From the INTRODUCTORY. Errors of Observation. 1. A quantity of which the magnitude is to be determined is either directly measured, or, as in the more usual case, deduced by calculation from quantities which are directly measured. The result of a direct measurement is called an observation. Observations of the kind here considered are thus of the nature of readings upon some scale, generally attached to an instrument of observation. The least count of the instrument is the smallest difference recognized in the readings of the instrument, so that every observation is recorded as an integral multiple of the least count. 2. Repeated observations of the same quantity, even when made with the same instrument and apparently under the same circumstances, will nevertheless differ materially. An increase in the nicety of the observations, and the precision of the instrument, may decrease the discrepancies in actual magnitude; but at the same time, by diminishing the least count, their numerical measures will generally be increased; so that, with the most refined instruments, the discrepancies may amount to many times the least count. Thus every observation is subject to an error, the error being the difference between the observed value and the true value; an observed value which exceeds the true value is regarded as having a positive error, and one which falls short of it as having a negative error. 3. An error may be regarded as the algebraic sum of a number of elemental errors due to various causes. So far as these causes can be ascertained, their results are not errors at all, in the sense in which the term is here used, and are supposed to have been removed by means of proper corrections. Systematic errors are such as result from unknown causes affecting all the observations alike. These again are not the subjects of the "theory of errors," which is concerned solely with the accidental errors which produce the discrepancies between the observations.