470 ADDRESS DELIVERED AT A MEETING OF 



It is now twenty years since the Rev. Mr. Warren of Cambridge* 

 showed that the ordinary imaginary symbol (v/ - 1) had a geome- 

 trical significancy, and denoted a right line whose length was equal 

 to unity, to be measured, not on the axis of the real units, but on an 

 axis at right angles to it. It followed from this, and from another 

 principle respecting the symbolical meaning of the sign +, as ap- 

 plied to lines, that the ordinary binomial imaginary, whose real 

 parts, or constituents, are multiplied by unity and v/ - 1, respec- 

 tively, may be taken to represent both the length and direction of 

 a right line in a given plane ; the square root of the sum of the 

 squares of the constituents being the length of the line, and their 

 quote, or ratio, the tangent of the angle which it forms with the 

 axis on which the first of them is measured. These quantities 

 have been denominated the modulus and the amplitude of the ima- 

 ginary binomial. 



Now, if two such binomials, or couplets, be added together ,fthe 

 sum is a binomial of a similar form, or a couplet whose constituents 

 are the sums of the constituents of the original couplet. And if two 

 couplets be multiplied together, the product is likewise a couplet ; 

 and the relation of the product to the factors is such, that the 

 modulus of the product is the product of the moduli of the factors, 

 and the amplitude of the product is the sum of the amplitude of the 

 factors. From these algebraical properties of couplets, combined 

 with their geometrical significancy, it follows that right lines in a 

 plane, having direction as well as magnitude, may be operated 

 upon according to certain simple algebraical conditions, and the 

 direction and amplitude of the resultant lines obtained by certain 

 simple algebraical rules. 



It was in the effort to generalize the theory of Couplets, and to 

 extend their properties to right lines in space, that Sir "William 

 Hamilton was led to the construction of his theory of Quaternions. 

 " Since," he says, " ^/ - 1 is, in a certain well- known sense, a line 



* Since the delivery of this Address the attention of the writer has heen directed by 

 Sir William Hamilton to the earlier steps of the inquiry. The first appears to have 

 been made by M. Buec, in a Paper published in the Philosophical Transactions for 

 1806, in which he lays down the principle that the symbol V^i, as applied to lines, 

 denoted perpendicularity. A further step was made by M. Argand, in a memoir pub- 

 lished at Paris in the same year, in which he sho'ws that the sum of two lines, 

 estimated in direction as well as magnitude, is the diagonal of the parallelogram con- 

 structed upon them. The subject was resumed and more fully developed by M. 

 Francois, in a memoir published in the Annales des Mathematiques for 1813. 



