CHAP. III.] ANALYTICAL OUTLINE. 93
own or another's reasoning, in syllogisms at full Logic tends
~ -i- -11 i -i to cultivate
length; yet a familiarity with logical principles habits of
tends very much (as all feel, who are really well de:irreason -
ing.
acquainted with them) to beget a habit of clear
and sound reasoning. The truth is, in this as
in many other things, there are processes going The habit
fixed, we
on in the mind (when we are practising any naturally foi-
thing quite familiar to us), with such rapidity
as to leave no trace in the memory ; and we
often apply principles which did not, as far as
we are conscious, even occur to us at the time.
69. Let it be remembered, that in every conclusion
c . i-ii follows from
process of reasoning, logically stated, the con- twoantece _
elusion is inferred from two antecedent propo- dent P ren -
sitions, called the Premises. Hence, it is man-
ifest, that in every argument, the fault, if there Fallacy, if
any, either in
be any, must be either, the
1st. In the premises; or,
2d. In the conclusion (when it does not follow orconciu-
~ sion, or both.
irom them) ; or,
3d. In both.
In every fallacy, the conclusion either does or
does not follow from the premises.
When the fault is in the premises; that is, when m the
when they are such as ought not to have been pre '
assumed, and the conclusion legitimately follows
from them, the fallacy 's called a Material Fal-
94
LOGIC.
[BOOK i
lacy, because it lies in the matter of the argu-
ment.
when in the Where the conclusion does not follow from
lon ' the premises, it is manifest that the fault is in
the reasoning, and in that alone : these, there-
fore, are called Logical Fallacies, as being prop-
erly violations of those rules of reasoning which
it is the province of logic to lay down.
When the fault lies in both the premises and
reasoning, the fallacy is both Material and Logical
when in
Rules for
gument.
tt Rule.
ad Rule.
3d Rule.
serve as
guides,
70. In examining a train of argumentation,
to ascertain if a fallacy have crept into it, the
following points would naturally suggest them-
selves :
1st. What is the proposition to be proved?
On what facts or truths, as premises, is the ar-
gument to rest? and, What are the marks of
truth by which the conclusion may be known ?
2d. Are the premises both true ? If facts, are
they substantiated by sufficient proofs ? If truths,
were they logically inferred, and from correct
premises ?
3d. Is the middle term what it should be, and
the conclusion logically inferred from the prem-
ises?
These general suggestions may serve as guides
. . .
m examining arguments for the purpose of de-
CHAP. III.] ANALYTICAL OUTLINE. 95
tecting fallacies ; but however perfect general to detect
rules may be, it is quite certain that error, in
its thousand forms, will not always be separated
from truth, even by those who most thoroughly
understand and carefully apply such rules
CONCLUDING REMARKS.
71. The imperfect and irregular sketch which Logic
correspond!
nas here been attempted 01 deductive logic, may witn tn e
suffice to point out the general drift and purpose reaa g " u
of the science, and to show its entire correspond-
ence with the reasonings in Geometry. The
analytical form, which has here been adopted, Analytical
is, generally speaking, better suited for introdu-
cing any science in the plainest and most inter-
esting form ; though the synthetical is the more synthetical
regular, and the more compendious form for sto-
ring it up in the memory.
72 It has been a matter about which wri- induction:
i ff i i i does it form
ters on logic have differed, whether, and in con- a part of
formity to what principles, Induction forms a Lo ^ lc?
part of the science ; Archbishop Whately main- whaujiy'i
. . . . . . ., . opinion
taming that logic is only concerned in inferring
truths from known and admitted premises, and
that all reasoning, whether Inductive or Deduc-
tive, is shown by analysis to have the syllogism
9U LOGIC. [BOOK i.
Mill's views, for its type ; while Mr. Mill, a writer of perhaps
greater authority, holds that deductive logic is
but the carrying out of what induction begins ;
that all reasoning is founded on principles of in-
ference ulterior to the syllogism, and that the
syllogism is the test of deduction only.
Without presuming at all to decide defini-
tively a question which has been considered and
Reasons for passed upon by two of the most acute minds of
the course
the age, it may perhaps not be out of place to
state the reasons which induced me to adopt
the opinions of Mr. Mill in view of the par-
ticular use which I wished to make of logic
Leading ob- 73. It was, as stated in the general plan,
Jectsof the
plan- one of my leading objects to point out the cor-
respondence between the science of logic and
the science of mathematics : to show, in fact,
TO show that that mathematical reasoning conforms, in every
mathemati- !_*".. r i j
cai reasoning respect, to the strictest rules oi logic, and is m-
conforms to j ee( j ^ ut j o ~ c ap pij ec i to tne abstract quantities,
logical rules.
