476 G. H. KNIBBS AND F. W. BARFORD. 
be group-results, that is to say, the ordinates may represent 
the total for a particular interval on the axis of abscissa, 
as for example the total number of persons in particular 
community between the ages of 0 and 5, 5 and 10, 10 and 
15, etc. Strictly, in such cases, the results should be 
indicated graphically by rectangles standing upon these 
intervals of the abscissae as bases, though for special pur- 
poses they may be otherwise shewn. The form of the data 
may be numerical or graphical: the numerical may be con- 
vertible into graphical by drawing and the graphical into 
numerical by scaling. 
3. Graphical Representation.—Since in a very large num- 
ber of cases graphic methods are not only convenient but 
essential to the proper understanding of the possible pre- 
cision of the relation, it will be indicated how numerical 
results can be graphically tested notwithstanding all diffi- 
culties as to the representation of large numbers on a 
limited scale. Poincaré’s dictum that ‘‘It is unprofitable 
to require a greater degree of precision from calculated 
than from observed results, but one ought not to demand 
a less,’’ may be accepted as a guiding principle.* 
Graphic methods greatly facilitate the recognition of the 
type of function which best represents any given curve. 
4. Principle governing adoption of particular curves.— 
Any curve represented graphically or indicated by a finite 
number of points, may be represented by an indefinitely 
large number of formule. The selection of a single formula 
should be guided by certain criteria which ordinarily depend 
upon two considerations, viz., (i) some rational view of the 
nature of the relation, i.e., one independent of the mere 
mathematics of the question, and (ii) the method by which 
the relation may be most simply expressed mathematically. 
In regard to (ii), it may be remarked that critical values 
1 H. Poincaré, Mécanique Celeste, Paris, 1892. 
