460 M. Cauchy on a New Formula for 
Thus, when viewed in its whole extent, the problem of in- 
terpolation consists in determining the coefficients or arbi- 
trary constants contained in the expression of the general 
laws of figures or phenomena, on the supposition that at least 
an equal number of points is given in the former or an equal 
number of observations or experiments made upon the latter. 
In a great number of questions these arbitrary constants enter 
only in the first degree into the equations containing them. 
This is precisely what happens when a function is capable of 
being developed in a converging series arranged according to 
the ascending or descending powers of an independent variable, 
or to the sines and cosines of the multiples of an arc. ‘Then 
the question is, to determine the coefficients of such of the 
terms of the series as cannot be disregarded without giving 
cause to fear that a sensible error in the values of the function 
may be the consequence. Among the small number of formulze 
that have been proposed for this purpose the most worthy of 
notice are,—that derived from the calculus of finite differences 
but applicable only when the different values of the indepen- 
dent variable are equi-different among themselves,—and that 
of Lagrange, which, whatever these values be, can be applied 
to series arranged according to the ascending powers of the 
independent variable. However, the latter formula itself be- 
comes more and more complicated in proportion as it is found 
desirable to retain a greater number of the terms of the series 
in which the function is developed; and what is still more 
annoying is, that the approximate values of the different orders 
corresponding with the different cases in which we should 
keep, first one term of the series, then two, then three, &c., 
are obtained by calculations almost independent of one 
another ; so that each new approximation, far from being ren- 
dered easier, is more tedious and laborious than those which 
precede. Struck with these inconveniences, and led by my 
investigations respecting the dispersion of light to turn my 
attention anew to the problem of interpolation, I have been 
so fortunate as to find for its solution a new formula, which, 
both in respect to the certainty of the results and the facility 
with which they are obtained, seems to me to possess such 
decided advantages over the others, that I have no doubt it 
will soon be generally employed by all persons devoted to the 
cultivation of the physical and the mathematical sciences. 
In order to give an idea of this formula, I suppose that a 
function of x represented by y is developed in a converging 
series arranged according to the ascending or descending 
powers of x, or according to the sines and cosines of the mul- 
tiples of an arc x, or, more generally, according to other func- 