Number and Space. In treating of space, about
which the science of Geometry is conversant, we
shall see that the reasoning rests mainly on the
Axioms, how axioms, and that these are established by induc-
tive processes. The processes of reasoning which
relate to numbers, whether the numbers are rep-
resented by figures or letters, consist of two parts.
CHAP. III.] ANALYTICAL OUTLINE. 97
1st. To obtain formulas for, that is, to express
in the language of science, the relations between
the quantities, facts, truths or principles, what- T *>*** <*
the reasoning
ever they may be, that form the subject of the process.
reasoning ; and,
2dly. To deduce from these, by processes
purely logical, all the truths which are implied
in them, as premises.
74. Before dismissing the subject, it may AIJ induc-
tion may be
be well to remark, that every induction may thrown into
be thrown into the form of a syllogism, by sup- o*^
plying the major premise. If this be done, we syiiogism,by
shall see that something equivalent to the uni- proper majoi
formity of the course of nature will appear as
the ultimate major premise of all inductions ;
and will, therefore, stand to all inductions in
the relation in which, as has been shown, the
major premise of a syllogism always stands to
the conclusion ; not contributing at all to prove
it, but being a necessary condition of its being
proved. This fact sustains the view taken by
Mr. Mill, as stated above ; for, this ultimate ma- HOW this
, . . f .. . . c major prem
jor premise, or any substitution tor it, is an inter- ^5
ence by Induction, but cannot be arrived at by
means of a syllogism.
7
ABSTRACT
g
CURRENCY.
WEIGHT.
H
M
H
P
a
BOOK II.
MATHEMATICAL SCIENCE.
CHAPTER I.
VTANTITY AND MATHEMATICAL SCIENCE DEFINED DIFFERENT KIXDg OP QTTAH-
TITT LANGUAGE OF MATHEMATICS EXPLAINED SUBJECTS CL A BSD-ISO UNIT
07 MEASURE DEFINED MATHEMATICS A DEDCCTITE SCONCE.
QUANTITY.
75. QUANTITY is any thing which can be Quantity
increased, diminished, and measured.
76. MATHEMATICS is the science of quan- Mthematfc
tity; that is, the science which treats of the
measures of quantities and of their relations to
each other.
77. There are two kinds of quantity ; Num- Kinds of
her and Space.
NUMBEB.
78. A NUMBER is a unit, or a collection Number
defined.
of units.
100 MATHEMATICAL SCIENCE. [BOOK II.
Abstract. AN ABSTRACT NUMBER is one whose unit is
not named; as, one, two, three, &c.
Denominate. A DENOMINATE NUMBER is One whose UUlt IS
named; as three feet, three yards, three pounds.
Such numbers are also called CONCRETE NUMBERS.
dow we ob- How do we acquire our first notions of num-
of number, bers ? By first presenting to the mind, through
the eye, a single thing, and calling it ONE.
Then presenting two things, and naming them
TWO ; then three things, and naming them
THREE; and so on for other numbers. Thus,
it is done by we acquire primarily, in a concrete form, our
oarison e l emen tary notions of number, by perception,
comparison,
and
reflection.
comparison, and reflection; for, we must first
perceive how many things are numbered; then
compare what is designated by the word one,
Reasons, with what is designated by the words two,
three, &c., and then reflect on the results of
such comparisons until we clearly apprehend
the difference in the signification of the words.
Having thus acquired, in a concrete form, our
conceptions of numbers, we can consider num-
bers as separated from any particular objects,
TWO axioms and thug fonn concep tion of them in the ab-
nessary for
the forma- gtract. We require but two axioms for the
tlon of num-
bers, formation of all numbers :
ut axiom. 1st. That one may be added to any number,
and that the number which results will be
CHAP. I.] DEFINITION'S. 101
greater by one than the number to which the
one was added.
2d. That one may be divided into any num- sa axiom,
ber of equal parts.
79. Under Number, we have four species, or Four kinds
of number.
subdivisions, each differing from the other three,
in the unit of its base: thus,
1. Abstract Number, when the base is the ab- Abstract,
stract unit one :
2. Number of Currency, when the base is a Currency,
unit of currency, as one dollar :
3. Number of Weight, when the base is a unit weight
of weight, as one pound :
4. Number of Time, when the base is a unit Time,
of time, as one day.
