a thermometer was devised there arose a science of heat, which was before impossible.  Such of our sensations as we have not yet found modes of measuring do not originate sciences.  We have no science of smells; nor have we one of tastes.  We have a science of the relations of sounds differing in pitch, because we have discovered a way to measure them; but we have no science of sounds in respect to their loudness or their timbre, because we have got no measures of loudness and timbre.

Obviously it is this reduction of the sensible phenomena it represents, to relations of magnitude, which gives to any division of knowledge its especially scientific character.  Originally men’s knowledge of weights and forces was in the same condition as their knowledge of smells and tastes is now—­a knowledge not extending beyond that given by the unaided sensations; and it remained so until weighing instruments and dynamometers were invented.  Before there were hour-glasses and clepsydras, most phenomena could be estimated as to their durations and intervals, with no greater precision than degrees of hardness can be estimated by the fingers.  Until a thermometric scale was contrived, men’s judgments respecting relative amounts of heat stood on the same footing with their present judgments respecting relative amounts of sound.  And as in these initial stages, with no aids to observation, only the roughest comparisons of cases could be made, and only the most marked differences perceived; it is obvious that only the most simple laws of dependence could be ascertained—­only those laws which, being uncomplicated with others, and not disturbed in their manifestations, required no niceties of observation to disentangle them.  Whence it appears not only that in proportion as knowledge becomes quantitative do its previsions become complete as well as certain, but that until its assumption of a quantitative character it is necessarily confined to the most elementary relations.

Moreover it is to be remarked that while, on the one hand, we can discover the laws of the greater proportion of phenomena only by investigating them quantitatively; on the other hand we can extend the range of our quantitative previsions only as fast as we detect the laws of the results we predict.  For clearly the ability to specify the magnitude of a result inaccessible to direct measurement, implies knowledge of its mode of dependence on something which can be measured—­implies that we know the particular fact dealt with to be an instance of some more general fact.  Thus the extent to which our quantitative previsions have been carried in any direction, indicates the depth to which our knowledge reaches in that direction.  And here, as another aspect of the same fact, we may further observe that as we pass from qualitative to quantitative prevision, we pass from inductive science to deductive science.  Science while purely inductive is purely qualitative:  when inaccurately quantitative it usually consists of part induction, part deduction:  and it becomes accurately quantitative only when wholly deductive.  We do not mean that the deductive and the quantitative are coextensive; for there is manifestly much deduction that is qualitative only.  We mean that all quantitative prevision is reached deductively; and that induction can achieve only qualitative prevision.