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The nature and utility of mathematics; with the best methods of instruction explained and illustrated online

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every property which belongs to one of a sub- 8ubs P ecies '

J r * and to every

species or class will be common to every indi- individual -
vidual of the class. For example : " the square Examples,
on the hypothenuse of a right-angled triangle is
equal to the sum of the squares described on
the other two sides," is a proposition equally
true of every right-angled triangle : and " every
straight line perpendicular to a chord, at the circle,
middle point, will pass through the centre," is
equally true of all circles.

MARKS OF WHAT MAY BE PROVED.

201. The characteristic properties of every character^
geometrical figure (that is, those properties with- l ' c t ^ per '

232 MATHEMATICAL SCIENCE. [BOOK II

out which the figures could not exist), are given

in the definitions. How are we to arrive at all

the other properties of these figures ? The

propositions implied in the definitions, viz. that

Marks: things corresponding to the words defined do or

may exist with the properties named ; and the

or what may self-evident propositions or axioms, contain the

be proved.

only marks of what can be proved ; and by a
HOW ex- skilful combination of these marks we are able

tended.

to discover and prove all that is discovered and
proved in Geometry.

Definitions and axioms, and propositions de-
* l8 ' duced from them, are major premises in each

The science: new demonstration; and the science is made up
consists. f the processes employed for bringing unfore-
seen cases under these known truths ; or, in syl-
logistic language, for proving the minors neces-
sary to complete the syllogisms. The marks
being so few, and the inductions which furnish
them so obvious and familiar, there would seem
to be very little difficulty in the deductive pro-
cesses which follow. The connecting together
of several of these marks constitutes Deductions,

Geometry, or Trains of Reasoning; and hence, Geometry

a Deductive . TA j * ci

science. ls a Deductive Science.

."HAP. 111.] GEOMETEY. 233

DEMONSTRATION.

2G2. As a first example, let us take the first
proposition in Legendre's Geometry :

"If a straight line meet another straight line, Proposition
the sum of the two adjacent angles will be equal
to two right angles."

Let the straight line DC

.1 i_. i- A T> / Foundation.

meet the straight line AB
at the point C, then will the
angle ACD plus the angle
DCB be equal to two right A
angles.

To prove this proposition, we need the defini- ' f *&

necessary Ic

tion of a right angle, viz. : P rove iL

When a straight line AB B

meets another straight line
CD. so as to make the ad-
jacent angles BAG and

O A D Definition*

each of those angles is called a RIGHT ANGLE, and

the line AB is said to be PERPENDICULAR to CD.

We shall also need the 2d, 3d, and 4th axioms, Axiom*.
ibr inferring equality,* viz. :

2. Things which are equal to the same thing second
are equal to each other.

* Section 109.

234

MATHEMATICAL SCIENCE.

OOK II

Third.

Fourth.

3. A whole is equal to the sum of all its
parts.

4. If equals be added to equals, the sums
will be equal.

Now before these formulas or tests can be ap-
Lmetobe plied, it is necessary to sup- E D

drawn.

pose a straight line CE to be
P>o f : drawn perpendicular to AB
at the point C : then by the
definition of a right angle, A C B

the angle ACE will be equal to the angle ECB.

By axiom 3rd, we have,

continued: ACD equal to ACE plus ECD: to each of
these equals add DCB ; and by the 4th axiom
\ve shall have,

ACD plus DCB equal to ACE plus ECD plus
DCB ; but by axiom 3rd,

ECD plus DCB equals ECB: therefore by
axiom 2d,

ACD plus DCB equals ACE plus ECB.
But by the definition of a right angle,
Conclusion. ACE plus ECB equals two right angles : there-
fore, by the 2d axiom,

ACD plus DCB equals two right angles,
m baa* It will be seen that the conclusiveness of the

proof results,

Fir*. 1st. From the definition, that ACE and ECB

are equal to each other, and each is called a

CHAP. III.] GEOMETRY. 235

right-angle : consequently, their sum is equal to
two right angles ; and,

2dly. In showing, by means of the axioms, that
ACD plus DCB equals ACE plus ECB; and
then inferring from axiom 2d, that, ACD plus
DCB equals two right angles.

