A

Abbreviation, arithmetical marks and signs used

for, 48. Addition, Simple, on the use and application of, 7.

of Compound quantities, 21; of Fractions, 29.

of Algebra, sign of, 162 ; illustrations of, 166; exercises in, 167, 168.

of Fractions in Algebra, 198; exercises in, 199.

ALGEBRA (Arabic al the, and gabron reduc

tion of fractions), general principles of, 161 et seq., signs of, 161; the study of, requires a previous knowledge of arithmetic, 161 ; definitions and explanations of, 162; exercises in, 164; operations in, 165.

Addition of, with examples for exercise, 166 et passim.

Subtraction of, 168.
Simple Equations, 172 et seq.

Transposition, 173; Clearing Fractions, ib.; questions for solution and exercise, 174, 175, 177.

Multiplication of, 179; Involution, 183.

Division of 187; Exponents, Roots, and Surds, 191 ; Evolution, or Extraction of Roots, 193; exercises in, 195.

FRACTIONS, 196; exercises in, 197 ; Addition and Subtraction of Fractions, 198 ; exercises in, 199; Multiplication of Fractions, 199; Division of Fractions, 200; exercises in, 201.

Solutions of Simple Equations, 201, 204, 206, 213; exercises in, 203, 206; Quadratic Equations, 209, 210, 213, 214; exercises in, 212, 215,

Homogeneous and Symmetrical Quadratics, 217.

Ratio and Proportion, 218.
Arithmetical Progression, 220.

Geometrical Progression, 222; exercises in, 225.

Arithmetic, the Rule of Reduction, 19.

Compound Quantities,-Addition, 21;
Subtraction, 22; Multiplication, 23; and Di-
vision, 25.

FRACTIONS, 26; Addition and Subtrac-
tion of, 29; Multiplication and Division of,
30; Proportion, 31; Rule of Three, 33.

DECIMALS, 36; Addition and Subtrac-
tion of, 37; Multiplication and Division of,
38 ; Extraction of the Square Root, 38.

general principles of, 40; table of fac-
tors, 41 ; symbols and signs of, introductory

to Algebra, 161, 261. See ALGEBRA.
Arithmetical Algebra, 261, 262,
Arithmetical values of quantities, 275.
Arithmetical progression in Algebra, 220.
Avoirdupois weight (Fr. avoir to have, du pois

some weight), 18.
Axioms (Gr. axioma authority) of Euclid, 47.

Convergent series, meaning of a, 264.
Cosines of angles (Lat. sinuo to diverge), nume-

rical value of, 315; to obtain the, 351.
Cotangents (Lat. co with, tangens touching),

calculation of, 352.
Cube root (Gr, kubos a six-sided die), of a com-
pound quantity, 235,

of decimals, exercises for finding the,
235.
Cycloid (Gr. kyklos a circle), construction of

the, 447.

D

B

Base line, its measurement in surveying, 415,

416.
Binomial (Lat. bis twice, and nomen a' name),

powers of a, 184; how to extract the square
root of a, 232; multiplies which render bino-

mial surds rational, 234.
Binomial theorem, how to state the, 266 ; how

to prove it, 266, 269.

DECIMALS (Lat. decim ten), beautiful contrir.

ance of, 6; principles and practice of, 36; re-
ductions of fractions to, 37.

Addition and Subtraction of, 37; Mul-
tiplication of, 38; Division of, 38.

exercises for finding the cube root of,
238.
Definitions of Eulid, 43, 85, 96, 116; of the prin-

ciples of Algebra, 162, 103; of plane trigono-
metry, 292, 294, 298, 299; of geometrical

planes, 242; of spherical geometry, 252.
Delambre's method of solving logarithmic sines

of small angles, 360..
Denominaton of fractions, 197.
Distances, the measurement of, 367.
Divergent Series, meaning of a, 264.
Dividend, the quantity so called, 187; when a

compound quantity, 189; exercises in th

working, 191.
Division, Simple, on the use and application of,
13; various problems in, 14-17.

of compound quantities, 25; of Frac. |
tions, 30; of Decimals, 38.

of Algebra, the sign of, 163; the Divi.
dend, the Divisor, and the Quotient, 187;
operations in, 189.

of Fractions in Algebra, 200; exercises
in, 201.
Divisor, the quantity so called, 187; when a

compound quantity, 189.
Double Rule of Three, principles and practice of,

с

34.

