296 MR EDWARD SANG ON THE TABULATION OF ALL FRACTIONS 
the subjoined table of all ratios represented by numbers up to 20, with their 
values in decimals to six places. In order to study the effects of the use of 
: ; d : yg © F ‘ 
such a minor list, let us insert between the fractions — , re first one intermediate 
by help of the multipliers p, g; and then another intermediate by help of 7 and 
s; these intermediate fractions are— 
pA+gqC rA+sC 
pa+gy and Ta—sy * 
The cross products of the members of these fractions are— 


prAatpsCa+t grAy + gsCy 
and prAa+psAy+qrCa+qsCy, 
which differ by (ps—qr) Ca—Ay), that is to say, by the product of the corre- 
PD 
‘ : i : A é 
sponding differences for the pairs of fractions —, 7 and - 15 Hence if each of 
these pairs be contiguous in the lists, the interpolated fractions are also con- 
tiguous; that is to say, no fraction of intermediate value can be in lower terms. 
By using for p and qg the successive pairs of values taken from such a table 
as the subjoined, we are almost enabled to dispense with the use of movable 
cards. 
If the two limiting fractions 2) - be not contiguous, that is, if aC—Ay be 
: : ae 2 hues 
other than unit, the interpolated fraction 5 may not be in its lowest terms, 
B 
Let us suppose that pA +gC and pa+qy have a common divisor 7, or that 
pA+qC=nB , pa+qy=nB. 
we find, on eliminating p, 
n(Ba—Ap) =9(Ca— Ay), 
and on eliminating g, 
n(CB—By)=p(Ca—Ay), 
wherefore, » must be a divisor of g (Ca—Ay) and also of p(Ca—Ay) ; now, p 
and g are always supposed to be prime to each other, wherefore must be a 
divisor of Ca—Ay. That is to say, a fraction interpolated between = and ° 
by means of two multipliers p and g prime to each other, can be reduced to 
lower terms by a common divisor, either Ca— Ay itself, or some of its factors. 
By taking note of this useful theorem, we are enabled to begin our opera- 
tions with any two fractions whatever. 
As an example, we may propose to compute those fractions which are inter- 
: 3 9 2 : 
mediate in value between 17 and ao where the difference between the cross 
products is 183=7:19. Proceeding by the addition of the members: of the 
