the next division beyond 40 on the right hand, 40 degrees 20 minutes is the angle. If with the second division beyond 40, then 40 degrees 40 minutes is the angle, and so in every other instance.

The second case is, when the index line does not coincide with any division on the limb. We are, in this instance, to look for a division on the nonius that shall stand directly opposite to one on the limb, and that division gives us the odd minutes, to be added to those pointed out by the index division: thus, suppose the index division does not coincide with 40 degrees, but that the next division to it is the first coincident division, then is the required angle 40 degrces 1 minute. If it had been the second division, the angle would have been 40 degrees 2 minutes, and so on to 20 minutes, when the index division coincides with the first 20 minutes from 40 degrees. Again, let us suppose the index division to stand between 30 degrees and 30 degrees 20 minutes, and that the 16th division on the nonius coincides exactly with a division on the limb, then the angle is 30 degrees 16 minutes. Further, let the index division stand between 35 degrees 20 minutes and 35 degrees 40 minutes, and at the same time the 12th division on the nonius stands directly opposite to a division on the arc, then the angle will be 35 degrees 32 minutes.

A GENERAL RULE FOR KNOWING THE VALUE OF EACH DIVISION, ON ANY NONIUS WHATSOEVER.

1. Find the value of each of the divisions, or subdivisions, of the limb to which the nonius is applied. 2. Divide the quantity of minutes or seconds thus found, by the number of divisions on the nonius, and the quotient will give the value of the nonius division. Thus, suppose each subdivision of the limb be 30 minutes, and that the nonius has 15 divisions, then gives 2 minutes for the value of the nonius. If

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 right 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.

Extracts from Dr. Hooke, on the Difficulty, &c. of Drawing Arcs of Great Circles. "This thing, says he, is so difficult, that it is almost impossible, especially where exactness is required, as I was sufficiently satisfied by the difficulties that occurred in striking a part of the arc of a circle of 60 feet for the radius, for the gage of a tool for grinding telescope glasses of that length; whereby it was found, that the beam compasses made with all care and eircumspection imaginable, and used with as great care, would not perform the operation; nor by the way, an angular compass, such as described by Guido Ubaldus, by Clavius, and by Plagrave, &c.

"The Royal Society met; I discoursed of my instrument to draw a great circle, and produced an instrument I had provided for that purpose; and therewith, by the direction of a wire about 100 feet long, I shewed how to draw a circle of that radius, which gave great satisfaction, &c. Again, at the last meeting I endeavoured to explain the difficulties there are in making considerable discoveries either in nature or art; and yet, when they are discovered, they often seem so obvious and plain, that it seems more difficult to give a reason why they were not sooner discovered, than how they came to be detected now; how easy it was, we now think, to find out a method of printing letters, and yet, except what may have happened in China, there is no specimen or history of any thing of that kind done in this part of the world. How obvious was the vibration of pendulous bodies? and yet, we do not find that it was made use of to divide the spaces of time, till Galileo discovered its isochronous motion, and thought of that proper motion for it, &c. And though it may be difficult enough to find a way before it be shewn, every one will be ready enough to say when done, that it is easy to do, and was obvious to be thought of and invented."

To illustrate this, the Doctor produced an instrument somewhat similar to that described, fig. 5, plate

11, as appears from the journal of the Royal Society, where it is said, that Dr. Hooke produced an instrument capable of describing very large circles, by the help of two rolling circles, or truckles in the two ends of a rule, made so as to be turned in their sockets to any assigned angle. In another place he had extended his views relative to this instrument, that he had contrived it to draw the arc of a circle to a centre though at a considerable distance, where the centre cannot be approached, as from the top of a pole set up in the midst of a wood, or from the spindle of a vane at the top of a tower, or from a point on the other side of a river; in all which cases the centre cannot be conveniently approached, otherwise than by the sight. This he performed by two telescopes, so placed at the truckles, as thereby to see through both of them the given centre, and by thus directing them to the centre, to set the truckles to their true inclination, so as to describe by their motion, any part of such a circle as shall be desired.

METHODS OF 'DESCRIBING ARCS OF CIRCLES OF LARGE MAGNITUDE. BY J. PRIESTLEY, ESQ. OF BRADFORD, YORKSHIRE.

In the projection of the sphere, perspective and architecture, as well as in many other branches of practical mathematics, it is often required to draw arcs of circles, whose radii are too great to admit the use of common, or even beam compasses; and to draw lines tending to a given point, whose situation is too distant to be brought upon the plan. The following essay is intended to furnish some methods, and describe a few instruments that may assist the artist in the performance of both these problems.

K