• KEY

ΤΟ

RAY'S ALGEBRA,

PARTS FIRST AND SECOND:

CONTAINING

STATEMENTS AND SOLUTIONS OF QUESTIONS,

WITH

REMARKS AND NOTES.

ALSO, AN APPENDIX, CONTAINING

INDETERMINATE AND DIOPHANTINE ANALYSIS, PROPERTIES
OF NUMBERS, AND SCALES OF NOTATION.

BY JOSEPH RAY, M. D.,

PROFESSOR OF MATHEMATICS IN WOODWARD SCHOOL.

CINCINNATI:

WILSON, HINKLE & CO.
PHIL'A: CLAXTON, REMSEN & HAFFELFINGER.
NEW YORK: CLARK & MAYNARD.

لامان

THE BEST AND CHEAPEST

MATHEMATICAL COURSE.

ARITHMETIC.

RAY'S ARITHMETIC, PART FIRST; very simple lessons for little
learners.

RAY'S ARITHMETIC, PART SECOND; a complete text-book i
Mental Arithmetic, by induction and analysis.

RAY'S ARITHMETIC, PART THIRD; for schools and academies ;
a full and complete treatise, on the inductive and analytic methods.
KEY TO RAY'S ARITHMETIC, containing solutions to the questions.
With an appendix embracing Slate and Black Board exercises.
RAY'S HIGHER ARITHMETIC; designed for advanced pupils
embracing the higher applications of the science of numbers, with
an extensive course of commercial calculations. Preparing.

ALGEBRA.

RAY'S ALGEBRA, PART FIRST; for commen schools and academies; a simple, progressive, and thiorough elementary treatise. RAY'S ALGEBRA, PART SECOND; An Analytical Treatise, embracing Sturm's Theorem, and Horner's Method of Approximation. KEY TO RAY'S ALGEBRA, PARTS FIRST and SECOND; Complete in one volume, 12 mo.

PREPARING FOR PUBLICATION,

ELEMENTS OF GEOMETRY; embracing Plane and Solid Geome try, with numerous interesting and practical exercises. TRIGONOMETRY AND MENSURATION; containing Logarithmic Computations, Plane and Spherical Trigonometry, with their applications, and Mensuration of Planes and Solids, with Logarithmic and other Tables.

SURVEYING AND NAVIGATION; containing Surveying and Leveling, Navigation, Barometric hights, &c.

The above will be followed by works on Analytical Geometry: Mechanical Philosophy: - Differential and Integral Calculus: - and Mathematical Astronomy.

PROFESSOR RAY, during a long period devoted to actual instruction in these several branches, has prepared much of the material requisite for the respective volumes. They will appear as rapidly as a due regard to their careful publication will permit.

Entered according to act of Congress, in the year Eighteen Hundred and Fifty. Two, by WINTHROP B. SMITH, in the Clerk's Office of the District Court of the United States, for the District of Ohio.

STEREOTYPED BY A. C JAMES.

HARVARD COLLEGE LIBMACY

GIFT OF

GORGE ARTHUR PLIMPTON

JANUARY 25, 1924

KEY

ΤΟ

RAY'S ALGEBRA, PART FIRST.

The numbers in parentheses, as seen in the margin of this KEY, refer to the corresponding number of example, under the same article in the Algebra.

TO TEACHERS. As these exercises, except, perhaps, a few in Lesson XIV, can be readily solved without the aid of Algebra, by pupils having a good knowledge of Mental Arithmetic, it is unnecessary to occupy space with their solution.

Some instructors who use the Algebra, pay no attention to the intellectual exercises, but permit their pupils to begin with the preliminary definitions and principles. This course is proper with pupils of considerable maturity of mind, and who possess a good knowledge of arithmetic; but in the case of learners generally, and especially those who are young, the intellectual exercises should be thoroughly studied.

the numbers are 12, 4, and 28.