Property of Vulgar Fractions. 333 



not exceeding 100, up to \ ; and the ' Table of Circles ' which 

 accompanies it only extends to 100, and occupies but one page. 

 The ' First Centenary ' itself * contains tables for the conver- 

 sion of vulgar fractions into decimals, arranged by denomi- 

 nators instead of in order of magnitude as in the ' Tabular 

 Series.' 



§11. The arguments in the c Tabular Series ' of 1823 afford 

 a beautiful illustration of the properties stated in § 1; and it 

 is from this work that the examples in § 5 have been taken. 

 It is very interesting to apply the reasoning of § 4 to groups 

 of fractions as they stand in this Table ; of course in § 5 it 

 was convenient to select examples in which only a few frac- 

 tions were involved. Mr. Groodwyn writes in the preface to 

 the ' Tabular Series ' of 1823 (p. iv) :— " The Computer would 

 draw the attention of the curious in such matters to the fol- 

 lowing remarkable property of the fractions which form the 

 Series : viz. In any three consecutive vulgar fractions in the 

 table if the numerators of the extremes and the denominator's 

 respectively be added together, the sum will form the numerator 

 and the denominator of a fraction equal to the mean." 



On pp. iv-v of the ' Tabular Series ' attached to the i First 

 Centenary ' of 1818, Mr. Groodwyn enunciates both properties. 

 He explains that the fractions do not form an arithmetical 

 progression, and proceeds: — " In fact, the law of the increase 

 is such, that each Fraction exceeds that which immediately 

 precedes it by a part equal to unity divided by the product of 

 their two denominators : so that the increment is anything but 

 constant. The law, however, is invariable ; and from it a 

 ready method is derived for verifying the arrangement of the 

 Tabular, or any similar, Series." 



He then refers to the mode of arrangement of the fractions 

 (which are printed zigzag fashion in two columns), and con- 

 tinues : — " At the same time, it serves to illustrate this other 

 law in the succession of the terms of the entire Series; namely, 

 that the numerator of any Fraction in it is always the same 

 aliquot fart, or submultiple, of the sum of the numerators of 

 the Fractions immediately preceding and immediately following 

 it, which its denominator is of the sum of their denominators : 

 ... It [the 'Tabular Series'] maybe curtailed, either by striking 

 out all the Fractions that are of a given denominator, or by 

 obliterating all those of which the denominators exceed a 

 given denominator ; but, while, in the latter case, both the 



* It is to be observed that the title of the tract of 1818 is " The First 

 Centenary .... to which is now added a Tabular Series . . . . " (see title 

 in § 9) : so that the 'First Centenary ' of 1818 contains both the < First 

 Centenary ' itself and also the ' Tabular Series.' 



Phil. Mag. S. 5. Yol. 7. No. 44. May 1879. 2 D 



