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are enumerated by him as eight in number. 1. To define nothing which cannot be expressed in clearer terms than those in which it is already expressed. 2. To leave no obscure or equivocal terms undefined. 3. To employ in the definition no terms not already known. 4. To omit nothing in the principles from which we argue, unless we are sure it is granted. 5. To lay down no axiom which is not perfectly evident. 6. To demonstrate nothing which is as clear already as we can make it.
7. To prove every thing in the least doubtful, by means of self-evident axioms, or of propositions already demonstrated. 8. To substitute mentally the definition instead of the thing defined. Of these rules he says, “the first, fourth, and sixth are not absolutely necessary to avoid error, but the other five are indispensable; and though they may be found in books of logic, none but the geometers have paid any regard to them.”
If we consider the nature of Geometrical and Algebraical reasoning, it will be evident that there is a marked distinction between them. To comprehend the one, the whole process must be kept in view from the commencement to the conclusion ; while in Algebraical reasonings, on the contrary, the mind loses the distinct perception of the particular Geometrical magnitudes compared; the attention is altogether withdrawn from the things signified, and confined to the symbols, with the performance of certain mechanical operations, according to rules of which the rationale may or may not be comprehended by the student. It must be obvious that greater fixedness of attention is required in the former of these cases, and that habits of close and patient observation, of careful and accurate discrimination will be formed by it, and the purposes of mental discipline more fully answered. In these remarks it is by no means intended to undervalue the methods of reasoning by means of symbolical language, which are no less important than Geometry. It appears, however, highly desirable that the provinces of Geometrical and Algebraical reasoning were more definitely settled than they are at present, at least in those branches of science which are employed as a means of mental discipline. The boundaries of Science have been extended by means of the higher analysis ; but it must not be forgotten that this has been effected by men well skilled in Geometry and fully able to give a geometrical interpretation of the results of their operations; and though it may be admitted that the higher analysis is the more powerful instrument for that purpose, it may still be questioned whether it be well suited to
form the chief discipline of ordinary intellects without a previous knowledge of the principles of Geometry, and some skill in their application. Though the method of Geometrical analysis is very greatly inferior in power to the Algebraical, yet as supplementary to the Elements of Euclid, it is of great importance. It may be added, that a sound knowledge of the ancient geometry is the best introduction to the pursuits of the higher analysis and its extensive applications. On this subject the judgment of Sir Isaac Newton has been recorded by Dr Pemberton, in the preface to his view of Sir Isaac Newton's Discoveries.
Newton censured the handling of geometrical subjects by algebraical calculations. He used to commend the laudable attempt of Hugo d'Omerique (in his Analysis Geometrica Nova et Vera,') to restore the ancient analysis, and very much esteemed the tract of 'Apollonius De Sectione Rationis,' for giving us a clearer notion of that analysis than we had before. The taste and mode of geometrical demonstration of the ancients he professed to admire, and even censured himself for not having more closely followed them than he did : and spoke with regret of his mistake, at the beginning of his mathematical studies, in applying himself to the works of Descartes and other algebraical writers, before he had considered the Elements of Euclid with the attention they deserve."
Regarding the study of Geometry as a means of mental discipline, it is obviously desirable that the student should be accustomed to the use of accurate and distinct expressions, and even to formal syllogisms. In most sciences our definitions of things are in reality only the results of the analysis of our own imperfect conceptions of the things; and in no science, except that of number, do the conceptions of the things coincide 80 exactly (if we may use the expression) with the things themselves, as in Geometry. Hence, in geometrical reasonings, the comparison made between the ideas of the things, becomes almost a comparison of the things themselves. The language of pure Geometry is always precise and definite. The demonstrations are effected by the comparison of magnitudes which remain unaltered, and the constant use of terms whose meaning does not on any occasion vary from the sense in which they were defined. It is this peculiarity which renders the study so valuable as a mental discipline : for we are not to suppose that the habits of thought thus acquired, will be necessarily confined to the consideration of lines, angles, surfaces and solids. The process of deduction pursued in Geometry from certain admitted principles and possible constructions to their consequences, and the rigidly exact comparison of those consequences with known and established truths, can scarcely fail of producing such habits of mind as will influence most beneficially our reasonings on all subjects that may come before us.
In support of the views here maintained, that Geometrical studies form one of the most suitable and proper introductory elements of a scientific education, we may add the judgment of a distinguished living writer, the author of “ The History and Philosophy of the Inductive Sciences," who has shewn, in his
Thoughts on the Study of Mathematics,” that mathematical studies judiciously pursued, form one of the most effective means of developing and cultivating the reason: and that “the object of a liberal education is to develope the whole mental system of man;—to make his speculative inferences coincide with his practical convictions;—to enable him to render a reason for the belief that is in him, and not to leave him in the condition of Solomon's sluggard, who is wiser in his own conceit than seven men that can render a reason.' To this we may subjoin that of Mr John Stuart Mill, which he has recorded in his invaluable System of Logic, (Vol. 11. p. 180) in the following terms. “The value of Mathematical instruction as a preparation for those more difficult investigations (physiology, society, government, &c.) consists in the applicability not of its doctrines, but of its method. Mathematics will ever remain the most perfect type of the Deductive Method in general; and the applications of Mathematics to the simpler branches of physics, furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education, and regarding, with Plato, one who is áryewuétontos, as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy.”
October 1, 1845.
Great care has been taken in the correction of the proofs, and it is believed that the Student will not meet with any errata of importance in the text to impede his progress : the following, however, have been discovered on revising the sheets :
3-2 125 23 3
3 172 In the diagram, Prop. III. A is misplaced. 177 In the diagram, Prop. IX. D and Ê are interchanged. 192 In the diagram, Prop. XXVI. for Dread C. 242 13 AE
AQ. 323 46 arc
circumference. 328 1 centre
circumference. 341 of
In the Geometrical Exercises, a few repetitions have occurred, and a few of the problems, perhaps, with more propriety, might have been arranged under a different book.
In the Index a few references are omitted, as the Examination Papers, from which the questions have been taken, were without dates; besides some four or five others, which were lost.