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XXXIV. Concerning the Analytical and Synthetical Modes 

 of Reasoning made use of in Mathematics and other 

 Sciences. By tlie Rev. John Toplis, A. M. 



To Mr. Tilloch. 



SIR, 



J-p you think the following worth insertion in the Philoso- 

 phical Magazine, it is at your service. I have almost wholly 

 extracted it from a long discourse read before the Philomatic' 

 Society at Paris by S. F. Lacroix. I am, sir. 



Your sincere well-wisher, 

 Arnold, Norn, JOHN ToPLIS. 



October 25, i!J04. _____ 



There are two methods of reasoning made use of in the 

 inathematical sciences ; the synthetical and the analytical. 

 By referring back to the Greek origin of the words syn- 

 thesis and analysis, we shall find that one signifies compo- 

 aition, and the other resolution or decomposition. Nothing 

 appears clearer at the fust glance than these denominations ; 

 and we easily conceive that the mtthods they denote are the 

 inverse one of the other : nevertheless, it appears to me that 

 we do not pay sufficient attention to the difference between 

 the proceedings of synthesis and analysis, nor form always 

 distinct notions of them. I therefore thought it necessary 

 to search into the writings of the antients for examples of 

 composition and resolution, in order to fix my ideas upon 

 this important pointy from which the following reflections 

 have arisen : 



The Elements of Euclid are according to the synthetical 

 method. That author having formed certain axioms, and 

 made certain demands, advances his propositions, which he 

 proves by means of what precedes; and thus continually 

 passes from simple to composite, which is an essential cha- 

 racter of synthesis. 



At the origin of geometry we mccl with traces of the ana- 

 lytical method ; for it is not correct to suppose that algebra 

 constitutes exclusively analysis: it serves likewise to facili- 

 tate synthetical demonstrations ; for it is at the bottom only 

 an abridged and regular method of writing, by means of 

 which we represent all the relations which magnitudes can 

 have with each other: and I shall remark upon this subject, 

 that Condiilac, when he shows in his Logic that algebra is a 

 language, merely repeats what Clairaut asserted and proved 

 in his "Elcmins d'Alnilre, printed in the year 1748. 



The first usage of analysis in geometrical researches is 

 Vol. 20. No. 79. Dec. 1804. N attributed 



