divisions of themselves mean nothing. It is only by placing the thermometer in liquids of the intervening temperatures and directly recording the height of the column at each temperature that we could get a definite graduation. As this latter method is too cumbersome, the marks are made at intervals by the dividing machine, and then the actual value of each mark is determined by sending the mercury up to it and noting the temperature required to do so. Thus each mark on the thermometer means that at some previous occasion of a certain character the mercury column reached to that point; when we now make a measurement of temperature we simply compare the length of the column at present with the record of its length at some previous time. The same is true of the galvanometer, the clock, and all apparatus in which the graduation is in units of length; exactly similar processes are used to arrive at other scales.
Exactness in Measurement.–Since we always measure physical quantities by means of a psychological judgment as to the agreement of two sensations or sets of sensations, we must so arrange matters that in a given case the psychological judgment introduces only a small uncertainty into the measurement. Since all psychological and physical measurements are made by means of apparatus, the error of the apparatus must be sufficiently small in comparison with the quantity measured. For example, in measuring the time between two successive culminations of the same star, the uncertainty introduced into the results by the variations of our judgments in the eye-and-ear or the graphic method are too small to be of importance for most physical purposes, the length of the sidereal day being determinable to within .05 seconds, or 510000 of 1%. In measuring mental times an outside limit of error of 11000 of a second is beyond the needed accuracy; the length of the time measured seldom is less than 1001000 of a second; we can thus allow an outside limit of error of 1%. We can therefore use a fork vibrating 100 times per second, whose accuracy has been determined to within 1%, that is, one whose vibrations during a sidereal day amount to 8,616,400 ± 3,200. The
