" Of little use to vary forms of, without changing the

principles of construction, 240.

Experiment, In what sense used, 25 (Note).

EXPONENT An expression to show how many equal factors are

employed, 297.

Extremes. Subject and predicate of a proposition, 38, 67.

FACT ... Any thinir which has been or is, 24.

" Knowledge of, how derived, 25.

" In what sense used, 25.



404 INDEX.



FACT regarded as a genus, Section 25.

Factories, Value of science in, 408.

FALLACY Any unsound mode of arguing which appears to de-
mand our conviction, and to be decisive of the ques-
tion in hand, when in fairness it is not, 68.

" Illustration of, 53.

" Example and analysis of, 59, 60.

" Material and Logical, 69.

" Rules for detecting, 70.

FIGURE A portion of space limited by boundaries, 82.

" Each geometrical, stands for a class, 281.

Figures In Arithmetic show how many times a unit is taken,

132.

" do not indicate the kind of unit, 132.

" Laws of the places of, 133, 134.

" have no value, 135, 205.

" Methods of reading, 137 ; of writing, 203.

Definitions of, 205, 206.

" should be early used in Arithmetic, 223.

First Arithmetic, what should be taught in it, 230.

" Faculties to be cultivated by it, 218.

" Construction of the lessons, 218-222.

" Lesson in Fractions, 224-228.

" Tables of Denominate Numbers Examples, 229

Fractions Come from the unit one, 139.

" should be constantly compared with one, 170.

Definitions of, 208.

" Lessons in, in First Arithmetic, 224-228

FKACTIONAL units, 163 ; orders of, 164; language of, 164-167, 201.

" " three things necessary to their apprehension,

168.

" " advantages of, 169.

" " two things necessary to their being equal, 169.

Galileo, Imprisoned in the 17th century, 393.

GENERALiZATlON...The process of contemplating the agreement of
several objects in certain points, and giving to all
and each of these objects a name applicable to
them in respect to this agreement, 14.
" implies abstraction, 14.

GENUS The most extensive term of classification, and conse-



I X D E X . 405



quently the one involving the fewest particulars,
Sections 16, 17.

GENUS HIGUEST. That which cannot be referred to a more

extended classification, 19.

" SUBALTERN. A species of a more extended classifi-

cation, 18.

Geometrical Magnitudes, three classes of, 242, 277.

" " do not involve matter, 251.

" " their boundaries or limits, 251.

" " each luis its unit of measure, 256.

" " analysis of comparison, 274, 27").

" "to what the examination of properties

has reference, 277.
" Proportion, 171 ; Ratio, 171 ; Progression, 178.

GEOMETRY Treats of space, and compares portions of space with

each other, for the purpose of pointing out their
properties and mutual relations, 241.
" Why a deductive science, 261.

" First notions of, how acquired, 368-370.

Practical utility of, 407.

" Origin of the science, 410.

* Its place in a course of instruction, 359.

* ANALYTICAL, Examines the properties, measures, and

relations of the Geometrical Magni-
tudes by means of the analytical
symbols, 285, 286.
" " originated with Descartes, 285.

* " difference between it and Calculus, 288.
" " its importance, extent, and methods,

188.

* DESCRIPTIVE. That branch of mathematics which

considers the positions of the Geo-
metrical Magnitudes as they may
exist in space, and determines these
positions by referring the magni-
tudes to two planes called the
Planes of Projection, 362.
" how regarded in France, 362.

Governor, Functions of, in machinery, 408.

Grammar Defined, 120.

Gravitation, I^aw of, 32, 394.



406 INDEX.



Hall, Captain's, voyage from San Bias to Rio Janeiro, Section 404.
Harlem river, Bridge over, and width, 412.
Herscliel, Sir John, Quotation from, 27, 372, 395,409.
Hull of the steamship, how formed, 409.

Illative Conjunctions, 48.

ILLICIT PROCESS..When a term is distributed in the conclusion which

was not distributed in one of the premises, 67.
Indefinite Propositions, 62.

Index Of a root, 299.

INDUCTION Is that part of Logic which infers truths from facts,

30-33.

" Logic of, 30.

