16 J. H. Alexander on a New Formula for Interpolations* 



but have sometimes treated the subject as if curves were essen- 

 tially different from straight lines. Hence, what are really lines 

 of the second order, have been termed by some, curves of the 

 first degree; and so on. It hardly need be said that this is in so 

 far a defect in the generality of the method ; and that it imposes 

 the necessity of a subordinate and at least cumbrous discrimina- 

 tion between degrees and orders, the first being the appellations 

 of the equations, and the second, of the curves dependent upon 



them. 



For the present aim, such discriminations would but tend to 



confuse. And the geometrical aspect which the formula affects, 



is that of the hypothenuse of a triangle or the arc of a curve, 

 having a general characteristic equation of the form y n =^ax (and 

 therefore parabolic), but divisible into as many different degrees 

 or orders as there are assignable values of n. The origin of the 

 co-ordinates is placed at the apex of the triangle, and at the pole 

 of the curve, in. the respective cases ; in the last the characteris- 



i 



tic equation assumes the form y = ax n , which brings the ordinates 

 to the linear scale and makes the base on which they are meas- 

 ured, tangent to the curve at the pole ; while the different ab- 

 scissae, a;', x", x'", etc., are in fact portions of diameters intercept- 

 ed between the arc itself and points in the base of the ordinates 

 corresponding to the values of y\ y", y"', etc. 



But geometrical illustrations are, in general, intelligible with 

 difficulty, unless accompanied by delineations; I prefer, there- 

 fore, to place the explanation of the formula upon the theory of 



figurate numbers; to which, as already said, the doctrine of in- 



terpolations belongs. Of course the derivation of the formula 

 will be equally satisfactory, whether it be made to flow from ex- 

 tension or from numbers; and the numerical deduction will have 

 the advantage of being of the same kind with the cases that 

 occur most abundantly in practice. 



These figurate numbers, then, are the terms of various artifi- 

 cial series, constructed from and corresponding to the natural 

 series of numbers in such wise that if the general term of this 



n 



last be called y , the general term of the others will be expressed 



n n+1 n n + 1 w + 2 n ft-fl n+2 ra + 3 

 Thus for instance, substituting in the expression, 7 . — 5— , the 



successive 



v j , ? , - -j — ;/ 



we have a figurate series, as under ; 



natural series : 1. 2. 3. 4 5. &c. 



** figurate series : L 3. 6. 10. 1 &c. 



1 















