_When a series of_ progressive _numbers is given as the result of observation, it may generally be reduced to law by combinations of arithmetical and geometrical progressions._ APHORISM XL. _A true formula for a progressive series of numbers cannot commonly be obtained from a_ narrow range _of observations._ APHORISM XLI. Recurrent _series of numbers must, in most cases, be expressed by circular formulæ._ APHORISM XLII. _The true construction of the conception is frequently suggested by some hypothesis; and in these cases, the hypothesis may be useful, though containing superfluous parts._ I. IN speaking of the discovery of laws of nature, those which depend upon _quantity_, as number, space, and the like, are most prominent and most easily conceived, and therefore in speaking of such researches, we shall often use language which applies peculiarly to {196} the cases in which quantities numerically measurable are concerned, leaving it for a subsequent task to extend our principles to ideas of other kinds. Hence we may at present consider the Construction of a Conception which shall include and connect the facts, as being the construction of a Mathematical Formula, coinciding with the numerical expression of the facts; and we have to consider how this process can be facilitated, it being supposed that we have already before us the numerical measures given by observation. 2. We may remark, however, that the construction of the right Formula for any such case, and the determination of the Coefficients of such formula, which we have spoken of as two separate steps, are in practice almost necessarily simultaneous; for the near coincidence of the results of the theoretical rule with the observed facts confirms at the same time the Formula and its Coefficients. In this case also, the mode of arriving at truth is to try various hypotheses;--to modify the hypotheses so as to approximate to the facts, and to multiply the facts so as to test the hypotheses. The Independent Variable, and the Formula which we would try, being once selected, mathematicians have devised certain special and technical processes by which the value of the coefficients may be determined. These we shall treat of in the next Chapter; but in the mean time we may note, in a more general manner, the mode in which, in physical researches, the proper formula may be obtained.