( 653 ) 
THE AREANGEMENT OF SUCCESSIVE CONVERGENTS IN 
OEDEE OF ACCUEACY. 
By Alexander Brown. 
(From the Applied Mathematics Laboratory, South African College.) 
(Eead September 15, 1915.) 
§ 1. One of the most important uses of simple continued fractions is for the 
solution of the problem to find the fraction, whose denominator does n<d exceed 
a given integer, ivhich shall most closely apjjroximate to a given nrimher com- 
mensurable or i7icommensurahle. A practically complete solution was pro- 
vided by Lagrange in 1769 in his paper " Sur la Eesolution des Equations 
Numeriques " * and his treatment, involving the use both of principal con- 
vergents (fractions principales) and of intermediate convergents (fractions 
serondaires), has become the classic exposition in text-books of algebra. 
His results give the fraction nearest in defect, and the fraction nearest in 
excess, satisfying the conditions. He does not consider the question of 
deciding which of these two fractions is nearest in absolute value to the 
given number. 
Chrystal, in his Algebra, f vol. ii, p. 424, Ex. 10, gives a rule " enabling 
us in most cases to save calculation in deciding between the closeness" of the 
fraction in defect and that in excess, but the rule is not easy to apply ; while 
Serret, in his Cours d'Alghbre 8uperieure,X p. 24, leaves the question as 
Lagrange dealt with it. The present paper was written to give an easy 
method of deciding between the two fractions, and to arrange the successive 
convergents in such a scheme that the nearest in absolute value satisfying 
the stated condition could be at once picked out. Afterwards a short note 
by Muir § was found On Convergents, i\\ which the same method of dis- 
crimination is given without proof. This note seems to have been overlooked 
by later writers. 
* "Memoires de PAcademie royale des Sciences et Belles-Lettrcs de Berlin," t. xxiii, 
1769. "(Euv^res de Lagrange" (Serret), 1868, vol. ii, p. 568. 
t "Algebra" (Chrystal), 1859^ part ii. 
X Sixth edition, 1910. 
§ Eeport of the British Association for the Advancement of Science for 1876. 