" supposes necessary observations accurately made, 32.

" Example of, Blakewell, 32 ; of Newton, 32.

" based upon the relation of cause and effect, 33.

" Reasoning from particulars to generals, 34.

" its place in Logic, 72.

" how thrown into the form of a syllogism, 74, 103.

" Truths of, verified by Deduction, 385, 386.

Inertia proportioned to weight, 272.

INFINITY, The limit of an increasing quantity, 306-310.

Integral Numbers, why easier than fractions, 170.

" constructed on a single principle, 235.

INTUITION Is strictly applicable only to that mode of contempla-
tion, in which we look at facts, or classes of facts,
and immediately apprehend their relations, 27.
Iron, different ideas attached to the word, 372.

JUDGMENT Is the comparing together in the mind two of the

notions (or ideas) which are the objects of appre-
hension, and pronouncing that they agree or dis-
agree, 8.
" is either Affirmative or Negative, 8.

Kant, Quotation from, 21.

KNOWLEDGE Is a clear and certain conception of that which is

true, 23.
" facts and truths elements of, 25.

of facts, how derived, 25.
" some possessed antecedently to reasoning, 29.



INDEX.



407



KNOWLEDGE,



the greater part matter of inference, Section 29.
two ways of increasing, 3?:!.
cannot exceed our ideas, 373.

the increase of, renders classification necessary,
page 20.



LANGUAGE



Affords the signs by which the operations of the
mind are recorded, expressed, and communicated,
10.

" Every branch of knowledge has its own, 11.

" of numbers, 91 ; of mathematics, 90.

" of mathematics must be thoroughly learned. 88.

" " " its generality, 89.

" for fractional units, 164, 167, 207.

" Arithmetical, 196-203.

" exact, necessary to accurate thought, 209.

* of Arithmetic, its uses, 223.

* of Algebra, the first thing to which the pupil's wind

should be directed, 294.

" Culture of the mind by the use of exact, 373.

" of Calculus, 346.

Laws of Nature, Science makes them known, 21, 319.

" " refers individual cases to them, 55.

" generalized facts, 55.

" include all contingencies, 372.

" every diversity the effect of, 396.

Newton's, 329-337.
one dimension of space, 80.
in First Arithmetic, how arranged, 218.
" " " their connections, 222.

may stand for all numbers, 280.
represents things in general, 281.
.Th application of the principles of Trigonometry to
the determination of the difference between the
distances of any two points from the centre of the
earth, 361.

Its practical uses, 351.
Definition of, 32'2.
of discontinuous quantity, 323.
of continuous quantity, 325.
differently defined, 339-343.



Lemma,

Length

Lessons

Letter
LEVELLING ..



Limit,



408 INDEX.



LINE One dimension of space, Sections 82, 243.

" A straight line does not change its direction, 82

243, 368.
* Curved line, one which changes its direction at every

point, 82, 243.

" Axiom of the straight, 243.

Lines, limits of, 251.

" Auxiliary, 263.

Liquid Measure, Its units and scale, 153.
" Local value of a figure," has no significance, 135, 205.
Locke, Quotation from, 373.

LOGIC Takes note of and decides upon the sufficiency of the

evidence by which truths are established, 29.
" Nearly the whole of science and conduct amenable

to, 29.

" of Induction, its nature, 30.

" Archbishop Whately's views of, 72.

Mr. Mill's views of, 72.
Logical Fallacy, 69.

Machinery of factories arranged on a general plan, 408.

" of the steamship, 409.

Major Premise, often suppressed, cannot be denied, 46.
" ultimate, of Induction, 74, 102.

Major Premises, of Geometry, 241, 261.

Mansfield, Mr., Quotation from, 375, 377.

MAKE The evidence contained in the attributes implied in

a general name, by which we infer that any thing
called by that name possesses another attribute or
set of attributes. For example : " All equilateral
triangles are equiangular." Knowing this general
proposition, when we consider any object possess-
ing the attributes implied in the term "equila-
teral triangle," we may infer that it possesses the
attributes implied in the term " equiangular ; "
thus using the first attributes as a mark or evi-
dence of the second. Hence, whatever possesses
any mark possesses those attributes of which it is
a mark, 101,231,263.

