TABLE OF CONTENTS. PAGE Various Geometrical Problems in "Triangles, Quadrilaterals, and Par- allels," analyzed by Geometry, 64 in number Geometrical Problems involving Properties of "the Circle," analyzed Theorem and Problem proposed by Francis Miller Problem to illustrate the Mode of "Discussing a Problem” Construction of Algebraic Equations of one unknown quantity of the Three Theorems believed to be original. A Theorem and Singular Property deducible from these A Problem in Tree-Planting deduced from Theorem X. . Problem in Tree-Planting, proposed in The Agriculturist An Attempt at simple Illustrations of some Principles and Problems in Having given two Equations of two unknown quantities of the First Power, to find the values of the unknown quantities by To construct the Circle from its Equation Section on Curves, showing the manner of constructing several of The Hyperbola.-To determine Points in the Hyperbola geomet- rically, and find its Equation when referred to its Asymptotes Having the Equation of the Hyperbola referred to its Asymptotes, INTRODUCTORY NOTE. IN an engagement of nearly forty years' duration in teaching Mathematics, first in the boarding-schools of Fair Hill, Maryland, and Westtown, Pennsylvania, and afterwards in a boarding-school of my own, for boys and young men, at Alexandria, Virginia, I became convinced that the analytic or algebraic method of Descartes, Delambre, and Laplace, while it is a most efficient instrument in the hands of a mathematician, is not so well adapted as the geometric or Greek method, to impart to the student a knowledge of mathematical principles, or to inspire such student with an affection and taste for the science. The mind of the young is less capable than that of an older person of abstract thought, and it needs assistance to the concentration of its ideas, such as is afforded by a mathematical diagram; and with this aid continued for some time, the faculties of perception and conception become cultivated and strengthened, till the mind of the student can readily grasp, without a diagram, a problem of considerable intricacy, and be able to apply to it Descartes' method efficiently. So entirely has the analytic method taken the place of the geometric in our prominent schools and colleges, that no work on pure Geometrical Analysis has, to my knowledge, ever been published in this country. Thomas Simpson, F.R.S., and Professor of Mathematics in the Royal Academy of Woolwich, England, in his "Select Exercises. for Young Proficients in the Mathematics," which was published in London in 1752, gave a number of geometrical problems, with the method of constructing them. And, in a treatise on Geometry, by the same valued author, published in London in 1760, the fifth and sixth books are entirely devoted to the Construction and Demonstration of Geometrical Problems; and nearly fifty problems, of great variety, and some of them of much elegance and beauty, are constructed and demonstrated at the end of the volume. (9) To his Algebra, also, which was published in England about the same time, and reprinted in Philadelphia, by Mathew Carey, in 1809, "from the Eighth London Edition," Thomas Simpson added an "Appendix, containing the Construction of Geometrical Problems, with the Manner of resolving them numerically." This work was of very great value to the mathematical student, and of much service to my revered preceptor, John Gummere, in the preparation of his admirable treatise on Surveying, which was published a few years after Carey's edition of Simpson's Algebra was issued. But in all these works, including the problems in Gummere's Surveying, the mode of construction was arbitrarily given, without the least clue to the line of thought that had led to it; and the thoughtful student would naturally inquire, and even wonder, in a problem like the seventy-sixth or seventy-seventh of the Appendix to Simpson's Algebra (Problems six and eight in "Algebraic Analysis" in this book), how the author ever came to think of so complicated a method; and he would feel discouraged, in view of its complexity, from an apprehension that his own mind would never be endowed with a penetration sufficient to accomplish a result of such intricacy and depth. Whereas, had the problems been analyzed, and the student shown that it was all done by taking one step at a time, and that an easy one, gradually acquiring a knowledge of what we do not know by means of what we do know, which Doctor Johnson says is the only way it can be done, he would have felt increased confidence in his own powers, and been prepared to make an effort himself— which is a great point--in a similar direction. In 1821, Professor John Leslie, of Edinburgh College, published, in Edinburgh, a very valuable treatise on "Geometrical Analysis, and Geometry of Curve Lines." The "Geometrical Analysis" had been previously published, annexed to his "Elements of Geometry," and it greatly "conspired to advance the study of Geometry, by reviving the fine models bequeathed by the Greeks." As has been said with great truth of this work of Professor Leslie, "the study of such a digest, appears admirably fitted to improve the intellect, by training it to habits of precision, arrangement, and close investigation. The spirit of Geometrical Analysis may be carried, with the happiest effect, into the domains of Inductive Philosophy, which are to be explored by a similar procedure." But, about the time of the publication of this work of Professor Leslie, the algebraic method of Descartes and Leibnitz came rapidly |