Review of O. Gregory^s Treatise on Mechanics. 77 



fective in producing mental illumination, or a complete con" 

 viction of the truth, and is therefore improper in a work 

 calculated for learners. If, after the discovery of a mathe- 

 matical truth, a demonstration be necessary at all, it is ne- 

 cessary that the reasoning should be clear, and evident, at 

 every step ; but the analytical process is the very reverse 

 of this, it consisting of mechanical maneuvres of symbols 

 and abstract quantities, the perception of whose connexion 

 in the chain of reasoning is wholly lost: sometimes it goes 

 farther than this; its first principles, instead of being intui- 

 tive and elementary truths, on which the pure and legiti- 

 mate reasoning of the mathematics rests, are drawn from 

 the metaphysical and refined doctrine of ultimate and van- 

 ishing quantities, which are considered as difficult of con- 

 ception even by mathematicians, and wholly unintelligible 

 to learners. Such are all those pretended demonstrations 

 by the differential calculus, generally used by the continen- 

 tal mathematicians of Europe, and now without judgment 

 attempted to be introduced among the English population 

 throughout the world. To us there appears as much of 

 sanity in this new fangled mathematics for demonstrations, 

 as in endeavouring to lay the loundation of a structure at its 

 top, or to prove obvious truths by deductions from those 

 which arie the most remote and recondite. It was against 

 this system principally that the learned and acute Berkley 

 raised his voice. He was an admirer of mathematics in its 

 purity, as cultivated and delivered by the ancients, and very 

 much regretted the vitiation of its logical demonstrations by 

 the obscurity of the modern analytics. These, it must be 

 allowed, are excellent tools in the hands of the mathema- 

 tician, for the investigation of new truths, since they save 

 much time, and enable him to proceed to vastly greater 

 extent than he otherwise could do : but in delivering and 

 demonstrating to learners, propositions of the mathematics 

 already known, the process which leads to the result must 

 be explained step by step, whether we assume the analyti- 

 cal or synthetical method of reasoning. It is not against 

 either of these methods logically considered, that we con- 

 tend, but against what may be called the algorithm or com- 

 plicity of symbolical terms and expressions, maneuvered 

 according to the rules of algebra, and assumed as mathe- 

 matical reasoning and demonstratior^^ All problems purely 



