376 Cambridge Philosophical Society : — 



By Professor De Morgan " On the Early History of the Signs + 

 and — ." 



An account is given of the work on arithmetic of John Widman, 

 printed in 1489, in which the signs + and — are used to denote 

 more and less. The use made is twofold: a-\-b signifying that b 

 more than a is wanted, infers a direction to add b to a. But a-\-b 

 in the old rule of false position is used to signify that the assumption 

 of a for the answer gives b too much in the solution. This last 

 usage was continued by many writers through the greater part of the 

 sixteenth century. 



Some account is given of the Die Coss of Chr. Rudolf, which 

 passes for the first work in which + and — are used. The first 

 edition of his work being lost, a question is raised as to how far 

 the second edition, edited by Stifel, is a fair reprint of the first. A 

 Latin translation of this first edition is said to be in the Imperial 

 Library at Paris. 



From the mannner in which Widman introduces his signs, Mr. 

 De Morgan thinks there is some ground to suspect that they were 

 originally warehouse marks, indicating the scale into which smaller 

 weights were introduced to make the balance, when the nearest 

 number of larger weights had been put in. This point and others 

 require the examination of older works, print and manuscript. 



February 13, 1865. — Communications were made by Professor 

 Cayley " On Abstract Geometry." 



By Professor Clifton," Note on the Early History of the Signs + 

 and -." 



February 27.— Mr. Alfred Newton, M.A., F.L.S., communicated 

 some " Notes on Spitzbergen," of which the following is an abstract. 



The author stated that last summer he accompanied Mr. Edward 

 Birkbeck on a voyage to Spitzbergen, in that gentleman's yacht, the 

 'Sultana,' R.S.Y.C, After giving a slight sketch of some of the 

 principal voyages which had been made to that country, he pro- 

 ceeded to say that the 'Sultana'' left Hammerfest on the 3rd of 

 July, in company with a Norwegian sloop which had been specially 

 fitted to encounter the ice, and chartered to attend upon the yacht. 

 On the 6th they arrived at the entrance of Stor Fjord, which was 

 found to be entirely blocked by the ice, Horn Sound and Bell 

 Sound were subsequently discovered to be in the same condition. 

 They then made for Ice Sound, and anchored in Safe Haven on the 

 9th. Ice Sound was described as a very much larger inlet than it is 

 represented in the charts to be, extending at least fifty miles into 

 the interior. All the valleys on the north side, and consequently 

 having a southern aspect, are completely occupied by large glaciers, 

 which, with one exception, are only terminated by the sea. The 

 single exception consists of a small but remarkable glacier suspended 

 on a hill-side, some 360 yards from the beach, resting conformably 

 on its own moraine, and having no apparent means of discharge. 

 The author supposes this last was effected by filtration through its 

 bed. The south side of Ice Sound contains several bays of consi- 

 derable size j and the valleys opening upon it, and therefore having 



