.ifter I had found by experience, that the expansion of the instruments to be divided, occasioned by the increasing heat of the sun, or a contraction by a decrease thereof, was the grand difficulty with which I had to struggle, especially when two or three hours were required to lay off the princi pal points ; I immediately set about contriving how to lay them off in the least time possible, i. e. before any expan sion or contraction could take place ; and as the heat of three or four persons in the room may produce the same ef fect as the sun, I never admit more than one as an assistant. ../l•either must any fire be suffered in the room, till the princi pal points are done.
The above being understood,it was easy to conceive, that, having all the cho?ds brfore-mentioned computed and mea sured, the evening before they were to be laid off, I should be enabled to perform in a few minutes, what by trials would require same hours ; and as too much caution cannot be used, it is proper to lay off the principal points before sun rise, or else choose a cloudy morning.
The method of cutting the divisions as described above, is to prevent any inequality that would arise from the ex pansion of the beam compass by the heat of the hands, espe cially if the beam be of metal. Wooden beams will also alter, probably from a small bending ; but in this method.
if the beam should alter 16 of an inch or more, it would not cause any sensible inequality. As the points of the nonius divisions cannot be divided upon the nonius plate without inconvenience, it is best to use the method formerly de scribed, holding the beam compass a small while in the hand previous to the cutting. The points being but few in num ber, the divisions may be cut before any expansion can sen sibly take effect.
Sextants, or octants, for observing the distance of the moon from the fixed stars, should be divided by the fore going method, great accuracy being required. If instead of dividing sextants to every 20' upon the limb, as is commonly done, they should be divided to 15', a chord of 64° might be laid off, and divided by continual bisections. This would, in some measure, crowd the limb with divisions ; but it would shorten the nonius ; for 15, instead of 20, would shew one minute.
In dividing either arches or straight lines, a number (which will divide continually by 2) greater than is requir ed upon the arc or line, is the best to begin with, and may be used in dividing a circle, by laying off the chord of the difference. Suppose it was required to divide a circle into
54 equal parts, it would be 64 —54 = 10 = 10° 30' ; the chord of which laid off must be added to 360°, and it will be 360+10° 30', to be divided into 64 equal parts, 54 of which will complete the circle. if the arc of 10° 30' be laid off from a dividing plate, it will answer the Same pur pose.
Analogous to the foregoing method, my scale of equal Marts was divided. I took W of an inch in a beam com pass, laid the scale which I took it from, the brass scale to be divided, and the beam compass in a room facing the north, where they lay the whole night. Early next morning, after correcting the lengths, the above were laid off three times (the brass being long enough to take it in ;) then, having in other compasses 256, 128, and 64, 1 bisected the three spaces of 512 with all the expedition 1 could. Hav ing now only 64 inches in the last beam compass, any par tial or unequal expansion was not to be feared; therefore worked by continual bisections till I had done. The linear divisions were cut from the points with a beam compass, as before described.
The nonius divisions of this scale contains of an inch, which were divided into 100 in the following manner : i00 : 101 : : 256 : 258.56 tenths of an inch, the integer in this case being Suppose the scale to be numbered at every inch from left to right ; then exactly against to the left of 0, was made a fine point, front which was laid off 238.56 to the right hand. This was taken from a scale 3 feet in length, which was divided after the common me thod ; hut the error was so small as to vanish at the other extremity of the nonius, when divided continually by 2. • Whoever undertakes to divide a scale of the above kind, not being furnished with one long enough to lay off 258.56, may take W from that before him, to which he may add 8 56 A:ken from a diagonal scale, that may be made at a small ex pence.
To prove the expedience of the above methods of divid ing astronomical instruments, Sic. I need only to mention the following particulars, taken from the Nautical Almanac for this present year 1767, page 152.