232 



and that the one, as well as the other, consists of vertebrae modified 

 to meet the requirements of the parts in which they are found. He 

 concluded by stating that the greater number of those anatomists to 

 whose observations we are indebted for most of our knowledge of 

 the development of the skull and of the trunk, are agreed that the 

 differences between the mode of formation of the segments in the 

 two form no real argument against the vertebral character of either ; 

 and he thought stronger reasons must be adduced than had yet been 

 shown before the anatomists could be called upon to abandon the 

 vertebral theory of the skull. 



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 -f- 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, 1865. 



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- 



