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for it is either within the figure, as in the circle ; or in its perimeter, ai

in the semicircle ; or without the figure, as in certain conic lines."

Def. xxiv-xxix. Triangles are divided into three classes, by reference

to the relations of their sides ; and into three other classes, by referencÂ«

to their angles. A farther classification may be made by considering

both the relation of the sides and angles in each triangle.

In Simson's definition of the isosceles triangle, the word only must b<

omitted, as in the Cor. Prop. 5, Book i, an isosceles triangle may b<

equilateral, and an equilateral triangle is considered isosceles in Prop. 15

Book IV. Objection has been made to the definition of an acute-angle<

triangle. It is said that it cannot be admitted as a definition, that all th<

three angles of a triangle are acute, which is supposed in Def. 29. Ii

may be replied, that the definitions of the three kinds of angles point ou

and seem to supply a foundation for a similar distinction of triangles.

Def. xxx-xxxiv. The definitions of quadrilateral figures are liable t(

objection. All of them, except the trapezium, fall under the genera

idea of a parallelogram ; but as Euclid defined parallel straight linei

after he had defined four- sided figures, no other arrangement could b

adopted than the one he has followed ; and for which there appeared t(

him, without doubt, some probable reasons. Sir Henry Savile, in hi

Seventh Lecture, remarks on some of the definitions of Euclid, **Ne

dissimulandum aliquot harum in manibus exiguum esse usum in Ge

metria." A few verbal emendations have been made in some of them.

A square is a four-sided plane figure having all its sides equal, anc

one angle a right angle : because it is proved in Prop. 46, Book i, that if j

I)arallelogram have one angle a right angle, all its angles are righi

angles.

NOTES TO BOOK I. 45

An oblong, in the same manner, may be defined as a plane figure of

four sides, having only its opposite sides equal, and one of its angles a

right angle.

A rhomboid is a four- sided plane figure having only its opposite sides

equal to one another and its angles not right angles.

Sometimes an irregular four- sided figure which has two sides pai'allel,

is called a trapezoid.

Def. XXXV. It is possible for two right lines never to meet when pro-

duced, and not be parallel.

Def. A. The term parallelogram literally implies a figure formed by

parallel straight lines, and may consist of four, six, eight, or any even

number of sides, where every two of the opposite sides are parallel to one

another. In the Elements, however, the term is restricted to four-sided

figures, and includes the four species of figures named in the Definitions

XXX â€” XXXIII.

The synthetic method is followed by Euclid not only in the demon-

1 strations of the propositions, but also in laying down the definitions. He

j commences with the simplest abstractions, defining a point, a line, an

! angle, a superficies, and their different varieties. This mode of proceed-

( ing involves the difficulty, almost insurmountable, of defining satisfac-

torily the elementary abstractions of Geometry. It has been observed,

that it is necessary to consider a soli 1, that is, a magnitude which has

length, breadth, and thickness, in order to understand aright the defini-

tions of a point, a line, and a superficies. A solid or volume considered

apart from its physical properties, suggests the idea of the surfaces by

which it is bounded : a surface, the idea of the line or lines which form

its boundaries : and a finite line, the points which form its extremities.

A solid is therefore bounded by surfaces ; a surface is bounded by lines ;

and a line is terminated by two points. A point marks position only : a

line has one dimension, length only, and defines distance : a superficies

has two dimensions, length and breadth, and defines extension : and a

solid has three dimensions, length, breadth, and thickness, and defines

some portion of space.

It may also be remarked that two points are sufficient to determine

the position of a straight line, and three points not in the same straight

line, are necessary to fix the position of a plane.

ON THE POSTULATES.

The definitions assume the possible existence of straight lines and

circles, and the postulates predicate the possibility of drawing and of

producing straight lines, and of describing circles. The postulates form

the principles of construction assumed in the Elements ; and are, in fact,

problems, the possibility of which is admitted to be self-evident, and to

require no proof.

It must, however, bo carefully remarked, that the third postulate only

admits that when any line is given in position and magnitude, a circle

may be described from either extremity of the line as a center, and with

a radius equal to the length of the line, as in Euc. i, 1. It does not

admit the description of a circle with any other point as a center than

one of the extremities of the given line.

Euc. I. 2, shews how, from any given point, to draw a straight line

equal to another straight line which is given in magnitude and position.

t

46 Euclid's elements.

ON THE AXIOMS.

Axioms are usually defined to be self-evident truths, -whicli cannot be

rendered more evident by demonstration ; in other words, the axioms of

Geometry are theorems, the truth of vt'hich is admitted without proof.

It is by experience we first become acquainted with the different forms

of geometrical magnitudes, and the axioms, or the fundamental ideas of

their equality or inequality appear to rest on the same basis. The con-

ception of the truth of the axioms does not appear to be more removed

from experience than the conception of the definitions.

These axioms, or first principles of demonstration, are such theorems

as cannot be resolved into simpler theorems, and no theorem ought to be

admitted as a first principle of reasoning which is capable of being de-

monstrated. An axiom, and (when it is convertible) its converse, should

both be of such a nature as that neither of them should require a formal

demonstration.

