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.

EXPLANATION OF SOME SYMBOLS AND ABBREVIATIONS EMPLOYED.

L, angle.
Zs, angles.

A, triangle.

As, triangles.

or perp., perpendicular to, or at right angles to.

Il, parallel to.

..., therefore.

Hyp., hypothenuse.

< placed between two quantities, implies that the one which comes before the symbol is less than the one which follows it. Thus, ▲ A < ▲ B is to be understood and read, "the angle A is less than the angle B."

> placed between two quantities denotes that the one which precedes is greater than the one which follows it. Thus, ▲ B > ZA, is to be read, "the angle B is greater than the angle A." It may assist the memory to observe that the greater quantity is always on the side of the open part of the symbol, and the less quantity on the side of the closed part.

etc.

A', B', x', y', etc. are read, A prime, B prime, x prime, y prime,

A', B'', x", y', etc. are read, A second, B second, a second, y second, etc.

A'' is read, A third; Aiv, A fourth; B', B fifth, etc.

EXPLANATION OF SOME SYMBOLS AND ABBREVIATIONS EMPLOYED.

L, angle. Zs, angles. A, triangle.

As, triangles.

or perp., perpendicular to, or at right angles to.

I, parallel to.

..., therefore.

Hyp., hypothenuse.

< placed between two quantities, implies that the one which comes before the symbol is less than the one which follows it. Thus, LA ▲ B is to be understood and read, "the angle A is less than the angle B."

> placed between two quantities denotes that the one which precedes is greater than the one which follows it. Thus, B>

▲ A, is to be read, "the angle B is greater than the angle A.” It may assist the memory to observe that the greater quantity is always on the side of the open part of the symbol, and the less quantity on the side of the closed part.

etc.

A', B', x', y', etc. are read, A prime, B prime, x prime, y prime,

A", B", x'', y', etc. are read, A second, B second, x second, y second, etc.

A'' is read, A third; Aiv, A fourth; B', B fifth, etc.