Masts, of the steamship, how placed, 409.

Material Fallacy, 69.



INDEX.



Mathematical Reasoning conforms to logical rules, Section 73.

" " every truth established by, is developed

by a process of Arithmetic, Geometry, or
Analysis, or a combination of them, 96.
MATHEMATICS . .The science of quantity, 76.

" Pure, embraces the principles of the science, 98 to 104.

* " on what based, 100.

" Mixed, embraces the applications, 104.

" Primary signification, 99.

" Language of, 90.

" " Exact science," 100.

* Logical test of truth in, 100.

* a deductive science, 100, 101.

" concerned with number and space, 73, 76, 108.

* What gives rise to its existence, 103.

* Why peculiarly adapted to give clear ideas, 374-376,

379.

* a pure science, 379.

* considered as furnishing the keys of knowledge, 381.

* Widest applications are in nature, 384.

* Effects on the mind and character, 378, 390.

* Guidance through Nature, 390.
" Its necessity in Astronomy, 391.
Results reached by it, 399, 400.

* Practical advantages of, 405.

" What a course of, should present, and how, 866

" Reasonings of, the same in each branch, 349.

" Faculties required by, 351.

" Necessity of, to the philosopher, page 16.

MEASURE A term of comparison, lOo.

" Unit of, should be exhibited to give ideas of num-

bers, 140.

" " for lines, surfaces, volumes, 253.

" of a magnitude, how ascertained, 253.

Middle Term , distributed when the predicate of a negative proposi-
tion, 64.

" When equivocal, 67.

Mill, Mr., hia views of Logic, 72, 74.

Mind, Operations of, in reasoning, 6.

" Abstraction a faculty, process, and state of, 18

Processes of, which leave no trace, 68.
18



410



INDEX.



Mind,



Minus sign,

Motion

Multiplication,



Multiplication,



Names,



Faculties of, cultivated by Arithmetic, Section 188.
Thinking faculty of, peculiarly cultivated by mathe-
matics, 375, 376.

Power of, fixed by definition, 301.
proportional to force impressed, 272.
Readings in, 130 ; examples in, 161.
What the definition of, requires, 185.
Combinations in, 199.

All operations in, governed by one principle, 236.
in Algebra, illustrations of, 303-305.



Definitions are of, 1.
" given to portions of space, and defined in Geometry.

242.
Naturalist determines the species of an animal from examining

a bone, 383.
Negative premises, nothing can be inferred from, 67.

" demonstration, its nature, 267-269 ; illustration of

268.
Newton, his method of discovery, 32.

" changed Astronomy from an experimental to a de-

ductive science, 387-389.
" Lemmas, 329-337.

" in harmony with Leibnitz, 341.

Non-distribution of terms, 61.

" Word " some " which marks, not always expressed, 62.

NUMBER A unit, or a collection of units, 78.

" How learned, 78.

" Axioms for forming, 78, 308.

" Three ways of expressing, 113.

" Ideas of, complex, 115.

* Two things necessary for apprehending clearly, 117.
" Simple and Denominate, 118.

" Examples of reading Simple, 137.

" Two ways of forming from OXE, 138.

" first learned through the senses, 140, 366.

" Two ways of comparing, 171.

" compared, must be of the same kind, 179-183.

" Definitions of, 205, 206.

" must be of something, 279.

may stand for all things, 280.



INDEX. 411



NUMBER First lessons in, impress the first elements of mathe-
matical science, Section 352.

Olmsted's Mechanics, quotation from, 273.

Optician, Illustration, 216.

Oral Arithmetic, its inefficiency without figures, 223.
Order of subjects in Arithmetic, 190.

PARALLELOGRAM...A quadrilateral having its opposite sides, taken two

and, two parallel, 246.

" regarded as a species, 17 ; as a genus, 18.

" Properties of, 260.

Particular proposition, 62.

" premises, nothing can be proved from, 67.

Pendulum, the standard for measurement, 257.

Philosophy, Natural, originally experimental, 387.

" " has been rendered mathematical, 887.

Place, idea attached to the word, 81.

