S4 The Three Sections, Tangencies, 



is real^ is applicable only in particular states of the data. In 

 the Notes will be found a genuine ancient porism, from Avliich 

 the problem originated. 



I might also mention that Francis Van Schooten^ Professor 

 of Mathematics at Leyden^ published a restoration of some of 

 the particular cases of this problem, in 1657 ; and that a like 

 task was performed in an algebraic form by Fermat^ Coun- 

 cillor to the Parliament of Toulouse, in his Opera Yaria 

 Mathematica_, published in the year 1679. 



It is scarcely necessary to remark^ that in the present 

 improved state of Algebraic Geometry^ it would be an easy 

 matter to solve the general case of this problem; but^ to arrive 

 at a construction of the Locus, such as I give, would be 

 impossible without introducing other geometrical considera- 

 tions than those to be found in ordinary Algebraic Elements ; 

 besides, the complete discussion would present difficulties 

 which none but experienced analysts could overcome. 



The "Porisms" in the present paper, with those in the 

 Transactions for 1859, belong to the most numerous and 

 useful system in the whole range of elementary theorems. 

 Some few of them — as is evident from Professor Davies^ con- 

 tributions to the Mathematician — have been already noticed 

 by Mr. ISIark Noble, and by Professors Playfair and Wallace ; 

 but their number is so few^ that when they occur I 

 will make no scruple of reproducing them amongst the 

 classes to which they belong. I have already given proof of 

 their efficiency in the solution of difficult problems. 



They are most probably but restitutions of a part of the 

 lost treatise of Euclid, known as his Second Elements — com- 

 posed when his geometrical knowledge was fully matured, and 

 which, there is strong reason to suspect, contained all the 

 principles developed in the elementary writings of Gergonne, 

 Poncelet and Chasles. 



Having said thus much relating to the substance of my 

 paper, I think it' right, before closing these preliminary re- 

 marks, to explain the nature of the improved ideas and 

 theorems on which the spirit of my investigations is mainly 

 dependent. 



To do this, I may at once state that all the great masters 

 of Logic have observed that there are two points which must 

 be rigorously attended to in correct systems of reasoning. 



First : — That the propositions employed as premises are 

 unambiguous, and correctly understood. 



Second ; — That the steps (the auxiliary operations and 



