David Eugene Smith

“ Architecture should still be artistic and construction should still be substantial, else a building can never endure. So geometry must still exemplify good logic and must still bring to the pupil a feeling of exaltation, or it will perish and become a mere relic in the museum of human culture.
What, then, shall the propositions of geometry be, and in what manner shall they answer to the challenge of the industrial epoch in which we live? In reply, they must be better adapted to young minds and to all young minds than Euclid ever intended his own propositions to be. ”

“ It may seem strange that geometry is unable to define the terms which it uses most frequently, since it defines neither movement, nor number, nor space,—-the three things with which it is chiefly concerned. But we shall not be surprised if we stop to consider that this admirable science concerns only the most simple things, and the very quality that renders these things worthy of study renders them incapable of being defined. ”

“ While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid. ”

“ First, then, and foremost as a reason for studying geometry has always stood, and will always stand, the pleasure and the mental uplift that comes from contact with such a great body of human learning, and particularly with the exact truth that it contains. The teacher who is imbued with this feeling is on the road to success, whatever method of presentation he may use; the one who is not imbued with it is on the road to failure, however logical his presentation or however large his supply of practical applications. ”

“ The principle that the writer has so often laid down in other works, that whatever pretends to be practical should really be so, applies with much force to these exercises. When we can find the genuine application, if it is within reasonable grasp of the pupil, by all means let us use it. But to put before a class of girls some technicality of the steam engine that only a skilled mechanic would be expected to know is not education,—it is mere sham. There is a noble dignity to geometry, a dignity that a large majority of any class comes to appreciate when guided by an earnest teacher ”

“ In general, let it be said that any exercise that pretends to be real should be so, and that words taken from science or measurements do not necessarily make the problem genuine. To take a proposition and apply it in a manner that the world never sanctions is to indulge in deceit. On the other hand, wholly to neglect the common applications of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil, to arouse new interest, and to give a meaning to it that is otherwise wanting. ”

“ If we say that a tangent to a circle is an unlimited straight line that touches the circle in one point, and only one, we do not assert that it is possible to have such a line; that is a matter for proof. Not in all cases, however, can this proof be given, as in the existence of the simplest concepts. We cannot, for example, prove that a point or a straight line exists after we have defined these concepts. We therefore tacitly or explicitly assume (postulate) the existence of these fundamentals of geometry. ”

“ The study of grammar does not make an accurate writer, nor the study of rhetoric an orator, nor the study of meter a poet, nor the study of pedagogy a teacher. The study of geography in the school does not make travel particularly easier, nor does the study of biology tend to populate the earth. So we might pass in review the various subjects that we study and ought to study, and in no case would we find utility the moving cause, and in every case would we find it difficult to state the one great reason for the pursuit of the subject in question,—and so it is with geometry. ”

“ What teacher sees in biology all the grandeur of the evolution of the race, or imparts to his pupils the noble lessons of life that the study of this subject should suggest? What teacher of Latin brings his pupils to read the ancient letters with full appreciation of the dignity of style and the nobility of thought that they contain? ”

“ We thus come to the modern distinction between axiom and postulate, and say that a general statement admitted to be true without proof is an axiom, while a postulate in geometry is a geometric statement admitted to be true, without proof. For example, when we say "If equals are added to equals, the sums are equal," we state an assumption that is taken also as true in arithmetic, in algebra, and in elementary mathematics in general. ”

“ In America it is not alone the boy who is interested in education in general, or in mathematics in particular, who studies geometry, and who joins with others of like tastes in this pursuit, but it is often the boy and the girl who are not compelled to go out and work, and who fill the years of youth with a not over-strenuous school life. It is therefore clear that we cannot hold the interest of such pupils by the study of Euclid alone. Geometry must, for them, be less formal than it was half a century ago. ”

“ Let the home, the workshop, physics, art, play,—all contribute their quota of motive to geometry as to all mathematics and all other branches. But let us never forget that geometry has a raison d'être beyond all this, and that these applications are sought primarily for the sake of geometry, and that geometry is not taught primarily for the sake of these applications. ”

“ Pupils do not object to work if they like a subject, but they do object to aimless and uninteresting tasks. Geometry is particularly fortunate in that the feeling of accomplishment comes with every proposition proved; and, given a class of fair intelligence, a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that brings the most pleasure, and that seems the most profitable of all that are studied in the first years of the high school. ”

“ In every generation it behooves the open-minded, earnest, progressive teacher to seek for the best in the way of improvement, to endeavor to sift the few grains of gold out of the common dust, to weigh the values of proposed reforms, and to put forth his efforts to know and to use the best that the science of education has to offer. ”

“ A pupil has a passing interest in geometric design. He should learn to use the instruments of geometry, and [Pg 77] he learns this most easily by drawing a few such patterns. But to keep him week after week on questions relating to such designs of however great variety, and especially to keep him upon designs relating to only one or two types, is neither sound educational policy nor even common sense. That this enthusiastic teacher or that one succeeds by such a plan is of no significance; it is the enthusiasm that succeeds, not the plan. ”

“ We cannot first weigh a pupil in an intellectual or moral balance, then feed him geometry, and then weigh him again, and then set back his clock of time and begin all over again with the same individual. There is no "before taking" and "after taking" of a subject that extends over a year or two of a pupil's life. We can weigh utilities roughly, we can estimate the pleasure of a subject relatively, but we cannot say that geometry is worth so many dollars, and history so many, and so on through the curriculum. ”

“ It was pointed out by Aristotle, long before Euclid, that it is not a satisfactory procedure to define a thing by means of terms that are strictly not prior to it, as when we attempt to define something by means of its opposite. Thus to define a curve as "a line, no part of which is straight," would be a bad definition unless "straight" had already been explicitly defined; and to define "bad" as "not good" is unsatisfactory for the reason that "bad" and "good" are concepts that are evolved simultaneously. ”

“ There is no more reason for any human being on God's earth to be acquainted with the binomial theorem or the solution of triangles, unless he is a professional scientist,—when he can begin to specialize in mathematics at the same age as the lawyer begins to specialize in law or the surgeon in anatomy,—than for him to be expert in Choctaw, the Cabala, or the Book of Mormon. ”

“ He claims that to begin Latin and Greek at the same time has always proved to be confusing, and that the same is true of algebra and geometry. He grants that unified knowledge is desirable, but he argues that when the fine arts of music and color work fuse, and when the natural sciences of chemistry and physics are taught in the same class, and when we follow the declension of a German noun by that of a French noun and a Latin noun, and when we teach drawing and penmanship together, then it is well to talk of mixing algebra and geometry. ”

“ God is a circle of which the center is everywhere and the circumference nowhere.—Rabelais.
Without mathematics no one can fathom the depths of philosophy. Without philosophy no one can fathom the depths of mathematics. Without the two no one can fathom the depths of anything.—Bordas-Demoulin.
We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly understood of all truths, are on this account the most susceptible of ready application in reasoning.—D'Alembert. ”

“ It is good educational policy for the teacher to commend at least as often as he finds fault when criticizing a recitation at the blackboard and when discussing the pupils' papers. Optimism, encouragement, sympathy, the genuine desire to help, the putting of one's self in the pupil's place, the doing to the pupil as the teacher would that he should do in return,—these are educational policies that make for better geometry as they make for better life. ”