The first and most simple idea, derived from experience is, that every

magnitude fills a certain space, and that several magnitudes may succes-

sively fill the same space.

All the knowledge we have of magnitude is purely relative, and the

most simple relations are those of equality and inequality. In the com-

parison of magnitudes, some are considered as given or known, and the

unknown are compared with the known, and conclusions are syntheti-

cally deduced with respect to the equality or inequality of the magnitudes

under consideration. In this manner we form our idea of equality,

which is thus formally stated in the eighth axiom : " Magnitudes which

coincide with one another, that is, which exactly fill the same space, are

equal to one another."

Every specific definition is referred to this universal principle. With

regard to a few more general definitions which do not furnish an equality,

it will be found that some hypothesis is always made reducing them to

that principle, before any theory is built upon them. As for example,

the definition of a straight line is to be refe-rred to the tenth axiom ; the

definition of a right angle to the eleventh axiom ; and the definition of

parallel straight lines to the twelfth axiom.

The eighth axiom is called the principle of superposition, or, the

mental process by which one Geometrical magnitude may be conceived

to be placed on another, so as exactly to coincide with it, in the parts

which are made the subject of comparison. Thus, if one straight line be

conceived to be placed upon another, so that their extremities are coin-

cident, the two straight lines are equal. If the directions of two lines

which include one angle, coincide with the directions of the two lines

which contain another angle, where the points, from which the angles

diverge, coincide, then the two angles are equal : the lengths of the lines

not affecting in any way the magnitudes of the angles. When one plane

figure is conceived to be placed upon another, so that the boundaries of

one exactly coincide with the boundaries of the other, then the two

plane figures are equal. It may also be remarked, that the converse of

this proposition is not universally true, namely, that when two magni-

tudes are equal, they coincide with one another : since two magnitudes

may be equal in area, as two parallelograms or two triangles, Euc. i. 35,

37 ; but their boundaries may not be equal : and, consequently, by

superposition, the figures could not exactly coincide : all such figures,

however, having equal areas, by a different arrangement of their parts,

may be made to coincide exactly.

NOTES TO BOOK I. 47

This axiom is the criterion of Geometrical equality, and is essentially

different from the criterion of Arithmetical equality. Two geometrical

magnitudes are equal, when they coincide or may be made to coincide :

two abstract numbers are equal, when they contain the same aggregate

of units ; and two concrete numbers are equal, when they contain the

same number of units of the same kind of magnitude. It is at once ob-

vious, that Arithmetical representations of Geometrical magnitudes are

not admissible in Euclid's criterion of Geometrical Equality, as he has not

fixed the unit of magnitude of either the straight line, the angle, or the

superficies. Perhaps Euclid intended that the first seven axioms should

be applicable to numbers as well as to Geometrical magnitudes, and this

is in accordance with the words of Proclus, who calls the axioms, co7mnon

notions^ not peculiar to the subject of Geometry.

Several of the axioms maybe generally exemplified thus :

Axiom 1. If the straight line ABhe equal ^ B

to the straight line CD ; and if the straight C D

line EF he also equal to the straight line CD ; E

then the straight line AB is equal to the

straight line EF.

Axiom II. Ifthe line J.5 be equal to the line 4

CD ; and if the line EF be also equal to the

line GH: then the sum of the lines AB and EF ^

is equal to the sum of the lines CD and GH.

Axiom III. If the line AB be equal to the A

line CD ; and if the line EF\)q also qqual to the

line GH; then the difference of AB and EF, E

is equal to the difference of CD and GH.

Axiom IV. admits of being exemplified under the two following forms :

1. If the line ABhe equal to the line CD ; a B

and if the line EF be greater than the line GH ;

then the sum of the lines AB and EF is greater E F

than the sum of the lines CD and GH.

2. If the line AB be equal to the line CD ; a B

and if the line EF he less than the line GH ;

then the sum of the lines AB and EF is less e F

than the sum of the lines CD and GH.

Axiom V. also admits of two forms of exemplification.

1. If the line AB be equal to the line CD ; a B

and if the line EF he greater than the line GH ;

then the difference of the lines AB and EF is E F

greater than the difference of CD and GH.

2. If the line ABhQ equal to the line CD ; :^ ?

and if the line EF he less than the line GH;

then the difference of the lines AB and EF is ? 1"

less thanthe difference of the lines CD and GH.

The axiom, "Ifunequals be taken from equals, the remainders are

unequal," may be exemplified in the same manner.

Axiom VI. If the line yl-B be double of the A B

line CD ; and if the line EF be also double of C p

the line CD; E F

then the line AB is equal to the line EF.

Axiom VII. If the line AB be the half of A B

the line CD ; and if the line EF be also the C D

half of the line CD ; E F

then the line AB is equal to the line EF.

c

D

G

H

C

D

G

H

wing

C

forms :

D

G

H

C

D

G

H

C

D

G

n

G

D

H

â–

48 Euclid's elements.

It may be observed that when equal magnitudes are taken from un-

equal magnitudes, the greater remainder exceeds the less remainder by

as much as the greater of the unequal magnitudes exceeds the less.

