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IN DE X;

Glossarial, Explanatory, and Referential.

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.

Algebra, Proportion and Progression, 225.

Extraction of Roots,-of the Square Root of a Polynomial, 228; exercises 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; logarithmic sines of, 357, 358; Delambre's method for solving, 360.

spherical triangle, sines of the, 402. Apothecaries' weight (Gr. upotheca 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 polygon, 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.

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.

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E

Ellipse (Gr. elleipsis deficiency), definition and

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 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.

illustration of the, 439.
Equality, algebraic sign of, 163.
Equation (Lat. æquo to equal), to find the roots

of the, 340.

F

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, 296, 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.

Problems in practical geometry, 423–
447.
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.

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.

[blocks in formation]

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,

ductory remarks on, I 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 Earth work, to find the solid content of

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.

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.

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),

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

368.
Triangles, relations between the sides and the

angles, 322; methods of treating, 418, 419.

on the study of, 3.

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