ledge, ever since they were brought to light, have been taught as a necessary part of the instruction of welleducated men. The geocentric theory of the solar system, held by the ancients, has, in modern times, been superseded by the heliocentric or Copernican theory; and Astronomy in this form is, for additional reasons, a necessary part of a Liberal Education; namely, not only as a beautiful example of the mode in which common phenomena are made the subject of mathematical demonstration and calculation; but also as a most conspicuous instance of the progress of science, and the full establishment of Truths which at first sight seem opposed to the evidence of the senses. It is highly desirable that Astronomy should form a part of a Liberal Education to the extent just stated, the Doctrine of the Sphere, and the Doctrine of the Solar System; to which may be added Dialling, as an application of the Doctrine of the Sphere in a special and familiar case.

38 In Optics, as a Mathematical Science, the Greeks made little progress, though even in this they made a beginning. The Science had its rise for the most part, in the sixteenth century. Its main problems, the determination of the foci and images which spherical surfaces produce by reflection and refraction, deserve to be introduced into a Liberal Education on the ground already assigned in the case of Astronomy;-namely, their being beautiful and simple examples of common phenomena, made the subject of Mathematical demonstration; and a portion of knowledge, commonly diffused among educated men. Perspective, which may be considered as a branch of Optics, may be made part of Education on more general grounds. It was cultivated by the ancients as well as the moderns; and a knowledge of it is necessarily. assumed in judging of paintings; a subject in which all educated persons feel a lively interest.

39 I have now completed the enumeration of the branches of Mathematics which belong to a Liberal Education in general. The mathematical part of such an education may, of course, be carried further, if the mathematical aptitude of the student make such an exercise of it easy and satisfactory to his teachers and to him. The Principia of Newton, indeed, is so extensive a work that the student may find an abundant store of employment in the extension of his studies from one part of that book to another; and he will everywhere find in it what is admirable in its mathematical character, philosophically important, and historically interesting. He may also pursue the Differential Calculus, and the other portions of Algebra, to an unlimited extent; and especially the solutions which more modern analysts, by means of that calculus, have given of problems such as those which Newton solved by his geometrical method. These solutions must not be allowed to supersede those of Newton as permanent studies; for they have neither the mathematical instructiveness, nor the permanent interest of the Newtonian propositions. They are mere examples of a method of symbolical calculation. But they are proper and necessary objects of attention to a person who wishes to share in the progressive mathematical studies of his time: and the speculations which belong to this kind of mathematics are so beautiful an exercise of abstraction, generalization, and ingenuity, that we cannot wonder at their being pursued with an exclusive enthusiasm by many of those who attend to mathematical science for its own sake. With us, who have here to regard mathematical science only as an instrument of education, the case is, of course, different. We take parts of the geometry of Newton as a standard portion of our educational course; but we cannot consider, as containing anything essential to our object, the beautiful symbolical reasonings of the Mécanique

Analytique of Lagrange, or the Mécanique Céleste of Laplace; nor can we accept, as substitutes for our simpler forms of the science of Mechanics, any of the Elementary Treatises in which the like symbolic generalizations are presented in a more elementary manner. As we have already said, the main purpose of mathematical studies, in a liberal education, is, not to familiarize the student with symbolical abstraction and generalization, but with rigorous demonstrations, which exercise the reason, and which have long been accepted and referred to by educated men, as examples of solid reasoning.

SECT. 5. Of Analytical Mathematics as an Educational Study.

40 The question of the kind of Mathematics which is to be used in the course of Education, has not been exempt from difference of opinion in modern times. According to what we have said, the mathematical studies which belong generally to a Liberal Education must be portions of Mathematics which are established among mathematicians in a permanent form, and in which the reasoning is intelligible without learning a new mathematical language. But the beauty of the symbolical reasonings of modern times, and the interest which belongs to them as a part of the progressive science of our own day, have, as we have intimated, excited an enthusiasm in many persons; and these persons have often been disposed to substitute such symbolical forms of mathematics in the place of the geometrical modes of treating the subject, which we have represented as essential to the object of our Higher Education. Thus, proofs of the properties of Conic Sections, by means of symbolical combinations of co-ordinates, have been proposed as substitutes for the geometrical demonstrations of the like properties; and the relations between the curves described

by moving bodies and the forces which act upon them, have been investigated by applying the differential calculus to the co-ordinates of the curves; instead of establishing those relations in the way in which Newton did. Such branches of mathematical science are often called Analytical, and Analysis is often opposed to Geometry. This opposition is not very exact; for Geometry has also its Analysis. We demonstrate a Theorem, or solve a Problem, by means of Geometrical Analysis, when we suppose the Theorem proved, or the Problem solved, and trace the consequences of this supposition into a known theorem, or a problem obviously soluble. In solving a question by means of Algebra we, in like manner, express the given supposition in algebraical language, using symbols of unknown, as well as of known quantities, and by tracing the consequences of this expression, we can often find the value of the unknown quantities. And this analytical process is so much more common in Algebra than in Geometry, that processes conducted by means of algebraical symbols are commonly called Analytical. I will adopt this phraseology, though, as I have said, it is not very exact, while I make a few remarks on the use of Analytical Mathematics as an instrument of Education*. I have already spoken on this subject, in my Thoughts on the Study of Mathematics as a part of a Liberal Education, published in 1835; and all subsequent experience appears to have confirmed the truth of the views there expressed.

* The reader may observe that the reasons here given for adopting geometrical rather than analytical modes of reasoning in our elementary university course of mathematics, apply rather against analytical generalities and special calculus, than against the introduction of simple algebraical processes. For example, these reasons would not exclude from Mechanics the method of finding the space fallen through in a given time, by means of assuming an arithmetical series and taking its limit: and the like processes.

41

The recommendation of the geometrical branches of mathematics, as parts of education, is, as we have seen, that they are an effectual discipline of the reason, and have always been familiar as such among educated men. On the other hand, the recommendations of analytical forms of mathematics are such as these ;-their supplying easier solutions of the problems with which the mathematician has to deal ;the symmetry and generality of their processes;-—and their having, in consequence of these qualities, superseded geometrical methods in the mathematical literature of modern times. These merits of analytical processes have been shown, in a most striking manner, in the works of many great mathematicians of modern times; who have given to such processes great completeness and beauty, and have solved, by means of them, problems which had foiled the attempts of previous calculators. And these great works have been accompanied by many elementary works, which expound the like methods in a more limited form, accessible to common students, and applicable to simpler problems. We have to consider the advantages and disadvantages of employing in our Higher Education such Analytical Elementary Treatises, to the exclusion of geometrical modes of treating the same subjects.

42 The first reason which we have mentioned, why Mathematics, in the shape of Geometry, holds its place as an element of great and incomparable value among the permanent studies of a Liberal Education, is this: that it offers to us examples of solid and certain reasoning, by which the reasoning powers, and the apprehension of demonstrative proof, may be exercised, unfolded, and confirmed. This is eminently true of the Geometrical Forms of Elementary Geometry, Trigonometry, Conic Sections, Statics, and Dynamics. It is not true to the same extent, and hardly at all, of the Analytical Methods of treating