If unequals be taken from unequals, the remainders are not always

unequal ; they may be equal : also if unequals be added to unequals the

wholes are not always unequal, they may also be equal.

Axiom IX. The whole is greater than its part, and conversely, the

part is less than the whole. This axiom appears to assert the contrary

of the eighth axiom, namely, that two magnitudes, of which one is

greater than the other, cannot be made to coincide with one another.

Axiom X. The property of straight lines expressed by the tenth

axiom, namely, " that two straight lines cannot enclose a space," is ob-

viously implied in the definition of straight lines ; for if they enclosed a

space, they could not coincide between their extreme points, when the

two lines are equal.

Axiom XI. This axiom has been asserted to be a demonstrable theo-

rem. As an angle is a species of magnitude, this axiom is only a parti-

cular application of the eighth axiom to right angles.

Axiom XII. See the notes on Prop. xxix. Book i.

ON THE PROPOSITIONS.

Whenever a judgment is formally expressed, there must be some-

thing respecting which the judgment is expressed, and something else

which constitutes the judgment. The former is called the subject of the

proposition, and the latter, the predicate, which may be anything which

can be affirmed or denied respecting the subject.

The propositions in Euclid's Elements of Geometry may be divided

into two classes, problems and theorems. A proposition, as the term

imports, is something proposed ; it is a problem, when some Geometrical

construction is required to be effected : and it is a theorem when some Geo-

metrical property is to be demonstrated. Every proposition is natu-

rally divided into two parts ; a problem consists of the data, or things

given; and the qucesita, or things required: a theorem, consists of the

subject or hypothesis, and the conclusion, ox predicate. Hence the distinction

between a problem and a theorem is this, that a problem consists of the

data and the qugesita, and requires solution : and a theorem consists of

the hypothesis and the predicate, and requires demonstration.

All propositions are affirmative or negative ; that is, they either assert

some property, as Euc. i. 4, or deny the existence of some property, as

Euc. I. 7 ; and every proposition which is affirmatively stated has a con-

tradictory corresponding proposition. If the affirmative be proved to be

true, the contradictory is false.

All propositions may be viewed as (1) universally affirmative, or uni-

versally negative ; (2) as particularly affirmative, or particularly negative.

The connected course of reasoning by which any Geometrical truth is

established is called a demonstration. It is called a direct demonstration

when the predicate of the proposition is inferred directly from the pre-

misses, as the conclusion of a series of successive deductions. The de-

monstration is called indirect, when the conclusion shows that the intro-

duction of any other supposition contrary to the hypothesis stated in the

proposition, necessarily leads to an absurdity.

It has been remarked by Pascal, that " Geometry is almost the only

subject as to which we find truths wherein all men agree ; and one cause

of this is, that Geometers alone regard the true laws of demonstration."

KOTES TO BOOK I. 49

These are enumerated by him as eight in number. * * 1 . To define nothing

â€¢which cannot be expressed in clearer terms than those in which it is

already expressed. 2. To leave no obscure or equivocal terms undefined.

3. To employ in the definition no terms not already known. 4. To

omit nothing in the principles from which we argue, unless we are sure

it is granted. 5. To lay down no axiom which is not perfectly evident.

6. To demonstrate nothing which is as clear already as we can make it.

7. To prove every thing in the least doubtful by means of self-evident

axioms, or of propositions already demonstrated. 8. To substitute

mentally the definition instead of the thing defined." Of these rules, he

says, "the first, fourth and sixth are not absolutely necessary to avoid

error, but the other five are indispensable ; and though they may be found

in books of logic, none but the Geometers have paid any regard to them."

The course pursued in the demonstrations of the propositions in

Euclid's Elements of Geometry, is always to refer directly to some ex-

pressed principle, to leave nothing to be inferred from vague expressions,

and to make every step of the demonstrations the object of the under-

standing.

It has been maintained by some philosophers, that a genuine defini-

tion contains some property or properties which can form a basis for

demonstration, and that the science of Geometry is deduced from the

definitions, and that on them alone the demonstrations depend. Others

have maintained that a definition explains only the meaning of a term,

and does not embrace the nature and properties of the thing defined.

If the propositions usually called postulates and axioms are either

tacitly assumed or expressly stated in the definitions ; in this view, de-

monstrations may be said to be legitimately founded on definitions. If,

on the other hand, a definition is simply an explanation of the meaning

of a term, whether abstract or concrete, by such marks as may prevent a

misconception of the thing defined ; it will be at once obvious that some

constructive and theoretic principles must be assumed, besides the defini-

tions to form the ground of legitimate demonstration. These principles

we conceive to be the postulates and axioms. The postulates describe

constructions which may be admitted as possible by direct appeal to our

experience ; and the axioms assert general theoretic ti'uths so simple

and self-evident as to require no proof, but to be admitted as the assumed

first principles of demonstration. Under this view all Geometrical

reasonings proceed upon the admission of the hypotheses assumed in

the definitions, and the unquestioned possibility of the postulates, and

the truth of the axioms.

