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the side of the equilateral triangle DBA to meet the circumference in G :

next, with center D and radius DO, describing the circle GKL^ and then

producing DA to meet the circumference in L.

By a similar construction the less of two given straight lines may be

produced, so that the less together with the part produced may be equal

to the greater.

Prop. III. This problem admits of two solutions, and it is left unde-

termined from which end of the greater line the part is to be cut off.

By means of this problem, a straight line may be found equal to the

um or the difierence of two given lines.

Prop. IV. This forms the first case of equal triangles, two other cases

re proved in Prop. viii. and Prop. xxvi.

The term base is obviously taken from the idea of a building, and the

ame may be said of the term altitude. In Geometry, however, these

erms are not restricted to one particular position of a figure, as in the

ase of a building, but may be in any position whatever.

Prop. V. Proclus has given, in his commentary, a proof for the

quality of the angles at the base, without producing the equal sides.

The construction follows the same order, taking in AB one side of

he isosceles triangle ABC, a point D and cutting off from AC o. part

VE equal to AD, and then joining CD and BE.

A corollary is a theorem which results from the demonstration of

proposition.

Prop. VI. is the converse of one part of Prop. v. One proposition

D 2

52 Euclid's elements.

is defined to be the cotiverse of another when the hypothesis of the

former becomes the predicate of the latter ; and vice versa.

There is besides this, another kind of conversion, when a theorem

has several hypotheses and one predicate ; by assuming the predicate

and one, or more than one of the hypotheses, some one of the hypotheses

may be inferred as the predicate of the converse. In this manner,

Prop. VIII. is the converse of Prop. iv. It may here be observed,

that converse theorems are not universally true : as for instance, the

following direct proposition is universally true; **If two triangles have

their three sides respectively equal, the three angles of each shall be

respectively equal." But the converse is not universally true ; namely,

"If two triangles have the three angles in each respectively equal,

the three sides are respectively equal." Converse theorems require,

in some instances, the consideration of other conditions than those

which enter into the proof of the direct theorem. Converse and contrary

propositions are by no means to be confounded ; the contrnry proposition

denies what is asserted, or asserts what is denied, in the direct pro-

position, but the subject and predicate in each are the same. A contrary

-proposition is a completely contradictory proposition, and the distinction

consists in this â€” that two contrary propositions may both be false, but

of two contradictory propositions, one of them must be true, and the

other false. It may here be remarked, that one of the most common

intellectual mistakes of learners, is to imagine that the denial of a

proposition is a legitimate ground for affirming the contrary as true :

whereas the rules of sound reasoning allow that the affirmation of a

proposition as true, only affords a ground for the denial of the contrary

as false.

Prop. VI. is the first instance of indirect demonstrations, and they

are more suited for the proof of converse propositions. All those pro-

positions which are demonstrated ex absurdo, are properly analytical

demonstrations, according to the Greek notion of analysis, which first

supposed the thing required, to be done, or to be true, and then shewed

the consistency or inconsistency of this construction or hypothesis

with truths admitted or already demonstrated.

In indirect demonstrations, where hypotheses are made which are

not true and contrary to the truth stated in the proposition, it seems

desirable that a form of expression should be employed diff'erent from

that in which the hypotheses are true. In all cases therefore, whether

noted by Euclid or not, the words if possible have been introduced,

or some such qualifying expression, as in Euc. i. 6, so as not to leave

upon the mind of the learner, the impression that the hypothesis

which contradicts the proposition, is really true.

Prop. VIII. When the three sides of one triangle are shewn to

coincide with the three sides of any other, the equality of the triangles

is at once obvious. This, however, is not stated at the conclusion ot

Prop. VIII. or of Prop. xxvi. For the equality of the areas of two

coincident triangles, reference is always made by Euclid to Prop. iv.

A direct demonstration may be given of this proposition, and Prop.

VII. may be dispensed with altogether.

Let the triangles ABC, DEF be so placed that the base BC may

coincide with the base EF, and the vertices A, D may be on opposite

sides of EF. Join AD. Then because EAD is an isosceles triangle,

the angle EAD is equal to the angle EDA; and because CD A is an

isosceles triangle, the angle CAD is equal to the angle CDA. Hence

NOTES TO BOOK I.

