12S

PROJECT FOR A NEW

TRACT 25

every ten seconds of the quadrant, cannot be of so muchuse as it might seem at first sight; as the very near coin-cidence of the numbers in the new and old divisions appearrer}* seldom to happen. I find indeed, that our arc 1309answers nearly to 45 minutes, that arc exceeding 45' byonly -00632363 or part of a second nearly, and so inproportion for their equimultiples. But though this degreeof coincidence may be sufficient for checking the corre-sponding values of the arcs, in the first and second co-lumns, we are not thereby authorised to consider the sine,tangent, or secant, of 1309, as accurately equal to that of45', in all the seven places of figures, but differing from itby nearly the part of the difference corresponding towhich is about A of a unit in the sines and tangents,though next to nothing in the secants. This therefore,though ft makes no sensible difference in this particular case,will cause a difference that must not be neglected, in theequi-multiples of 1309 and 45', the sines and tangents ofwhich will differ by half a unit or more, and therefore willnot be expressed by the same number, but will have somesmall difference in the seventh or last figure. And the samewill happen in almost all the other arcs; so that generallythe sines, See, which are exact for the arcs in the first co-lumn, will not be quite so for those in the second, whenexpressed in whole seconds only, since these will sometimesdiffer by the part corresponding to almost half a second.However, in this, or any other case, where the difference isexactly known, we may profitably make use of the numbersin the old tables, for constructing or verifying those of thenew, by taking in the proportional part of the difference.Let, therefore, all the sines, &c, of every 1309 be com-puted from the old tables, and entered in the new, by add-ing to the sine, See , of the corresponding multiple of 45',the like multiple of the part of the proportional differ-ence for 1''. This will give about 120 sines, 8tc, to serveas a verification of the computations by the more generalmethods. But if the second column be exactly constructed,