320 PROCEEDINGS OF THE AMERICAN ACADEMY. 



column III, gives the mean percentage deviation for each table in 

 column VI, or the mean difference between the computed record time 

 and the actual record time in percentage of the latter. Reasons 

 have been given in connection with each table why certain records 

 should be left out of consideration. The remainder are discussed in 

 columns VII, VIII, IX, and X. These contain what may be called 

 the net results, while columns III, IV, V, and VI contain the gross 

 results. 



In the net results, horses come out the best, and nearly equally weU 

 for trotting, running, or pacing, viz., 1.9 per cent mean deviation. The 

 horses come out much better than the men in this comparison. It 

 should be observed, however, that the ranges of distances covered by 

 the horse-races are relatively small, — 20, 6.4, and 4, — whereas with 

 men the ranges are successively 1320, 120, 6, 185, 300, — much greater. 

 Perhaps if the ranges covered in the horses' performances had been 

 similar to those covered in the men's performances, the disparity in 

 precision would disappear. In the men's performances the net average 

 deviation is about 4 per cent, except in skating, where it is 7.5 per 

 cent. The net average deviation of all the 207 records is 3.9 

 per cent. 



Considering the gross results of columns III, IV, V, and VI, the 

 lowest deviation is found in trotting horses (2.43 per cent) followed by 

 men swimming (3.52 per cent). The greatest deviation is in skating 

 (13.26 per cent). The mean deviation of the whole series of 257 

 records, rejecting none, is 7.05 per cent. 



It is submitted that the summary in Table XI demonstrates the 

 proposition that the records in races of men and of horses approxi- 

 mately follow straight lines when plotted on logarithm paper ; because 

 the average percentage deviation of all the records is only 7 per cent 

 from the line, and excluding 50 of the records as unreliable for reasons 

 assigned, this average deviation falls to 4 per cent. The record time 

 that should belong to any given distance within the usual limits, and 

 for any of the events considered, except bicycling, can thus be assigned 

 with these probable degrees of accuracy. 



It is not so remarkable that the records of any one event, such as 

 men running, should approximately conform to a logarithmic straight 

 line ; but it is remarkable that the straight lines should be parallel, or 

 substantially parallel, in all of these eight classes of events, including 

 three gaits in quadrupeds and three gaits in bipeds, besides motion 

 in the water and over ice. Figure 1 5 collects all of the logarithmic 

 straight lines on one sheet to a reduced scale. The ascending parallel 

 straight lines are time-distance lines. The descending parallel straight 



