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Proceedings of the Royal Society of Edinburgh. [Sess. 
mistake. The idea that the subject has reached its present condition by 
the labours of many workers, largely obscure, is very helpful to learners, 
and gives a humanistic trend to the study of geometry. A Key to the 
Elements was published in 1885. 
It is almost needless to say that Dr Mackay did not view with favour 
the departure from the Euclidean sequence. He held that some logical 
sequence is necessary, and that Euclid’s is superior to any more recent 
innovations. Signs are not wanting that his views are now being shared 
by a growing number of mathematicians, who detect in our present 
system too much looseness and slovenliness. He was requested to write 
a text-book of geometry in accordance with the recent movement ; and 
although he complied with the request and produced his Plane Geometry , 
books i-iii in 1904, and books iv-v in 1905, they naturally have not 
the characteristic features of the earlier work. His Arithmetic Theoretical 
and Practical appeared in 1899, and forms one of the soundest and 
most illuminating books we have on the subject. 
This short account of his work will show the great service Dr Mackay 
rendered to mathematical learning, and the loss the scientific world has 
sustained by his death. 
