266 
On the Systematic Fitting of Curves 
Introductory Note. 
One of the most frequent tasks of the statistician, the physicist, and the 
engineer, is to represent a series of observations or measurements by a concise and 
suitable formula. Such a formula may either express a physical hypothesis, or on 
the other hand be merely empirical, i.e. it may enable us to represent by a few well 
selected constants a wide range of experimental or observational data. In the 
latter case it serves not only for purposes of interpolation, but frequently suggests 
new physical concepts or statistical constants. 
In any given case the formula or curve to be fitted to the data is : 
(i) Directly given by physical theory ; 
(ii) Chosen on the basis of a physical hint ; 
(iii) Although purely empirical, suggested by experience of goodness of fit 
in like cases ; 
(iv) Quite unknown and to be chosen solely by examination of the material. 
Now, as I hope to indicate in this paper, half the difficulty of curve-fitting in 
practice lies in the choice of a suitable curve. Especially in Case (iv) it is only a 
very considerable experience in curve-fitting that can lead to a suitable choice 
among all the possible algebraic, exponential and trigonometrical curves that 
suggest themselves. 
The hasty assumption of some physicists and many engineers that a parabola 
of the form 
y = Co + CiX + c^x^ + c^x^ + ... 
is always a good thing is to be deprecated, as may be seen at once by considering 
what a poor fit is obtained iu this way to material really expressed by such curves 
as 
y = yo6"~''^^ y — y^ ^i'^ y ( ^ + c) = h^, etc. 
To assume a curve of this form we must show the rapid convergency throughout 
the proposed range of the Maclaurin Expansion, and this is not always feasible. 
The present paper does not concern itself with the choice of suitable curves, 
but only with the determination of the constants, when the form of the curve has 
been selected. This I readily allow to be the easier half of the task. 
So far I have not, however, been able to find any systematic treatise on curve- 
fitting. It is usually taken for granted that the right method for determining the 
constants is the method of least squares. But it is left to the unfortunate physicist 
or engineer to make the discovery that the equations for the constants found in 
this manner are in nine cases out of ten insoluble, or a solution so laborious that it 
cannot pi'ofitably be attempted. 