8 263. The difficulty in the geometrical rea- ""CM "

J the demon-

soning consists mainly in showing that the prop- strauonu.
osition to be proved contains the marks whici
prove it. To accomplish this, it is frequently
necessary to draw many auxiliary lines, forming Auxiliaries
new figures and angles, which can be shown to
possess marks of these marks, and which thus
become connecting links between the known connecting

and the unknown truths. Indeed, most of the
skill and ingenuity exhibited in the geometrical
processes are employed in the use of these auxil-
iary means. The example above affords an illus-
tration. We were unable to show that the sum HOW used.
of the two angles possessed the mark of being
equal to two right angles, until we had drawn a
perpendicular, or supposed one drawn, at the
point where the given lines intersect. That be-
ing done, the two right angles ACE and ECB conclusion.
were formed, which enabled us to compare the
sum of the angle ACD and DCB with two right
angles, and thus we proved the proposition.

23G

MATHEMATICAL SCIENCE. [BOOK IL

Proposition. 264. As a second example, let us take the

following proposition :
Enunciation. If two straight lines intersect each other, the

opposite or vertical angles will be equal.
Let the straight line AB

intersect the straight line

Diagram. ED ^ ^ po j nt g . then

will the angle ACD be
equal to the opposite an-
gle ECB ; and the angle ACE equal to the an-
gle DCB.

Principles To prove this proposition, we need the last
proposition, and also the 2d and 5th axioms, viz. :

" If a straight line meet another straight line,
the sum of the two adjacent angles will be equal
to two right angles."

" Things which are equal to the same thing
are equal to each other."

" If equals be taken from equals, the remain-
ders will be equal."

Now, since the straight line AC meets the
straight line ED at the point C, we have,

ACD plus ACE equal to two right angles.

And since the straight line DC meets the
straight line AB, we have,

ACD plus DCB equal to two right angles :
hence, by the second axiom,

ACD plus ACE equals ACD plus DCB : ta-

Axioms.

Proof.

(HAP. III. J GEOMETRY. 23?

king from each the common angle ACD, we conclusion,
know from the fifth axiom that the remain-
ders will be equal ; that is, the angle ACE
equal to the opposite or vertical angle DCB.

265. The two demonstrations given above
combine all the processes of proof employed in De monstr a-
every demonstration of the same class. When U(
any new truth is to be proved, the known tests
of truth are gradually extended to auxiliary Uaeof auxil .
quantities having a more intimate connection Iar yi uanU-

tl6S

with such new truth than existed between it and
the known tests, until finally, the known tests,
through a series of links, become applicable to
the final truth to be established : the interme-
diate processes, as it were, bridging over the
space between the known tests and the final
truth to be proved.

266. There are two classes of demonstra- Direct dem
tions, quite different from each other, in some
lespects, although the same processes of argu-
mentation are employed in both, and although
the conclusions in both are subjected to the
same logical tests. They are called Direct, or

* Negative,

Positive Demonstration, and Negative Demon- or

Reductio .id

stration, or the Reductio ad Absurdum.

238 MATHEMATICAL SCIENCE. [BOOK IL

Difference. 267. The main differences in the two
methods are these : The method of direct demon-
Direct Dem- stration rests its arguments on known and ad-

ouslralioa

mitted truths, and shows by logical processes

that the proposition can be brought under some

previous definition, axiom, or proposition : while

Negative the negative demonstration rests its arguments

Demonstra-
tion, on an hypothesis, combines this with known pro-
positions, and deduces a conclusion by processes

conclusion: strictly logical. Now if the conclusion so de-
duced agrees with any known truth, we infer

with vt'vu t ^ at tne hypothesis, (which was the only link in

compared, * * '

the chain not previously known), was true ; but
if the conclusion be excluded from the truths
previously established ; that is, if it he opposed
to any one of them, then it follows thai <he hy-
pothesis, being contradictory to such truth, must
Determines fa f a j se j n t ne negative demonstration, there-

whether the

hypothesis is fore, the conclusion is compared with the truths

trueorfalie.

known antecedently to the proposition in ques-
tion : if it agrees with any one of them, the hy-
pothesis is correct ; if it disagrees with any one
of them, the hypothesis is false.

Proof fe y 268. We will give for an illustration of this

Negative

Demonstra- method, Proposition XVII. of the First Book of
Legendre : " When two right-angled triangles
have the hypothenuse and a side of the one equal

CHAP. III.* GEOMETRY. 239

to the hyp )thenuse and a side of the other, each Enunciation
to each, the remaining parts will be equal, each to
each, and the triangles themselves will be equal."

