406 PHILOSOPHICAL TRANSACTIONS. [aNNO 1708. 



30 years ago they made lime of the shells, and manured their lands with it ; but 

 a jMDor countryman, who out of laziness or poverty had not provided to make 

 lime, threw the shells unburnt on his land, and his crop proved as good as his 

 neighbours, and the second and third crop better; from this, all took the hint, 

 and have used them so ever since. Where shells are not to be procured, sea- 

 wrack or sand supply the want of them, but are not so good: sea- wrack lasts 

 but 3 years, and sand not much longer. 



It is certain that Ireland has formerly been better inhabited than it is at pre- 

 sent : mountains, that now are covered with bogs have formerly been ploughed ; 

 for on digging 5 or 6 feet deep, they discover a proper soil for vegetables, and 

 find it ploughed into ridges and furrows. This is observable in the wild moun- 

 tains between Ardmah and Dundalk, where a redoubt is built, and likewise on 

 the mountains of Altmore ; the same, it seems, has been observed in the 

 county of Londonderry and Donegal ; a plough was found in a very deep bog 

 in the latter, and a hedge with wattles standing, under a bog that was 5 or 6 

 feet deep above it. I have seen the stump of a large tree in a bog 10 feet deep 

 at Castle-Forbes ; the trunk had been burnt, and some of the cinders and ashes 

 lay still on the stump. I have also seen large old oaks grow on land, that had 

 the remains of ridges and furrows. And I am informed, that on the top of a 

 high mountain in the north, there are still remaining the streets and traces 

 of a large town ; and indeed there are few places which do not visibly, when 

 the bog is removed, show marks of the plough ; which must prove that the 

 country was well inhabited. It is likely that the Danes first, and then the 

 English, destroyed the natives, and the old woods seem to be about 3 or 400 

 years standing; which was near the time that Courcey and the English subdued 

 the north of Ireland ; and it is likely made havoc of the people that remained, 

 after the Danes were driven out of Ireland. 



Of the Length of Curve Lines. By the Rev. John Craig. N°314, p. 64. 



Translated from the Latin. 



Lemma. To divide the Sum of two Squares into two other Squares. — Let z^ 



and u* be two given squares, whose sum i^ -|- m* is to be divided into two 



other squares .r% y"^ ; also let m and n be any two numbers taken at pleasure. 



Now, from the condition in the problem, it is .r^ -\- y^ z=. i^ -j. t^* j hence, as 



r T>- u L • (mm — nny z -f 2mnu j • (nn — mm) u + 3mnai 



appears from Diophantus, .r = = . , and y = ^ jj*_-t_*»^ 



'^^ * mm + nn ' '^ mm + nn 



Problem. To find innumerable Curves, of the same Length with any pro^ 

 posed Curve, whether Algebraical or Transcendental. — Let z, u denote the co-or- 

 dinates of the proposed curve, and x, y the ordinates of the required curve. 



