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PAGE.

PAGE.

Geometrical Progression,

403

SECTION VIII.

Infinite Series,

410

PERCENTAGE.

Simple Percentage,

239

SECTION XII.

General Formulas,

245

Profit and Loss,

246 HIGHER PERCENTAGE.

Commission,

251

Compound Interest,

411

Stocks and Dividends. 255 Annuities,

416

Par, Premium, and Discount, 259

Contingent Annuities, 424

Brokerage,

263

Insurance,

427

Income from Investments, 268

Life Insurance,

435

General Taxes,

274 Building Associations, 442

Simple Interest,

278

Interest on Daily Balances,. 285

SECTION XIII.

Promissory Notes,

286

PROPERTIES OF NUMBERS,

453

Annual Interest, ·

289

Composite Numbers, 454

Partial Payments,

291

Prime Numbers,

457

True Discount,

295

Even and Odd Numbers, 461

Bank Discount and Banking, 296 Perfect and Imperfect Num-

Savings Bank Accounts, . 303

bers,

463

Investments with Interest, . 305 Properties of the Number 9, 464

Exchange,

307

Arbitration of Exchange,

Properties of the Number 11, 465

318

Properties of the Number 7, 466

Duties,

321

Excess of 9's and 11's, . 467

Scales of Notation,

470

SECTION IX.

RATIO AND PROPORTION.

SECTION XIV.

Ratio,

328 MENSURATION.

Simple Proportion,

Mensuration of Surfaces, 473

Compound Proportion, 339

The Triangle,

474

Partitive Proportion,

343 The Quadrilateral, 475

Conjoined Proportion, 346

Polygons,

476

Medial Proportion, ·

348

The Circle,

477

Partnership,

355 The Ellipse,

479

Bankruptcy,

Mensuration of Volumes, 480

Equation of Payments

The Prism,

480

Averaging Accounts,

366

The Pyramid,

481

Settlement of Accounts, 369

The Cylinder,

482

Account Sales,

370

The Cone,

482

The Frustums,

483

SECTION X.

The Sphere, :

The Spheroid,

485

INVOLUTION AND EVOLUTION.

Irregular Bodies, .

485

Involution,

371

Gauging,

435

Squaring Numbers, . 373

Lumbermen's Rule,

486

Cubing Numbers,

374

Evolution,

376

SECTION XV.

Square Root,

378

Right-angled Triangles,

382 ARITHMETICAL ANALYSIS, 197

Similar Figures,

38+

Cube Root,

385

SECTION XVI.

Similar Volumes, 394 MISCELLANEOUS PROBLEMS, . 492

Higher Roots, .

395

APPENDIX.

SECTION XI.

Tables of Comp. Interest, 509

ARITHMETICAL AND GEOMET-

Table of Annuities,

511

RICAL SERIES.

Table of Fire Insurance, 513

Arithmetical Progression,

397 Table of Life Insurance, . 514

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484

PREFACE.

THE

HE cordial reception given to my Normal Mental and Written

Arithmetics immediately created a demand for a Higher Arithmetic, written upon the same general plan. This demand has become more and more pressing each year, as the publication of the work, which was known to be in preparation, was delayed. The care of a large institution, in connection with many other professional duties, has occupied so much of my time during the last few years, that it was impossible to have the work ready for the press at an earlier day. Indeed, it would have been much longer delayed had it not been for the assistance of Miss Deborah P. Atherton, a former pupil, who has rendered most valuable aid in its preparation.

As now presented, the work is, as its title indicates, a Higher Arithmetic. The object has been to give quite a full treatise upon the science of numbers and its most extensive applications. Especial pains have been taken to exhibit the logical relations of the science ; to present clear and concise definitions and solutions; to state the rules and principles in a brief, exact, and comprehensive form ; and to make an extensive application of its methods to the business practices of the country at the present time. Developed in accordance with this plan, it is thought that the work will commend itself to public favor on account of both its scientific and practical character. Attention is briefly called to a few of its striking peculiarities in both of these respects.

SCIENTIFIC.— The object has been, as stated above, to present a scientific treatise upon the science of numbers. Formerly arithmetic was treated mainly as an art, and the pupil was drilled in mechanical processes, without any conception of the interesting relations of the science and the simplicity of its reasoning processes. A great improvement has been made in this respect within the last quarter of a century. Arithmetic has risen to the dignity of a science, and is beginning to stand in logical completeness beside its sister science, Geometry. I have endeavored to carry out the spirit of modern arithmetic by presenting its principles in logical order, and by making such contributions as I thought would more fully accomplish this object. With respect to this general feature of the work, attention is invited to the following points :

1. The Logical Outline of Arithmetic, in which it is assumed that all arithmetical processes are embraced under Synthesis, Analysis, and Comparison.