In the two right-angled triangles BAG and
EDF (see next figure), let the hypothenuse AC Enunciation
be equal to DF, the side BA to the side ED : b:
then will the side BC be equal to EF, the angle
A to the angle D, and the angle C to the angle F.
To prove this proposition, we need the follow-
ing, which have been before proved ; viz. :

Prop. X. (of Legendre). "When two triangles prevjoug
have the three sides of the one equal to the three truth9 nece *

sary.

sides of the other, each to each, the three an-
gles will also be equal, each to each, and the
triangles themselves will be equal."

Prop. V. " When two triangles have two imposition
sides and the ii eluded angle of the one, equal
to two sides and the included angle of the other,
each to each, the two triangles will be equal."

Axiom I. " Things which are equal to the Anoma.
same thing, are equal to each other."

Axiom X. (of Legendre). "All right angles
are equal to each other."

Prop. XV. " If from a point without a straight Propc*iik
line, a perpendicular be let fall on the line, and
oblique lines be drawn to different points,

1st. "The perpendicular will te shorter than
any oblique line ;

240 MATHEMATICAL SCIENCE. [BOOK II.

2d. " Of two oblique lines, drawn at pleasure,
that which is farther from the perpendicular will
be the longer."

Now the two sides BC and
Beginning of EF are either equal or un-

the demon-
stration, equal. If they are equal,

then by Prop. X. the remain-
ing parts of the two trian- c G B F
gles are also equal, and the triangles themselves
are equal. If the two sides are unequal, one of
them must be greater than the other: suppose
BC to be the greater.

construction On the greater side BC take a part BG, equal
' to EF, and draw AG. Then, in the two trian-
gles BAG and DEF the angle B is equal to the
angle E, by axiom X (Legendre), both being
right angles. The side AB is equal to the side
DE, and by hypothesis the side BG is equal to the
side EF : then it follows from Prop. V. that the
side AG is equal to the side DF. But the side

Demonsta- D p j g ft j iQ ^ gide A Q . he fe ax j om j

Uon.

the side AG is equal to AC. But the line AG
cannot be equal to the line AC, having been
shown to be less than it by Prop. XV. : hence,
the conclusion contradicts a known truth, and is
therefore false ; consequently, the supposition (on
which the conclusion rests), that BC and EF are
unequal, is also false ; therefore, they are equal

THAP. III.] GEOMETRY. 241

269. It is often supposed, though erroneous- Negative
Jy, that the Negative Demonstration, or the dem- e n . "
onstration involving the " reductio ad absurdurn,"
is less conclusive and satisfactory than direct or conclusive,
positive demonstration. This impression is sim-
ply the result of a want of proper analysis. For
example : in the demonstration just given, it was Reasons,
proved that the two sides BC and EF cannot
be unequal; for, such a supposition, in a logi-
cal argumentation, resulted in a conclusion di- conclusion
rectly opposed to a known truth ; and as equality
and inequality are the only general conditions

known truth

of relation between two quantities, it follows
that if they do not fulfil the one, they must the
other. In both kinds of demonstration, the
premises and conclusion agree ; that is, they are Agreement
both true, or both false ; and the reasoning or
argument in both is supposed to be strictly logi-
cal.

In the direct demonstration, the premises are
known, being antecedent truths; and hence,
the conclusion is true. In the negative demon- Differences iu
stration, one element is assumed, and the con- ^^
elusion is then compared with truths previously
established. If the conclusion is found to agree
with any one of these, we infer that the hy- when the
pothesis or assumed element is true ; if it con-
tradicts any one of these truths, we infer that

16

242

MATHEMATICAL SCIENCE. [flOOKJI.

when false, the assumed element is false, and hence that its
opposite is true.

General

Remarks.

Measured: 270. Having explained the meaning of the
luon. " term measured, as applied to a geometrical mag-
nitude, viz. that it implies the comparison of a
magnitude with its unit of measure ; and having
also explained the signification of the word Prop-
erty, and the processes of reasoning by which,
in all figures, properties not before noticed are
inferred from those that are known ; we shall
now add a few remarks on the relations of the
geometrical figures, and the methods of compar-
ing them with each other.

PROPORTION OF FIGURES.

