Re: C.S. Peirce • New Elements of Ge ometry Based on Benjamin Peirce's Works and Teachings (MS 94)

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Note 2

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Peircers,

Here is the Preface to MS 94.

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NEM 2, pp. 235–237
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PREFACE

That elementary geometry is in a disgracefully antiquated condition is an opinion in which the leaders of mathematical thought seem at one.  Its 
superficial arrangement, its inaccurate logic, and far worse yet its dearth of mathematical life render it unsuitable as an introduction to 
mathematics and put a drag upon all other mathematical exposition which has to be accommodated to what the current text books of Elementary Geometry 
have taught.

Those of us who are getting into the sere and yellow leaf were in our early Youth when Cayley made his famous discovery that the metrical geometry is 
but a special case of the projective;  and even those who have [at] best a very slight and secondhand acquaintance with modern mathematics are 
accustomed to join the chorus of admiration at the deep truth of this doctrine and its illuminative value.  Yet during all these years there has been 
hardly any attempt at all to reflect some portion of the light from the Sun of geometry upon the path of the beginner.  Theoretically everybody says 
that metrics is founded in graphics;  but very little has been done to put this creed into practice.

That projective geometry in its turn ought to be founded upon topology is a proposition not so often repeated, but as firmly believed by the 
comparatively small number of those who have occupied themselves with topological problems.

No doubt what has postponed a reform generally pronounced necessary has been the knowledge that most of the teachers are so miserably overworked that 
they cannot be expected to teach a book which would necessarily contain propositions they had not met with before.  But there are now unmistakable 
signs that the postponement cannot last much longer;  and the editors of this book felt that they would not be doing justice to their father's memory, 
if they allowed another edition to appear without some sign of consciousness that we are living in a new mathematical age when a Euclidean way of 
reasoning will not pass even with the general public for true geometrical thinking.

Not only is topology entitled logically to the first place, but it is particularly well adapted to give the first training to the geometrical 
imagination, without taxing too much the powers of abstract concentration.  A similar remark applies to the assignment of the second place in the 
order of exposition to perspective and its family of geometrical operations.  While calling for a good deal more thought than does topology, it is 
vastly easier than the metrical geometry into which it has been the custom to force the student at the very outset.  Projective geometry is easier 
than metrical geometry could be made;  but when we consider that metrical geometry is usually taught while expecting the unfortunate youth to pick up 
by instinct a highly artificial and not very sound logical system at the same time, and that it is taught by teachers who, however accomplished, 
cannot be expected to possess the compound of psychological knowledge, logical knowledge, and mathematical knowledge required to enable them to see 
wherein their students' difficulties consist, teachers worn down to the very raw of their patience and their powers, so far from wondering that a good 
many of the students fail to “catch on” to geometry, we ought rather to admire the adaptedness of the human mind as displayed in so large a number of 
them making out what is meant.

Without any doubt elementary geometry gives many students an impression which they carry through life that they have no capacity for mathematics, when 
it is really because they have a natural turn for real mathematics that the elements of geometry have filled them with disgust.  We are assured by 
actual experiment that the method of bringing the subject to the apprehension of the student here employed and of training his different powers of 
imagination, of generalization, and of concentration will in many cases convert the least hopeful of the geometrical dunces into bright and strong 
intellects.

The proper course of instruction consists in beginning by awakening the mind of the pupil to mathematical thought.  The teacher and pupil must be 
prepared to go on for a good while without manifest results.  But the effect will be that after a time the student will “catch on” to the kind of 
thought required, — a most important power for all his after life, — and will suddenly make a start.  It is best to leave the book on the fundamental 
properties of space to be referred to as required and begin with topology.  Here is something really easy, but so presented that the pupil will, after 
patient trial, be able to ascertain whether or not he has a natural imagination for mathematical images.  There is nothing in which minds differ more. 
  A musical ear is less often wanting than a mathematical imagination.  But let a strenuous effort be made;  for mathematical thought is by far the 
best of all intellectual exercises.  In the traditional mode of teaching, many and many a mind, really capable of gaining this inestimable advantage, 
lost it merely for want of comprehension of the excessively artificial and almost sophistical logic of the Elements, and because his mind was really 
never tested upon modern mathematics.

At the same time, the book has so been put together that those teachers who prefer to begin in the usual way have only to put their pupils at the 
start into the second part of the book of Metrics and reserve the rest for a later period.

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Regards,

Jon

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