44 
Proceedings of the Royal Society 
which continued for some time after the experiment was made. The 
effect of this rotating apparatus might form an interesting study in 
connection with the important investigation at present conducted by 
Professor Alex, Crum Brown on the function of the semicircular 
canals of the ear. 
3. On the Principles of the Logical Algebra; with Applications. 
Part I. By Alexander Macfarlane, M.A., D.Sc. 
(Read 16th December 1878.) 
(Abstract.) 
In this memoir I examine the principles of the logical calculus of 
Boole, as laid down in his celebrated treatise on the “ Laws of 
Thought,” and also the criticisms which have been published con- 
cerning these principles. I bring forward a new theory of the 
operation of the mind, founded upon an analysis of language 
and the nature of mathematical reasoning, which enables me to 
correct these principles, to place them on a clear, rational, and 
generalised basis, and to show that there is a logical algebra which 
coincides with the ordinary algebra when its symbols are integral, 
but is a generalised form of the ordinary algebra when its symbols 
are fractional. Hence all the theorems in ordinary algebra when 
generalised properly are true in the logical algebra. I show the 
analytical meaning of the axioms of logic and their relation to the 
algebraic axioms of operation. By means of this algebra I inves- 
tigate the theory of immediate inference, and also the conclusions 
and numbers of conclusions of different kinds which can be deduced 
from premises of certain given forms. The memoir also professes 
to prove a great number of new theorems in the theory of necessary 
and probable inference. 
4. Note on Ulodendron and Halonia. By Mr D’Arcy 
Wentworth Thompson. Communicated by Sir Wyville 
Thomson. 
