192 



On the ORIGIN and 



and to remark that, in one of them, it became indeterminate ; 

 and that, by a curious coincidence, this happened in the 

 only cafe which could be fuppofed applicable to the aftronomi- 

 cal problem above mentioned ; in other words, he found, that 

 in the ftate of the data, which mull there always take place, 

 innumerable lines might be drawn, that would be all cut 

 in the fame ratio, by the four lines given in pofition. This 

 he demonstrated in a differtation publifhed at Rome in 1749, 

 and fince that time in the third volume of his Opufcula. A 

 demonstration of it, by the fame author, is alfo inferted at the 

 end of Castillon's Commentary on the Arithmetica Univer- 

 salis, where it is deduced from a construction of the general 

 problem, given by Mr Thomas Simpson, at the end of his Ele- 

 ments of Geometry*. The propofition, in Boscovich's words, is 

 this: " Problema quo qugeritur recta linea quas quatuor rectas po- 

 " fitione datas ita fecet, ut tria ejus fegmenta {int invicem in ra- 

 " tionedata, evadit aliquandoindeterminatum, ita ut per quod- 

 " vis punctum cujufvis ex iis quatuor rectis duci pofTit recta. 

 " linea, quse ei conditioni faciat fatis f." 



It is needlefs, I believe, to remark, that the propofition thus 

 enunciated is a Porifm, and that it was difcovered by Bosco- 

 vich, in the fame way, in which I have fuppofed Porifms to 

 have been firft difcovered by the geometers of antiquity. I 

 lhall add here a new analyfis of it, conducted according to 

 the method of the preceding examples, and to which the fol- 

 lowing lemma is fubfervient. 



LEMMA 



* Elements, p. 243. Edit. 3. Simpson's folution is remarkably elegant, but no men- 

 tion is made in it, of the indt ttrminait cafe. 



} Jos. Boscovich Opera, BafTani. torn. 3. p. 331. 