Deductive reasoning is generally delivered in the form of an enthymeme,

or an argument wherein one enunciation is not expressed, but is readily

supplied by the reader : and it may be observed, that although this is the

ordinary mode of speaking and writing, it is not in the strictly syllogistic

form ; as either the major or the minor premiss only is formally stated

before the conclusion : Thus in Euc. i. 1.

Because the point A is the center of the circle BCD ;

therefore the straight line AB is equal to the straight line AC.

The premiss here omitted, is : all straight lines drawn from the center

of a circle to the circumference are equal.

In a similar way may be supplied the reserved premiss in every enthy-

meme. The conclusion of two enthymemes may form the major and minor

premiss of a third syllogism, and so on, and thus any process of reasoning

is reduced to the strictly syllogistic form. And in this way it is shewn

i

50 Euclid's elements.

that the general theorems of Oeometry are demonstrated by means of

syllogisms founded on the axioms and definitions.

Every syllogism consists of three propositions, of which, two are called

the premisses, and the third, the conclusion. These propositions contain

three terms, the subject and predicate of the conclusion, and the middle

term which connects the predicate and the conclusion together. The

subject of the conclusion is called the minor, and the predicate of the con-

clusion is called the major term, of the syllogism. The major term appears

in one premiss, and the minor term in the other, with the middle term,

which is in both premisses. That premiss which contains the middle

term and the major term, is called the major premiss; and that which

contains the middle term and the minor term, is called the minor premiss

of the syllogism. As an example, we may take the syllogism in the demon-

stration of Prop. 1, Book 1, wherein it will be seen that the middle term is

the subject of the major premiss and the predicate of the minor.

Major premiss: because the straight line y^J? is equal to the straight line AC\

Minor premiss : and, because the straight line ^C is equal to the straight

line AB ;

Conclusion : therefore the straight line BC is equal to the straight line AC.

Here, BC is the subject, and AC the predicate of the conclusion.

BC is the subject, and AB the predicate of the minor premiss.

AB is the subject, and AC the predicate of the major premiss.

Also, AC is the major term, ^C the minor term, and AB the middle term

of the syllogism.

In this syllogism, it may be remarked that the definition of a straight

line is assumed, and the definition of the Geometrical equality of two

straight lines ; also that a general theoretic truth, or axiom, forms the

ground of the conclusion. And further, though it be impossible to make

any point, mark or sign (o-tj/ueloi/) which has not both length and breadth,

and any line which has not both length and breadth ; the demonstrations

in Geometry do not on this account become invalid. For they are pursued

on the hypothesis that the point has no parts, but position only : and the

line has length only, but no breadth or thickness : also that the surface

has length and breadth only, but no thickness : and all the conclusions

at which we arrive are independent of every other consideration.

The truth of the conclusion in the syllogism depends upon the truth

of the premisses. If the premisses, or only one of them be not true, the

conclusion is false. The conclusion is said to follow from the premisses;

whereas, in truth, it is contained in the premisses. The expression must

be understood of the mind apprehending in succession, the truth of

the premisses, and subsequent to that, the truth of the conclusion ;

so that the conclusion follows from the premisses in order of time

as far as reference is made to the mind's apprehension of the whole

argument.

Every proposition, when complete, may be divided into six parts, as

Proclus has pointed out in his commentary.

1 . The proposition, or general emmciation, which states in general terms

the conditions of the problem or theorem.

2. The exposition, or particular eiiunciation, which exhibits the subject

of the proposition in particular terms as a fact, and refers it to some

diagram described.

3. The determination contains the predicate in particular terms, as it

is pointed out in the diagram, and diiects attention to the demonstration,

by pronouncing the thuig sought.

NOTES TO BOOK I. 51

4. TJie constniction applies tlie postulates to prepare the diagram for

the demonstration.

5. The demotistration is the connexion of syllogisms, which prove the

truth or falsehood of the theorem, the possibility or impossibility of the

problem, in that particular case exhibited in the diagram.

6. The conclusion is merely the repetition of the general enunciation,

wherein the predicate is asserted as a demonstrated truth.

Prop. I. In the first two Books, the circle is employed as a me-

chanical instrument, in the same manner as the straight line, and the use

made of it rests entirely on the third postulate. No properties of the

circle are discussed in these books beyond the definition and the third

postulate. When two circles are described, one of which has its center in

the circumference of the other, the two circles being each of them partly

within and partly without the other, their circumferences must intersect

each other in two points ; and it is obvious from the two circles cutting

each other, in two points, one on each side of the given line, that two

equilateral triangles may be formed on the given line.

Prop. II. When the given point is neither in the line, nor in the line

, produced, this problem admits of eight dififerent lines being drawn from

j the given point in different directions, every one of which is a solution

[of the problem. For, 1. The given line has two extremities, to each of

which a line may be drawn from the given point. 2. The equilateral

triangle may be described on either side of this line. 3. And the side

BD of the equilateral triangle ABB may be produced either way.

But when the given point lies either in the line or in the line pro-

duced, the distinction which arises from joining the two ends of the line

with the given point, no longer exists, and there are only four cases of

the problem.

