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The great Utility of a complete System of pure and mixed Mathematics, calculated to be put into the hands of a youth, as soon as he has become acquainted with the rules of Arithmetic, is too obvious to be insisted upon. Hitherto, no well-adapted volume for such general purpose has existed. A separate treatise of Algebra limits the energies of the Student to that science, and he appears to be pursuing it for its own sake; but all the branches of Mathematics are connected, and many advantages will result from the Books of Euclid being read at the same time that the student is engaged in Algebra. Again, the extension and application of these become apparent in the subsequent chapters, and an incentive is created to go through the course.

The collected price of separate books in various branches of Mathematics far exceeds the cost of this volume; and distinct works on each subject are generally too full in their details, and too prolix, for the purposes of students. These, and many other reasons, will be apparent to all intelligent persons; and incalculable advantages must accrue to the rising generation from the power, thus possessed by every Tutor, of bringing the whole of these sciences at one view before his Pupils. Nor will the work be without its uses as a Text and Reference Book in our Universities, among whose Students it will serve, at least, as a chain connecting all the branches of mathematical study, and, in its several parts, will often assist in rendering obvious the sense of abstruse treatises in particular branches of these noble sciences.

Every mathematician, of the present day, must be aware of the existence of a work, published upwards of a century ago, under the title of “ WARD's YOUNG MATHEMATICIAN's Guide;" and many of them, particularly the self-taught, will readily acknowledge their obligations to it in the commencement of their studies. Nevertheless, Ward's book is an imperfect and limited series; and, since the great improvements in the principles of analysis, it has become altogether obsolete. It is a book which suggested the idea of the present volume; but it has by no meanis served as its model either in plan or execution.

Ward's Guide commenced with the principles of Arithmetic; but the present system supposes that the Student has previously rendered himself familiar with that science. His book, also, was limited to practical Geometry, with little com no theory; whereas, the present work includes, not on the whole of Euclid, printed literally from Dr. Simsce i

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edition, but it also includes Conic Sections, and the entire doctrine of Curves, with their application to various branches of science and philosophy. The Algebra of Ward is, also, not only altogether obsolete, but did not extend beyond Quadratic Equations; while these Elements include the solution of all manner of Equations, together with Fluxions, and the Differential Calculus, according to the most approved theories and practices.

As an introduction to Mathematics, for the use of Schools, and Students in general, the present volume possesses, therefore, original pretensions to the favour of the public. Other systems, of more recent date than Ward's, have appeared; but, on comparison, it will be found that no one is more comprehensive in its objects, or more compact in its details; while, in regard to the various improvements in Mathematics, which have been made by the great analists of France, England, and Germany, it will, as a general ele. mentary book, be considered, by enlightened Mathemati. cians, not inferior to any similar work in the English or any modern language.

By means of economy in printing, the Author has been enabled to attempt more, within the same number of pages, than has previously been effected by works of greater mag. nitude, but printed in a larger type. He has thus been empowered to present to the public a complete body of Mathematics, at a price which accords with the often-limnited resources of students.

Of course, in such a work, it has been less the object of the Editor to invent, than to compile with discretion, and arrange with judgment. At the saine time, he trusts it will be discovered, that, in every part where originality was required, he has supplied many deficiences, and corrected the errors of some of his predecessors. A few of the chapters have been adapted to the purposes of this publication from separate tracts of his own, and others have been originally compiled from materials which at present lie scattered in various expensive works.

To confer every possible perfection on this work, a series of correct Logarithmic and Trigonometrical Tables have been Bubjoined; and a collection of upwards of two bundred Miscellaneous Questions have been introduced as exercises on the various subjects discussed through the volume.

For the use of Tutors, a separate Key has been printed, which contains Answers, worked at length: not only to the Questions alluded to, but also to all the otti Questions and Iroblems scattered throughout the volume. Claremont - Place, Drunswick-Square.

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Page

ELEMENTS OF EUCLID-Book XI.

283

-Book XII.

318

PLANE TRIGONOMETRY

345

Theory and Arithmetic of Sines

ib.

Calculation of the Tables of Sines, by Series

340*

Solution of the Cases of Plane Triangles

348

Triigonometry applied to Heights and Distances 361

SPHERICAL TRIGONOMETRY

361

Miscellaneous Examples on Heights and Distances 367

Principles and Proportions for the Solution of Sphe-

rical Triangles

375

Application of the preceding Principles and Pro-

portions

379

Example of Right-Angled Spherical Triangles - 383

Examples of the cases of Oblique-Angled Spherical

Triangles

393

CONIC SECTIONS

397

Origin and General Equation of the Conic Sections 404

OF THE PARABOLA

407

OF THE ELLIPSE

410

OF THE HYPERBOLA

418

OF SEVERAL OTHER CURVES

426

1. The Conchoid of Nicomedes

ib.

II. The Cissoid of Diocles

428

III. The Logarithmic Curve

429

IV. The Cycloid

430

V. The Quadratrix of Dinostrates

431

VI. The Spiral of Archimedes

433

VII. The Parabolic Spiral

434

VIII. The Hyperbolic or Reciprocal Spiral 435

IX. The Logarithmic Spiral

436

FLUXIONS

437

Rules for finding the Fluxions of any Proposed

Functions

440

Of Logarithmic and Exponential Fluxions 443

Of the Fluxions of Sines, Cosines, &c. and other

Circular Functions-

445

Application of Fluxions to the Theory of Curves 447

Of Involute and Evolute Curves

452

Of Vanishing Fractions

464

OF THE INVERSE METHOD OF FLUXIONS

467

Of Quantities Susceptible of an Exact Integration ib.

On the Integration of Rational Fractions

474

Of the Integration of Logarithmic and Exponential

Fluxions

486

On the Integration of Fluxions containing Sines, Co-

sines, &c.

491

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