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 circumspection 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 Blagrave, &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 instru ment 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, Esa. 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 ares 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 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 circumspection 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 Blagrave, &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 7 11, as appears from the journal of the Royal Society, where it is said, that Dr. Hooke produced an instru ment 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, Esa. 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 ares 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 OF FINDING POINTS IN, AND DESCRIBING ARCS OF LARGE CIRCLES. The methods and instruments I shall propose for this purpose, will chiefly depend on the following propositions, which I shall premise as principles. Principle 1. The angles in the same segment of a circle, are equal one to another. Let A CD B, fig. 1, plate 10, be the segment of a circle: the angles formed by lines drawn from the extremities A and B, of the base of the segment, to any points C and D of its arc, as the angles ACB, ADB, are equal. This is the 31st proposition of Euclid's third book of the Elements of Geometry. Principle 2. If upon the ends A B, fig. 2, plate 10, of a right line A B as an axis, two circles or rollers CD and E F be firmly fixed, so that the said line shall pass through the centres, and at right angles to the plains of the circles; and the whole be suffered to roll upon a plain without sliding. 1. If the rollers CD and E F be equal in diameter, the lines described upon the plain by their circumferences, will be parallel right lines; and the axis A B, and every line D F, drawn between contemporary points of contact of the rollers and plain, will be parallel among themselves. 2. If the rollers CD and EF be unequal, then lines formed by their circumferences upon the plain will be concentric circles; and the axis AB, and also the lines D F, will, in every situation, tend to the common centre of those circles. Principle 3. If there be two equal circles or rollers A and B, fig. 3, plate 10, each separately fixed to its own axis, moveable on pivots; and these axes placed in a proper frame, so as to be in the same plain, and to maintain the situation given them with respect to each other; and if the apparatus be rolled upon a plain without sliding: |