into vogue in this country through the popularity of some eminent French writers upon Mathematics, and Professor Leslie's work was never, as far as I know, published in the United States, or, to much extent, placed within the reach of the American student.

The want that I experienced, in my engagement of teaching, of a suitable text-book upon Geometrical Analysis, induced me to make a collection of problems, for the benefit of my students, analyzing, constructing, demonstrating, and giving a method of calculation;~solving in this way all I could meet with or frame myself, and difficult ones forwarded to me for solution, through a number of years, by former students, and many other persons, known and unknown, so that the number of such problems became considerable, from which those in the present volume have been selected, as best adapted to supply the want I had experienced.

These problems were given to the more advanced mathematical students as "extras," or 'extras," or "trial questions;" and in their solution, much ingenuity was frequently manifested, and also gratifying evidence of progress, in the skill with which the student would proceed in the different steps of the analysis; and it was from the solicitations of many of my former students who had become teachers, to publish the problems from which they had derived, as they believed, great benefit, that the preparation of the present work was undertaken.

The practical teaching of young persons consists of two parts: -instructing them how to do something; and giving the reason for doing it in that way. Now, it accords with reason and sound philosophy to do one thing at a time. Hence these two parts of teaching, as a general rule, should, with the young, be kept as separate as possible. Youth should first learn, well, the practical part, how to do; then the reason In acquiring a knowledge of the primary rules of Arithmetic, for instance, to undertake to give a child the reason for every step he takes in the processes of subtraction, multiplication, division, etc. would but confuse him. It would be too complicated. He needs that the ideas he is to acquire shall be placed, with well-defined outline, in the simplest possible light, and one at a time, so that his mind can clearly and concentratedly comprehend the truth to be imparted. And when this is effected, the sparkling eye and animated countenance will attest the pleasurable sensations of the soul, from the conscious acquisition of a truth previously unknown to him.

The practical part-how to do--first; then the why. We must know a fact, before we care about the reason for it. And this is

the natural mode of the mind's progress in knowledge. Children learn to use words, before they learn the definitions of them. They form phrases, before they are able to "construe" or "parse" them. And the more nearly a teacher keeps to this natural process, the more successful will he be in developing the minds of his students, and in pleasurably educating them.

Accordingly, whenever practicable, in teaching young persons, objects, models, maps, globes, diagrams, apparatus, specimens, and other means of illustration of the truths designed to be taught, should be employed, as an aid to assist in concentrating their thoughts, and imparting a clear idea of the subject. In children, the perceptive faculties are most prominent and active, and must principally be brought into requisition; and thus the reflective powers will gradually develop and strengthen, when there will be less need of this educational machinery.

Mathematics is an elevated study. Its constant aim is the discovery of truth. There is no new truth. Every fact in science, every mathematical principle, every property of the triangle or the circle, is eternal. No matter when or by whom it was discovered, it pre-existed, and is the living embodiment of a divine thought. All the mathematician or scientist does in this way is to discover what was previously unknown, although eternally existing. When Dr. Herschel and Prof. Leverrier discovered each a new planet," as it was termed, in Uranus and Neptune, they only saw for the first time what had existed from the period of Creation. The discovery was new, not the object discovered. So of every truth, principle, or property throughout the whole range of mathematics and all science.

It seems remarkable, and, as I get older, the astonishment does not in the least abate, that the five plane figures formed by the cutting of a cone by a plane in different positions - the Triangle, the Circle, Parabola, Ellipse, and Hyperbola-should possess such a great number and diversity of singular and interesting properties, many of which are embodied in the following pages, and which, although eternally existing, have been gradually disclosing themselves. to the patient research of mathematicians for over two thousand years, and we are by no means at liberty to suppose that all of them are yet known.

For instance, that in any triangle, of whatever size or shape, the three perpendiculars let fall from each angular point upon the opposite sides (the sides and the perpendiculars being produced if necessary) should all pass through the same point. Also, that the three

lines bisecting the angles respectively, and the three lines drawn from the angular points to the middle point of the opposite sides respectively, should in each case all pass through the same point. (See Scholiums 1, 2, 3 to "Theorem" 6, in the following pages.)

Of the same singular character are the properties of the circle and triangle combined, in "Theorems" 11, 12, and 13 of this work, which were original discoveries with the author, although they may have. been made by some one before me, whose writings I have not met with.

It is deemed proper, in this Introductory Note, to give some explanation of the arrangement of the work, and of the mode in which it is believed it may be used to the greatest advantage.

Each problem is first analyzed, then constructed, demonstrated, and the method of calculation by Plane Trigonometry clearly indicated. In many cases the limits are shown within which the problem is restricted.