Hence, in number, we have four kinds of Four kinds
of units.
units: Abstract Units ; Units of Currency; Units
of Weight ; and Units of Time.
SPACE.
80. SPACE is indefinite extension. We ac- space
defined.
quire our ideas of it by observing that parts
of it are occupied by matter or bodies. This
enables us to attach a definite idea to the word
place. We are then able to sav, intellisriblv,
J ' Place :
that a point is that which has place, or posi-
102 MATHEMATICAL SCIEHCE. [BOOK II.
tion in space, without occupying any part of
it. Having conceived a second point in space,
we can understand the important axiom, "A
Axiom con- straight line is the shortest distance between
traighfiine. two points;" and this line we call length, or a
dimension of space.
81. If we conceive a second straight line
to be drawn, meeting the first, but lying in a
direction directly from it, we shall have a sec-
Breadth
defined, ond dimension of space, which we call breadth.
If these lines be prolonged, in both directions,
they will include four portions of space, which
make up what is called a plane surface, or plane :
A plane hence, a plane has two dimensions, length and
breadth. If now we draw a line on either side
of this plane, we shall have another dimen-
space has s i n f space, called thickness : hence, space has
th ^ en " three dimensions length, breadth, and thick-
ness.
Figure 82. A portion of space limited by bounda-
ries, is called a Figure. If such portion of
Line de- space have but one dimension, it is called a line,
and may be limited by two points, one at each
TWO kinds extremity. There are two kinds of lines, straight
straighTand anc * curve <l. A straight line, is one which does
curved. no cna nge its direction between any two of its
CHAP. I.]
SPACE.
103
Plane,
curved.
Difference.
points, and a curved line constantly changes its
direction at every point.
83. A portion of space having two dimen- surface:
sions is called a surface. There are two kinds
of surfaces Plane Surfaces and Curved Sur-
faces. With the former, a straight line, having
two points in common, will always coincide,
however it may be placed, while with the latter
J Boundaries
it will not. The boundaries of surfaces are of a surface,
lines, straight or curved.
84. A limited portion of space, having three volumes,
dimensions, is called a Volume. All volumes Boundaries
are bounded by surfaces, either plane or curved.
Angle?.
85. AN ANGLE is the amount of divergence
of two lines, of two planes, or of several planes,
meeting at a point ; and is measured, like other
magnitudes, by comparing it with its unit of
measure. Hence, in space, we have four units, measure,
differing in kind:
1. Linear Units, for the measurement of Linear,
lines ;
2. Units of Surface, for the measurement of surface,
surfaces ;
3. Units of Volume, for the measurement of Volume,
volumes ; and
104 MATHEMATICAL SCIENCE. [BOOK II.
Angle. 4. Units of Angles, for the measurement of
angles.
Bight unite. 86. Besides the eight kinds of units, four
of number and four of space, embraced in the
above classification, and in which the units of
each class are connected by known laws, there
are yet isolated denominate numbers, such as
five chairs, six horses, seven things, &c., which
Unite with- do not admit of classification, because they have
out law.
no law of formation. Neither does this classi-
nnite" 1 " 1 fication include the Infinitesimal Units, which
are specially treated of in Chapter V., Book II.,
and which are the elements of a very important
branch of Mathematical Science.
Language 87. The language of Mathematics is mixed.
mathema- Although composed mainly of symbols, which
are defined with reference to the uses whicli
are made of them, and therefore have a pre-
cise signification; it is also composed, in part,
of words transferred from our common lan-
1)018 guage. The symbols, although arbitrary signs,
are, nevertheless, entirely general, as signs and
instruments of thought ; and when the sense in
which they are used is once fixed, by definition,
Reuse un- they preserve throughout the entire analvsis
changed.
precisely the same signification. The meaning
CHAP. I.] LANGUAGE OF MATHEMATICS. 105
of the words borrowed from our common vocab- words bor-
ulary is often modified, and sometimes entirely common
changed, when the words are transferred to the a
8ense -
language of science. They are then used in a and n^i '
* technical
particular sense, and are said to have a technical
signification.
