Hydrostatics were reduced by Archimedes to a shape in which they contained propositions deduced by the severest processes of demonstration from manifest fundamental principles; and these studies might have been used, as Geometry was used, as a discipline of the Reason. At first, however, this was not done. The speculative powers of men in general did not take firm hold of these demonstrated truths, when they were first brought to light. After the time of Archimedes, they were again let slip as speculative truths; and were not recovered, and ranged among the results of demonstrated mathematical reasoning, till comparatively modern times. Having, however, now taken this place, they may be used, along with Geometry, as a means of educating the Reason. And besides the direct effect of the study of the mathematical sciences of Mechanics as an intellectual discipline, the introduction of this study into a liberal education, opens the way to the student's possessing a clear and steady comprehension of the greatest of the achievements of man in the domain of material knowledge, the Newtonian system of the universe. At the present day (as I have elsewhere attempted to show*) no education can be called good, which leaves the pupil in ignorance, or with a loose and merely verbal knowledge, of these discoveries. Mechanics then, as teaching the principles, both of machines, and of the mechanism of the heavens, should be one of the Permanent Studies which belongs to our Higher Education. Other branches of Mathematics have grown up in modern times, which may also be used with advantage as parts of a liberal education; but these, for the most part, have the character of the progressive, rather than the permanent, educational studies. * Philosophy of the Inductive Sciences, Book xIII. Chap. 3. 18 But before I proceed to speak of those progressive studies, I may remark, with regard to the mode of educating the Reason, that there has been, in this case, as in the education of the faculty of Language, not only a direct study of the great models of excellence, but an attempt to analyse the means of success. The Greeks employed themselves in discovering, and stating in a technical form, the conditions under which Reasoning is rigorously demonstrative. They had, here also, a critical as well as an exemplary branch of study; Logic, as well as Mathematics. In the work of reasoning, as in the work of literary composition, the reflex tendencies of their minds came into play. As they laid down rules excluding Solecisms in Grammar, they laid down also rules excluding Fallacies in Logic. And Logic has, from their times, held its place in a Liberal Education, along with Grammar and Rhetoric. SECT. 3. Of Progressive Educational Studies. 19 Besides the permanent results of human thought, which, once brought into being, remain ever after as subjects of human study, there are other works of the intellect which from time to time change their form. There are progressive portions of literature and knowledge;-new sciences, or new methods of science, new forms of criticism and philosophy. And as the permanent subjects of educational study educe in men those human faculties and those ideas by which they are connected with the past condition of humanity, so these progressive subjects of study connect men with the condition of humanity in their own time, and give them their share in the future intellectual prospects of the race. Even among the ancient Greeks, these progressive subjects of study were already cultivated. Astronomy had been brought into being as a mathematical science at the time of Plato; and the subject made some remarkable advances soon afterwards. Optics and Harmonics were also cultivated as mathematical sciences among the Greeks. And these sciences, especially Astronomy, were employed as portions of the Higher Education. Such sciences derive their fundamental principles from observation of the material world; and deduce results from these principles by mathematical processes. Besides these portions of knowledge, thus obtained in various ways, the Greeks employed themselves in speculating concerning the nature of knowledge in general, and the mode in which man may and must acquire it. Such speculations formed a large portion of their Philosophy; and such Philosophy has occupied every succeeding generation up to the present time; and most, the generations of greatest intellectual activity. 20 In this portion of human knowledge, which has thus been progressive, it naturally follows that the subjects are expanded, transformed, and multiplied by the successive steps of progress. The Science and Philosophy of modern times differ from, or at least extend beyond, those of the ancient world. Even those Sciences which had begun to exist among them, have so changed their aspect and enlarged their boundaries, that the ancient portion is the smallest part of them. Our Mechanics, and our Hydrostatics, are much more extensive and profound than those of the Greeks. Our Astronomy has undergone revolutions which have made it belong eminently to modern times, although the ancient foundations laid by the Greek geometers have not even at this day lost their validity or importance. But in addition to these ancient sciences, others have sprung up, which did not exist at all, or at least in any scientific form, among the Greeks and Romans. Such are the Classificatory Sciences, Botany and Zoology. Such are those Sciences which I have elsewhere* termed Palætiological, and which explore the past history of the world by studying the causes of change; among which we may especially notice Geology, the History of the Material Earth, and Ethnography, or Glossology, the History of Languages. Such, again, are the Sciences which consider bodies according to the elements of which they are composed: Chemistry, which analyses them, and Mineralogy, which classifies them with a view to their analysis. Such sciences, finally, are those which attend to the structure, the symmetry, and the functions of living beings; Anatomy, Comparative Anatomy, Morphology, Biology. On these subjects, whatever sparks and gleams of intelligence we may discover in ancient authors, the broad light of science was not shed, till the human mind, in the course of its movements, arrived at its modern period of activity. These are the subjects with which a person must acquaint himself, who wishes fully to appreciate the progress which man has made and is making in the pursuit of truth. And though it may not be possible for any one to give his attention to the whole of these; and though it is not necessary, for educational purposes, that a man should attempt to acquaint himself with any large portion of them; yet it is requisite, as a part of a Liberal Education, that a person should so far become acquainted with some portion of this body of accumulated and imperishable knowledge, as to know of what nature it is, what is the evidence of its reality, by what means additions to it are made from time to time, and what are the prospects which it opens to the present generation of mankind. The Progressive Sciences, to this extent, ought to enter into the scheme of a Liberal Education. * History of the Inductive Sciences, Book xVIII. [PT. I.] duction. Intro 3 21 The Sciences just mentioned derive their new truths in part from experiment and observation, and are progressive in virtue of new facts, as well as new reasonings, which they incorporate into their texture. But even those sciences which consist altogether of reasonings are progressive, and require to be noticed under this aspect. Mathematics must be studied in the character of a Progressive, as well as of a Permanent Science. For the Mathematics of modern times involves processes unknown to the ancients. Results are now deduced from principles by combinations of symbols of number and quantity, rather than by reasoning upon the relations of space. And thus, in addition to the Elementary Geometry and Conic Sections of the Greeks, and the calculation of numerical questions directly; we have the calculation of such questions by symbols (Algebra), and the calculation of the properties of curves by the symbols of their coordinates (the Algebra of Curves), and by the symbols of the changes of such quantities (the Differential Calculus); and these modes of calculation form additions to the body of mathematics, which may overlay, and almost put out of sight, the original form of mathematical sciences. 22 Again, as the study of the exact reasoning of the ancients pointed to Logic, which defines the methods of strict reasoning; so the sciences which are, as we have said, derived from facts, direct us to the study of those processes which determine the methods of obtaining truth from facts. To obtain consequences from principles is Deduction; to obtain general truths from particular facts is Induction. The Logic of Induction, or at least a Philosophy which includes Induction within its scope, is a necessary accompaniment of the progressive sciences; and such a Philosophy ought also to make a part of our Liberal Education. 23 I have said that a portion of the Sciences |