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XXXV.— On the Third Co-ordinate Branch of the Higher Calculus. 



By Edward Sang, Esq. 



(Read 15th January 1866.) 



That part of universal arithmetic which relates to the changes of variable 

 quantities, and which is known under the titles — Fluxions, The Higher Calculus, 

 The Infinitesimal Calculus, The Theory of Functions, has been divided into two 

 branches, called respectively the Differential and the Integral Calculus ; the one 

 of these being regarded as the converse of the other; and every problem connected 

 with variation has been supposed to require either or both of the processes known 

 as differentiation and integration. 



This two-sided view of the higher calculus arose naturally in the course of its 

 development. When we study the changes of a variable quantity, our attention 

 is called to its differences and to the velocity of the change. The first branch of 

 the subject is thus that which teaches us to pass from the variable quantity 

 to its differential or differential coefficient; and, as our attention has been 

 engrossed by the mutual relations of the variable and the differential, the con- 

 verse problem " to pass back again from the differential to the integral " seems to 

 be its complete complement. 



Just so, in the progress of algebra proper, attention was drawn to the squares 

 and cubes of numbers and to their higher powers. Thereafter it was directed 

 to the inverse problem " from the power to compute the number or root," and 

 thus algebra was divided into two parts treating, respectively, of Involution and 

 Evolution ; these two processes seeming to be the complete converses of each 

 other. 



However, later arithmeticians, regarding the subject more comprehensively, 

 have considered that there are three related things, the root, the index, and 

 the power, and that, therefore, there are three branches of the subject or three 

 problems — the first, when the root and index are given and the power sought ; the 

 second, when the index and the power are given and the root sought ; and the 

 third, when, the root and the power being both given, the index is sought. The 

 last of these problems was resolved by Nepair; it is the exponential or 

 Logarithmic problem. 



In the present paper it is my object to indicate that the Theory of Variables 

 has, like Algebra proper, a third co-ordinate branch bearing to the differential 

 and integral calculi relations somewhat analogous to those which the Theory of 

 Logarithms bears to Involution and Evolution. 



VOL. XXIV. PART III. 7 A 