53

the angle EAF is equal to the angle EDF, (ax. 2 or 3) : or the angle

BDC is equal'to the angle EDF.

Prop. IX. If BA, ^C be in the same straight line. This problem

then becomes the same as Prob. xi, which may be regarded as drawing

a line which bisects an angle equal to two right angles.

If FA be produced in the fig. Prop. 9, it bisects the angle which

is the defect of the angle B AC from four right angles.

By means of this problem, any angle may be divided into four,

eight, sixteen, &c. equal angles.

Prop. X. A finite straight line may, by this problem, be divided

into four, eight, sixteen, &c. equal parts.

Prop. XI. When the point is at the extremity of the line ; by

the second postulate the line may be produced, and then the construction

applies. See note on Euc. III. 31.

The distance between two points is the straight line which joins

the points ; but the distance between a point and a straight line, is

the shortest line which can be drawn from the point to the line.

From this Prop, it follows that only one perpendicular can be drawn

from a given point to a given line ; and this perpendicular may be

shewn to be less than any other line which can be drawn from the

given point to the given line : and of the rest, the line which is nearer

to the perpendicular is less than one more remote from it : also only

two equal straight lines can be drawn from the same point to the line,

one on each side of the perpendicular or the least. This property

is analogous to Euc. iii. 7, 8.

The corollary to this proposition is not in the Greek text, but

was added by Simson, who states that it "is necessary to Prop. 1,

Book XI., and otherwise."

Prop. XII. The third postulate requires that the line CD should

be drawn before the circle can be described with the center C, and

radius CD.

Prop. XIV. is the converse of Prop. xiii. " Upon the opposite sides

of it." If these words were omitted, it is possible for two lines to make

with a third, two angles, which together are equal to two right angles, in

such a manner that the two lines shall not be in the same straight line.

The line BE may be supposed to fall above, as in Euclid's figure,

or below the line BD^ and the demonstration is the same in form.

Prop. XV. is the development of the definition of an angle. If the lines

at the angular point be produced, the produced lines have the same incli-

nation to one another as the original lines, but in a different position,

The converse of this Proposition is not proved by Euclid, namely : â€”

If the vertical angles made by four straight lines at the sajne point

be respectively equal to each other, each pair of opposite lines shall

be in the same straight line.

Prop. XVII. appears to be only a corollary to the preceding pro-

position, and it seems to be introduced to explain Axiom xii, of which

it is the converse. The exact truth respecting the angles of a triangle

is proved in Prop, xxxii.

Prop, xviii. It may here be remarked, for the purpose of guarding

the student against a very common mistake, that in this proposition

and in the converse of it, the hypothesis is stated before the predicate.

Prop. XIX. is the converse of Prop, xviii. It may be remarked,

that Prop. XIX. bears the same relation to Prop, xviii., as Prop. vi.

does to Prop. v.

64 Euclid's elements.

Prop. XX. The following corollary arises from this proposition: â€”

A straight line is the shortest distance between two points. For

the straight line J5C is always less than BA and AC, however near

the point J may be to the line BC.

It may be easily shewn from this proposition, that the difference

of any two sides of a triangle is less than the third side.

Prop. XXII. When the sum of two of the lines is equal to, and

when it is less than, the third line ; let the diagrams be described,

and they will exhibit the impossibility implied by the restriction laid

down in the Proposition.

The same remark may be made here, as was made under the first

Proposition, namely: â€” if one circle lies partly within and partly without

another circle, the circumferences of the circles intersect each other

in two points.

Prop. XXIII. CD might be taken equal to CE, and the construction

effected by means of an isosceles triangle. It would, however, be less

general than Euclid's, but is more convenient in practice.

Prop. XXIV. Simson makes the angle EBG at D in the line ED,

the side which is not the greater of the two ED, DF ; otherwise, three

different cases would arise, as may be seen by forming the different

figures. The point G might fall below or upon the base EF produced

as well as above it. Prop. xxiv. and Prop. xxv. bear to each other

the same relation as Prop. iv. and Prop. viii.

Prop. xxvi. This forms the third case of the equality of two tri-

angles. Every triangle has three sides and three angles, and when

any three of one triangle are given equal to any three of another, the

triangles may be proved to be equal to one another, whenever the

three magnitudes given in the hypothesis are independent of one another.

