RAY'S ALGEBRA, PART FIRST: ON THE ANALYTIC AND INDUCTIVE METHODS OF INSTRUCTION: WITH NUMEROUS PRACTICAL EXERCISES. DESIGNED FOR COMMON SCHOOLS AND ACADEMIES. BY JOSEPH RAY, M. D. PROFESSOR OF MATHEMATICS IN WOODWARD COLLEGE. VAN ANTWERP, BRAGG & CO., 137 WALNUT STREET, CINCINNATI. 28 BOND STREET, NEW YORK. Educ T 128.75, 730 ECLECTIC EDUCATIONAL SERIES. RAY'S MATHEMATICS. EMBRACING A Thorough, Progressive, and Complete Course in Arithmetic, Algebra, and the Higher Mathematics. Ray's Primary Arithmetic. Ray's Test Examples in Arith. Ray's Higher Arithmetic. Ray's New Elementary Algebra. Ray's Plane and Solid Geometry. By ELI T. TAPPAN, A. M., Pres't Kenyon College. 12mo., cloth, 276 pp. Ray's Geometry and Trigonometry. By ELIT. TAPPAN, A. M., Pres't Kenyon College. 8vo., sheep, 420 pp. By GEO. H. HOWISON, A. M., Prof. in Mass. Institute of Technology. Ray's Elements of Astronomy. By S. H. PEABODY, A. M., Prof. in the Chicago High School. Handsomely and profusely illustrated. 8vo., sheep, 336 pp. Ray's Surveying and Navigation. With a Preliminary Treatise on Trigonometry and Mensuration. By A. SCHUYLER, Prof. of Applied Mathematics and Logic in Baldwin University. 8vo., sheep, 4o3 pp. Ray's Differential and Integral Calculus. Elements of the Infinitesimal Calculus, with numerous examples and applications to Analysis and Geometry. By JAS. G. CLARK, A. M., Prof. in William Jewell College. 8vo., sheep, 440 pp. Entered according to Act of Congress, in the year 1848, by WINTHROP B. SMITH, in the Clerk's Office of the District Court of the United States for the District of Ohio. MARVARD COLLEGE LIBRARY GIFT OF THE GRADUATE SCHOOL OF YOUCATION PREFACE. THE object of the study of Mathematics, is two fold—the acquisition of useful knowledge, and the cultivation and discipline of the mental powers. A parent often inquires, "Why should my son study mathematics? I do not expect him to be a surveyor, an engineer, or an astronomer." Yet, the parent is very desirous that his son should be able to reason correctly, and to exercise, in all his relations in life, the energies of a cultivated and disciplined mind. This is, indeed, of more value than the mere attainment of any branch of knowledge. The science of Algebra, properly taught, stands among the first of those studies essential to both the great objects of education. In a course of instruction properly arranged, it naturally follows Arithmetic, and should be taught immediately after it. In the following work, the object has been, to furnish an elementary treatise, commencing with the first principles, and leading the pupil, by gradual and easy steps, to a knowledge of the elements of the science. The design has been, to present these in a brief, clear, and scientific manner, so that the pupil should not be taught merely to perform a certain routine of exercises mechanically, but to understand the why and the wherefore of every step. For this purpose, every rule is demonstrated, and every principle analyzed, in order that the mind of the pupil may be disciplined and strengthened so as to prepare him, either for pursuing the study of Mathematics intelligently, or more successfully attending to any pursuit in life. Some teachers may object, that this work is too simple, and too easily understood. A leading object has been, to make the pupil feel, that he is not operating on unmeaning symbols, by means of arbitrary rules; that Algebra is both a rational and a practical subject, and that he can rely upon his reasoning, and the results of his operations, with the same confidence as in arithmetic. For this purpose, he is furnished, at almost every step, with the means of testing the accuracy of the principles on which the rules are founded, and of the results which they produce. .Throughout the work, the aim has been, to combine the clear, explanatory methods of the French mathematicians, with the practical exercises of the English and German, so that the pupil should acquire both a practical and theoretical knowledge of the subject. While every page is the result of the author's own reflection, and the experience of many years in the school-room, it is also proper to state, that a large number of the best treatises on the same subject, both English and French, have been carefully consulted, so that the present work might embrace the modern and most approved methods of treating the various subjects presented. With these remarks, the work is submitted to the judgment of fellow laborers in the field of education. WOODWARD COLLEGE, August, 1848. SUGGESTIONS TO TEACHERS. It is intended that the pupil shall recite the Intellectual Exercises with the book open before him, as in mental Arithmetic. Advanced pupils may omit these exercises. The following subjects may be omitted by the younger pupils, and passed over by those more advanced, until the book is reviewed. Observations on Addition and Subtraction, Articles 60-64. The greater part of Chapter II. Supplement to Equations of the First Degree, Articles 164-177. Properties of the Roots of an Equation of the Second Degree, Articles 215-217. In reviewing the book, the pupil should demonstrate the rules on the blackboard. The work will be found to contain a large number of examples for practice. Should any instructor deem these too numerous, a portion of them may be omitted. To teach the subject successfully, the principles must be first clearly explained, and then the pupil exercised in the solution of appropriate examples, until they are rendered perfectly familiar. CONTENTS. PAGES. CHAPTER I-FUNDAMENTAL RULES. Preliminary Definitions and Principles Definitions of Terms, and Explanation of Signs Division of Monomials-Rule of the Signs To reduce Fractions to a Common Denominator To reduce a Quantity to a Fraction with a given Denominator To multiply one Fractional Quantity by another To divide one Fractional Quantity by another Equations of the First Degree, containing one Unknown Quan- |