124 COLLIGATIVE RELATIONS AND SCIENTIFIC LAWS 



earth. Everybody who attaches the same denotation to the concept 

 of Keplerian orbit— that is, everybody who in this sense speaks the 

 same language— will arrive at the same orbit. As with pressure, the 

 orbit is computed in an agreed-upon way from observations on 

 which all agree, too. Continuing in the same fashion, by a deter- 

 minate set of mathematical manipulations we pass from observations 

 of the positions of the planet in the sky to an orbit for it plotted on 

 paper— much as we proceed from observations of a J-tube to a plot of 

 pV versus p. And now to prediction! Given Boyle's law and the plot 

 of pV versus p we readily derive predictions of what we should ( and 

 ordinarily do) observe under specified conditions. Just so, given Kep- 

 ler's laws and the plotted orbits, the inverse of the procedure by 

 which past observations were rendered as "points" on paper yields us 

 predictions of where in the sky a given planet will be observable at 

 specified times in the future. These predictions are generally found 

 quite respectably ( though not perfectly ) sound. 



Functioning as a typical colligative relation, the Keplerian laws 

 seem to differ from the other relations so far considered only in the 

 magnitude of the paper-and-pencil operation which intervenes be- 

 tween observations made and observations predicted. But this mag- 

 nitude may be no more than an incidental reflection of the state of 

 scientific development. Thus years ago the derivation of densities 

 ( from observed weights and volumes ) , speeds ( from observed times 

 and distances ) , and pressures ( from the observed heights of mercury 

 columns ) demanded paper-and-pencil operations today wholly elim- 

 inated by the use of such devices as the hydrometer, speedometer, 

 and Bourdon gage. From such devices we can read "density," 

 "speed," and "pressure" directly, and can more directly predict what 

 will so be read. V/ith primitive instrumentation an elaborate compu- 

 tational operation may be required; with more elaborate instrumen- 

 tation, on the other hand, little or no such operation may be needed 

 even to use Kepler's laws. Provided the specifications of the paper- 

 and-pencil operation are clearly understood and generally acknowl- 

 edged—as they are in the case of Kepler's laws— the conceptual deno- 

 tations remain firmly established in the one case as in the other. And 

 by such anchoring colligative relations are secured from what might 

 otherwise prove an irresistible drift into the status of conventions. 



Each and every colligative relation, sooner or later, is challenged 

 by some datum or data at variance with it. Lest science dissolve into 