Proportion. 271. Proportion is the relation which one

geometrical magnitude bears to another of the

same kind, with respect to its being greater or

less. The two magnitudes so compared are called

its measure, terms, and the measure of the proportion is the

quotient which arises from dividing the second

Ratio. term ty' the first, and is called their Ratio. Only

Quantities of quantities of the same kind can be compared

the same together, and it follows from the nature of the

kind com

relation that the quotient or ratio of any two
terms will be an abstract number, whether the
terms themselves be abstract or concrete

CHAP. III.] GEOMETRY. 243

272. The term Proportion is defined by most Proportion:

, . ,, . , ,. how defined.

authors, " An equality of ratios between four
numbers or quantities, compared together two
and two." A proportion certainly arises from
such a comparison : thus, if

T T\

__ _ Example.

A~C'
A : B : : C : D

is a proportion.

But if we have two quantities A and B, which True defini-

tion.
may change their values, and are, at the same

time, so connected together that one of them
shall increase or decrease just as many times as
the other, their ratio will not be altered by such
changes ; and the two quantities are then said t ^ 1 p 1 ^
to be in proportion, or proportional. Ues -

Thus, if A be increased three times and B
three times, then,

3J3 = A
3A~~B ;

that is, 3 A and 3 B bear to each other the same
proportion as A and B. Science needed a gen- Term need

ed.

eral term to express this relation between two
quantities which change their values, without
altering their quotient, and the term "propor-
tional," or "in proportion," is employed for that HOWIWXL
purpose.

244 MATHEMATICAL SCIENCE. [BOOK IL

Reasons for As some apology for the modification of the
definition of proportion, which has been so long
accepted, it may be proper to state that the term
has been used by the best authors in the exact

Use of the sense here attributed to it. In the definition of

term.

the second law of motion, we have, "Motion,
or change of motion, is proportional to the force
impressed ;"* and again, " The inertia of a body
is proportioned to its weight."f Similar exam-
ples may be multiplied to any extent. Indeed,
symbol used there is a symbol or character to express the

to represent

proportion, relation between two quantities, when they un-
dergo changes of value, without altering their
ratio. That character is oc, and is read " pro-
portional to." Thus, if we have two quantities
denoted by A and B, written,

Example. A OC B,

the expression is read, " A proportional to B."
Another Hnd 273. There is yet another kind of relation

of propor-
tion, which may exist between two quantities A and

B, which it is very important to consider and
understand. Suppose the quantities to be so
connected with each other, that when the first
is increased according to any law of change, the
second shall decrease according to the same law ;
and the reverse.

* Olmsted's Mechanics, p. 28. f Ibid. p. 23.

CHAP. III.]

GEOMETRY

245

For example : the area of a rect-
angle is equal to the product of its
base and altitude. Then, in the
rectangle ABCD, we have

Area = AB x BC.

Take a second rectangle EFGH, having a
longer base EF, and a less altitude FG, but such Exwn P l0 -
that it shall have an equal
area with the first : then we
shall have

Area = EF x FG.

Now since the areas are equal, we shall have
AB x BC = EF x FG ;

and by resolving the terms of this equation into
a proportion, we shall have

AB : EF : : FG : BC.

It is plain that the sides of the rectangle ABCD
may be so changed in value as to become the
sides of the rectangle EFGH, and that while
they are undergoing this change, AB will in-
crease and BC diminish. The change in the Relation* of

the quantt-

values of these quantities will therefore take place ties:
according to a fixed law : that is, one will be di-
minished as many times as the other is increased,

246 MATHEMATICAL SCIENCE. [BOOK II.

since their product is constantly equal to the
area of the rectangle EFGH.
Expressed by Denote the side AB by x and BC bv y, ana

letters.

the area of the rectangle EFGH, which is known,
by a; then

xy = a;

and when the product of two varying quantities

is constantly equal to a known quantity, the two

Reciprocal quantities are said to be Reciprocally or Inverse-

or

inverse Pro- fy proportional ; thus x and y are said to be in-
poruon. versely proportional to each other. If we divide
1 by each member of the above equation, we
shall have

;1 J,

xy a '

Reductions a nd by multiplying both members by x, we shall

of the
Equations, have

I -

ya>

and then by dividing both numbers by x, we have
1

Final form. y _

- = a'
x

that is, the ratio of a: to - is constantly equal to - ;
that is, equal to the same quantity, hov* ever x or

CHAP. 111.]

GEOMETRY.

24?

V may vary , for, a anu consequently - does not
change. Hence,

Two quantities, which may change their values,
are reciprocally or inversely proportional, when Proportion

defined.

one is proportional to unity divided by the other,
and then their product remains constant.

