REPORT ON CERTAIN BRANCHES OF ANALYSIS. 191 



titles as having a real existence in algebra, however vain might 

 be the attempt to interpret their meaning. 



Both Mr. Frend and Baron Maseres were sensible of the con- 

 sequences of admitting the truth of this theory of the compo- 

 sition of equations as far as their system was concerned, and it 

 must be allowed that they have struggled against it with con- 

 siderable ingenuity: they admitted the possibility of multiple 

 real, that is, positive roots, and which are all equally congruous 

 to the problem whose solution was required through the medium 

 of the equation, indicating an indetermination in the problem 

 proposed : but it would be easy to propose problems leading to 

 equations whose roots were real and positive, and yet not con- 

 gruous to the problem proposed, whose existence must be ad- 

 mitted upon their own principles ; and if so, why not admit the 

 existence of other roots, whether negative or impossible, to 

 which the algebraical solution of the problem might lead, though 

 they might admit of no very direct interpretation, in conformity 

 with the expressed conditions of the problem * ? 



pliy, which were overthrown by a more accurate investigation of nature ; and 

 if the name Ptolemy can no longer support his epicycles, nor that of Des 

 Cartes his vortices, Newton's dereliction of the principles of reasoning cannot 

 establish the fallacious notion, that every equation has as many roots as it has 

 dimensions." 



" This notion of Newton and others is founded on precipitation. Instead of 

 a patient examination of the subject, an hypothesis which accounts for many 

 appearances is formed; where it fails, unintelligible terms are used; in those 

 terms indolence acquiesces : much time is wasted on a jargon which has the 

 appearance of science, and real knowledge is retarded. Thus volumes upon 

 volumes have been written on the stu])id dreams of Athanasius, and on the im- 

 possible roots of an equation of n dimensions." 



This work of Mr. Frend, though containing many assertions which show 

 great distrust of the results of algebraical science which were in existence at 

 the time it was written, presents a very clear and logical view of the principles 

 of arithmetical algebra. 



The voluminous labours of Baron Maseres are contained in his Scriptores 

 Logarithmici, and in a thick volume of Tracts on the Resolution of Cubic and 

 Biquadratic Equations. He seems generally to have forgotten that an}' change 

 had taken place in the science of algebra between the age of Ferrari, Cardan, 

 Des Cartes, and Harriot, and the end of the 18th century ; and by considering 

 all algebraical formulee as essentially arithmetical, he is speedily overwhelmed 

 by the same multiplicity of cases (which are all included in the same really al- 

 gebraical formula) which embarrassed and confounded the first authors of the 

 science. 



* Thus, in the solution of the following problem : " Sold a horse for 24?., 

 and by so doing lost as much per cent, as the horse cost me : required the 

 prime cost of the horse ?" we arrive at the equation 



100 X —x^ =1 2400 ; 

 if we subtract both sides of this equation from 2500, we get 

 2500 — 100 a; -|- x^ = 100, 

 or «;2 _ 100 X + 2500 = 100, 

 iaashruch as the quantities upon each side of the sign = are in both cases 



