36 Mr. Ivory in repljj io the Hisiu) ical Sketch 



cussion, what I then advanced requires to be qualified in any 

 respect. Precision requires that we consider separately the 

 infinite series, after the manner of Brook Taylor, and the 

 finite formula which expresses the density by the four first 

 powers of the refraction. 



I made no observation on the infinite series, except that it 

 does not converge ; and I proved the want of convergency in 

 a particular instance. This brought upon me a torrent of 

 vehement writing not confined within the limits of bienseance. 

 Since that time, mathematicians, whom Dr. Young dares not 

 treat with so little ceremony, have come to the same conclu- 

 sion. There is now so great a relaxation in asserting the 

 convergency, that it is not necessary to add any thing more on 

 this head. 



In the Historical Sketch the finite formula of four terms is 

 called an approximatory method. I cannot find any good rea- 

 son why this name is given to it. As the coefficients are in 

 some respects arbitrary, it certainly cannot approximate to 

 any one determinate thing. But it may be thought to have a 

 pre-eminent title to the appellation, because it will approxi- 

 mate to every thing. I called it empirical. This again ex- 

 posed me to much violent writing, at the same time that the 

 author, with an inconsistency not unusual to him, allowed that 

 it was partly empirical. He now says that the coefficients are 

 to be empirically modified. We are therefore likewise agreed 

 on this head. I shall drop my offensive epithet, and use the 

 phraseology of the author, which will equally serve my pur- 

 pose. 



The only explanation of the table of refractions in the 

 Nautical Almanack that has ever been laid before the public, 

 is the formula for calculation with its coefficients assigned in 

 numbers. But in order to render this satisfactory, the author 

 should at the same time have disclosed the considerations by 

 which he determined what is arbitrary in the coefficients. 

 Every astronomer can compute the table by means of the 

 formula ; but no one can construct the formula itself. There 

 is no similar difficulty with regard to my table and the for- 

 mula for its construction given in Dr. Young's paper in the 

 Philosophical Transactions 182 1. For here the table is known; 

 and the empirical modifications mean nothing more than adjust- 

 ing the coefficients so as to reproduce the numbers of the table. 

 By the by, this making a table and a formula fit one another 

 is very singular and curious, and may not unaptly be called 

 playing at calculation. But in the other case, the table is the 

 thing unknown ; and we can attain a knowledge of it only by 



exploring 