" designates the unit of a number, 206.

PLANE That with which a straight line, having two points in

common, and anyhow placed, will coincide, 244.
" First idea of, how impressed, 369.

PLANE FIGURE. .Any portion of a plane bounded by lines, 244.
Plane Figures in general, 247.

PorNT That which has position in space without occupying

any part of it, 80.

Points, extremities or limits of a line, 243.

Practical Rules in Arithmetic, 185, 186.

The true, 211, must be the consequent of science, 233.

" Popular meaning of, 401, 403.

" Questions with regard to, 401, 402.

" Consequences of an erroneous view of, 404.

" True signification of, 404.

Practice precedes theory, but is improved by it, 42.

" without science is empiricism, page 13.

PREDICATE That which is affirmed or denied of the subject, 38.

" Distribution, 63.

" Non-distribution, 63.

sometimes coincides with the subject, 63.

PREMISE Each of two propositions of a syllogism admitted to

be true, 40.



412



INDEX.



PREMISE MAJOR PREMISE The proposition of a syllogism

which contains the predicate of the conclusion, Sec-
tion 40.
MINOR PREMISE The proposition of a syllogism

which contains the subject of the conclusion, 40.
Pressure, a law of fluids, 414.

Principle of science applied, 22.

" on which valid arguments are constructed, 52.

" Value of a, greater as it is more simple, 54.

" Aristotle's Dictum, a general, 55.

" the same in the ground rules for simple and denomi-

nate numbers, 159-162, 236.
" of science and rule of art, 187.

Principles should be separated from applications, 194, 195.

" of science are general truths, 212.

" of Arithmetic, how taught, 212.

" should precede practice, 233.

" of Mathematics, deduced from definitions and axioms,

301.

Process of acquiring mathematical knowledge, 366-370

Product of several numbers, 296.

Progression, Geometrical, 178.
Property of a figure, 260.

PROPORTION . . . .The relation which one quantity bears to another with

respect to its being greater or less, 171, 271-273
" Arithmetical and Geometrical, 171.

" Reciprocal or Inverse, 273.

" of geometrical figures, 274-277.

PROPOSITION . . . A judgment expressed in words, 35.

" All truth and all error lie in propositions, also answers

to all questions, 36.

" formed by putting together two names, 37.

" consists of three parts, 38.

" subject and predicate, called extremes, 38.

" Affirmative, 39 ; Negative, 39.

" Three propositions essential to a syllogism, 40.

" Universal, 62.

" Particular, 62.

QUADRILATERAL..A portion of a plane bounded by four straight lines,
246.



INDEX. 413



QUADRILATERAL regarded as a genus, Section 17.

" Different varieties of, 246.

Quality of a proposition refers to its being affirmative or

negative, 63.

Quantities only of the same kind can be compared, 271.

" Two classes of, in Algebra, 293, 317.

" " " " in the other branches of Analysis,

286, 287, 317.

* compared, must be equal or unequal, 109, 311.
QUANTITY Is a general term applicable to every thing which can

be increased or diminished, and measured, 75, 371.
" Abstract, 75, 107.

Concrete, 107.

" Propositions divided according to, 62.

" presented by symbols, 89.

" consists of parts which can be numbered, 280.

* Constant, 286.

" Variable, 286.

" Six operations can be performed on, 292, 299.

* represented by six signs, 293.

" Nature of, not affected by the sign, 294, 300.

Questions known, when all propositions are known, 36.

" Analysis of, 183, 184.

" with regard to methods of instruction, 353.

Quotations from Kant, 21 ; Sir John Herschel, 27, 372, 391. 409;

Cousin, 188; Olmsted's Mechanics, 272; Locke,
370; Mansfield's Discourse on Mathematics, 375,
377 ; Lord Bacon, 378 ; Dr. Barrow, 378, 390.

Railways, Problem presented in, 411.^

Rainbow, Illustration, 372.

RATIO The quotient arising from dividing one number or

quantity by another, 171, 271.

" Discussion concerning it, 173-179.

" Arithmetical and Geometrical, 17L

- How determined, 173.

An abstract number, 271, 276.