The construction of this problem assumes a neater form, by first de-

scribing the circle CGH with center B and radius BC, and producing Z)Z?

in the semicircle ; or without the figure, as in certain conic lines."

Def. xxiv-xxix. Triangles are divided into three classes, by reference

to the relations of their sides ; and into three other classes, by referencÂ«

to their angles. A farther classification may be made by considering

both the relation of the sides and angles in each triangle.

In Simson's definition of the isosceles triangle, the word only must b<

omitted, as in the Cor. Prop. 5, Book i, an isosceles triangle may b<

equilateral, and an equilateral triangle is considered isosceles in Prop. 15

Book IV. Objection has been made to the definition of an acute-angle<

triangle. It is said that it cannot be admitted as a definition, that all th<

three angles of a triangle are acute, which is supposed in Def. 29. Ii

may be replied, that the definitions of the three kinds of angles point ou

and seem to supply a foundation for a similar distinction of triangles.

Def. xxx-xxxiv. The definitions of quadrilateral figures are liable t(

objection. All of them, except the trapezium, fall under the genera

idea of a parallelogram ; but as Euclid defined parallel straight linei

after he had defined four- sided figures, no other arrangement could b

adopted than the one he has followed ; and for which there appeared t(

him, without doubt, some probable reasons. Sir Henry Savile, in hi

Seventh Lecture, remarks on some of the definitions of Euclid, **Ne

dissimulandum aliquot harum in manibus exiguum esse usum in Ge

metria." A few verbal emendations have been made in some of them.

A square is a four-sided plane figure having all its sides equal, anc

one angle a right angle : because it is proved in Prop. 46, Book i, that if j

I)arallelogram have one angle a right angle, all its angles are righi

angles.

NOTES TO BOOK I. 45

An oblong, in the same manner, may be defined as a plane figure of

four sides, having only its opposite sides equal, and one of its angles a

right angle.

A rhomboid is a four- sided plane figure having only its opposite sides

equal to one another and its angles not right angles.

Sometimes an irregular four- sided figure which has two sides pai'allel,

is called a trapezoid.

Def. XXXV. It is possible for two right lines never to meet when pro-

duced, and not be parallel.

Def. A. The term parallelogram literally implies a figure formed by

parallel straight lines, and may consist of four, six, eight, or any even

number of sides, where every two of the opposite sides are parallel to one

another. In the Elements, however, the term is restricted to four-sided

figures, and includes the four species of figures named in the Definitions

XXX â€” XXXIII.

The synthetic method is followed by Euclid not only in the demon-

1 strations of the propositions, but also in laying down the definitions. He

j commences with the simplest abstractions, defining a point, a line, an

! angle, a superficies, and their different varieties. This mode of proceed-

( ing involves the difficulty, almost insurmountable, of defining satisfac-

torily the elementary abstractions of Geometry. It has been observed,

that it is necessary to consider a soli 1, that is, a magnitude which has

length, breadth, and thickness, in order to understand aright the defini-

tions of a point, a line, and a superficies. A solid or volume considered

apart from its physical properties, suggests the idea of the surfaces by

which it is bounded : a surface, the idea of the line or lines which form

its boundaries : and a finite line, the points which form its extremities.

A solid is therefore bounded by surfaces ; a surface is bounded by lines ;

and a line is terminated by two points. A point marks position only : a

line has one dimension, length only, and defines distance : a superficies

has two dimensions, length and breadth, and defines extension : and a

solid has three dimensions, length, breadth, and thickness, and defines

some portion of space.

It may also be remarked that two points are sufficient to determine

the position of a straight line, and three points not in the same straight

line, are necessary to fix the position of a plane.

ON THE POSTULATES.

The definitions assume the possible existence of straight lines and

circles, and the postulates predicate the possibility of drawing and of

producing straight lines, and of describing circles. The postulates form

the principles of construction assumed in the Elements ; and are, in fact,

problems, the possibility of which is admitted to be self-evident, and to

require no proof.

It must, however, bo carefully remarked, that the third postulate only

admits that when any line is given in position and magnitude, a circle

may be described from either extremity of the line as a center, and with

a radius equal to the length of the line, as in Euc. i, 1. It does not

admit the description of a circle with any other point as a center than

one of the extremities of the given line.

Euc. I. 2, shews how, from any given point, to draw a straight line

equal to another straight line which is given in magnitude and position.

t

46 Euclid's elements.

ON THE AXIOMS.

Axioms are usually defined to be self-evident truths, -whicli cannot be

rendered more evident by demonstration ; in other words, the axioms of

Geometry are theorems, the truth of vt'hich is admitted without proof.

It is by experience we first become acquainted with the different forms

of geometrical magnitudes, and the axioms, or the fundamental ideas of

their equality or inequality appear to rest on the same basis. The con-

ception of the truth of the axioms does not appear to be more removed

from experience than the conception of the definitions.

These axioms, or first principles of demonstration, are such theorems

as cannot be resolved into simpler theorems, and no theorem ought to be

admitted as a first principle of reasoning which is capable of being de-

monstrated. An axiom, and (when it is convertible) its converse, should

both be of such a nature as that neither of them should require a formal

demonstration.

