The divisions hitherto mentioned being only very fine points in a fine arch a b d, scarce descernible by the naked eye, it was necessary, as usual, to strike lines perpendicular to the arch, through every one of them. But since it is very difficult, and tedious too, to draw lines exactly through every point by the edge of a ruler, the following method was judged more accurate and expeditious. It was pro posed then to divide any other concentric arch, f h t, by cross strokes, into similar parts to those in the given arch aegebd. Take a small beam compass, and having once fixed its points at any convenient interval ; upon the cen tres e, g, &c. being the given points of the divided arch, strike the small arches,/ i, h k, &c. cutting the undivided arch in f, h, &c.; then will the intercepted arches, as f, h, &c. be similar to the arches e, g, &c. that is, they will sub tend the same angles at their common centre o For join ing e f, g h, and also o f, o h, o e, o g, the triangles e o f, g o h, will be similar and equal to each other ; every side in one being respectively equal to every side in the other. Therefore, by taking away the common angle e o I,, from the equal angles, e o f, g o h, the angles e o f o h, that remain, will also be equal.
If the triangles e f o, g h o, &c. be right angled at f and h ; the dividing strokes f i, h k, kc. will cut the quadrantal arch f h t at right angles, also at f and h, &T." Respecting the means employed by Graham in the di vision of the arc of 90°, there is nothing in them, except what has already been noticed, that evinces superior con trivance ; it is to his accurate execution, and the invention of the continually bisectional arc, that he owes the well-de served reputation which he holds among those who have distinguished themselves in the art of graduating astrono mical instruments.
Before the time that Bird flourished, there is not a word to be found in the history of graduation about the dilatation and contraction of metals in different temperatures ; and it may be supposed, that as Graham made his quadrant of iron, and the limb of brass, he was unacquainted with the different expansions of those metals. Our doubts, how ever, respecting the accuracy of that instrument, from this circumstance, have been partly removed, by the writer of this part of our article having heard Troughton say, that lie had looked at it with an eye to this matter; that he is of opinion, from estimating the elasticity of those metals at 8 or 10 times their difference of expansion, in such tempera tures as they are exposed to in the observatory, that the iron will command the brass, and little or no error will en sue. The same artist also observes, that as the iron quad
rant has long borne an additional arc of 96°, divided by Bird, and as there is no proof of its having changed its figure since that arc was put on, as the brass one has cer tainly done to the amount of 7 or 8 seconds, for aught that is known to the contrary, the iron one may be the better in strument; but as both of them are now superseded by the new mural circle, this question may, perhaps, remain for ever at issue, and its discussion is now rendered of far less interest than it would have been 30 years ago.
Graham laid it down as a principle or maxim, " that it is possible, practically, to bisect an arch or a right line, but not to trisect, quinquisect, &c." Bird not only acted upon this maxim, to the full scope of what had been clone by the artist who conceived it, but extended it even to the division of the arc of 90°. We will not, however, anticipate the me thod of Bird, which is in many respects original and inge for every one who dips into these matters will de rive the most satisfactory information from perusing his own account of it, as published by order of the Board of Longitude in 1767, the whole of which is subjoined.
" The following method of dividing astronomical instru ments, &cc. is collected principally from the experience which I have gained in the space of thirty-four years ; and, in some parts, from the instructions which I received from the late Mr Jonathan Sisson.
What I call my own, I have distinguished by Italic cha racters. If any other instrument makers have used the same method, it is unknown to me ; and shall, therefore, pay no regard to any pretensions unsupported by evidence ; —I mean, pretensions, without producing astronomical in struments superior, or, at least, equal to those which I have made.
How far the lunar theory bath been improved by the ob servations of the late Dr Bradley and Mr Mayer, I leave to the decision of those who have tried it by observations, in order to find the longitude at sea, &c. 1 cannot help, however, being fully of opinion, that a still more perfect knowledge of the motion of the heavenly bodies may be ob tained by future observations, skilfully made, with accurate instruments.
I have been favoured with so distinguished a mark* of approbation from the Commissioners of Longitude, that nothing on my part shall be wanting, that may, in the least, contribute to so desirable an end :"—So far the preface.
" It will be sufficient, for my purpose, to make use of no more lines and arcs than are represented in Fig. 2. Plate CCLXXXII.