Aphorism XXXIX.
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.
1. 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.