302 On the Principles of Motion, tyc. 



errors may be committed ; for example, if it were required 

 to inscribe any chord given in magnitude within a given cir- 

 cle, x denoting the versed sine of the arch, which will be 

 subtended by the given chord, a the diameter of the circle, 



b the chord, x will be = — ; therefore whatever be the giv- 

 en magnitudes of b and a, there may always be found a val- 

 ue of x, which will satisfy the algrebraic equation, yet it is 

 manifest, that if 6 be greater than «, the value of a?, derived 

 from the equation will be of no use for solving the equation. 

 In pure Algebra where the numerical values only of the 

 terms are considered, negative quantities independently con- 

 sidered, are absurd, and impossible ; and because the rules 

 of the science are derived from their connected, or relative 

 effect on positive, or affirmative quantities, it is, that the op- 

 erations of negative quantities never produce negative even 

 powers as -a 2 , the roots of such quantities are said to be 

 impossible, but this is not otherwise the case, than that — a 

 itself is impossible in pure Algebra. The first is accounted 

 so because there is no reversion of rules, which will produce 

 its root, in the same manner as the impossibility in the irre- 

 ducible case in Cardan's theorem, arises from the imper- 

 fection of the assumptions in the composition of the rules, 

 being grounded on a very restricted condition. Pure Alge- 

 bra, or Analysis, therefore is imperfect. It is only in geome- 

 try, that the use of negatives affords to it the greatest evi- 

 dence, and universality. 



In drawing conclusions from pure analytical expressions, 

 without reference, to the nature, condition, or restrictions 

 of the problem, we may commit the greatest mistakes, and fall 

 into the greatest absurdities ; some of the most egregious in 

 the writings of Euler have been detected by Mr. Robins, 

 which may be seen in the tracts of the latter vol. 2, p. 209, &c. 



Our remarks were intended to show the connexion, har- 

 mony, and dependence of the two great branches of the 

 mathematics, and that the fundamental principles of both 

 are identical, although differently represented. 



Proclus. 



