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With this view, as well as in compliance with the wishes of several intelligent teachers, I have also been led to subjoin to it, by way of an Appendix, a small tract on the application of Algebra to the solution of Geometrical Problems; which, it is hoped, will prove acceptable to such classes of students as may not have an opportunity of consulting more voluminous and expensive works on this interesting branch of the science.

JOHN BONNYCASTLE.

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TO

THE SECOND NEW-YORK EDITION.

It would be superfluous to advance any thing in commendation of " Bonnycastle's Introduction to Algebra," as the number of European editions, and the increase of demand for it since its publication in this country, are sufficient proofs of its great utility.

But to make it universally useful both to the tutor and scholar, I have given in this edition, the answers that were omitted by the Author in the original.

In the course of the work, particularly in Addition, Subtraction, Multiplication, Division, Fractions, Simple Equations, and Quadratics, I have added a great variety of practical examples, as being essentially necessary to exercise young students in the elementary principles.

Several new rules are introduced, those of principal note are the following: Case 12. Surds, containing two rules for finding any root of a Binomial Surd, the Solution of Cubics by Converging Series, the Solution of Biquadratics by Simpson's and Euler's methods:

all these rules are investigated in the plainest manner possible, with notes and remarks, interspersed throughout the work, containing some very useful matter.

There is also given all the Diophantine Analysis, contained in Bonnycastle's Algebra, Vol. I. 8vo. 1820., being a methodical abstract of this part of the science, which comprehends most of the methods hitherto known for resolving problems of this kind, and will be found a ready compendium for such readers as may acquire some knowledge of the ANALYTIC ART.

New-York, Jan. 1, 1822.

JAMES RYAN.

ALGEBRA.

ALGEBRA is the science which treats of a general method of performing calculations, and resolving mathematical problems, by means of the letters of the alphabet.

Its leading rules are the same as those of arithmetic ; and the operations to be performed are denoted by the following characters:

+ plus, or more, the sign of addition; signifying that the quantities between which it is placed are to be added together.

Thus, a+b shows that the number, or quantity, represented by b, is to be added to that represented by a ; and is read a plus b.

minus, or less, the sign of subtraction; signifying that the latter of the two quantities between which it is placed is to be taken from the former.

Thus ab shows that the quantity represented by bis to be taken from that represented by a; and is read a minus b.

Also, ab represents the difference of the two quantities a and b, when it is not known which of them is the greater.

X into, the sign of multiplication; signifying that the quantities between which it is placed are to be multiplied together.

Thus, a Xb shows that the quantity represented by a is to be multiplied by that represented by b; and is read a into b.

The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word.

B

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