We express this reciprocal or inverse relation
thus:

Aocl

A is said to be inversely proportional to B : the
symbols also express that A is directly proper- HOW ex

pressed.

tional to -^. If we have
B

Generally,

we say, that A is directly proportional to B, and
inversely proportional to C.

The terms Direct, Inverse or Reciprocal, ap-
ply to the nature of the proportion, and not to
the Ratio, which is always a mere quotient and
the measure of proportion. The term Direct ap- Direct and
plies to all proportions in which the terms in-

torms not

crease or decrease together ; and the term In- applicable to
verse or Reciprocal to those in which one term
increases as the other decreases. They cannot,
therefore, properly be applied to ratio without
changing entirely its signification and definition.

Kutio.

248 MATHEMATICAL SCIENCE. [BOOK II.

COMPARISON OF FIGURES.

Geometrical 274. In comparing geometrical magnitudes,

magnitudes ......

compared, by means ot their quotient, it is not the quotient
alone which we consider. The comparison im-
plies a general relation of the magnitudes, which
is measured by the Ratio. For example : we
Example, say that " Similar triangles are to each other as
the squares of their homologous sides." What
do we mean by that ? Just this :
Formula of That the area of a triangle

Is to the area of a similar triangle

As the area of a square described on a side of

the first,

To the area of a square described on an ho-
mologous side of the second.
Thus, we see that every term of such a pro-
changes of portion is in fact a surface, and that the area
how affected ^ a triangle increases or decreases much faster
than its sides ; that is, if we double each side of
a triangle, the area will be four times as great :
if we multiply each side by three, the area will
Results, be nine times as great ; or if we divide each
side by two, we diminish the area four times, and
so on. Again,
circles com- The area of one circle

Is to the area of another circle,

As a square described on the diameter of the first

CHAP. III.] GEOMETRY. 249

To a squar* described on the diameter of the

second.
Hence, if we double the diameter of a circle, How their

area* change*

the area of the circle whose diameter is so in-
creased will be four times as great : if we mul-
tiply the diameter by three, the area will be nine
times as great ; and similarly if we divide the
diameter.

275. In comparing volumes together, the comparison
same general principles obtain. Similar volumes
are to each other as the cubes described on their
homologous or corresponding sides. That is,

A prism

Formal*.

Is to a similar prism,

As a cube described on a 'ide of the first,

Is to a cube described on an homologous side

of the second.
Hence, if the sides of a prism be doubled, the con- HOW the

volumes

tents of volume will be increased eight-fold. Again, change.
A sphere

Is to a sphere, Sphere:

As a cube described on the diameter of the first,
Is to a cube described on a diameter of the

second.
Hence, if the diameter of a sphere be doubled, HOW it

volume

its contents of volume will be increased eight- change*
fold; if the diameter be multiplied by three, the

250 MATHEMATICAL SCIENCE. [BOOK II.

contents of volume will be increased twenty-seven
fold : if the diameter be multiplied by four, the
contents of volume will be increased sixty-four
fold; the contents of volume increasing as the
cubes of the numbers 1, 2, 3, 4, &c.

Eatio: 276. The relation or ratio of two magnitudes

an abstract to each other, maybe, and indeed is, expressed

by an abstract number. This number has a

whenhav- fixed value so long as we do not introduce a

ing a fixed

value, change in the contents of the figures; but if
we wish to express their ratio under the sup-
position that their contents may change accord-
ing to fixed laws (that is, so that the volumes
HOW varying shall continue similar), we then compare them
compared, with similar figures described on their homol-
ogous or corresponding sides; or, what is the
same thing, take into account the corresponding
changes which take place in the abstract num-
bers that express their contents.

BECAPITULATION.

General 277. We have now completed a general out-
line of the science of Geometry, and what has
been said may be recapitulated under the follow-
ing heads. It has been shown,

Geometry: 1st. That Geometry is conversant about space,

CHAP. III.] GEOMETRY. 251

or those limited portions of space which are to what it

relates.

called Geometrical Magnitudes.

2d. That the geometrical magnitudes embrace
tour species :

1st. Lines straight and curved ; Lines.

2d. Surfaces plane and curved ; surface*.

3d. Volumes bounded either by plane sur- volumes,
faces or curved, or both ; and,

4th. Angles, arising from the positions of Angles,
lines with each other; or of planes with each
other the lines and planes being boundaries.
3d. That the science of Geometry is made up science:
)f those processes by means of which all the w u T a '
properties of these magnitudes are examined and
developed, and that the results arrived at con-
stitute the truths of Geometry.

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