" Terms direct, inverse, or reciprocal, not applicable to,

273.
Reading in Addition, 123, 124 ; advantages of, 125.

* in Subtraction, 127.



414



INDEX.



Reading in Multiplication, Section 129.

" in Division, 130.

of figures, its aid in practical operations, 334.
Reason, To make use of arguments, 42.

A premise placed after the conclusion, 48.

REASONING The act of proceeding from certain judgments to

another, founded on them, 9.

" Three operations of the mind concerned in, 6.

" Process, sameness of the, 42, 43, 45, 318.

" processes of mathematics consist of two parts, 73.

" in Analysis is based on the supposition that we are

dealing with things, 282.
Reciprocal or Inverse Proportion, 270.

RECTANGLE A parallelogram whose angles are right angles, 246.

Remarks, Concluding subject of Arithmetic, 240.

Reservoirs, Croton, description of, 412.
Right angle, Definition of, 262.
Roman Table, when taught, 219.
Root, Symbol for the extraction of, 299.

Rule of Three, Solution of questions in, 177.
" Comparison of numbers, 194.

" should precede its applications, 195.

Rules, Every thing done according to, 21.

" of reasoning analogous to those of Arithmetic, 45.

. " Advantages of logical, 50.

" for teaching, 194.

How framed, 301.

Scale of Tens, Units increasing by, 131-137, 165, 191.

SCIENCE In its popular sense means knowledge reduced to

order, 21, 276.
" In its technical sense means an analysis of the laws

of nature, 21.

" contrasted with art, 22.

" of Arithmetic, 180.

Principles of, 204, 212.
" 'Methods of, must be followed in Arithmetic, 232.

* of Geometry, 241, 252. 261.

" Objects and means of pure, 372.

* should be made as much deductive as possible, 386.

* Deductive and experimental, 387.



IHDEX.



SCIENCE when experimental, Sections 388, 389 ; when deduc-
tive, 388, 389.

" What it has accomplished, 398.

" Practical value of, in factories, 408.

" " " " in constructing steamships, 409.

* " " " in laying out and measuring land,

410.

" " " " in constructing railways, 411.

" Its power illustrated in Croton aqueduct, 412.

" What constitutes it, 354.

Second Arithmetic, its place and construction, 231-334
Sextant, its uses in Navigation, 409.

SHADES, SHADOWS, AND PERSPECTIVE An application of Descriptive

Geometry, 363.

SIGXIFICATE . . . .An individual for which a common term stands, 15.
Signs, Six used to denote operations on quantity, 293.

" How to be interpreted, 294.

" do not affect the nature of the quantity, 294, 300.

" indicate operations, 300.

Solution of all questions in the Rule of Three, 177.

" of an equation in Algebra, 312.

SPACE. Is indefinite extension, 80.

" has three dimensions, length, breadth, and thickness,

80-82.

" Clear conception of, necessary to understand Geome-

try, 242.

SPECIES One of the divisions of a genus in which the charac-
teristic is less extensive, but more full and com-
plete, 16, 17.

SUBSPECIES One of the divisions of a species, in
which the characteristic is less extensive, but more
full and complete, 16, 19.

LOWEST SPECIES A species which cannot be regard-
ed as a genus, 17.
Spelling, 120 ; in Addition, &c., 122-130.

SQUARE A quadrilateral whose sides are equal and angles

right an<rles, 249.
Statement of a proposition in Algebra, 312.

" in what it consists, 313.

Steamship, an application of science, 409.



416 IKDEX.



SUBJECT The name denoting the person or thing of which

something is affirmed or denied, Section 38.
Subjects, How presented in a text-book, 213-216.

Subtraction, Eeadings in, 127.
" Examples in, 160.

" Combinations in, 198.

" All operations in, governed by one principle, 236.

" in Algebra, illustration of, 302.

'Suggestions, for teaching Geometry, 277.

" for teaching Algebra, 319.

Sam, Its definition, 207.

SURFACE A portion of space having two dimensions, 83, 244*

369.

" Plane and Curved, 83, 244.

Surfaces, Curved, 249.

" " of Elementary Geometry, 249.

" Limits of, 251.

