To elucidate the principle merely of the manner in which scales are compared, we must first show how it is that very small lengths can be measured. A screw can be very accurately constructed, say with threads one-twentieth of an inch apart : if this screw be the axis of a circular plate, which turns with it, and the edge of the plate be divided into 100 parts, each of these parts will be very perceptible, if the plate be three-quarters of an inch or more in diameter, and it will not be difficult to estimate the half or quarter of one of the divisions. Let there be an index attached to the frame, which does not move with the screw, by which it may be seen, when the plate (and with it the screw) is turned, how many divisions it is turned through, New since a whole turn of the screw moves the end of it forward through one-twentieth of an inch, a motion of the plate which passes one of the divisions over the index, or the hundredth part of a turn, sends the end of the screw forward through only one two-thousandth of an inch, and a quarter of a division answers to one eight-thousandth of an inch. Suppose a couple of such screws, each of which is attached to a pointer, as in the following diagram, in which the pointers only are inserted, and one of the scales which are to be compared ; the screws which move the pointers, and all the frame-work, being omitted. Observe also that this is not the apparatus employed, but only a con venient illustration of it.
It is supposed that A and is can be moved, by the screw motion, in such a manner that a motion so small as the eight-thousandth of an inch may be given to either. The scale at present used is E F, on which
are two points, c and D, which are, or are supposed to be, exactly a yard asunder. Let the screws be moved until the ends of the pointers, which all but touch the scale, are exactly over o and r• ; then if the scale be removed, the length C D is retained in the distance between the points of the pointers. Now let another scale be introduced, and let its points be brought as near as may be, conveniently, to the pointers : it is supposed that the distances 'CD and off are very nearly equal, for workmen used to the construction of mathematical instru ments never fail in making two yard measures agree within a fiftieth of an inch. Perhaps the reader will say the point 0 might be brought exactly under the pointer A, and then the pointer 13 alone would show whether the present scale is shorter or longer than its predecessor: but as the pointer is much less cumbrous than the scale, it is easier and safer to put the scale in a convenient position than to attempt to place it in one exactly given. This being done, move the pointer A from C to a, and observe how many turns, or how much of a turn, of the screw, is required to do it : say it makes 874 divisions of the plate pass the index. Also move the pointer R from D to which makes, say, 971 divisions of the plate pass the index. Now we ob viously have