Review of the Cambridge Course of Mathematics. 321 



Me, on account of its connexion with the solution of equa- 

 tions of the higher decrees. 



After disposing of fractions, he resumes equations of the 

 first degree, and discusses those cases in which two or more 

 unknown quantities enter into them. This he does, by re- 

 solving several problems at great length, and seizing every 

 opportunity that is presented, in the progress of the solu- 

 tion, to give important theoretical and practical instruction. 

 In this way, he takes occasion to explain the nature of in- 

 sulated negative quantities, (what was shewn before un- 

 der the simple rules, having related to negative, quantities 

 combined in expressions with positive,) and he has demon- 

 strated, a priori, that they follow the same rules as other 

 quantities. This was necessary, as the theory of negative 

 quantities is, at the same time, one of the most important 

 and difficult of Algebra, and ought, therefore, to be estab- 

 lished by rigorous reasonings. I.ideed, it appears from 

 the history of Algebraic science, that this theory especially 

 in what relates to negative solutions of problems, was but 

 little understood before the tirneof DeCartes.*(Essais,258 ) 



The signification of the phrase infinite quantities in mathe- 

 matics, is deduced from a fractional expression in which 

 the numerator remains constant, while the denominator is 

 continually diminished. The ultimate point towards which 

 this diminution advances is zero, whence the expression 

 - is naturally the symbol of infinity ; and mathematical in- 

 finity is a negative idea, and signifies merely the exclusion 

 of all limit either in smallness or greatness. We arrive at 

 the idea, therefore, by a series of negations, and infinity 

 is that of which we affirm the limits cannot be attained by 

 any conceivable magnitude whatever it may be.* The 



* Assez sourent, says Lacroix, on i substitue le mot indejini au mot infi- 

 ni, croyant par ]k eluder les difficnltes que faisait naitre ce dernier ; mais 

 je ne vois en cela qu'une faute d' expression ; car 1' indefini pent avoir des 

 iimites, mais on en fait abstraction pour le moment, tandis que Tinfini est 

 afeoessairement ce dont on affirme que les Iimites ne peuvent ctre attainted 

 par quelque grandeur concevable que ce soit." Traite du calc. Diff. &c. 

 Pref. xix. 



Another way more plain but less rigorous, of obtaining the idea ofiiifinity 

 given above, is this ; any quantity m divided by a quantity much smaller 

 than itself, gives a quotient much greater than itself, whence since the 

 values of fractions vs^hose numerators are constant, are inversely v>s their 

 denominators, m divided by a quantity very small, will give a quotient very 

 large ; therefore, m divided by zero, gives a quotient greater than any fiuitf 

 quantity. 



