259 



Wallis* and of Heinrich Kiihnf , Professor at Dantzig. The re- 

 searches of these writers upon the geometrical representation of 

 imaginary quantities were not known to Mr. Warren. A paper by 

 M. Buee, containing some partially-developed views on the meaning 

 and application of algebraic signs, was printed in the Philosophical 

 Transactions for 1806, and a work by M. Mourey on the true 

 Theory of Negative and Imaginary Quantities appeared at Paris in 

 1828. The former was unknown to Mr. Warren till his Treatise 

 was in the press, and the latter was not seen by him before De- 

 cember 1828. It cannot therefore be said that he was indebted for 

 his views to preceding writers ; in fact, the work carries with it 

 evident marks of originality, and has received honourable mention as 

 well from Continental as from English mathematicians, who have 

 since written on the same subject. The names of Buee, Warren, 

 and Mourey are generally associated as having taken the lead in a 

 department of mathematics, which in the present day J has received 

 remarkable elucidations, developments and accessions at the hands 

 of Gauss, Sir W. R. Hamilton, Professors Peacock, the late D. F. 

 Gregory, De Morgan, C. Graves, and others. 



The title of Mr. Warren's Treatise hardly conveys an exact idea of 

 its main object. He proposes to represent every kind of quantity 

 geometrically by the intervention of symbolical expressions, which 

 involve the square roots of negative quantities, and designate lines 

 in position as well as magnitude. After laying down definitions of 

 addition, subtraction, multiplication, division, involution and evolu- 

 tion in the sense in which these operations must be taken when ap- 

 plied to quantity so represented, he proceeds to show the coincidence 

 of the symbolical results obtained from such definitions with the 

 ordinary results of arithmetical and symbolical algebra. He was 

 strongly convinced of the superiority of geometry, as a means of de- 

 monstration, above the use of mere symbols of quantity, and enter- 

 tained the opinion that the obscurity attaching to the proofs of some 

 of the fundamental rules of algebraic and analytical operations, might 

 be removed by adopting a geometrical representation of quantity, 

 such as that proposed in his Treatise. 



On Feb. 19, 1829, a paper by Mr. Warren, entitled " Considera- 

 tion of the objections raised against the geometrical representation 

 of the square roots of negative quantities," was read before the Royal 



* 'Treatise of Algebra/ chapters Ixvii.-lxix. fol. Oxford, 1685, cited by Ben. 

 Gompertz, ' The Principles and Application of Imaginary Quantities,' book ii. 4to. 

 London, 1818. 



t Commercium Mathematico-Petropolitanum, anno 1736. Meditationes de 

 Quaniitatibus Imaginariis constrnendis, et Radicihus Imaginariis exhibendis, in 

 Nov. Comment. Acad. Scient. Imper. Petrop. pp. 170-223, ad annos 1750 et 1751. 

 Petrop. 1753. 



% See George Peacock, ' Report on the recent Progress and present State of 

 certain Branches of Analysis,' in ' Reports of the British Association for the Ad- 

 vancement of Science,' vol. iii. pp. 228-30. An account of several recent works 

 upon this subject may be found in Wilhelm Matzka, Versuch einer richtigen Lehre 

 von der Realitdt der Vorgeblich imaginiiren Grossen der Algebra, §§ 132-139, 4to 

 Prag, 1850. 



