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P. R. E. F. A C E .

IN the preparation of the present edition of the Geometry of A. M. LEGENDRE, the original has been consulted as a model and guide, but not implicitly followed as a standard. The language employed, and the arrangement of the arguments in many of the demonstrations, will be found to differ essentially from the original, and also from the English translation by DR. BREWSTER.

In the Original work, as well as in the translation, the propositions are not enunciated in general terms, but with reference to, and by the aid of the particular diagrams used for the demonstrations. It is believed that this departure from the method of Euclid has been generally regretted. The propositions of Geometry are general truths, and as such, should be stated in general terms, and without reference to particular figures. The method of enunciating them by the aid of particular diagrams seems to have been adopted to avoid the difficulty which beginners experience in comprehending abstract propositions. But in avoiding this difficulty, and thus lessening, at first, the intellectual labor, the faculty of abstraction, which it is one of the primary objects of the study of Geometry to strengthen, remains, to a certain extent, unimproved. .

The methods of demonstration, in several of the Books, have been entirely changed. By regarding the circle as the limit of the inscribed and circumscribed polygons, the demonstrations in Book V. have been much simplified; and the same principle is made the basis of several im. portant demonstrations in Book VIII.

The subjects of Plane and Spherical Trigonometry have been treated in a manner quite different from that employed in the original work. In Plane Trigonometry, especially, important changes have been made. The separation of the part which relates to the computations of the sides and angles of triangles from that which is purely analytical, will, it is hoped, be found to be a decided improvement. - .

The application of Trigonometry to the measurement of Heights and Distances, embracing the use of the Table of Logarithms, and of Logarithmic Sines; and the application of Geometry to the mensuration of planes and solids, are useful exercises for the Student. Practical examples cannot fail to point out the generality and utility of abstract science.

FISHKILL LANDING, |
July, 1851.

A P P E N ID I X.

PAGE

Note A, ------------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 245

The Regular Polyedrons, --------------------------------------- 247

Application of Algebra to the Solution of Geometrical Problems, - - - - - 249

PLAN E TRIG ON O METR Y.

Logarithms Defined, ------------------------------------------- 255

Logarithms, Use of --------------------------------------- - - - - 256

General Principles,------------- ------------------------------- 256

Table of Logarithms, ------------------------------------------ 257

To Find from the Table the Logarithm of a Number, - - - - - - - - - - - - - - - 258

To Find from the Table the Number corresponding to a Given Loga-

rithm, --------------------------------------------------- 260

Multiplication by Logarithms, ----------------------------------- 261

Division by Logarithms, --------------------------- - - - - - - - - - - - - 262

Arithmetical Complement, -------------------------------------- 263

To find the Powers and Roots of Numbers, by Logarithms, - - - - - - - - - - 265

Geometrical Constructions, ------------------------------------- 266

Description of Instruments, -------------------------- - - - - - - - - - - 266

Dividers, ---------------------------------------------------- 266

Ruler and Triangle, ------------------------------------------- 266

Problems, ---------------------------------------------------- 267

Scale of Equal Parts, ----------------------------------------- 268

Diagonal Scale of Equal Parts, - - - - - - - - - - - - --------------------- 268

Semicircular Protractor, --------------------------------------- 270

To Lay off an Angle with a Protractor,. ------------------------ 270

Parts of a Plane Triangle, -------------------------------------- 271

Plane Trigonometry, Defined,----------------------------------- 271

Division of the Circumference, - - - - - - - - - - - - - - -- * * * * * * * * * * * * * * * * *s o is 271

Measures of Angles,------------------------------------------ 271

Complement of an Are, ---------------------------------------- 271

Definitions of Trigonometrical Lines, - - - - - - - - - - - - - - - - - - - - - - - - - - - - 272

Table of Natural Sines, --------------------------------------- 273

Table of Logarithmic Sines, ------------------------------------ 274

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PAGE.

Circular Functions, ------------------------------------------- 297

Analytical Plane Trigonometry, Defined, - - - - - - - - - - - - - - - - - - - - - - - - - 297

Quadrants of the Circumference, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 298

Versed-Sine, ------------------------------------------------- 298

Relations of Circular Functions, -------------------------------- 299

Table I, of Formulas, --------------------------------------- 301

Algebraic Signs of the Functions,-------------------------------- 301

Table II., of Formulas, ---------------------------------------- 306

General Formulas, -------------------------------------------- 307

Homogeneity of Terms,. -------------------------------------- 3.13

Formulas for Triangles.--------------------------------------- 315

Construction of Trigonometrical Tables, - - - - - - - - - - - - - - - - - - - - - - - - - - 317

SPHE RIC A.T. TRIG O N O METRY.

Spherical Triangle, Defined,------------------------------------ 321

Spherical Trigonometry, Defined, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 321

First Principles,---------------------------------------------- 321

Napier's Analogies, ------------------------------------------- 329

Napier's Circular Parts, ---------------------------------------- 329

Theorems, ------------------------------------ --------------- 330

Solution of Right-Angled Spherical Triangles, by Logarithms, - - - - - - - 333

Of Quadrantal Triangles,-------------------------------------- 335

Solution of Oblique-Angled Triangles, by Logarithms, - - - - - - - - - - - - - - 338

MENS U R A TION OF SUR FA C E S.

Area, or Contents of a Surface, - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - 347

Unit of Measure for Surfaces, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 347

Area of a Square, Rectangle, or Parallelogram,-- - - - - - - - - - - - - - - - - - 347

Area of a Triangle------------------------------------------- 348

Area of a Trapezoid, ------------------------------------------ 350

Area of a Quadrilateral,--------------------------------------- 351

Area of an Irregular Polygon,---------------------------------- 351

Area of a Long and Irregular Figure bounded on One Side by a Right

Line.---------------------------------------------------- 351

Area of a Regular Polygon,------------------------------------ 353

To Find the Circumference or Diameter of a Circle, - - - - - - - - - - - - - - 354

To find the Length of an Are, -------------------------- - - - - - - - - 355

Area of a Circle.--------------------------------------------- 356

Area of a Sector of a Circle, ----------------------------------- 356

Area of a Segment of a Circle, --------------------------------- 356

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