Re: C.S. Peirce • Elements of Mathem atics (MS 165)

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Note 3

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Peircers,

Here is another variant of the preface to MS 94 that Carolyn Eisele attached to the "lost-and-found" MS 165.

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NEM 2, pp. 4–6
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PREFACE (2) (94a)

Benjamin Peirce's _Elementary Treatise on Geometry_ was published in 1837, and since he had been teaching the subject in Harvard College (beginning 
two years before his graduation) for ten years, it is safe to infer that the substance of the work had been in his mind for a long time.

The whole aspect of the science has been metamorphosed since say 1830 or 1835.  In the first place, the nature of the hypotheses (that is, the Axioms 
and Postulates, together with other propositions virtually taken for granted in the old books without explicit statement) is differently conceived. 
We all see, now, that geometry has two parts;  the one deals with the _facts_  about real space, the investigation of which is a physical, or perhaps 
a metaphysical, problem, at any rate, outside of the purview of the mathematician, who accepts the generally admitted propositions about space, 
without question, as his _hypotheses_, that is, as the ideal truth whose consequences are deduced in the second, or mathematical, part of geometry.

In the second place, Listing and others have created the topical branch of geometry, which studies the connection of places.  This branch deals with 
only a portion of the hypotheses accepted in other parts of geometry;  and for that reason, as well as because of its relative simplicity, it should 
be studied before the others.  Moreover, it is most desirable that, before the scholar comes to the difficulties which in the old system meet him at 
the threshold of geometry, he should have had some previous training leading up to the mental effort which he is then called upon to exert.

In the third place, although in 1830 not a little had really been done by individuals toward restoring that _graphical_, or projective, or 
intersectional, branch of geometry, which formed so prominent a part of the ancient science, yet it remained, for the general world of mathematicians, 
a closed book.  It has since become the most prominent part, not only of geometry, but of all mathematics.  For more than forty years, now, geometers 
have clearly perceived that metrical geometry is but a special problem of graphical geometry.  There is reason to suspect that in the school of Euclid 
some instruction in this branch preceded the study of the Elements.  At any rate, in its modern development, some slight treatment of it, so far, at 
least, as to explain the practically important principles of Linear Perspective, is a needed propaedeutic to metrical geometry.

In the fourth place, the whole conception of metrical geometry has been revolutionized.  In 1837, the "Doctrine of Parallels" formed an urgent but 
unsolved problem.  The earnest and persistent efforts of Legendre and of many other eminent mathematicians had been powerless to clear up its 
difficulties.  Benjamin Peirce, in his treatise, thought to conquer them by the introduction of the idea of a _difference of direction_.  There can be 
no intelligent question that this idea brings strong help to mathematical inquiry.  Soon after its introduction by Peirce, it was taken up by Hamilton 
and by Grassmann with such effect as might have been anticipated.  But Professor Peirce himself subsequently admitted, with the rest of the 
mathematical world, that there was no solution for the question of parallels except from the idea which the pupils of Gauss, Bolyai, Lobatchewsky, 
Riemann, derived through their master, from Lambert, and ultimately from the Italian Jesuit, Saccheri.  The idea was that it is simply a question for 
observation of nature whether the sum of the angles of a triangle is less than, or more than, or possibly equal to two right angles.  Subsequently, 
Cayley (in 1854) and Klein (more fully, in 1873) showed that metrical geometry is simply the geometry of the _firmament_, or _absolute_, or infinitely 
distant part of space, which constitutes a surface which is one or another quadric surface, according to the system of measurement adopted, that is, 
according to the way in which rigid bodies move.  In the course of this inquiry, a fallacy in Euclid's 16th proposition was brought to light that had 
remained undetected for two thousand years.

In the fifth place, Georg Cantor and others have succeeded in analyzing the conceptions of infinity and of continuity, so as to render our reasonings 
concerning them far more exact than they had previously been;  and the fundamental researches that have largely occupied mathematicians of late years, 
into the theory of functions, do much to render geometrical reasonings more exhaustive and precise.

All these intellectual movements ought, in the opinion of very many mathematicians, to have their effects upon the system of teaching the elements of 
the subject.

But this is not all.  Pedagogy is an art which has come in the last sixty years to be based more and more upon modern scientific psychology, and upon 
modern views of logic.  If diligent and intelligent youths find difficulty in understanding mathematics, teachers no longer deem it becoming to flog 
them or to objurgate them, as they used to do, in the days when the second theorem of Euclid received the name of the "Asses' Bridge."  On the 
contrary, they consider the fact that a considerable percentage of the best minds imagine themselves to be utterly unable to comprehend mathematical 
reasoning to constitute an emphatic condemnation of that old system of teaching which had such a result.

Teachers now see that the difficulties of the first steps in mathematics must be divided and conquered, one by one.  The three functions of the mind 
that are exercised in mathematics are exact reasoning, mathematical imagination, and complex generalization.  The first of these, the logical part, is 
best acquired in the study of the theory of number, because that subject involves little other difficulty.  It calls for but slight efforts of 
imagination and of generalization.  Topology, or connective geometry, is the best field for the growth of imagination, demanding little logic and not 
very much generalization.  Graphics, or projective geometry, carried far enough, affords good training in generalization.  When some familiarity with 
the business of the mathematician has been acquired by such studies, metrical geometry may be taken up without fear that the student's mind will be 
confounded by its aggregation of various difficulties.

This volume is intended to contain all the mathematics (except practical arithmetic) which is necessary for a man with a good common school education, 
and at the same time, to give the thoughts of the student such training as may prepare him for a study of the higher mathematics.

Most of the text books of geometry have contained some algebra.  Euclid's Elements is more than half devoted to that branch.  Benjamin Peirce's work 
has a brief algebraical introduction.  The present volume gives all the algebra which is indispensable to an ordinary man.

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