EducT/48.78.457

Sept. 13,1941

Miss Ellen L. Wentworth

Entered according to Act of Congress, in the year 1871, by

HARPER & BROTHERS,

In the Office of the Librarian of Congress, at Washington.

9

PREFACE.

THE only apology necessary for adding another to the many text-books with which the market is overstocked is that this little work on Geometry is very much needed. The greater portion of it was prepared while engaged in teaching the subject, and while all the difficulties of presenting it in a practical manner were fresh in the author's mind. Several friends-themselves superior instructors-strongly advised its publication, in the hope that it would render the study of the elementary principles of Geometry more simple and easy of comprehension.

In teaching Geometry, it appeared, at first, singular that students could master the difficulties of Arithmetic and Algebra, and yet fail to comprehend the relations of magnitudes which appealed to the sense of sight. A little closer observation, however, revealed the fact that many students accomplished little more than the committing to memory variations of "A," "B," "C," and “1," "2," "3." Of correct geometrical reasoning they had hardly a conception. Even the intelligent pupils were found unable to apply the principles to new matter; and the solution of problems not in the book was almost an impossibility. Geometry, if properly taught and thoroughly understood, is just as flexible as Arithmetic or Algebra.

Geometry was truly a "rope of sand" to whole classes. The necessary examination completed, the

subject was abandoned and forgotten. The principal cause for this was a wide departure by many recent writers from the rigid system of Euclid. For example, Euclid commences with the simple problem, "On a given straight line to construct an equilateral triangle." By means of the postulate or problem, whose solution. is self-evident, that "A circle may be described with any point as centre and any length of line as radius," how simple, beautiful, and satisfactory to the mind of the learner the construction becomes! Besides, the compasses and ruler are placed in the hand of the student from the very beginning; he does something for himself; sees its truth, and assimilates it with his intellect. Nearly all the Geometries in use in the schools commence with a theorem. The pupils are told to erect a perpendicular from a given point in a given line. By what authority? By a postulate! Then it is a problem whose solution is self-evident! If so, why after using it to establish, link by link, a chain of truths extending through three books, does the author proceed to demonstrate it? It reminds us of a man who, building a superstructure on a false foundation, is forced to pause in his work when he has completed his third story, and reconstruct a true foundation to prevent the whole edifice from toppling over. It is a problem whose solution is self-evident that "A line may be bisected." Why not, also, that it may be trisected? Five postulates are subsequently demonstrated. First, they are self-evident; second, they are not self-evident, and require solution! Is it any wonder that the youthful mind is shocked ab initio ? Imagine Euclid asking his auditors: "I beg that you will grant that a straight line may be drawn through a given point parallel to a given line." His auditors would have granted no such thing, and would have

told him he was begging too much. It is really more "self-evident" that "if one straight line cut another straight line, the opposite or vertical angles are equal." Why not beg this too? or, indeed, beg the whole subject?

Three other works on Geometry contain no postulates whatever! If Geometry be founded on definitions, axioms, and postulates, it is certainly a violation of rigid geometrical reasoning to omit any necessary step in the process. If the first fifteen or twenty propositions are thoroughly taught and perfectly mastered, the subsequent study of geometry is comparatively simple. The aim of the teacher should be to train the scholar rigidly -to take nothing for granted, unless really self-evident or previously demonstrated. The why and the wherefore of every step must be stated. Each link in the chain must be as strong as any other link.

A great evil has arisen from the attempts to shorten the demonstrations of the propositions. Important omissions are likely to occur, and haste and inaccuracy frequently follow. It is much better to make the proof complete and satisfactory, so that the student will not be obliged to review again and again. In every geometrical demonstration so much and no more is necessary. It is as bad to omit as to add; and great care should be exercised in giving just enough. Circumlocution is tedious; but lack of thoroughness often vitiates the truth. Besides its practical utility, the study of Geometry imparts a love for truth for its own sake; it strengthens the reasoning faculties more than the study of any of the other mathematical sciences; and, unless carried to too great an extent, cultivates clearness, precision, and brevity of expression.

The present volume is intended only for beginners, for those who are preparing for college, and for inter

mediate and high schools generally. The Geometry of Planes and Solids is omitted. Nearly all the works hitherto published on this subject contain in addition appendices on Plane and Spherical Trigonometry and Logarithms. The vast majority of students rarely advance beyond the geometry of lines, angles, and plane figures. The work in its present form will be cheap, and will exhaust the first and most important department of geometrical study. Any pupil wishing to make further progress can readily do so by taking up any other work on the subject. Should the present volume, however, accomplish its mission, the author will publish a second volume containing the higher departments.

We claim for this little work on Geometry-1. That it commences aright; 2. That it contains more problems, solved and unsolved, than any other volume of its size extant; 3. That it is more practical than the works generally in use; 4. That it contains an appendix on Mensuration of Surfaces, which furnishes a useful application of Arithmetic to the Geometry previ ously studied.