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Keviezv of the CamhruJge Conrse of Mathematics. 32 1 



ble, on accovint of its conaexion with the solution of equa- 

 tions ofthe higher dep;rees. 



After disposing of fractions, he resumes equations ofthe 

 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, havu)g 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 dffficult of Alijebra, and ought, therefore, to be estab- 

 hshed by rigorous reasonings. Indeed, it appears from 

 the history of Algebraic science, that this theorj especially 

 m what relates to negative solutions of problems, was but 

 little understood before the timeofDeCartes.*(Essais,258 ) 

 The signification of thepbrase infinite quantities in mathc- 

 tnarics, is deduced from a fractional expression in which 

 the numerator remains constant, while the denominator is 

 continually diminished. The ultimate point towards which 

 ^hls diminution advances is zero, whence the expression 



is naturally the symbol of infinity ; and mathematical in- 

 finity is a negalive 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 

 >s that of which we affirm the limits cannot be attained by 

 3ny conceivable magnitude whatever it may be.* The 



* Assez sou vent, says Lncroix, on a substitue le mot indejini au mot zn/J- 

 ^h proyant par la elnder les difficnUes qne faisait naitre ce dernier ; mais 

 Je ne vois en cela qu'une faute d' expression ; car !' indejini pecU avoir dea 

 ^'P^^P^jmais on en fiiit abstraction pour le moment, tanJIs que J'iDfini e-t 

 '^''essairenient ce dont on aifirme que les limites ne peuvent etre attainte? 

 P^>^ quclque gi-audeur concevable que ce soitl" Traite du calc. Diff. Arc. 



Another wav more plain but less rigorous, of obtaining the idea ofiofinrfy 

 I'^^en ahov-e, is this ; any quantity m divided by a qu:»1}tif? oiucli smaller 

 ^ban itself, ^ivos a quotient much greater than itself, whence since ihe 

 values effractions whose numerators are constant, are inversely as tlieir 

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

 ^^rgo ; therefore, m divided bv zero, ^ives a qnotient greater than a«y 6aite 



m 