The greatest number of problems are analyzed by pure Geometry. Where they are analyzed by Algebra, or the Differential Calculus, the constructions and demonstrations are by pure Geometry, and the method of calculation is given by Plane Trigonometry, showing the harmonious results by the different modes of investigation.

*

The references in the text (as III. 8* in the first problem of the circle) are to Davies's edition of Legendre's Geometry,--the III. 8 meaning the 8th Proposition of the Third Book of Legendre, and the referring to a marginal note or foot-note where the Proposition is given in which the same property is demonstrated in Playfair's Euclid. In this example, the note at the bottom of the page is III. 14, meaning the 14th Proposition of the Third Book of Playfair's Euclid. So of other cases.

The design of the work is principally as an aid to professors and teachers, to supply a want which the author had severely experienced. The problems are distributed through the volume without much reference to their intricacy or difficulty of solution, in order that the instructor may have a wide field from which to select extra or trial problems, so as to give different problems to different classes in successive years.

The importance of the student's constructing the different problems given him for solution, by means of the Scale and Dividers, with the greatest possible precision, can scarcely be too much insisted upon, as a very improving engagement in training to accuracy, and one of the best preparations for draughting and other duties in the Coast

Survey or Civil Engineering; or for the business of a machinist, or any mechanical vocation requiring neatness and precision. If this constructing is done thoughtfully, the eye will soon become practiced in judging of distances and positions, so that drawing by hand can be more rapidly and accurately effected.

The solutions of the problems in relation to surfaces and solids, in the brief article on the Differential and Integral Calculus, and some other portions of that article, if read by a student or a class before entering upon, or while pursuing, that study, it is believed will be of material assistance in mastering the subject satisfactorily.

The same may be said of the article on Analytical Geometry. As an exercise or lesson preparatory to using a standard text-book upon the subject, it will be a great assistance, and all the problems there given should be carefully constructed, and the results measured with the Scale and Dividers.

Or, the teacher or professor may make the explanations, and give these or similar problems, orally, to his class. This method possesses great advantages. The students generally understand and retain what they hear from their instructor, better than what they read in their text-books. Besides, their respect and regard for their instructor are increased when the students discover that the contents of the text-book are not the limits of his knowledge of the subject they are studying, and see his willingness thus to render the knowledge he possesses advantageous to them.

In like manner, before entering upon the study of the Properties. of the Parabola, in a treatise on Conic Sections, let the three pages on that Curve, in this volume, be read, or equivalent ideas be imparted by the instructor, the student constructing accurately the three problems given, or their equivalents proposed by the teacher or professor.

The same course is recommended in regard to the Ellipse and Hyperbola. The student will thus acquire a practical acquaintance with the Axes, Ordinates, Abscissas, Asymptotes, Opposite Hyperbolas, etc., which, as these will then be familiar to him, will prove of the greatest advantage in gaining a knowledge of the properties of the Curve, by leaving his mind more free to concentrate its powers upon the particular idea or truth to be acquired.

Also, in regard to the remaining Curves,--the Conchoid, Cissoid, Quadratrix, Spiral, etc.,-the student will find it to his decided advantage to construct each Curve by the rules herein given, accurately

with the Scale and Dividers, previous to entering upon the study of its properties in his text-book.

A Table of Square Roots, carried to two and three places of decimals, of numbers from 1 to 200, is added at the close of the volume, to facilitate the construction of Curves from their Equations, by means of points.

The author does not expect or desire any pecuniary return from his work. It is a labor of love for the youth of our country, and of interest and sympathy for those to whom their education may be intrusted, in their most arduous and responsible engagement.

If the work shall prove of some service to the student, and lighten the labor of the instructor, the highest aim of the author will be reached. The more good it does, and the more service it performs, the more fully will his object be attained.

Should anything in this Introductory Note appear like dictating or prescribing to teachers a particular mode of proceeding, the author trusts it will be excused, and understood as only a suggestion from one who has now completed two more than his "threescore years and ten," and who, although he has not been practically engaged in that vocation for a number of years past, still feels a deep interest in the noble profession of Teaching, and those actively concerned in it, and who regards the preparation of this work, in which he has spent this, his seventy-second birthday, as the closing and crowning labor of his life in that direction.

BENJAMIN HALLOWELL.

SANDY SPRING, MARYLAND, 8mo. 17, 1871.

Postscript.—It seems only proper to add, that my son, HENRY C. HALLOWELL, read the whole work carefully in manuscript, corrected several clerical errors, and, in some instances, where there appeared to be abstruseness, suggested one or two intermediate steps in the process, in order that the idea might be more readily comprehended by the student.

Note. For the gratification of his former students, who are widely scattered over our country, some of whom have not seen him for many years, a likeness of the author is prefixed to the volume, which is accompanied by his kindest remembrances.