No. ILI. 
February 10, 1845. (See page 64.) 
The spirit of Sir William Hamilton’s communication, which 
was designed as a further illustration from geometry of the 
author’s theory of algebraic quaternions, consisted in re- 
garding operations on such quaternions as admitting of being 
ultimately interpreted as operations on straight lines ; each 
line being considered as having not only a determinate length, 
but also a determinate direction. The quotient ” obtained by 
the division. of one such line (b) by another (a), is, generally, 
in the author’s view, a quaternion ; it depends, in general, on 
Jour distinct elements, of which one, namely, the modulus, is 
a positive or absolute number expressing the relative magni- 
tude of the dividend and divisor lines, while the three other 
elements serve jointly to express the relative direction of 
those two lines. Of the three latter, one is the amplitude, and 
marks the inclination of one line to the other, or the magni- 
tude of the angle which they include; while the two others 
_ determine the plane of that angle, and are what have been 
called, in a former communication, the directional coordi- 
nate, such as the longitude and colatitude of the quaternion. 
In this comparatively geometrical view, as in the more alge- 
braical view which was formerly stated to the Academy, the 
_ consideration of these four elements, modulus, amplitude, lon- 
gitude, and colatitude, presents itself, therefore, naturally. We 
_ may also speak of the avis of a quaternion, meaning thereby 
_ the axis perpendicular to the plane of the two straight lines of 
which that quaternion is a quotient; and may say, that such 
an axis is itself positive or negative, or that it is taken in the 
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