The first and most simple idea, derived from experience is, that every

magnitude fills a certain space, and that several magnitudes may succes-

sively fill the same space.

All the knowledge we have of magnitude is purely relative, and the

most simple relations are those of equality and inequality. In the com-

parison of magnitudes, some are considered as given or known, and the

unknown are compared with the known, and conclusions are syntheti-

cally deduced with respect to the equality or inequality of the magnitudes

under consideration. In this manner we form our idea of equality,

which is thus formally stated in the eighth axiom : " Magnitudes which

coincide with one another, that is, which exactly fill the same space, are

equal to one another."

Every specific definition is referred to this universal principle. With

regard to a few more general definitions which do not furnish an equality,

it will be found that some hypothesis is always made reducing them to

that principle, before any theory is built upon them. As for example,

the definition of a straight line is to be refe-rred to the tenth axiom ; the

definition of a right angle to the eleventh axiom ; and the definition of

parallel straight lines to the twelfth axiom.

The eighth axiom is called the principle of superposition, or, the

mental process by which one Geometrical magnitude may be conceived

to be placed on another, so as exactly to coincide with it, in the parts

which are made the subject of comparison. Thus, if one straight line be

conceived to be placed upon another, so that their extremities are coin-

cident, the two straight lines are equal. If the directions of two lines

which include one angle, coincide with the directions of the two lines

which contain another angle, where the points, from which the angles

diverge, coincide, then the two angles are equal : the lengths of the lines

not affecting in any way the magnitudes of the angles. When one plane

figure is conceived to be placed upon another, so that the boundaries of

one exactly coincide with the boundaries of the other, then the two

plane figures are equal. It may also be remarked, that the converse of

this proposition is not universally true, namely, that when two magni-

tudes are equal, they coincide with one another : since two magnitudes

may be equal in area, as two parallelograms or two triangles, Euc. i. 35,

37 ; but their boundaries may not be equal : and, consequently, by

superposition, the figures could not exactly coincide : all such figures,

however, having equal areas, by a different arrangement of their parts,

may be made to coincide exactly.

NOTES TO BOOK I. 47

This axiom is the criterion of Geometrical equality, and is essentially

different from the criterion of Arithmetical equality. Two geometrical

magnitudes are equal, when they coincide or may be made to coincide :

two abstract numbers are equal, when they contain the same aggregate

of units ; and two concrete numbers are equal, when they contain the

same number of units of the same kind of magnitude. It is at once ob-

vious, that Arithmetical representations of Geometrical magnitudes are

not admissible in Euclid's criterion of Geometrical Equality, as he has not

fixed the unit of magnitude of either the straight line, the angle, or the

superficies. Perhaps Euclid intended that the first seven axioms should

be applicable to numbers as well as to Geometrical magnitudes, and this

is in accordance with the words of Proclus, who calls the axioms, co7mnon

notions^ not peculiar to the subject of Geometry.

Several of the axioms maybe generally exemplified thus :

Axiom 1. If the straight line ABhe equal ^ B

to the straight line CD ; and if the straight C D

line EF he also equal to the straight line CD ; E

then the straight line AB is equal to the

straight line EF.

Axiom II. Ifthe line J.5 be equal to the line 4

CD ; and if the line EF be also equal to the

line GH: then the sum of the lines AB and EF ^

is equal to the sum of the lines CD and GH.

Axiom III. If the line AB be equal to the A

line CD ; and if the line EF\)q also qqual to the

line GH; then the difference of AB and EF, E

is equal to the difference of CD and GH.

Axiom IV. admits of being exemplified under the two following forms :

1. If the line ABhe equal to the line CD ; a B

and if the line EF be greater than the line GH ;

then the sum of the lines AB and EF is greater E F

than the sum of the lines CD and GH.

2. If the line AB be equal to the line CD ; a B

and if the line EF he less than the line GH ;

then the sum of the lines AB and EF is less e F

than the sum of the lines CD and GH.

Axiom V. also admits of two forms of exemplification.

1. If the line AB be equal to the line CD ; a B

and if the line EF he greater than the line GH ;

then the difference of the lines AB and EF is E F

greater than the difference of CD and GH.

2. If the line ABhQ equal to the line CD ; :^ ?

and if the line EF he less than the line GH;

then the difference of the lines AB and EF is ? 1"

less thanthe difference of the lines CD and GH.

The axiom, "Ifunequals be taken from equals, the remainders are

unequal," may be exemplified in the same manner.

Axiom VI. If the line yl-B be double of the A B

line CD ; and if the line EF be also double of C p

the line CD; E F

then the line AB is equal to the line EF.

Axiom VII. If the line AB be the half of A B

the line CD ; and if the line EF be also the C D

half of the line CD ; E F

then the line AB is equal to the line EF.

c

D

G

H

C

D

G

H

wing

C

forms :

D

G

H

C

D

G

H

C

D

G

n

G

D

H

â–

48 Euclid's elements.

It may be observed that when equal magnitudes are taken from un-

equal magnitudes, the greater remainder exceeds the less remainder by

as much as the greater of the unequal magnitudes exceeds the less.

