298 
THE “ OUTLINES OP QUATERNIONS. 
scription wliicli was awarded the Prize Medal of the United Service 
Institution. Colonel Hime has always the courage of his opinions and 
his literary motto is “ L’audace, Vciudcice, tonjours Vaudace” 
In the present work he has let the light in on another facet of his 
many-sided intellect and appears as a disciple of Sir William Rowan 
Hamilton in that abstruse field of Pure Mathematics known as the 
“ Theory of Quaternions.” 
He, like Colonel Hime, was a graduate of Trinity College, Dublin. 
Some sixty years ago his attention was drawn to the attempts that had 
been made by many geometers to give a geometrical interpretation to 
the imaginary quantity symbolized by the square root of negative unity 
s/ — 1. Descartes, Newton and Euler had discussed the quantity, bat 
merely from an algebraic stand point. H. Kuhn, of Dantzic, 1750, 
appears to have been the first to associate the symbol with geometrical 
perpendicularity. He regarded a V — 1 as representing a line per¬ 
pendicular to a line a and equal to a in length and was followed by 
Argand who interpreted the complex number a+b V — 1. Sir William 
Hamilton based himself on the conception of Kiihn. The notion of a 
“ Vector,” a straight line which has both magnitude and direction, next 
occupied his mind and he was led, as also was Grassmann about the same 
time, to the theory of the geometric addition of vectors in space. The 
great step made by Hamilton was the passage from vector algebra to 
the formation of an operational calculus. A quaternion is an operator 
which turns any one vector into another. It operates in two ways (1), 
by tension, positive or negative (2), by torsion or version and is found 
to involve a knowledge of four numbers. From this occurrence of the 
number four the name “quaternion” is derived. Hamilton elaborated 
the theory to an extraordinary extent and showed the power of the new 
calculus by extensive applications both to Pure Mathematics and to 
Physics. His “Lectures” appeared in 1852 and his “Elements” in 
1866. The great work given to the world in these two portly volumes 
was composed at the Dunsink Observatory when Hamilton was Royal 
Astronomer of Ireland. The reading of these works presents great 
difficulties even to professional mathematicians. Easier treatises have 
appeared of recent years, but none of them are on the same lines as the 
book before us. 
Colonel Hime adopts Clifford’s more general notion of a “ Vector,” 
defining it to be “ any quantity which has both magnitude and direc¬ 
tion,” and from the point of view of the numerous phj sical applications, 
this is no doubt preferable. 
After an exposition of the subtraction and addition of vectors in 
spaces of two and three dimensions, he proceeds in Part IT. to explain 
the nature of a “quaternion.” This is as clear and precise a state¬ 
ment of the first principles of this difficult-subject as has ever appeared 
in print. The elementary algebra and differentiation of quaternions 
is succeeded by an account of scalar and linear vector equations and 
the volume concludes w r ith numerous well-chosen illustrations of the 
power of the quaternion analysis in elementary geometry. 
In the columns of “ Nature ” and elsewhere a controversy has been 
