19 



Art. I. — The " Three Sectioaif," the " Tangencies,'' and a 

 "Loci Problem'^ of ApoUonliis, and Porismaiic Develop- 

 menls. By ]Maiitix GaudixeRj C. E. (formerly Science 

 Scholar, Queen's College, Gahvay), 



[Read before the Royal Society, Jime 4, 18G0. ] 

 PRELIMINARY OBSERVATIONS. 



In the Transactions for 1859 I promised solutions to the 

 celebrated problems of the Greek and French schools, and 

 the present paper is the first instalment towards the fulfil- 

 ment of that promise. 



I commence with the problems of Apollonius, known as 

 his " Three Sections/' and " Tangencies," and the principal 

 problem of his treatise on Loci ; but I propose also the con- 

 tinuation of the development of interesting " Porisms.'' 



The problems of the Three Sections are famous from the 

 number of geometers who have assayed their solutions. 



"Willebrord Snel (the first person who measured the length 

 of an arc of the meridian by means of a geodetic survey), who 

 was born at Oudewater, in Holland, in the year 1590, was 

 the first geometer of eminence to restore the Section of Ratio. 

 His solution was published at Leydeu, in 1608. Early in the 

 eighteenth century. Dr. Halley discovered an Arabic manu- 

 script in the Bodleian Library containing distinctinvcstigations 

 to the numerous subdivisions of the Section of Ptatio, a Latin 

 edition of which he published at Oxford in the year 1706; 

 but there is no evidence as to whether this relic is a transcript 

 from the original of Apollonius, or merely a string of solutions 

 to its various cases by some other geometer; it covers 138 

 pages. Since then the principal solution is that by Reuben 

 Burrow, vv^hich was published about the year 1780 in his 

 " Apollonius." An application of the problem may be seen 

 in David Gregory's Astronomy. 



The Section of Space received an original solution from 

 Dr. Halley, which is similar to that given in Leslie's Geome- 

 trical Analysis. Other solutions may be found scattered 

 througli mathematical periodicals ; but as they are all similar 

 and incomplete, they deserve no particular notice. 



Indeed, q^n unaccountable neglect has besn shown to this 

 problem by the geometers who attempted the other " Sec- 



c2 