2. The logical presentation of the language of arithmetic, showing the principle of Numeration and its true relation to Notation.

3. The use of the word term in the Fundamental Rules, so as to avoid the error of confounding the words figure and number.

4. The comprehension of the several processes following the Fundamental Rules, under the head of Secondary Processes of arithmetic.

5. The treatment of Greatest Common Divisor and Least Common Multiple, and the extension of these processes to Decimals and Denominate Numbers.

6. A New Method of Cube Root, as previously presented in my Elementary Algebra.

7. Important modifications of definitions, as in Multiplication, Division, Fractions, Denominate Numbers, Ratio, Similar Repetends, etc.

8. The logical division of subjects into cases, especially valuable to one preparing to teach arithmetic.

9. Interesting historical notes introduced throughout the entire work.

PRACTICAL. — Though the science of arithmetic is important to the teacher and scholar, the practice of arithmetic, to the man of business, is none the less important. The large majority of those who study arithmetic, need to use it in the transaction of the practical affairs of life ; hence a text-book on the subject should be especially practical. It should, so far as is possible, represent the actual business methods of the times. This has been the especial aim in the preparation of this work. While a few of the problems will be found to be mainly intended to illustrate some principle of the science, or to prepare for the more intricate business problems, by far the greater number are inserted for the purpose of showing the application of the science to the actual business transactions of the day.

In this practical character it is believed that the book will be found especially strong and reliable. It has been the aim to represent all the leading business methods and practices of the times. This idea has been carried all through the book, and constitutes one of its most prominent features. In its terms, names of articles, practical examples, methods, forms, etc., it will be found, it is thought, to be an actual reflection of all the great leading lines of business in this intensely practical and busy age. As examples, we call attention to many of the Practical Problems in the Fundamental Rules, the application of Decimals, the forms of Bills and Accounts, the examples in Denominate Numbers, the varied and extensive applications of Percentage. The articles on Exchange, Custom House Business, Partnership, Insurance, Building Associations, etc., were prepared from material obtained directly from those connected with these various lines of business, and represent the actual business transactions of the present day. The subject of Building Associations is here for the first time presented in a work of this kind.

In this application of the science, it is of course not possible, nor is it desirable, to represent the minor details of every known business interest in the country, since this would require several volumes instead of a single work. The object has been to represent the processes and methods used in all the leading forms of business, so that a pupil trained in these general methods shall be able to apply his knowledge readily to any particular form. In this manner the young man goes out into the business world, not an imitative parrot, capable only of following a particular routine, but an intelligent person, with ability to adopt or originate any process that may be regarded as best for the special case which arises.

These are the principles by which I have been guided in the preparation of this work. Realizing that theory alone renders a man unpractical in life, I have endeavored not to restrict this work to the mere theory of numbers. Realizing also that practice alone gives a person no power to adapt old or originate new processes in particular cases that may arise, I have not confined myself to the presentation of the merely mechanical methods of the counting-house and the market. The object has been to find the golden mean in this respect, and to give that union of theory and practice which shall result in the best mental discipline and the most thorough training. The motto has been, Theory and Practice, properly combined, give the desirable results of Mental Power and Business Capacity.

The work does not aim at novelties, but is based upon that system of arithmetic which has grown up in our schools under the wisdom and experience of the best teachers of the last half century. The ambition has been the improvement of the established system, rather than the futile attempt to create a new one. Neither has the work been shaped to meet that spirit of superficiality in arithmetical instruction which is now quite popular among a certain class of educators, but it presents a full and thorough course of instruction in the science. It is designed, not for superficial, but for thorough teachers of arithmetic,—for teachers who realize that there is no royal road to mathematics, but that all solid attainments come by hard work, and that one of the most important elements of an education is the acquisition of the habit of persistent and self-reliant labor. Above all, the wants of the class-room have been kept constantly in view, and the effort has been continually made to realize and meet the wants of good teachers and earnest students.

With a profound sense of the responsibility of one who attempts to write text-books for the intellectual training of the rising generation, and an earnest desire to measure up to the high demands of this responsibility, this work has been written and is now presented for the consideration of teachers, educators, and directors of public instruction, to whom is intrusted the development of the intellect of the present and future generations.

EDWARD BROOKS.
STATE NORMAL SCHOOL,

Jan. 16, 1876.

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