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deducible truth, it is equally a suitable element of geometrical reasoning. But this identity of direction in all its parts, is that peculiar property of the straight line, which enters into every consideration of angles and parallels; and the neglect of which has been the cause of most of the embarrassment that has been felt in discussing the doctrine of parallel lines. The definition of an angle here adopted, is that which was given by Euclid. It was also adopted by Clairaut and Laplace; and seems not to be susceptible of improvement. The inclination of the two lines is precisely the property which is used in all discussions in which an angle is an element. . The above definition of parallel lines is adopted because it is believed to characterize the relation of the parallels to each other, more precisely than those definitions which make the parallelism of the lines consist in their not meeting or their being throughout at the same distance from each other. Parallel lines are throughout at the same distance from each other, and cannot meet. These are truths which result from the property or rather the relation of parallelism ; that is, from their having the same direction. This identity of direction is what constitutes the parallelism of the lines. And this notion of parallels should be, at first, presented to the contemplation of the learner; for this is the simple principle from which result all those propositions that make up what is called the doctrine of parallel lines. * The fundamental truth from which are deduced the various propositions respecting parallels, whatever may be the form in which it is stated, is essentially this :—Parallel lines are equally inclined to a straight line meeting them. It seems to be pretty generally acknowledged that this proposition has never yet been geometrically proved, in any of the elementary treatises of geometry, though it is a fundamental principle of a great part of the system. Yet Laplace said of this proposition, that “its mere enunciation produces the fullest conviction; we ought not, therefore, in elementary teaching, to insist upon what may be wanting to the perfect rigor of the proofs which are given of it; this discussion should be left to metaphysical geometers, at least until it has been so far elucidated as to leave no obscurity in the minds of beginners.” It is gratifying to be able to give so high an authority for such an opinion. The student should at first have presented to him those truths only which he can clearly comprehend, and concerning which he will have no doubt. He may go back and discuss the difficulties of the subject, when he has acquired knowledge and power which will enable him to surmount them with ease. That the mere enunciation of the above proposition produces the fullest conviction, is no doubt true ; but this conviction does not result from the definition of parallels given in common books ; it results rather from the common notion formed of parallel lines before we read of them in books. It is hardly credible that the authors themselves, in using parallel lines in the various demonstrations in which they occur, usually think of them as not meeting. They contemplate them merely as having the same direction, and mentally derive their results from this property. This is certainly true of those who read their books.
It may be well to consider the three definitions given above, in their connexion with each other.
The straight line has the same direction in every part. An angle is the inclination of one straight line to another ; that is, the inclination to each other of these two directions. Two parallel straight lines have the same direction. Therefore, a straight line (which has but one direction in every part), meeting two straight lines which have but one
direction in all their parts, must have the same inclination to both. That is, When a straight line meets two parallel straight lines, the angles which it makes with the one are equal to those which it makes with the other. Clearer evidence of the truth of this proposition cannot be desired.
For those propositions which are generally proved by the method of indivisibles, or the reductio ad absurdum, or both, the method here employed is essentially the method of limits or ultimate ratios, used by NEwToN in his Principia, “to avoid,” as he says, “the tediousness of deducing perplexed demonstrations ad absurdum.” Carnot says of the method of limits, that “it is of very great importance, as it relieves us from the necessity of using the reductio ad absurdum, the most troublesome of those operations which constitute the method of exhaustion.” It may be added that, after the student has gone through the labor of committing perfectly to memory one of these troublesome arguments, there is frequently reason to doubt whether he comprehends the entire force of the process.
A second reason for preferring the method of limits to the argument ad absurdum, is, that throughout the whole of the process, it approaches directly the truth sought; and is substantially that process of the mind by which the truth may be supposed to have been discovered.
A third reason for adopting this method in preference to the others will be given in the words of Laplace. “The method of limits is the basis of the infinitesimal calculus. To facilitate the student's understanding of this calculus, it is useful to point out its earliest germs in elementary truths, which should always be demonstrated by methods the most general. The student thus gains, at the same time, knowledge and the means of increasing it. In his subsequent studies he merely follows the path which has been traced for him, and in which he has become accustomed to walk ; and thus his advancement in it will grow much less difficult. Moreover a system of knowledge is best preserved and extended by connecting its parts by a uniform method. In teaching, therefore, prefer the most general methods, endeavour to present them in the most simple manner, and you will find, at the same time, that they are almost always the most easy.”
In discussing those subjects in which the three dimensions of space are concerned, the terms solid and solid angle are not used. The term solid is calculated to mislead the learner ; as, in its common signification, it expresses a property of which geometry takes no cognizance.
Instead of the term solid angle, the recent method of designating the different angles formed by any number of planes, is here adopted. This nomenclature is useful, not only on account of its definiteness, as presenting to the mind distinct and precise objects of contemplation, instead of the more general notion; but also as an introduction to modern treatises on geometry of three dimensions.
It may be proper to remark, that in the whole course of the Elements, the author has left something for the learner to do. He has asked questions which he has left unanswered, proposed problems to be solved and truths to be proved, as an exercise of the learner's ingenuity; and has stated certain things as evident, which may require a moment's reflection from the young reader : but by this it is believed, his progress will be rather facilitated than impeded, as it secures his attention, and gives him the habit of deducing truths from premises without the assistance of another.
“If I must apologize,” says Bézout, “for neglecting the use of the words Aziom, Theorem, Lemma, Corollary, Scholium, &c., two reasons have determined me; the first is, that the use of these words adds nothing to the clearness of a demonstration ; the second is, that this apparatus of terms may frequently divert the attention of beginners from the truth in question, by leading them to suppose that a proposition invested with the name of Theorem must be a proposition as remote from their knowledge, as the name is from terms with which they are familiar. But, to the end that those of my readers who shall open other books of Geometry may not imagine that they have fallen upon an unknown region, I think it proper to inform them that, “Aaiom signifies a self-evident proposition. “Theorem, a proposition which makes a part of the science in question, but whose truth, in order to be perceived, requires a course of reasoning called a demonstration. “Lemma is a proposition which is not necessarily a part of the theory in question, but which serves to facilitate the transition from one proposition to another. “Corollary is a consequence which is drawn from a proposition that has just been established. “Scholium is a remark upon what precedes, or a recapitulation of what precedes.”
The latter part of the volume contains an Introduction to Descriptive Geometry, a science whose practical connexion with so many liberal pursuits gives it a claim to be reckoned one of the regular branches of academical instruction; aside from the peculiar kind of discipline which it affords to the mind, the interesting nature of many of the topics which it discusses, and their effect upon the taste of the student. go
By the study of Descriptive Geometry, the mind sees bodies and their parts in all their relations of position, mag