of 15, gives 15°, which stepped from the points already found, divides the quadrant to every fifth degree. The computed chord of 6° being laid off, divides 30 degrees into five parts; and set off from the other divisions, subdivides the quadrant into single degrees. Thus with five extents of the beam compasses, and none of them less than six degrees, the quadrant is divided into 90 degrees. Fifteen degrees bisected, gives 7° 30', which set off from the other divisions, divides the quadrant into half degrees. The chord of 6° 40′ divides 20° into three parts, and set off from the rest of the divisions, divides the whole instrument to every ten minutes. The chord of 10° 5' divides the degrees into 12 parts, each equal to five minutes of a degree. Method 2. The chords are here supposed to be computed as before, and taken off from the nonius scale. 1. Radius bisected divides the quadrant into three parts, each equal to 30 degrees. 2. The chord of 10° gives nine parts, each equal to 10 degrees. 3. Thirty degrees bisected and set off, gives 18 parts, each equal to five degrees. 4. Thirty degrees into five, by the chord of 6°; then set off as before gives 90 parts, each equal to Įdegree. 5. The chord of 6° 40′ gives 270 parts, each equal to 20 minutes. 6. The chord of 7° 30′ gives 540 parts, each equal to 10 minutes. 7. The chord of 7° 45′ gives 1080 parts, each equal to five minutes. Or Method 3. The computed chords supposed. 1. 60 into gives 2.30 + equal 15 3 parts, equal 30° 6 15° 7. 15, 30 equal 7° 45′ Thus may the practitioner vary his numbers for any division whatsoever, and yet preserve a sufficient extent between the points of his compasses. If the quadrant be divided as above, to every 15 degrees, and then the computed arc of 16 degrees set off, this are may be divided by continual bisection into single degrees. If from the arc of 45°, 2° 20′ be taken, or 1° 10' from 22° 30', we may obtain every fifth minute by continual bisection. If to the arc of 7° 10' be added the arc of 62 minutes, the arc of every single minute may be had by bisection. OF MR. BIRD'S METHOD OF DIVIDING, In 1767 the Commissioners of Longitude proposed an handsome reward to Mr. Bird, on condition, among other things, that he should publish an ac count of his method of dividing astronomical instruments; which was accordingly done: and a tract, describing his method of dividing, was written by him, and published by order of the Commissioners of Longitude in the same year; some defects in this publication were supplied by the Rev. Mr. Ludlam, one of the gentlemen who attended Mr. Bird to be instructed by him in his method of dividing, in consequence of the Board's agreement with him. Mr. Ludlam's tract was published in 1787, in 4to. I shall use my endeavours to render this method still clearer to the practitioner, by combining and arranging the subject of both tracts, MR. BIRD'S METHOD. 1. One of the first requisites is a scale of inches, each inch being subdivided into 10 equal parts. 2. Contiguous to this line of inches, there must be a nonius, in which 10.1 inches is divided into 100 equal parts, thus shewing the 1000ndth part of an inch. By the assistance of a magnifying glass of one inch focus, the 3000ndth part may be estimated. 3. Six beam compasses are necessary, furnished with magnifying glasses of not more than one inch focus. The longest beam is to measure the radius or chord of 60; the second for the chord of 42.40; the third for the chord of 30; the fourth for 10.20; the fifth for 4.40; the sixth for the chord of 15 degrees. 4. Compute the chords by the rules given, and take their computed length from the scale in the different beam compasses. 5. Let these operations be performed in the evening, and let the scale and the different beam compasses be laid upon the instrument to be divided, and remain there till the next morning. the 6. The next morning, before sun-rise, examine compasses by the scale, and rectify them, if they are either lengthened or shortened by any change in the temperature of the air. 7. The quadrant and scale being of the same temperature, describe the faint arc bd, or primitive circle; then with the compasses that are set to the ra-· dius and with a fine prick punch, make a point at a, which is to be the 0 point of the quadrant; see fig. 2, plate 9. 8. With the same beam compasses unaltered lay off from a to e the chord of 60°, making a fine point at e. 9. Bisect the arc a c with the chord of 30°. 10. Then from the point c, with the beam com passes containing 60, mark the point r, which is that of 90 degrees. 11. Next, with the beam compasses containing 15°, bisect the arc e r in n, which gives 75°, 12. Lay off from n towards r the chord of 10° 20', and from r towards n the chord of 4° 40'; these two ought to meet exactly at the point g of 85° 30': 13. Now as in large instruments each degree is generally subdivided into 12 equal parts, of five minutes each, we shall find that 85° 20' contains 10.24 such parts, because 20′ equal 4 of these parts, and 85 X 12 makes 1020; now 1024 is a number divisi ble by continual bisection. The last computed chord was 42° 40', with which ag was bisected into o, and a o, og, were bisected by trials. Though Mr. Bird seems to have used this method himself, still he thinks it more advisable to take the computed chord of 21° 20′, and by it find the point g; then proceed by continual bisections till you have 1024 parts. Thus the arc 85° 20′, by ten bisections, will give us the arcs 42° 40′, 21° 80', 10° 40′, 5° 20′, 2° 40′, 1° 20′, 40′, 20′, 10′, 5′. 14. To fill up the space between g and r, 85° 20′, and 90°, which is 4° 40', or 4 X 12 + 8 equal to 56 divisions; the chord of 64 divisions was laid off from g towards d, and divided like the rest by continual bisections, as was also from a towards b. If the work is well performed, you will again find the points 30, 45, 60, 75, and 90, without any sensible difference. It is evident that these arcs, as well as those of 15o, are multiples of the arc of 5'; for one degree contains 12 arcs of 5' each, of which 15° contains 180; the arc of 30° contains 360; the arc of 60°, 720; that of 75°, 900; and, therefore, 90° contains 1080. Mr. Graham, in 1725 applied to the quadrant divided into 90°, or rather into 1080 parts of five minutes each, another quadrant, which he divided into 96 equal parts, subdividing each of these into 16 equal parts, forming in all 1536. This arc is a se vere check upon the divisions of the other; but Bird says, that if his instructions be strictly followed, the coincidence between them will be surprising, and their difference from the truth exceedingly small. The arc of 96° is to be divided first into three equal parts, in the same manner as the arc of 90°; each third contains 512 divisions, which number is divisible continually by 2, and gives 16 in each 96th part of the whole. The next step is to cut the linear divisions from the points obtained by the foregoing rules. For this purpose a pair of beam compasses is to be used, both of whose points are conical and very sharp. Draw a tangent to the arc bd, suppose at e, it will intersect the arc x y in q, this will be the distance between the points of the beam compasses to cut the divisions nearly at right angles to the arc. The point of the beam compasses next the right hand is to be placed in the point r, the other point to fall freely into the are xy; then pressing gently upon the screw head which fastens the socket, cut the divisions with the point towards the right hand, proceeding thus till you have finished all the divisions of the limb. FOR THE NONIUS. 1. Chuse any part of the arc where there is a coincidence of the 90 and 96 arcs; for example, at e, the point of 60°. Draw the faint arc s t and ik, which may be continued to any length towards A; upon these the nonius divisions are to be divided in points. The original points for the nonius of the 90th arc are to be made upon the arc st; the original points for the nonius of the 96th arc are to be made upon the arc ik. Because 90 is to 96, as 15 to 16, there will be a coincidence at 15° and 16pts, 30° and 32pts, 45o and 48pts; 60° and 64pts, 90° and 96pts. 2. Draw a tangent line to the primitive circle as |