306 THE PRINCIPLES OF SCIENCE. [CHAP. 



Brahe suggested that if the elevation of any circumpolar 

 star were observed at its higher and lower passages across 

 the meridian, half the sum of the elevations would be the 

 latitude of the place, which is equal to the height of the 

 pole. Such a star is as much above the pole at its highest 

 passage, as it is below at its lowest, so that the mean must 

 necessarily give the height of the pole itself free from 

 doubt, except as regards incidental errors. The Pole-star 

 is usually selected for the purpose of such observations 

 because it describes the smallest circle, and is thus on the 

 whole least affected by atmospheric refraction. 



Whenever several causes are in action, each of which at 

 one time increases and at another time decreases the joint 

 effect by equal quantities, we may apply this method and 

 disentangle the effects. Thus the solar and lunar tides 

 roll on in almost complete independence of each other. 

 When the moon is new or full the solar tide coincides, or 

 nearly so, with that caused by the moon, and the joint 

 effect is the sum of the separate effects. When the moon 

 is in quadrature, or half full, the two tides are acting in 

 opposition, one raising and the other depressing the water, 

 so that we observe only the difference of the effects. We 

 have in fact 



Spring tide = lunar tide + solar tide ; 

 Neap tide = lunar tide solar tide. 

 We have only then to add together the heights of the 

 maximum spring tide and the minimum neap tide, and 

 half the sum is the true height of the lunar tide. Half 

 the difference of the spring and neap tides on the other 

 hand gives the solar tide. 



Effects of very small amount may be detected with 

 great approach to certainty among much greater fluctua- 

 tions, provided that we have a series of observations suf- 

 ciently numerous and long continued to enable us to 

 balance all the larger effects against each other. For this 

 purpose the observations should be continued over at least 

 ' one complete cycle, in which the effects run through all 

 their variations, and return exactly to the same relative 

 positions as at the commencement. If casual or irregular 

 disturbing causes exist, we should probably require many 

 such cycles of results to render their effect inappreciable. 

 We obtain the desired result by taking the mean of all the 



