4 K D. Preston — Measurement of the Peruvian Arc. 



nal one was added this ratio was changed, and 222 to 223 given. 

 The formula was still that of the sines squared. Later, an error 

 was discovered in this remeasurement. Picard had used a 

 toise for his base measures, which was too short by its one- 

 thousandth part. The introduction of this new value modified 

 the result so essentially that the law previously adopted no 

 longer satisfied the observations within admissible errors. The 

 formula was changed to one where the increments of the length 

 of the degree varied as the fourth power of the sines of the lat- 

 itude, and a ratio of 178 to 179 was given for the length of the 



axes. 



The introduction of a power of the sines higher than the square 

 was done reluctantly. But it was found, that in order to repre- 

 sent the curve by the second power of the function, supposing the 

 three arcs subject to the same error, it was necessary to increase 

 the degree in France by sixty-nine toises, and diminish the 

 other two by an equal amount. This would have re-established 

 the ratio 214 to 215 and would have been nearer the truth, as 

 we now know. But such large errors were not thought possi- 

 ble. In fact, reasoning from their accidental errors of observa- 

 tion, only an error of seventeen toises could be admitted for 

 the middle degree, and forty-four for the equatorial one. This 

 would necessitate subtracting one hundred and forty toises 

 from the northern one, which seemed beyond all reason. The 

 procedure, however, would make the meridian a perfect ellipse 

 and give a ratio of 250 to 251. 



When compared with Newton's theoretical value of the ellip- 

 ticity, it was remarked that this erred in defect about as much 

 as the previous conclusion had erred in excess. Therefore the 

 observations left the choice of only two suppositions : either 

 that of the fourth power of the sines, or that of some function 

 of the latitude itself. The arc of longitude which had been 

 measured was brought to bear on the decision, and it was 

 found that the measure would, by the first solution, be in error 

 by one hundred and fifty toises, whereas by the second the error 

 would be reduced to eighteen. This decided the question and 

 the law of the fourth powers, and the ratio 178 to 179 was 

 adopted. 



This is the result as given by Bouguer in his discussion . of 

 the Peruvian work. Of course it is far from being the truth ; 

 but the recapitulation shows to what extent the measures of 

 one hundred and fifty years ago were defective, and gives an 

 idea of the influence of this equatorial arc on the elements of 

 the ellipsoid that are used in all geodetic computations of the 

 present day. A later discussion improved this result, and now 

 there are so many middle arcs entering with great weight on 

 account of their increased accuracy, that the Peruvian arc has 



