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 ; Ad-
dition and Subtraction of Fractions, 198; ex-
ercises 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 Quad-

Algebra, Proportion and Progression, 225.

Extraction of Roots,-of the Square
Root of a Polynomial, 228; excrcises in, 231,
234; of the Square Root of a Binomial, 232;
of the cube root of a compound quantity, 235;
exercises, 235 et seq.

arithmetical and symbolical, 261, 262.
"Ambiguities" arising from the use of formulas,
311.

Angle, circular measure of an, 293; definition
of, 298.

Angles, ratios of, 295 et seq.; functions of, 301;
of a corresponding character, 303; sines and
cosines of, 315; the trigonometrical ratios of,
316; to obtain the cosines of, 351; methods of
correcting, 417, 419.

formulas for determining the relations,
304 et seq.

sines and cosines of, 310, 311; loga-
rithmic sines of, 357, 358; Delambre's method
for solving, 360.

spherical triangle, sines of the, 402.
Apothecaries' weight (Gr. apotheca a repository),

18.

Areas, on the mensuration of, 371.

to find the area of a rectangle, 371; of
a triangle, 372; of a parallelogram, 372; of a
trapezoid, 372; of a trapezium, 373; of a poly-
gon, 373; of an ellipse, 376; of a parabola,
377; of a plane figure bounded by a curve,
378; of a right prism, 380; of a right cylinder,
380; of the curved surface of a right cone, 381;
of the frustum of a cone, 381; of a portion of
the surface of a sphere, 382.

ARITHMETIC (Gr. arithmos number), intro-
ductory to geometry, 2; uses and objects of,
3, 4; the ten figures, 4; numeration table,
and the reading of numbers, 5; decimal system
of, 6; local value of figures, 6.

Simple Addition, 7; Subtraction, 8;
Multiplication, 10; and Division, 13.

Tables of Money, Time, Weight, and
Measures, 18.

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.

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.

C

Calculable logarithm, every number has a, 279.
Calculations facilitated by logarithms, 276.
Cask, to find the volume of a, 398.
Circle, to find the radius of a, 327; to find the
area of a quadrilateral inscribed in a, 328.
Quadrature of the, difficulty of solving
the problem, 126.

Circular measure of an angle, 293.
Cloth, measures of, 18.

Coefficient (Lat. co with, and efficio to work out),
the multiplier in algebra so called, 163; illus-
trated, 164.

Coins, gold and silver, 18.

Compasses used in geometry, 421.

Composite numbers (Lat. compositus compounded
sited), factors of the, 41.

Compound Quantities,-Addition of, 21; Sub-
traction of, 22; Multiplication of, 23; Division
of, 25.
Compound quantity, cube root of
a, 235.

Cone (Gr. konos, a top or pine-apple), to find the
volume of the frustum of a, 395.

Conic Sections, on the construction of, 438; the
ellipse, 439; the parabola, 444; the hyperbola,

446.

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,
34.

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

E

Ellipse (Gr. elleipsis deficiency), definition and
illustration of the, 439.

Equality, algebraic sign of, 163.

Equation (Lat. æquo to equal), to find the roots
of the, 340.

EQUATIONS, arithmetical and algebraical, 172 et
seq.; on the solution of, 172; different modes
of operating, 173; transposition, and clearing
fractions, ib.; how to solve a simple equation
containing only one unknown quantity, 174;
questions for solution and exercises, 175–179;
rules and operations for their solution with
unknown quantities, 201, 202, 204, 206, 208;
exercises in, 209, 210.

Quadratic, solutions of, with unknown
quantities, 209, 210, 213, 214; exercises in,
212.

Equivalent Forms, the permance of, 261.
EUCLID, elements of, and definitions, 43; his pos-
tulates, 47; his marks of abreviation, 48.

Propositions of Book I., 48-67; com-
ments on, 68-85; exercises on, 85.

Definitions and Propositions of Book
II., 86-92; remarks on, 93; exercises on
Books I. and II., 95.

of Book III., 96—112; remarks
on, 112; exercises on Books I., II., III., 115.
of Book IV., 116-124; re-
marks on, 124; exercises on the Four Books
of, 128.

his Plane Geometry, 241; his proposi-
tions on planes, Book XI., 243-250.

his Spherical Geometry, 251; defini-
tions and propositions, 252-260.

447.