88. It is of the first importance that the Language
elements of the language be clearly understood, undent Jid :
that the signification of every word or sym-
bol be distinctly apprehended, and that the con-
nection between the thought and the word or
symbol which expresses it, be so well estab-
lished that the one shall immediately suggest Mathemati
the other. It is not possible to pursue the sub- <* [ reason-
ings require
tie reasonings of Mathematics, and to carry out jt -
the trains of thought to which they give rise,
without entire familiarity with those means
which the mind employs to aid its investiga-
tions. The child cannot read till he has learned Cannotn80
the alphabet; nor can the scholar feel the deli- a "y lan -
guage well
cate beauties of Shakspeare, or be moved by the tiu we
J know it.
sublimity of Milton, before studying and learn-
ing the language in which their immortal
thoughts are clothed.
89. All Quantities, whether abstract or con- Quantities
crete, are, in mathematical science?, presented
5*
106 MATHEMATICAL SCIENCE. [BOOK II.
sentedby to the mind by arbitrary symbols. They are
symbols ;
S :ui are oper- viewed and operated on through these symbols
these sym- which represent them; and all operations are
indicated by another class of symbols called
signs, signs. These, combined with the symbols
What consti- which represent the quantities, make up, as
we have stated above, the pure mathematical
language ; and this, in connection with that
which is borrowed from our common language,
forms the language of mathematical science.
This language is at once comprehensive and
its nature, accurate. It is capable of stating the most gen-
eral propositions, and of presenting to the mind,
in their proper order, all elementary principles
what it ac- connected with their solution. By its gener-
ality it reaches over the whole field of the
pure and mixed sciences, and gathers into con-
densed forms all the conditions and relations
necessary to the development of particular facts
and universal truths. Thus, the skill of the
analyst deduces from the same equation the ve-
Extent and locity of an apple falling to the ground, and the
Analysis, verification of the law of universal gravitation.
LANGUAGE OF MATHEMATICS.
Language 90. The language of Mathematics embraces,
of
Mathema- ^ s ^ ^ ar ts of our written and spoken language ;
tlcl< 2d. The language of Figures ,
CHAA I.] LANGUAGE OF MATHEMATICS. 107
3d. The language of Lines straight and curved ; Lines.
4th. The language of Letters ; and these forms Letters.
of language determine the classification of the
branches of the Science of Mathematics.
LANGUAGE OF NUMBER ARITHMETIC.
91. The ten characters, called figures, are Language
of
the alphabet of the language of number. The Number,
various ways in which they are combined, form-
ing the exact and copious language of compu-
tation, will be fully explained under the head
of Arithmetic, in a chapter exclusively devoted
to the consideration of numbers, their laws and
their language.
LANGUAGE OF LINES GEOMETRY.
92. Lines, straight and curved, are the ele- Languag*
of
meuts of the pictorial language applicable to Geometry,
space. The definitions and axioms relating to
space, and all the reasonings founded on them,
make up the science of Geometry, which will
be fully treated under its proper head.
LANGUAGE OF LETTERS ANALYSIS.
93. ANALYSIS is a general term embracing Analysis.
all the operations which can be performed on
108 MATHEMATICAL SCIENCE. [BOOK II.
quantities when represented by letters. In this
branch of mathematics, all the quantities con-
Qmntities sidered, whether abstract or concrete, are rep-
by letters, resented by letters of the alphabet, and the
operations to be performed on them are indi-
cated by a few arbitrary signs. The letters
symbols, and signs are called Symbols, and by their
combination we form the Algebraic Notation
and Language.
Analysis, 94 - Analysis, in its simplest form, takes
AnaTyt^cai * ne name ^ Algebra. Analytical Geometry, the
Geometry. Differential and Integral Calculus, extended to
Calculus, include the Theory of Variations, are its higher
and most advanced branches.
Term Anaiy- 95. The term Analysis has also another sig-
sia defined.
nification. It denotes the process of separating
its nature, any complex whole into the elements of which
^etaed* ** is com P sed - It; is opposed to Synthesis, a
term which denotes the processes of first con-
sidering the elements separately, then combin-
ing them, and ascertaining the results of the
combination.
Analytical The Analytical method is best adapted to in-
vestigation, and the presentation of subjects in
synthetical their general outlines: the Synthetical method
method.
is best adapted to instruction, because it e-xhib-
CHAP. I.] INFINITESIMAL CALCULUS. 109
its all the parts of a subject separately, and in Analysis.
their proper order and connection. Analysis
deduces all the parts from a whole: Synthesis synthesis.
forms a whole from the separate parts.
