XXL] THEORY OF APPROXIMATION. 467 



perfect vacuum, where there is no resisting medium of any 

 kind ; the force of gravity must be uniform and act in 

 parallel lines ; or else the moving body must be either a 

 mere point, or a perfect centrobaric body, that is a body 

 possessing a definite centre of gravity. These conditions 

 cannot be really fulfilled in practice. The next great step 

 in the problem was made by Newton and Huyghens, the 

 latter of whom asserted that the atmosphere would offer a 

 resistance proportional to the velocity of the moving body, 

 and concluded that the path would have in consequence 

 a logarithmic character. Newton investigated in a general 

 manner the subject of resisting media, and came to the 

 conclusion that the resistance is more nearly proportional 

 to the square of the velocity. The subject then fell into 

 the hands of Daniel Bernoulli, who pointed out the enor- 

 mous resistance of the air in cases of rapid movement, 

 and calculated that a cannon ball, if fired vertically in a 

 vacuum, would rise eight times as high as in the atmo- 

 sphere. In recent times an immense amount both of 

 theoretical and experimental investigation has been spent' 

 upon the subject, since it is one of importance in the art 

 of war. Successive approximations to the true law have 

 been made, but nothing like a complete and final solution 

 has been achieved or even hoped for. 1 



It is quite to be expected that the earliest experimenters 

 in any branch of science will overlook errors which after- 

 wards become most apparent. The Arabian astronomers 

 determined the meridian by taking the middle point be- 

 tween the places of the sun when at equal altitudes on 

 the same day. They overlooked the fact that the sun has 

 its own motion in the time between the observations. 

 Newton thought that the mutual disturbances of the 

 planets might be disregarded, excepting perhaps the effect 

 of the mutual attraction of the greater planets, Jupiter 

 and Saturn, near their conjunction. 2 The expansion of 

 quicksilver was long used as the measure of temperature, 

 no clear idea being possessed of temperature apart from 

 some of its more obvious effects. Eumford, in the first 

 experiment leading to a determination of the mechanic^ 



1 Hutton's Mathematical Dictionary, vol. ii. pp. 287 292. 



2 Principia, bk. iii. Prop. 13. 



H H 2 