Drawing-pens used in geometry, 421.
Dry Goods, measures for, 19.

E

Ellipse (Gr. elleipsis deficiency), definition and

F

EQUATIONS, arithmetical and algebraical, 172 et

Factor (Lat. factor a working agent), meaning

of the term, 165.
Factors, operations of, when they are simple

quantities, 179; to express sin. x, in a series
of, 344.

of the Composite numbers, table of the,
41.
Figures of arithmetic, 4; local value of, 5, 6;

use of in arithmetic and algebra, 161.
Formulas (Lat. formula a rule or maxim), for

determining the relations of different angles,
304 et seq. ; relations between the four funda-
mental ones, 308; various formulas and ex-
pressions, 309; "ambiguities” arising from
the use of, 311; for demonstrating trigonome-
trical problems, 323; adapted for logarithmic
calculation, 325,

Fundamental, of sperical trigonometry,
403 et seq.
FRACTIONS (Lat. fractio a breaking into parts),

principles and practice of, 26; addition and
subtraction of, 29; multiplication and division
of, 30; reduction of to Decimals, 37.

in Algebra, operations of, 196; exer-
cises in, 196; how to reduce them to a common
denominator, 179.

Addition and Subtraction of,
198.

Multiplication of, 199; exer-
cises in, 200.

-- Division of, 200; exer in,
201.
Frustum (Lat. frustum a fragment) of a right

prism, to determine the volume of, 389.

G

Geodetical operations (Gr. ge land, and daio to

divide), on the formulas peculiar to, 414 et seq.
Geometrical progression in algebra, 222.
GEOMETRY (Gr. ge and metron land-measur-

ing), arithmetic introductory to, 2; illustra-
tions of, in the four first Books of Euclid,
43—128 (see EUCLID); general disquisition on,
68 et seq. ; to be combined with algebra, 161.

Plane, introduction to, 241 ; definitions
of, 242; propositions in, 243– 250.

Spherical, introduction to, 251 ; defini-
tions of, 252; propositions for solution, 253—
260.

Practical, general treatise on, 421 et
seq.; the instruments in general use, 421 ;
solutions of various problems, 423—447 ; con-
struction of conic sections, 438; the ellipse,
439; the parabola, 444; the hyperbola, 446;
the cycloid, 447.

Letters, use of in algebra, 161.
Lines and ratios, trigonometrical, 294, 297.
Liquids, measures for, 19.
LOGARITHMS (Gr. logos and arithmos a discourse

on numbers), treatise on, 261 et seq.; on the
calculations of, 275; principle on which they
may be used to facilitate calculations, 276;
every number has a calculable logarithm, 279;
numerical values of may be calculated, 279,
281; methods of finding any power of a num-
ber, 289.

P

M

MATHEMATICS (Gr. mathema learning), intro-

Parabola, definition of the, 432 ; illustrations of

the, 444,
Parallelipeds, equality of, 38.5, 386.
Pence table, 18.
Pencils used in geometry, 421.
Pens used in geometry, 422.
Plane geometry and trigonometry, introduction

to, 241 et seq.
Plane trigonometry, treatise on, 292 ct seq. (See

TRIGONOMETRY).
Planes, definitions of, 242; propositions in, 243

-250.
Polygon, to find the area of a, 329.
Polynomial (Gr. polus many; Lat. nomin names),

how to extract the square root of a, 228,
Positive values in algebra, 215.
Postulates of Euclid, 47; problems illustrative

of, 422.
Prismoid, to find the value of a, 391.
Prisms, various measurements of, 389—391.
Problems (Gr. problema a proposition requiring

solution) of Euclid, 48 et seq. ; 423-417; (see

PROPOSITIONS).
Progression, Arithmetical, 220; exercises in,

Proportion, rule of, 30.
PROPOSITIONS of Euclid, 48–67 ; 87-92; 97—
112; 117-124; 423—447.