96. Arithmetic, Algebra, and Geometry are Arithmetic,
the Elementary branches of Mathematical Sci- ^^ e ^.
ence. Every truth which is established by elementary
branches.
mathematical reasoning, is developed by an
arithmetical, geometrical, or analytical process,
or by a combination of them. The reasoning
in each branch is conducted on principles iden-
tically the same. Every sign, or symbol, or Reasoning
technical word, is accurately defined, so that to
each there is attached a definite and precise
idea. Thus, the language is made so exact and Language
exact.
certain, as to admit of no ambiguity.
INFINITESIMAL CALCULUS.
97. The language of the Infinitesimal Cal-
i
Calculus.
of the
cuhis is very simple. Its chief element is the
letter d, which, when written before a quantity,
denotes that that quantity increases or decreases
according to the law of continuity, and the ex-
pression thus arising is one link in that law.
Thus, dx denotes that the quantity represented What does
by x, changes according to the law of continuity,
and that dx is the unit of that change.
110 MATHEMATICAL SCIENCE. [BOOK II.
PURE MATHEMATICS.
Pare Mathe- 98. The Pure Mathematics embraces all the
matics.
principles of the science, and all the processes
by which those principles are developed from
Number and the abstract quantities, Number and Space. All
Space.
the definitions and axioms, and all the truths
deduced from them, are traceable to these two
sources.
Mathema- 99. Mathematics, in its primary significa-
nce, as need
by the an- tion, as used by the ancients, embraced every
acquired science, and was equally applicable to
all branches of knowledge. Subsequently it
was restricted to those branches only which
were acquired by severe study, or discipline, and
Embraced its votaries were called Disciples. Those sub-
all subjects
which were jects, therefore, which required patient investi-
difciplinary . .
in their na- gation, exact reasoning, and the aid ot the ma-
thematical analysis, were called Disciplinal or
Mathematical, because of the greater evidence
in the arguments, the infallible certainty of the
conclusions, and the mental training and de-
velopment which such exercises produced.
Pure Mathc- 100. The Pure Mathematics is based on
definitions and intuitive truths, called axioms,
What are its
foundations, which are inferred from observation and expe-
CHAP. I.J PUBE MATHEMATICS. Ill
rience ; that is, observation and experience fur- Premises.
nish the information necessary to such intui-
tive inductions.* From these definitions and
axioms, as premises, all the truths of the science Reasoning,
are established by processes of deductive reason-
ing; and there is not, in the whole range of iu tests of
mathematical science any logical test of truth,
but in a conformity of the conclusions to the What they
are.
definitions and axioms, or to such principles or
propositions as have been established from them.
Hence, we see, that the science of Pure Mathe- in what the
unities, which consists merely in inferring, by ^ e ^ e ui c01
fixed rules, all the truths which can be deduced
from given premises, is purely a Deductive is purely
Deductive.
Science. The precision and accuracy of the
definitions ; the certainty which is felt in the
truth of the axioms; the obvious and fixed re- Precision of
its language.
lation between the sign and the thing signified ;
and the certain formulas to which the reason-
ing processes are reduced, have given to mathe-
matics the name of " Exact Science." science.
101. We have remarked that all the rea- AII reason-
soilings of mathematical science, and all the d'eflnitionT
truths which they establish, are based on the and axiomfl -
definitions and axioms, which correspond to the
* Section 27.
112 MATHEMATICAL SCIENCE. [BOOK IL
major premise of the syllogism. If the resem-
blance which the minor premise asserts to the
Relations middle term were obvious to the senses, as it
)U8 ' is in the proposition, " Socrates was a man,"
or were at once ascertainable by direct observa-
tion, or were as evident as the intuitive truth,
" A whole is equal to the sum of all its parts ;"
Deductive there would be no necessity for trains of rea-
necegsary soning, and Deductive Science would not exist.
Trains of Trains of reasoning are necessary only for the
aing ' sake of extending the definitions and axioms to
what they other cases in which we not only cannot di-
wnpis . rec f.] v observe what is to be proved, but cannot
directly observe even the mark which is to
prove it.
syllogism, 102. Although the syllogism is the ultimate
the final test , , MJJJ- / j j j
of deduc- test in all deductive reasoning (and indeed in
tion- all inductive, if we admit the uniformity of the
course of nature), still we do not find it con-
venient or necessary, in mathematics, to throw
every proposition into the form of a syllogism.
Axioms and ^ ne definitions and axioms, and the propo-
deflnitions, B {tj ons established from them, are our tests of
testa of
truth, truth; and whenever any new proposition can