ij2 On the ORIGIN and 



have a given ratio to one another : the propofition becomes a 

 Porifm, and is the fame that has been juft investigated. 



Here it is evident, that the local theorem is changed into a 

 Porifm, by leaving out what relates to the determination of the 

 point D, and of the given ratio. But though all propositions 

 formed in this way, fiom the conversion of Loci, be Porifms, 

 yet all Porifms are not formed from the conversion of Loci. 

 The firft and fecond of the preceding, for inftance, cannot by 

 converfion be changed into Loci ; and therefore the definition 

 which defcribes all Porifms as being fo convertible, is not fuf- 

 ficiently comprehensive. Fermat's idea of Porifms, as has 

 been already obferved, was founded wholly on this definition, 

 and therefore could not fail to be imperfect. 



1 8. It appears, therefore, that the definition of Porifms 

 given above, (§ 16.) agrees with Pappus's idea of thefe propo- 

 sitions, as far at lead as can be collected from the imperfect 

 fragment which contains his general defcription of them. It 

 agrees alfo with Dr Simson's definition, which is this * : " Po- 

 " rifma eft propofitio in qua proponitur demonftrare rem ali- 

 u quam, vel plures datas efTe, cui, vel quibus, ut et cuilibet ex 

 " rebus innumeris,'non quidem datis, fed quae ad ea quse data 

 " funt eandem habent relationem, convenire onendendum eft 

 u affe(tionem quandam communem in propofitione defcrip- 

 " tarn." 



It cannot be denied, that there is a considerable degree of 

 obfcurity in this definition f j notwithftanding of which, it is 



certain, 



* Simson's Opera Reliqua, p. 323. 



\ The following tranflation will perhaps be found to remedy fome of the obfcurity 

 complained of. 



" A Porism is a propofition, in which it is propofed to demo ftrate. that one or more 

 thing* are given, between which and every one of innumerable othi-r thing-, not given, 

 but aflumed according to a given law, a certain relation, delcrjbed in the proportion, • 

 is to be fhewn to take place." 



It 