If unequals be taken from unequals, the remainders are not always

unequal ; they may be equal : also if unequals be added to unequals the

wholes are not always unequal, they may also be equal.

Axiom IX. The whole is greater than its part, and conversely, the

part is less than the whole. This axiom appears to assert the contrary

of the eighth axiom, namely, that two magnitudes, of which one is

greater than the other, cannot be made to coincide with one another.

Axiom X. The property of straight lines expressed by the tenth

axiom, namely, " that two straight lines cannot enclose a space," is ob-

viously implied in the definition of straight lines ; for if they enclosed a

space, they could not coincide between their extreme points, when the

two lines are equal.

Axiom XI. This axiom has been asserted to be a demonstrable theo-

rem. As an angle is a species of magnitude, this axiom is only a parti-

cular application of the eighth axiom to right angles.

Axiom XII. See the notes on Prop. xxix. Book i.

ON THE PROPOSITIONS.

Whenever a judgment is formally expressed, there must be some-

thing respecting which the judgment is expressed, and something else

which constitutes the judgment. The former is called the subject of the

proposition, and the latter, the predicate, which may be anything which

can be affirmed or denied respecting the subject.

The propositions in Euclid's Elements of Geometry may be divided

into two classes, problems and theorems. A proposition, as the term

imports, is something proposed ; it is a problem, when some Geometrical

construction is required to be effected : and it is a theorem when some Geo-

metrical property is to be demonstrated. Every proposition is natu-

rally divided into two parts ; a problem consists of the data, or things

given; and the qucesita, or things required: a theorem, consists of the

subject or hypothesis, and the conclusion, ox predicate. Hence the distinction

between a problem and a theorem is this, that a problem consists of the

data and the qugesita, and requires solution : and a theorem consists of

the hypothesis and the predicate, and requires demonstration.

All propositions are affirmative or negative ; that is, they either assert

some property, as Euc. i. 4, or deny the existence of some property, as

Euc. I. 7 ; and every proposition which is affirmatively stated has a con-

tradictory corresponding proposition. If the affirmative be proved to be

true, the contradictory is false.

All propositions may be viewed as (1) universally affirmative, or uni-

versally negative ; (2) as particularly affirmative, or particularly negative.

The connected course of reasoning by which any Geometrical truth is

established is called a demonstration. It is called a direct demonstration

when the predicate of the proposition is inferred directly from the pre-

misses, as the conclusion of a series of successive deductions. The de-

monstration is called indirect, when the conclusion shows that the intro-

duction of any other supposition contrary to the hypothesis stated in the

proposition, necessarily leads to an absurdity.

It has been remarked by Pascal, that " Geometry is almost the only

subject as to which we find truths wherein all men agree ; and one cause

of this is, that Geometers alone regard the true laws of demonstration."

KOTES TO BOOK I. 49

These are enumerated by him as eight in number. * * 1 . To define nothing

â€¢which cannot be expressed in clearer terms than those in which it is

already expressed. 2. To leave no obscure or equivocal terms undefined.

3. To employ in the definition no terms not already known. 4. To

omit nothing in the principles from which we argue, unless we are sure

it is granted. 5. To lay down no axiom which is not perfectly evident.

6. To demonstrate nothing which is as clear already as we can make it.

7. To prove every thing in the least doubtful by means of self-evident

axioms, or of propositions already demonstrated. 8. To substitute

mentally the definition instead of the thing defined." Of these rules, he

says, "the first, fourth and sixth are not absolutely necessary to avoid

error, but the other five are indispensable ; and though they may be found

in books of logic, none but the Geometers have paid any regard to them."

The course pursued in the demonstrations of the propositions in

Euclid's Elements of Geometry, is always to refer directly to some ex-

pressed principle, to leave nothing to be inferred from vague expressions,

and to make every step of the demonstrations the object of the under-

standing.

It has been maintained by some philosophers, that a genuine defini-

tion contains some property or properties which can form a basis for

demonstration, and that the science of Geometry is deduced from the

definitions, and that on them alone the demonstrations depend. Others

have maintained that a definition explains only the meaning of a term,

and does not embrace the nature and properties of the thing defined.

If the propositions usually called postulates and axioms are either

tacitly assumed or expressly stated in the definitions ; in this view, de-

monstrations may be said to be legitimately founded on definitions. If,

on the other hand, a definition is simply an explanation of the meaning

of a term, whether abstract or concrete, by such marks as may prevent a

misconception of the thing defined ; it will be at once obvious that some

constructive and theoretic principles must be assumed, besides the defini-

tions to form the ground of legitimate demonstration. These principles

we conceive to be the postulates and axioms. The postulates describe

constructions which may be admitted as possible by direct appeal to our

experience ; and the axioms assert general theoretic ti'uths so simple

and self-evident as to require no proof, but to be admitted as the assumed

first principles of demonstration. Under this view all Geometrical

reasonings proceed upon the admission of the hypotheses assumed in

the definitions, and the unquestioned possibility of the postulates, and

the truth of the axioms.