in Plane Geometry, 243—250.
in Spherical Gcometry, 253——260.

in Plane Trigonometry, 299, 305-307.
Pyramids, mensuration of, 389.

Quadratic Equations (Lat. quadratus fourfold),

solutions of with unknown quantities, 209,

210, 213, 214; exercises in, 212.
Quadratics, homogenecus and symmetrical, 216,

117.
Quadrature of the Circle (Lat. quadratura the

squaring of anything), problem of the, 126,

127.
Quadrilateral (Lat. quadratus fourfold, and

latera sides), to find its area inscribed in a

circle, 328.
Quantities, Simple, operation of the factors in,

179; in calculating the arithmetical values of,
275.

in Algebra, simple and compound, 163;
illustrations of, 164.

Unknown, solution of simple equations
with, 201, 204, 206, 208, 213; with quadratic

equations, 209, 213, 214.
Quantity, marks and symbols of, 161, 162.

Mixed, reduction of to an improper
fraction, 196.
Quotient (Lat. quoties so many times), the quan.

tity so called, 187,

Series, treatise on, 261 et seq. ; what is meant

by a Convergent and Divergent Series, 264.

(See LOGARITHMS).
Series and Tables of Trigonometry, 330 et seq.
Signs of operation in Algebra, 161, 162, 163;

practical illustrations of, 164.
Sines of Angles, numerical value of, 315; how to
calculate the value of, 349, 350.

Logarithmic, tables of, 355.

Natural, tables of, 353.
Solids, mensuration of, 19, 383, 393 et seq.
Sphere, to find the volume of the portion of a,

396.
Spherical Geometry, 251 et seq. (See GEOMETRY).
Spherical triangles, sines of the angles propor-

tional, 402; solution of the, 408 ; Napier's

rule for solving, 409.
Spherical Trigonometry. (See TRIGONOMETRY).
Spheroid, to find the volume of a portion of a,

398.
Square Root, extraction of the, 38, 39.

of a polynomial, 228.

various algebraic exercises for extract.
ing the, 231,

of a binomial, 232.
Squaring the Circle, problem of the, 126.
Subtraction, Simple, use and application of, 8;
various workings in, 9.

of Compound Quantities, 22; of Frac-
tions, 29; of Decimals, 37.

of Algebra, sign of, 162 ; illustrations
of, 169; exercises in, 170.

of Fractions in Algebra, 198; exercises
in, 199.
Surds in Algebra (Lat. surdus undistinguishable),

illustrations of, 191.
Surface, measurement of, 19.
Surveying a Country, operations of, 414, 416

R

et seq.

Radius of an inscribed circle, to find the, 327.
Railway Cutting, to find the solid content of a,

Symbolical Algebra, 261.
Symbols of Quantity in algebra, 162.
Symmetrical Quadratics, 217.

393.

T

Ratio and Proportion, in algebra, 218.
Ratios of Angles, 295, 297, 299, 300.

- Inverse Trigonometrical, explained, 318.
Reduction, rule of, 19; its rise and application,

19-21.
Roots in algebra, 191 ; extraction of, 193; rules

for, 194; exercises in, 195.
Rule of Three, principles and practice of, 33.

Double, illustrations of, 34.
Rulers used in geometry, 421.

Tangents, the calculation of, 352.
Tables of money, time, weights, and measures,

S

Science (Lat. scientia the knowledge of things),