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XLIII. On Vulgar Fractions. 



To Mr. Tilloch. 



Srn, — In vnur Magazine for May, are some remarks by Mr, 

 Farcv senior, upon wliat he considers a singular property of vul- 

 gar fractions, but which upon minute examination will be found 

 notiiins more than what is deducible from the rules of arithme- 

 tical pro])ortion. 



No fractions can be added, without first being reduced to a 

 common denominator, and then the numerator is only increased 

 in rjurmtity, the denominator remaining as before: but upon 

 Mr. F's proviso, both numerator and denominator are indivi- 

 dually added to the corresponding ones, and that without being 

 reduced to a common value ; consequently we must refer to the 

 integers which compose the fractions, and not to the fractions 

 collectively, for the properties Mr, F, has pointed out. 



Now, if any four numbers be in arithmetical proportion, the 

 sum of the means will always be equal the sum of the extremes, 

 e. g. If we take the numbers 5, 4, 3, 2, then will 5 + 2 = 4 -f- 3 : 

 or in the numbers 5, 4, 4, 3 (which are still in proportion, though 

 the two means are equal, for the difference between the 1 st and 3rd 

 is equal to the difference between the 2d and 4th), .•. 5 +3 = 



4 4-4 = 4x2. Now, if we take the fractions — , 4 > -3, 



which are a part of Mr. F's series, it is evident 5+3 = 4 x2j 



and the numerators being equal, 1 + 1 = 1x2; or t— — = 



1 X '^ 2 . . I 1 



— -^= — , which is of the same value as — , both terms being 

 multiplied by the like common number: or if we take the num- 

 bers — , — , — , the result still agrees with the principles laid 

 do\vn: for, if the fractions are reduced to a common denominator j, 



4 5 6 



the series will be — , — , — , where the numerators only will 

 be in progression, and the denominators equal: or, if we take 



^ . , . 15 28 13 „ 30 28 26 g . u .. 



part of the series -, -, -, &c. = --, -, -^, &c. both terms 



, c 30+26 28 X 2 56 , 



are in regular progression : therefore Y5qp^ = 5? xT = m ' ^"° 

 SiD on, for any other series laid down according to the rule 

 given. 



I am, sir, yours rcspectfuUyj, 

 Wakefield, August 31, 1816. ' S. A. 



XLIV. Oil- 



