Euclid's Elements of Geometry

PREFACE TO THE FIRST EDITION.

THIS new edition of Euclid's Elements of Geometry will be found to differ considerably from those at present in general use in Academical Education. The text is taken from Dr. Simson's approved edition, with occasional alterations; but so arranged as to exhibit to the eye of the student the successive steps of the demonstrations, and to facilitate his apprehension of the reasoning. No abbreviations or symbols of any kind are employed in the text. The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science. We may question the propriety of allowing a learner, at the commencement of his Geometrical studies, to exhibit Geometrical demonstrations in Algebraical symbols. Surely it is not too much to apprehend that such a practice may occasion serious confusion of thought. It may be remarked that the practice of exhibiting the demonstrations of Elementary Geometry in an Algebraical form, is now generally discouraged in this University. To each book are appended explanatory notes, in which especial care has been taken to guard the student against the common mistake of confounding ideas of number with those of magnitude. The work contains a selection of problems and theorems from the Senate-house and College Examination Papers, for the last forty-five years. These are arranged as Geometrical exercises to the several books of the Elements, and to a few only in each book the solutions are given. An Introduction is prefixed, giving a brief outline of the history and progress of Geometry.

The analysis of language, together with the sciences of number and magnitude, have been long employed as the chief elements of intellectual education. At a very early period, the study of Geometry was regarded as a very important mental discipline, as may be shewn from the seventh book of the Republic of Plato. To his testimony may be added that of the celebrated Pascal, (Euvres, Tom. I. p. 66,) which Mr. Hallam has quoted in his History of the Literature of the Middle Ages. "Geometry, Pascal observes, is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers álone regard the true laws of demonstration. These

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 self-evident. 6. To demonstrate nothing which is as clear already as it can be made. 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 requisite to avoid erroneous conclusions; but the other five are indispensable. He also remarks, that although they may be found in our ordinary books of logic, yet none but geometers have recognised their importance, or been guided by 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.
He says:
"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,') to restore
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 demon-
stration 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 mathe-
matical studies, in applying himself to the works of Descartes and
other algebraic writers, before he had considered the Elements of
Euclid with that attention which so excellent a writer deserves."

the

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 so 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. II. 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 ảyewμétpntos, as wanting in one of the most essential qualifications for the successful culti vation of the higher branches of philosophy."

TRINITY COLLEGE,

October 1, 1845.

R. P.