PREFACE.

In the present work is developed what is commonly known as the ancient or Euclidian Geometry, the ground covered being nearly the same as in the standard treatises of Euclid, Legendre, and Chauvenet. The question of the best form of development is one of such interest at the present time, among both teachers and thinkers, as to justify a statement of the plan which has been adopted.

It being still held in influential quarters that no real improvement upon Euclid has been made by the moderns, a comparison with the ancienț model will naturally be the first subject of consideration. The author has followed this model in its one most distinctive feature, that of founding the whole subject upon clearly enunciated definitions and axioms, and stating the steps of each course of reasoning in their completeness. By the common consent of a large majority of educators the discipline of Euclid is the best for developing the powers of deductive reasoning. If the work had no other object than that of teaching geometry, a more rapid and cursory system might have been followed; but where the general training of the powers of thought and expression is, as it should be, the main object, it becomes important to guard the pupil against those habits of loose thought and incomplete expression to which he is prone. This can be best done by teaching geometry on the time-honored plan.

Notwithstanding this excellence of method, there are several points in which the system of Euclid fails to meet modern requirements, and should therefore be remodeled. The most decided failure is in the treatment of angular magnitude. We find neither in Euclid nor among his modern followers any recognition of angles equal to

or exceeding 180°, or any explicit definition of what is meant by the sum of two or more angles. The additions to the old system of angular measurement are the following two:

Firstly. An explicit definition of the angle which is equal to the sum of two angles.

Secondly. The recognition of the sum of two right angles as itself an angle. The term "straight angle" has been adopted from the Syllabus of the English Association for the Improvement of Geometrical Teaching. Although not unobjectionable, it seems to be as good a term as our language affords. The term gestreckte Winkel, used by the Germans, is more expressive.

Another deficiency of the Euclidian Geometry is its restriction of the definition of plane figures to portions of a plane surface. In modern geometry figures are considered from a much more general point of view as forms of any kind, whether made up of points, lines, surfaces, or solids. The natural language of ordinary life corresponds to this by regarding figures as formed by the lines actually drawn. The definitions have therefore been framed from this modern standpoint instead of the ancient one. All unprejudiced teachers will, the author believes, be quite willing to renounce the teaching of a nomenclature which has to be changed and forgotten as soon as the pupil advances to analytic geometry. A discussion of the subject will be found in the Appendix.

The above are the most important innovations upon the ancient system. The other leading features of the work, which may be briefly pointed out, are the following:

I. The addition of an introductory book designed not only to present the usual fundamental axioms and definitions, but to practice the student in the analysis of geometric relations by means of the eye before instructing him in formal demonstrations. The exercises in sections 24 to 34 are first attempts in this direction, to which the teacher may add at pleasure until he finds that the pupil has thoroughly mastered the conceptions necessary for subsequent use.

II. The application of the symmetric properties of figures in demonstrating the fundamental theorem of parallels. This system has been adopted from the Germans.

III. After the second book, the analysis of the problems of con

struction, whereby the pupil is led to discover the construction by reasoning.

IV. The division of each demonstration into separate numbered steps, and the statement of each conclusion, where practicable, as a relation between magnitudes. It is believed that this system will make it much easier to carry the steps of the demonstration in mind.

Each step is, when deemed necessary, accompanied by a reference to the previous proposition on which the conclusion is founded, not, however, to encourage the too frequent habit of requiring the pupil to memorize the numbers, but simply to enable him to refer to the proposition. He should always be ready, if required, to cite the proposition, but its number in the book is not of such importance that his memory need be burdened with it. A reference has not been considered necessary after a few repetitions.

V. The theorems for exercise have been selected from native and foreign works with a view to present those best adapted, either by their elegance or their applications in the higher geometry, to interest the student. An attempt has been made to arrange those of each book in the order of their difficulty.

VI. Some of the first principles of conic sections have been developed for the purpose of enabling pupils who do not intend to study analytic geometry to have some knowledge of these curves. It is believed that a previous study of these principles will be a valuable preparation for the advanced treatment of conic sections.

VII. The most difficult subject to treat has been that of Proportion. The ancient treatment as found in Euclid is perfectly rigorous, but has the great disadvantages of intolerable prolixity, unfamiliar conceptions, and the non-use of numbers. The system common in our American works, of treating the subject merely as the algebra of fractions, has the advantage of ease and simplicity. But, assuming, as it does, that geometric magnitudes can be used as multipliers and divisors on a system which is not demonstrated, even for algebraic quantities, it is not only devoid of geometric rigor, but is not properly geometry at all. The author has essayed a middle course between these extremes which he submits to the judgment of teachers with

some reserve.

On the ancient system, magnitudes are compared with respect to