Deductive reasoning is generally delivered in the form of an enthymeme,

or an argument wherein one enunciation is not expressed, but is readily

supplied by the reader : and it may be observed, that although this is the

ordinary mode of speaking and writing, it is not in the strictly syllogistic

form ; as either the major or the minor premiss only is formally stated

before the conclusion : Thus in Euc. i. 1.

Because the point A is the center of the circle BCD ;

therefore the straight line AB is equal to the straight line AC.

The premiss here omitted, is : all straight lines drawn from the center

of a circle to the circumference are equal.

In a similar way may be supplied the reserved premiss in every enthy-

meme. The conclusion of two enthymemes may form the major and minor

premiss of a third syllogism, and so on, and thus any process of reasoning

is reduced to the strictly syllogistic form. And in this way it is shewn

i

50 Euclid's elements.

that the general theorems of Oeometry are demonstrated by means of

syllogisms founded on the axioms and definitions.

Every syllogism consists of three propositions, of which, two are called

the premisses, and the third, the conclusion. These propositions contain

three terms, the subject and predicate of the conclusion, and the middle

term which connects the predicate and the conclusion together. The

subject of the conclusion is called the minor, and the predicate of the con-

clusion is called the major term, of the syllogism. The major term appears

in one premiss, and the minor term in the other, with the middle term,

which is in both premisses. That premiss which contains the middle

term and the major term, is called the major premiss; and that which

contains the middle term and the minor term, is called the minor premiss

of the syllogism. As an example, we may take the syllogism in the demon-

stration of Prop. 1, Book 1, wherein it will be seen that the middle term is

the subject of the major premiss and the predicate of the minor.

Major premiss: because the straight line y^J? is equal to the straight line AC\

Minor premiss : and, because the straight line ^C is equal to the straight

line AB ;

Conclusion : therefore the straight line BC is equal to the straight line AC.

Here, BC is the subject, and AC the predicate of the conclusion.

BC is the subject, and AB the predicate of the minor premiss.

AB is the subject, and AC the predicate of the major premiss.

Also, AC is the major term, ^C the minor term, and AB the middle term

of the syllogism.

In this syllogism, it may be remarked that the definition of a straight

line is assumed, and the definition of the Geometrical equality of two

straight lines ; also that a general theoretic truth, or axiom, forms the

ground of the conclusion. And further, though it be impossible to make

any point, mark or sign (o-tj/ueloi/) which has not both length and breadth,

and any line which has not both length and breadth ; the demonstrations

in Geometry do not on this account become invalid. For they are pursued

on the hypothesis that the point has no parts, but position only : and the

line has length only, but no breadth or thickness : also that the surface

has length and breadth only, but no thickness : and all the conclusions

at which we arrive are independent of every other consideration.

The truth of the conclusion in the syllogism depends upon the truth

of the premisses. If the premisses, or only one of them be not true, the

conclusion is false. The conclusion is said to follow from the premisses;

whereas, in truth, it is contained in the premisses. The expression must

be understood of the mind apprehending in succession, the truth of

the premisses, and subsequent to that, the truth of the conclusion ;

so that the conclusion follows from the premisses in order of time

as far as reference is made to the mind's apprehension of the whole

argument.

Every proposition, when complete, may be divided into six parts, as

Proclus has pointed out in his commentary.

1 . The proposition, or general emmciation, which states in general terms

the conditions of the problem or theorem.

2. The exposition, or particular eiiunciation, which exhibits the subject

of the proposition in particular terms as a fact, and refers it to some

diagram described.

3. The determination contains the predicate in particular terms, as it

is pointed out in the diagram, and diiects attention to the demonstration,

by pronouncing the thuig sought.

NOTES TO BOOK I. 51

4. TJie constniction applies tlie postulates to prepare the diagram for

the demonstration.

5. The demotistration is the connexion of syllogisms, which prove the

truth or falsehood of the theorem, the possibility or impossibility of the

problem, in that particular case exhibited in the diagram.

6. The conclusion is merely the repetition of the general enunciation,

wherein the predicate is asserted as a demonstrated truth.

Prop. I. In the first two Books, the circle is employed as a me-

chanical instrument, in the same manner as the straight line, and the use

made of it rests entirely on the third postulate. No properties of the

circle are discussed in these books beyond the definition and the third

postulate. When two circles are described, one of which has its center in

the circumference of the other, the two circles being each of them partly

within and partly without the other, their circumferences must intersect

each other in two points ; and it is obvious from the two circles cutting

each other, in two points, one on each side of the given line, that two

equilateral triangles may be formed on the given line.

Prop. II. When the given point is neither in the line, nor in the line

, produced, this problem admits of eight dififerent lines being drawn from

j the given point in different directions, every one of which is a solution

[of the problem. For, 1. The given line has two extremities, to each of

which a line may be drawn from the given point. 2. The equilateral

triangle may be described on either side of this line. 3. And the side

BD of the equilateral triangle ABB may be produced either way.

But when the given point lies either in the line or in the line pro-

duced, the distinction which arises from joining the two ends of the line

with the given point, no longer exists, and there are only four cases of

the problem.

The construction of this problem assumes a neater form, by first de-

scribing the circle CGH with center B and radius BC, and producing Z)Z?

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