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

 cal significancy, and may denote a right line whose length is equal 

 to unity, 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 applied 

 to lines, that the ordinary binomial imaginary, whose real parts, or 

 constituents, are multiplied by unity and \/ ~ ^> respectively, 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 imaginary binomial. 



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

 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 mo- 

 dulus of the product is the product of the moduli of the factors, 

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

 the factors. From these algebraical properties of couplets, com- 

 bined 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 eflfort 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, '■'■ ^J -\ is, in a certain well-known sense, a line 



ciple, that the symbol y'ZT, as applied to lines, denoted perpendicularity. 

 A further step was made by M. Argand, in a memoir published at Paris in 

 the same year, in which he shows that the sum of two lines, estimated ia 

 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. Francais, in a memoir published in the Annales des Mathematiques for 

 1813. 



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