398 REPORT—1883. 
space closely connected with the theory of the linear complex, as has been shown by 
Cremona. 
I have mentioned already the ‘ Analytical Mechanics’ of Lagrange, which-is 
without any trace of geometry, although there is scarcely a branch of applied 
mathematics which is in its very nature more geometrical. In fact one part of it, 
now separated as kinematics, treats solely of changes in position and shape of 
geometrical quantities, and differs from pure geometry only in this, that the 
changes are considered as referring not to space alone, but also to time. 
What mechanics gains by introducing geometry to the full will be apparent 
to all who have become acquainted with modern Continental text-books on the 
subject. 
‘Tet us compare the analytical with the geometrical reduction of a system of 
forces acting on a rigid body, or, to use Clifford’s nomenclature, the reduction of a 
system of rotors, which may represent either forces or rotations, or any other 
quantities which have certain fundamental properties in common with those, so 
that they may be represented by rotors. In the analytical process the system is 
reduced to a rotor and a vector, that is a resultant force and acouple. In the 
geometrical treatment we see that this is only one way of reducing the rotors to 
two, viz. the one which is best fitted to be treated by analysis. But there is a 
multitude of other reductions. These all appear as of equal importance in the 
geometrical method. Furthermore, this method shows us in the simplest way 
possible how all the line pairs which may be the lines of action of two resultant 
rotors, although there are infinities of infinities of such pairs, are arranged in space, 
so that one gets a clear picture of all these reductions in one’s mind. 
Again, compare Mobius’ geometrical investigation of the rays of light passing 
through a system of lenses with that of Gauss, whose very name suggests simplicity 
and elegance. The celebrated ‘cardinal points’ appear in Gauss’ original paper as 
the result of a somewhat long though certainly elegant analysis, whilst by Mébius 
they are the natural outcome of his geometry, so that any student once started 
on this method is bound to come across these points, or rather across pairs of 
points, of which the cardinal points of Gauss are only one special case, The whole 
is, in fact, contained in the following easily proved proposition: the rays of light start- 
ing from a point in the axis of the system before entering the first lens, and after 
leaving the last, form two homographic pencils in perspective position. 
This is only one small part of the advantage which optics can derive from 
geometry. 
That the old-established mode of teaching the elements of geometry based on 
Euclid requires a thorough and fundamental change has been often acknowledged, 
among others, at Exeter and Bradford, by two of the most eminent mathematicians 
who have occupied this chair, and besides by the many teachers who constitute the 
Association for the Improvement of Geometrical Teaching, which itself grew out 
of the action of our section. I know therefore of no opportunity better suited to 
review the progress made in this direction than the present one, as the subject has 
on several occasions occupied the attention of our section. Nevertheless I have 
hesitated at entering on this somewhat delicate question, because I fear that I have 
little to offer but criticism, which might seem hostile to the Association just named. 
But I hope that the many earnest workers, who have devoted much time and 
thought to the drawing up of syllabuses on different parts of our subject, will excuse 
the remarks of one who has himself tried his hand at the same work, and who 
therefore may be supposed somewhat to know the difficulties that have to be 
overcome, 
When the syllabus on the elements of plane geometry appeared, I resolved to 
give it a thorough trial, and took the best means in my power to form an opinion 
on its merits, by introducing it into one of my classes. The fact that it did not 
quite satisfy me, and that I gave up its use again, does not, of course, prove that it 
fails also for use in schools, for which it was originally intended. 
Let me add that the more I have become acquainted with the difficulty of the 
whole subject, the greater has become my admiration for Euclid’s book, whilst my 
conviction of its unfitness as a school book has equally gained in strength, 
