carried a single wire very near the surface to be divided; this wire was moveable by a fine micrometer screw, and was viewed by a single lens inserted in the lower end of a tube, which, for the purpose of taking off the parallax, was four inches long. The greatest objec tion to this mode of constructing the apparatus is, that the wire being necessarily exposed, is apt to gather up the dust; yet it is preferable to the one now in use, in cases where any doubt is entertained of the accuracy of the plane which is to receive tl › divisions."

Various other methods by ingenious artists have been suggested for more expeditiously dividing instruments. Mr. Troughton candidly acknowledges that several persons proposed to him a new method of dividing by a roller, perambulating the edge of the circle to be divided, and that it also was considered as feasible a long time back by Hooke, Sisson, and others. Mr. Troughton,-as

that if any line be divided into equal parts, the length of each part will be greater, the fewer divisions there are in the original; on the contrary, the length of each division will be less in proportion, as the divi

sions are more numerous.

Thus, let us suppose the limb of Hadley's quadrant divided to every 20 minutes, which are the smallest divisions on the quadrant; the two extreme strokes on the nonius contain seven degrees, or 21

anable and very experienced artist, has availed himself of this principle, and has discovered that when "a roller is properly proportioned to the radius of the circle to be divided, and with its edge made a small matter conical, so that one side may be too great and the other too little, it may be adjusted so exactly that it may be carried several times round 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." With a framed apparatus and microscopic micrometer, constructed with this roller, he has invented a machine that very expeditiously punctuates dots upon the circle to be divided, and from tables of apparent errors of these dots first made, and another table of real errors subsequently calculated and corrected by a small sectoral instrument, he has disclosed his method of dividing circular instruments, which he states will require but about a fourth part of the time of that employed after the method of Mr. Bird, though this saving of time perhaps cannot be absolutely appreciated by future artists, unless much experience and dexterity, in an equal degree, in both methods have been used.

For further particulars I refer the reader to Mr. Troughton's "Account of a Method of Dividing Astronomical and other Instruments, by occular Inspection; in which the usual Tools for graduating are not employed, the whole Operation being so contrived that no Error can occur, but what is chargeable to Vision, when assisted by the best optical means of viewing and measuring inute Quantities. Published in the Philosophical Transactions, for 1809, and in the 34th Volume of the Philosophical Magazine. In the same volume of the Transactions, is printed an account of "An Improvement in the Manner of dividing Astronomical Instruments" by H. Cavendish, Esq. whose method consists in using a beam compass with only one point, and a microscope instead of the other, thus avoiding the setting of a point into a division. This may prove useful to a young artist.

This paper is succeeded by another "On a lethod of examining the Divisions of Astronomical Instruments, by Professor Lax, of Cambridge University. EDIT.

the nonius has 10 divisions, it would give three minutes; if the limb be divided to every 12 minutes, and the nonius to 24 parts, then 12 minutes, or 720 seconds divided by 24, gives 30 seconds for the required value.

OF INSTRUMENTS FOR DESCRIBING CIRCLES OF

EVERY POSSIBLE MAGNITUDE.

As there are many cases where arcs are required to be drawn of a radius too large for any ordinary compasses, Mr. Heywood and myself contrived several instruments for this purpose; the most perfect of these is delineated at fig. 5, plate 11. It is an instrument that must give great satisfaction to every one who uses it, as it is so extensive in its nature, being capable of describing arcs from an infinite radius, or a straight line, to those of two or three inches diameter. When it was first contrived, both Mr. Heywood and myself were ignorant of what had been done by that ever to be celebrated mechanician, Dr. Hooke.

Since the invention thereof, I have received some very valuable communications from different gentlemen, who saw and admired the simplicity of its construction; among others, from Mr. Nicholson, author of several very valuable works; Dr. Rotheram, Earl Stanhope, and J. Priestley, Esq. of Bradford, Yorkshire; the last gentleman has favoured me with so complete an investigation of the subject, and a description of so many admirable contrivances to answer the purpose of the artist, that any thing I could say would be altogether superfluous; I shall, therefore, be very brief in my description of the instrument, represented fig. 5, plate 11, that I may not keep the reader from Mr. Priestley's valuable essay, subjoining Dr. Hooke's account of his own contrivance to that of ours. Much is always to be gained from an attention to this great man; and I

am sure my reader will think his time well employed in perusing the short extract I shall here insert.

The branches A and B, fig. 5, plate 11, carry two independent equal wheels C, D. The pencil, or point E, is in a line drawn between the centre of the axis of the branches, and equidistant from each; a weight is to be placed over the pencil when in use. When all the wheels have their axes in one line, and the instrument is moved in rotation, it will describe an infinitely small circle; in this case the instrument will overset. When the two wheels C, D, · have their horizontal axes parallel to each other, a night line or infinitely large circle will be described; when these axes are inclined to each other, a circle. of infinite magnitude will be described.

The distance between one axis and the centre, (or pencil,) being taken as unity, or the common radius, the numbers 1, 2, 3, 4, &c. being sought for in the natural tangents, will give arcs of inclination for setting the nonii, and at which circles of the radii of the said numbers, multiplied into the common radius, will be described.