SUKVEXLKO The application of the principles of Trigonometry to

the measurement of portions of the earth's surface,
361.
" A branch of practical science, 410.

SYLLOGISM A form of stating the connection which may exist for

the purpose of reasoning, between three proposi-
tions, 40.

* A formula for ascertaining what may be predicated.

How it accomplishes this, 41.

* not meant by Aristotle to be the form in which argu-

ments should always be stated, 53.
not a distinct kind of argument, 54.
an argument stated at full length, 56.
Symbols used for the terms of, 56.
Rules for examining syllogisms, 67.
has three and only three terms, 67.
" " " " " propositions, 67.
test of deductive reasoning, 72, 102, 811.

SYMBOLS The letters which denote quantities, and the signs

which indicate operations, 93, 89, 300.
" used for the terms of a syllogism, 58.

* Advantages of, 57.

* Validity of the argument still evident, 58.

* Truths inferred by means of, true of all things, 381.



INDEX. 417



SYMBOLS regarded as things, Section 282.

" Two classes of, in analysis, 310.

" Abstract and concrete quantity represented by, 371.

SYNTHESIS The process of first considering the elements sepa-
rately, then combining them, and ascertaining the
results of combination, 95, 377.
Synthetical form, for what best adapted, 71, 95.

Tables of Denominate Numbers, 141-158.

TANGENT Tangent and Limit, 327, 328.

TECHNICAL Particular and limited sense, 90.

TERM Is an act of apprehension expressed in words, 15.

A singular term denotes but a single individual, 15.
A common denotes any individual of a whole class, 15.
" affords the means of classification, 16.
" Nature of, 20.

" No real thing corresponding to, 20.
" Why applicable to several individuals, 20.
MAJOR TtfRM The predicate of the conclusion, 40.
MINOR TERM The subject of the conclusion, 40.
MIDDLE TERM The common term of the two

premises, 40.
* DISTRIBUTED A term is distributed when it stands

for all its significates, 61.

NOT DISTRIBUTED When it stands for a part of ita

significates only, 61.

TERMS Two of the three parts of a proposition, 38.

The antecedent and consequent of a proportion, 172.

should always be used in the same sense, 178, 209.

TEXT-BOOK Should be an aid to the teacher in imparting instruo

tion, and to the learner in acquiring knowledge,
213.

THICKNESS A dimension of space, 81.

Third Arithmetic, Principles contained in, and method of construction

235-240.

Time, Measure, its units and scale, 155.
Topograpny, Its uses, 410.

TKAPK/OID A quadrilateral, having two sides parallel, 246.

TRIANGLE A portion of a plane bounded by three straight lino

245.

18*



418 ISTDEX.



TRIANGLE The simplest plane figure, Section 245.

" Different kinds of, 245.

" regarded as a genus, 260.

TRIGONOMETRY.. An application of the principles of Arithmetic, Alge-
bra, and Geometry to the determination of the
sides and angles of triangles, 360.
" Plane and Spherical, 360.

Troy Weight, Its units and scale, 144.

TRUTH An exact accordance with what has been, is, or shall

be, 24.

" Two methods of ascertaining, 24

" is inference from facts or other truths, 24, 25.

" regarded as a species. 25.

" How inferred from facts, 26.

" A true proposition. 36.

TRUTHS INTUITIVE OR SELF-EVIDENT Are such as become

known by considering all the facts on which
they depend, and apprehending the relations of
those facts at the same time, and by the same act
by which we apprehend the facts themselves, 27.
** LOGICAL Those inferred from numerous and com-

plicated facts ; and also, truths inferred from
truths, 28.

" of Geometry, 241.

" Three classes of, 241.

" Demonstrative, 241.

Unit fixed by the place of the figure, 134

" of the fraction, 168, 169.

" of the expression, 168.

Unities, Advantages of the system of, 157-162.
UNIT OP MEA8URE...The standard for measurement, 105.

" for lines, surfaces, volumes, 253.

" only basis for estimating quantity, 255.

UNIT ONE A single thing, 111.

" All numbers come from, 115, 116, 139, 157.

** Method of impressing its values, 140.

* Three kinds of operations performed upon, 190-194