Problems in practical geometry, 423-

Evolutions (Lat. evolutio the process of evolv-
ing), in algebra, 193; exercises, 195.
EXERCISES in the problems and theorems of Eu-
clid, Book I., 85; Books I. and II., 95; Books
I., II., and III, 115; on the four first Books,
128.

in the rules of ALGEBRA, 164 et seq.;
in addition of algebra, 168; in subtraction,
170; in equations, 175, 177, 203, 206; in multipli-
cation of algebra, 180; in involution, 187; in
division, 189; in extraction of roots, 195; in
fractions, 196; in addition and subtraction of
fractions, 198, 199; in multiplication of frac-
tions, 199, 200; in division of fractions, 200,
201; in the solution of simple equations, 201,
203, 206; in quadratic equations, 212; in arith-
metical and geometrical progressions, 222,
225; in extracting the square root of a poly-
nomial, 231; of a binomial, 234, 235; in the
cube root of decimals, 238.

Exponentials, to obtain sines, &c., in terms of,
335.

Exponents in algebra (Lat. expono to set forth),
illustrations of, 191.

Expression, meaning of the term in algebra, 163.
Extraction of roots, 193; rules for, 194; exer-
cises in, 195.

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.

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.

H.

Heights, the measurement of, 367.
Homogeneous Quadratics (Gr. homos the same,
and genea birth; Lat. quadra a square), defi-
nition and illustration of, 216.
Hyperbola (Gr. hyper over, and ballo to throw),
definition of the, 439; illustrations of the, 446.

Multiplication of Algebra, the sign of, 162, 163;
the rule of, 179; when the factors are simple
quantities, ib; exercises in, 180, 183.

of Fractions in Algebra, 199; exercises |

in, 200.
Multiplier, algebraically called the co-efficient, i
163; in algebra a factor, 165.

Multipliers, which render binomial surds ra- |
tional, 234.

Impossible Expressions, 262.

Indices (Lat. indico to indicate), theory of, 262.
Infinite series, arithmetical values of, 275.
Instruments used in geometry compasses,
ruler, pencil, 421; drawing pen, 422.
Involution (Lat. involutio the process of in-
volving), of simple quantities, 183; exercises
in, 184, 187; the powers of a binomial, 184.

L

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.

M

MATHEMATICS (Gr. mathema learning), intro-
ductory remarks on, 1 et seq.; different sub-
jects connected with, 3; on the study of, as a
science, 3; the general elements, problems,
and axioms of, 43 et seq. passim. (See EUCLID,
GEOMETRY, ALGEBRA, &c.)
Measures, tables of, 18, 19.

MENSURATION (Lat. mensuro to measure), trea-
tise on, 367 et seq.; heights and distances,
367-370; the measure of areas, 371-382; the
measure of solids, 383-400.

Military Earthwork, to find the solid content of
a, 394.

Moivre's trigonometrical theorem, 332.
Money, tables of, 18.

Monomial quantity (Gr. monos one, and Lat.
nomen a name), how to extract a proposed
root of a, 194; exercises, 195.

Multiplication, Simple, use and application of,
10; table of, ib.; various workings in, 11, 12.
of Compound Quantities, 23; of Frac-
tions, 30; of Decimals, 38.

Napier's analogies, 407.

N

Negative and positive values in algebra, 215.
Negative Angles, 299.

Negative sign, use of, to denote position, 298.
Number, methods of finding any power of a, 286

et seq.; use of a table of, 283, 284; method of
finding the characteristic, 285.

Numbers, reading of, 5.

Numeration table (Lat. numero to number), 5.
Numerator of fractions, 197.

Numerical solution of right-angled triangles,
361 et seq.

Numerical value of logarithms, 281; of sines
and cosines of angles, 315.

P

Parabola, definition of the, 432; illustrations of
the, 444.

Parallelipeds, equality of, 385, 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.; 423-447; (see
PROPOSITIONS).

Progression, Arithmetical, 220; exercises in,
222; Geometrical, 222.
Proportion in algebra, 218.

Proportion and Progression, questions in which
they are concerned, 225.

Proportional Parts, use of a table of, 283.

Proportion, rule of, 30.

PROPOSITIONS of Euclid, 48-67; 87-92; 97-
112; 117-124; 423-447.

in Plane Geometry, 243-250.

in Spherical Geometry, 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.

R

Radius of an inscribed circle, to find the, 327.
Railway Cutting, to find the solid content of a,
393.

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.

|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
et seq.

Symbolical Algebra, 261.

Symbols of Quantity in algebra, 162.
Symmetrical Quadratics, 217.

T

Tangents, the calculation of, 352.

Tables of money, time, weights, and measures,

18.

Term, meaning of the word in algebra, 163.
Theorems of Euclid (Gr. theorema a proposition
requiring demonstration), 49 et seq. (See PRO-
POSITIONS.)

Time, tables of, 18.

S

Science (Lat. scientia the knowledge of things),
on the study of, 3.

Tower, how to determine the height of a, 367
368.

Triangles, relations between the sides and the
angles, 322; methods of treating, 418, 419.