Reason they need not be possessed at all. They are fit matters of the study of the professed mathematician, when his general education is terminated. But of Geometry, on the other hand, it is not too much to say that it is a necessary part of a good education. There is no other study by which the Reason can be so exactly and rigorously exercised. In learning Geometry, as I have on a former occasion said*, the student is rendered familiar with the most perfect examples of strict inference; he is compelled habitually to fix his attention on those conditions on which the cogency of the demonstration depends; and in the mistakes and imperfect attempts at demonstration made by himself and others, he is presented with examples of the more natural fallacies, which he sees exposed and corrected. He is accustomed to a chain of deduction in which each link hangs from the preceding, yet without any insecurity in the whole; to an ascent, beginning from solid ground, in which each step, as soon as it is made, is a foundation for a further ascent, no less solid than the first self-evident truths. Hence he learns continuity of attention, coherency of thought, and confidence in the power of human Reason to arrive at the truth. These great advantages, resulting from the study of Geometry, have justly made it a part of every good system of Liberal Education, from the time of the Greeks to our own. 33 Arithmetic has usually been a portion of Education on somewhat different grounds; namely, not so much on account of its being an example of reasoning, as on account of its practical use in the business of life. To know and to be able familiarly to apply the rules of Arithmetic, is requisite on innumerable occasions of private and public business; and since this University Education, p. 139: Thoughts on the Study of ability can never be so easily or completely acquired as in early youth, it ought to be a part of the business of the boy at school. For the like reasons, Mensuration ought to be learnt at an early period; that is, the Rules for determining the magnitude, in numbers, of lines, spaces, and solids, under given conditions; a branch of knowledge which differs from Geometry, as the practical from the speculative; and which, like other practical habits, may be most easily learnt in boyhood, leaving the theoretical aspect of the subject for the business of the Higher Education which comes at a later period. There is another reason for making Arithmetic a part of the school-learning of all who are to have a Liberal Education: namely, that without a very complete familiarity with actual arithmetical cesses, none of the branches of Algebra can be at all understood. Algebra was, at first, a generalization and abstraction of Arithmetic; and whatever other shape it may take by successive steps in the minds of mathematicians, it will never be really understood by those students who do not go through this step. And, as we have already said, there is, in a general education, little or nothing gained by going beyond this. The successive generalizations of one or another New Calculus, may form subjects of progressive study for those whose Education is completed, but cannot enter into a general Education, without destroying the proportion of its parts. 34 I have spoken of Geometry as a necessary part of a Liberal Education. It may be asked, how far this Geometry extends? the Elements of Euclid, especially the first Six Books, are generally accepted as the essential portion of Geometry for this purpose. This portion of Mathematics is, however, insufficient fully to exercise the activity of the Reason, and to balance the influence of classical studies. If we consider what portions of mathematics may most properly be added to Elementary Geometry, the parts that offer themselves are Solid Geometry, Conic Sections, Mechanics, Hydrostatics, Optics and Astronomy. Of Solid Geometry, we have an Elementary portion in the Eleventh and Twelfth Books of Euclid, and which has often, and very suitably been used for purposes of Education. Conic Sections are a very beautiful extension of Elementary Geometry; and would probably have been made a part of a general Education more commonly than it has been, if we had inherited from the Greeks any Treatise on the subject, as perfect as the Elements of Euclid are on their subjects. The properties of the Conic Sections are not merely so many propositions added to those of Elementary Geometry: there are introduced, in this branch of Geometry, new geometrical conceptions; for instance, that of the Curvature of a curve at any point. The proofs of the properties of Conic Sections, discovered by the Greek geometers, have come down to us only in a fragmentary manner; and although there have appeared several modern treatises which are very good examples of geometry, no one of them has acquired a permanent and general place, as a part of a Liberal Education. This has arisen in part, at least in England, from the prevalence of a disposition among mathematical students in modern times to adopt the algebraical mode of treating these as well as other curves. But we may observe that the subject of Conic Sections, so treated, is of small comparative value as a portion of Education. If we make the Conic Sections merely examples of the application of Algebra to curves, they are of no more importance than Cissoids, Conchoids, or any other curves; and have little claim to be considered as a distinct part of our Educational Studies: while a geometrical system of Conic Sections is both a striking example of geometrical reasoning; a distinct member of an enlarged system of Geometry; [PT. I.] 4 and a necessary introduction to other mathematical studies, which may very fitly be brought before the student when he has mastered these. 35 Mechanics and Hydrostatics are subjects in some respects in the same condition as Conic Sections. The Greeks, especially Archimedes, established, with full evidence, some of the fundamental propositions of these mathematical Sciences; but they did not transmit the Sciences to us in a form in which they have retained a place in Educational Study; and no particular modern treatises have permanently and generally acquired such a place. Yet these Sciences, as a mathematical portion of our Higher Education, are eminently fitted to promote the developement of the reason; since they not only offer examples of reasoning as solid and evident as that of Geometry, but also show to the student that such reasoning is not confined, in its power, to Space and Number only, but may be extended to the ideas of Force and Motion. And there is another reason why Mechanics and Hydrostatics are valuable educational studies. The truths which they teach are perpetually exemplified in the external world, and serve to explain the practical properties of machines, structures, and fluids. If the Sciences of Mechanics and Hydrostatics be introduced into Education in a proper form, the habits thence acquired, of coherent and conclusive thought on such subjects, will be continually confirmed and extended, by the consideration of the mechanical problems which come before men in a practical form. 36 There is a further reason, as we have already (17) remarked, why Mechanics and Hydrostatics are studies which may be introduced into a Liberal Education. These sciences are the key to the understanding of all the great discoveries of modern times with regard to the constitution of the universe, and especially the discoveries of Newton. By these discoveries, the motions of the earth, planets, moons, and other cosmical bodies, are explained upon mechanical principles, and their apparent irregularities deduced from their mutual forces by mathematical reasoning. These discoveries also offer the best example which the world has yet seen of Induction, the inference of a true theory from phenomena by philosophical sagacity; as well as of the Deduction of consequences from hypotheses by direct reasoning. These discoveries of Newton are still so recent, and have been followed by so continuous a train of similar discoveries with regard to other parts of the system of the universe, that they may be considered as belonging to the Progressive Sciences;-as exemplifying a mental activity which is still going on, and as a portion of the subjects of intellectual research and promise which at present interest men. But yet Newton's exposition of his system has been so long before the world, is so complete in its reasonings, and is so familiar to all who have shared in the intellectual culture of recent times, that it may now be very properly used also as a portion of our Permanent Educational Studies;-a specimen of what such reasoning ought to be; and a ground-work on which all must build in doing more than is there done. The Propositions contained in the Principia of Newton are beautiful examples of mathematical combination and invention, following the course of the ancient geometry; and for the purpose of general education, a portion might be selected from this work without difficulty, which should be not too long or complex for the student, and which should come in a natural order after Conic Sections and Mechanics. 37 Astronomy is a subject in which the Greeks made some of the steps which form the foundation of the Mathematical Science; especially the Doctrine of the Sphere and the Doctrine of the relative motions of the Sun, Moon, and Planets. These portions of know |