PORISMS, 





PorUmt. In the abstract lie h " fiv<-n of ilic labours of 

 ~"- , ^ who prfHvdi-d lurn, lie 1m-. mentioned tin- three book* 

 on Porisms hy Kuclu^, and has given thirty-fight geo- 

 ric'il proportions iif no very c,n: ilty, 



ii for the- comprehension of the work itself. 

 .)gether with tin- imperfect definition, and an 

 example of a porism which refers to a figure that is 

 lo-,t, and which is so rem.trkably confused as almost to 

 n-nder its reconstruction impossible, arc all the data 

 that remained. From these materials modern geome-. 

 ters have attempted to restore the work of Euclid ; and 

 although the task is one of extreme difficulty, yet, wlu-n 

 executed, a very probable estimate may be formed of 

 its resemblance to thj original. 



Albert Cirard, in a work on trigonometry, printed 

 at the Hague in 1(>2{), mentions the lost books of po- 

 risms, and says that he had made a restoration of them : 

 and again, in an edition of the works of Stevinus, * he 

 declares that he had reinvented the porisms of Euclid, 

 and intended shortly to publish them. Unfortunately 

 he died before this intention was accomplished. As 

 the works of Stevinus were printed by his widow after 

 his death, possibly the manuscript may still exist in 

 some of the libraries in Holland. From the subsequent 

 discoveries of Dr. Simson, however, it appears that the 

 idea which Girard had of the species of propositions to 

 which he annexed the name of porisrns, was by no 

 means the same as that which the former writer has so 

 ably proved to have been attached to it by the ancient 

 geometers. 



After Girard, the next attempt to explain the nature 

 of porisms was made by Bullialdus ; t but this seems 

 to have been derived from a communication with Fer- 

 mat, to which distinguished mathematician we must 

 now advert. Amongst his posthumous works j is a short 

 paper, entitled, " Porismatum EudidcEorum rcnovata 

 tlnctrina," from which it appears that he had approach- 

 ed nearer than any of his predecessors to the true 

 meaning of this class' of propositions; and, in fact, se- 

 veral of those with which he illustrates his view of the 

 subject are in reality porisms : but he did not arrive at 

 any definition which should clearly separate porisms 

 from local theorems, nor did he even conjecture that 

 there existed some peculiar mode of analysis by which 

 Mich propositions might be discovered, nor attempt to 

 restore any of those of Euclid ; his promised restoration 

 of the whole of the three books never having been 

 published. 



Dr. Halley, who possessed an extensive and profound 

 acquaintance with the ancient geometry, made some 

 attempts to decypher the enunciation of the porism 

 given by Pappus. He had successfully restored the 8th 

 book of the conies of Appollonius, and the two books 

 of the same author, DC Seclione S/ialii; and had 

 achieved a still more difficult labour, that of translating 

 from the Arabic (a language with which he was un- 

 acquainted) the work of Apollonius De Seclione Ita- 

 tionis; yet he was baffled by the ob&curity which per- 

 vaded the mutilated description of Pappus, and ob- 

 serves, " Hactenus Porismatum descnptio nee mihi 

 intellecta nee lectori profutura." 



The failure of all who preceded in elucidating this 

 obscure subject, as well as the high rank which Pappus 

 assigned to these propositions, seems to have stimu- 

 lated the curiosity of one whose unabated perseverance 



has been rewarded by complete IUCCCM, Dr. Hot* 

 Simson has described the progress he made in thi ^^V^ 1 

 uhjert, in a way which cannot f.iil to interest the At- 

 tention of those who have devoted even a small portion 

 of their time to geometrical inquiries. " Pottqusm 

 vero apud I'.ippum legeram porismata Kuclici . cull- 

 tionem fuisse artifieio*iBimam multarum rerum, qua? 

 tnt ad analysin diHiciliorum et general ium proble- 

 inaturn, magno desiderio teni;bar, aliquid de ii cogno- 

 scendi; quare saepiu.s et multis variiwiue viis turn Pappi 

 propositionem generalem mancam et imperfectam, turn 

 primum, lib. 1. Porisma quod solum ex omnibus in tri- 

 bus libris integrum adhuc manet, intelligcre et resti- 

 tuere conabar; frustra tamen, nihil enim pronciebam. 

 Cumque cogitatjones de hac re multutn mihi temporis 

 consumpserint, atque molestac admodum evaserint, 

 firmiter animum induxi htcc nunquam imposterum in- 

 vestigare; pnesertim cum optimus geometra Halleius 

 spem omnem de iis intelligendis abjeciaset. Unde 

 quoties mente occurrebant, toties eas arcebam. Postea 

 tamen accidit, ut improvidum et prtepositi immemorera 

 invaserint, meque detinuerint donee tandem lux quae- 

 dam efFulserit, qua; spem mihi faciebat inveniendi sal- 

 tern Pappi propositionem generalem, quam quidem 

 multa investigatione tandem restitui. 



" Descriptio autem quam tradit ( Pappus ) porismatum 



adeo brevis est et obscura, et injuria temporis aut aliter 

 vitiata, ut nisi Dens benigne animum et vires dederit 

 in ea pertinaciter inquirere, in perpetuum fbrsan geo- 

 metris latuisset." Simsoni Opera Rcliqua, p. 51-3. 



Dr. Trail, in his life of Simson, gives the following 

 account of the discovery. 



" Dr. Simson maintained for some time his resolu- 

 tion of abstaining from all attempts at the rediscovery 

 of porisms; but happening one day to be walking 

 with some friends on the banks of the river Clyde at 

 Glasgow, and by accident being left behind his com- 

 pany, he inadvertently fell into a reverie respecting 

 porisms. 



" Some new ideas struck his mind, and with his chalk 

 having drawn some lines on an adjoining tree, at that 

 moment, for the first time, he acquired a just notion of 

 one of Euclid's porisms." 



The first publication of Simson on this subject, was 

 a paper inserted in the Philosophical Transactions for 

 the year 1723; it was not, however, until after his 

 death, that the whole of his investigations were made 

 public in the posthumous edition of his works, for 

 which the mathematical world is indebted to the muni- 

 ficence of the late Earl Stanhope. Some few years 

 after, this subject attracted the attention of Mr. Play- 

 fair, who has given a most philosophical account of the 

 origin of this class of propositions, and has removed 

 whatever obscurity remained attached to them. The 

 paper in which his views are explained, is indeed a 

 model of that peculiarly beautiful style of writing for 

 which he was so justly celebrated, and which is un- 

 fortunately so rarely met with in the literary produc- 

 tions of mathematicians. 



In the geometrical explanation of porisms. we shall 

 avail ourselves of the light which he has thrown on 

 the subject, and then endeavour to supply those obser- 

 vations which he promised respecting their algebraical 

 investigation. 



The definition of porisms which Simson has given, 



* (Euvrcs de Simon Stevin, par A. Girard. Lugd. Bat. 1634. 



f- Kxcrcitationes (ieometria?. Parisiis, 1667. 



J Format Opera Varia, p. 1 16. Tolos, 1679. 



Tr ansactlont of tfu Koyal Society of Edinburgh, 1793, vol. iii. 



