298 Mr DE morgan, on the general PRINCIPLES OF WHICH THE 



have been one of the postulated lines. The difficulties of angular section and quadrature of 

 the circle would never have existed : hut until it can be shown that some equally exciting 

 difficulties of construction would have opened into view from the position thus taken, it may 

 be almost affirmed that geometry would not have been the gainer. In the meanwhile, all the 

 struggles to arrive at general angular section and circular quadrature, so far as they were 

 made under Euclid's restrictions, may be described as attempts to combine translation and 

 rotation under the condition that translation and rotation are not to be combined. 



4. Velocity of translation and rotation. As these are merely, so far as measures are 

 concerned, the translations and rotations themselves defined as made in a given time, nothing 

 additional need be said on the working meaning of these terms. But I proceed to ask 

 whether the postulates are results of thought when applied to velocity, without reference to 

 its measure. 



Velocity without reference to its measure! Why, what is velocity but a measure .J* This 

 is sure to be the first question. I answer that if velocity be a measure, it must be a measure 

 of something: of wl:at.^ If of velocity, then it is no otherwise a measure than as every mag- 

 nitude is a measure of itself: and the word measure is superfluous. But the common sense of 

 mankind, which mathematicians have more than once endeavoured to stifle under a convention, 

 when a psychological difficulty would otherwise have demanded an investigation of its grounds, 

 recognises swiftness or quickness as a thing per se, magnitude in its nature, more or less, not 

 space, not time, not description of space in time, but a notion accompanying the description of 

 space in time, and not expressible by anything but of the same kind as itself. Whenever any 

 two magnitudes are continuously changed together, this notion arises: and what we treat 

 under the name of a differential coefficient was considered by the mediasval writers, but not 

 numerically, under the name of intension or remission, according as, in our language, it is 

 positive or negative. If we could suppose a particle of matter to have its changes of rapidity 

 consequent upon its own volition, and poetry bears witness that this is possible in thought, 

 what we call velocity would be the measure of the will to change state. Without going so far, 

 let it only be distinctly conceived that quickness admits of numerical measure somehow, 

 because quicknesses may be conceived under the relation of more and less : let it be conceived 

 that aggregation is possible, that it is independent of all but relative direction, that order of 

 aggregation is of no account, and that, in one and the same direction, quicknesses are aggre- 

 gated by addition. Suspend for a moment the question whether such abstraction be practicable 

 to us without the aid derived from a measure of quickness. It follows tliat the aggregation of 

 velocities of translation and rotation is established prior to the acquisition of a measure. 



Now it is to be observed that, in tiie matter of magnitude, as measurable by magnitude of 

 its own kind, independently of other magnitude, human reason is progressive, in a manner 

 which ought to check that disposition to put psychological thought to sleep which I have 

 adverted to as not uncommon among mathematicians. We are but just arrived at the full 

 notion of the angle, as to be expressed by angles only. When I was a student, works in 

 repute at Cambridge defined the angle as being an arc to the radius unity. A very modern 



