LECTURE II. GEOMETRY, THE FOUNDATION OF BY THOMAS HILL, OF WALTHAM, MASS. I PROPOSE as my thesis, that Geometry is the foundation of learning. It is said that Plato wrote over his school-room door, Let no one ignorant of Geometry enter here. And although the anecdote cannot be found in good Greek, and is, therefore, to be considered rather mythical, it deserves to have been true. It is the inscription, which is in fact written over all the higher schools of life. Geometry is required for admission into the high schools of nature, and is always taught in Nature's infant school. It has been sadly neglected by human teachers, since the invention of logarithms and other facilities for arithmetical computation; but it has remained the foundation of learning, and no man has ever arrived at any knowledge, until he first learned from Nature herself, unconsciously perchance, Geometry enough to build it upon. In order that you may not accuse me of overrating Geometry, and of underrating all other branches of knowledge, I will, before going further, give you a brief sketch of my views of a perfect education. A child is a spirit, a finite will, actuating a body, under the impulse of sentiment, appetite or passion, and by the guidance of reason. Hence he needs four sorts of education. For the spirit, or will, he needs religious guidance; for the body, he must have physical training; for the impulsive nature, a moral education; and for the reason, an intellectual education. So that the intellectual training of the schools is but one out of four indispensable branches of a true education. Take next the intellectual branch, and consider what studies are to be pursued. The grand circle of human science is divided into five sections: Theology, Psychology, History, Natural History, and Mathematics. Again,- Mathematics may be divided into Arithmetic, Algebra, and Geometry. So that Geometry is but one out of three branches of Mathesis, and Mathesis is but one out of five sections of human learning, but one of five courses of intellectual training, and intellectual training is but one of four indispensable branches of a true education. You see, therefore, that I do not overlook the rest of education, and allow Geometry to fill my whole horizon. But I nevertheless affirm, that Geometry is necessarily the first study of a finite mind; that we cannot conceive of a mind having a beginning and growth, that should not find in Geometry the only milk for its earliest intellectual nourishment. For if we take up the five divisions of Science which I have named as including all possible human knowledge, we shall find that they necessarily follow each other in the order in which I have placed them. Theology must necessarily be preceded by Psychology; we must know something of our own spiritual powers before we can know anything of Him, in whose image we were created. Psychology must be preceded by History; we must know something of the actions of men, something of the ways in which they have exercised their powers, and displayed their passions, before we can know what those powers and passions are. History must be preceded by Natural History; we must know something of the field. wherein men have acted, something of the materials whereon they have acted, before we can understand what they have done. Natural History must be preceded by Mathematics; we must know something of the laws of Space and Time before we can understand the phenomena subject to those laws. Mathesis, therefore, is, in order of time, the first of human sciences. The same conclusion would be reached, if we examined these five sciences in the light of the powers by which we apprehend them. We shall find that all knowledge rests on a double basis, of perception and conception; of sensation and consciousness. We shall find that of these powers the perceptive are first developed, the conceptive last. The infant only perceives, does not imagine nor reaHis powers of imagination and reasoning are developed through the exercise given by observation. Hence the natural order of education will be to teach, first, the sciences most dependent on observation; and son. 1 lastly, those most dependent on consciousness. Now this order will lead us first to Mathematics, in which consciousness plays the least important part, and so on to Natural History, History, Psychology, and Theology. The same conclusion that Mathematics is the first study to be pursued, will be attained if we look at the course which Divine Providence has pursued in the education of the race. Mathematics were the firstborn of human sciences, and the very name that they bear of Mathesis, or learning, shows that they date back to the time when there were no other sciences to divide the honor of that name with them. But of the three branches of Mathematics, Arithmetic, Algebra, and Geometry, which shall take precedence? Remember that I do not speak of precedence in importance, but of precedence in time. Arithmetic, the Science of Numbers; Algebra, of Time; Geometry, of Space, which comes first in the order of study. Beyond all controversy we must say Geometry. For although Arithmetic and Algebra are not directly dependent on Geometry, and the order of the three cannot thus be determined, yet by the other modes of inquiry the decision is very clear, Geometry is dependent almost entirely on sensation; Algebra, almost entirely on consciousness, and, therefore, Geometry should precede Algebra; while Arithmetic, being an abstraction, must necessarily depend either upon Algebra or Geometry; and, therefore, as Geometry precedes Algebra, Arithmetic cannot precede Geometry. The child begins to study Geometry as soon as it opens its eyes; and it distinguishes by the outline a |