Prop. IV. contains the first case, when the hypothesis consists of two

sides and the included angle of each triangle. Prop. viii. contains

the second, when the hypothesis consists of the three sides of each

triangle. Prop. xxvi. contains the third, when the hypothesis consists

of two angles, and one side either adjacent to the equal angles, or

opposite to one of the equal angles in each triangle. There is another

case, not proved by Euclid, when the hypothesis consists of two sides

and one angle in each triangle, but these not the angles included by

the two given sides in each triangle. This case however is only true

under a certain restriction, thus :

If two triangles have two sides of one of them equal to tico sides of the

other, each to each, and have also the angles opposite to one of the equal sides

in each triangle, equal to one another, and if the angles opposite to the other

equal sides he both acute, or both obtuse angles ; then shall the third sides

he equal in each triangle, as also the remaining angles of the one to the

remaijiing angles of the other.

Let ABC, DEF be two triangles which have the sides AB, AC equal

to the two sides DE, DF, each to each, and the angle ABC equal to the

angle DEF: then, if the angles ACB, DEF, be both acute, or both obtuse

angles, the third side BC shall be equal to the third side EF, and also

the angle BCA to the angle EFD, and the angle BJC to the angle EDF.

First. Let the angles ACB, DFE opposite to the equal sides AB,

DE, be both acute angles.

if BC be not equal to EF, let BC be the greater, and from BC, cut off

BG equal to EF, and join AG.

Then in the triangles ABG, DEF, Euc. i. 4. AG is equal to DF,

NOTES TO BOOK I. 55

and the angle AGE to DFE. But since AC i^ equal to DF, AG is, equal

to AC: and therefore the angle ACG is equal to the angle AGC^ which

is also an acute angle. But because AGC^ AGB are together equal

to two right angles, and that AGC is an acute angle, AGB must be

an obtuse angle ; which is absurd. Wherefore, BC is not unequal

to EF, that is, BC is equal to EF, and also the remaining angles of

one triangle to the remaining angles of the other.

Secondly. Let the angles ACB, DFE, be both obtuse angles. By

proceeding in a similar way, it may be shewn that BC cannot be

otherwise than equal to EF.

If ACB, DFE be both right angles: the case falls under Euc. i. 26.

Prop. xxviT. Alternate angles are defined to be the two angles

which two straight lines make with another at its extremities, but upon

opposite sides of it.

When a straight line intersects two other straight lines, two pairs of

alternate angles are formed by the lines at their intersections, as in the

figure, BEF, EEC are alternate angles as well as the angles AEF^ EFD.

Prop. XXVIII. One angle is called " the exterior angle," and another

"the interior and opposite angle," when they are formed on the same

side of a straight line which falls upon or intersects two other straight

lines. It is also obvious that on each side of the line, there will be two

exterior and two interior and opposite angles. The exterior angle EGB

has the angle GHD for its corresponding interior and opposite angle :

also the exterior angle FHD has the angle HGB for its interior and

opposite angle.

Prop. XXIX is the converse of Prop, xxvii and Prop, xxviii. â€¢

As the definition of parallel straight lines simply describes them

by a statement of the negative property, that they never meet ; it is

necessary that some positive property of parallel lines should be assumed

as an axiom, on which reasonings on such lines may be founded.

Euclid has assumed the statement in the twelfth axiom, which has

been objected to, as not being self-evident. A stronger objection

appears to be, that the converse of it forms Euc. i. 17; for both the

assumed axiom and its converse, should be so obvious as not to require

formal demonstration.

Simson has attempted to overcome the objection, not by any improved

definition and axiom respecting parallel lines ; but, by considering Euclid's

twelfth axiom to be a theorem, and for its proof, assuming two definitions

and one axiom, and then demonstrating five subsidiary Propositions.

Instead of Euclid's twelfth axiom, the following has been proposed

as a more simple property for the foundation of reasonings on parallel

lines ; namely, " If a straight line fall on two parallel straight lines,

the alternate angles are equal to one another." In whatever this may

exceed Euclid's definition in simplicity, it is liable to a similar objection,

being the converse of Euc. i. 27.

Professor Playfair has adopted in his Elements of Geometry, that

â€¢* Two straight lines which intersect one another cannot be both parallel

to the same straight line." This apparently more simple axiom follows

as a direct inference from Euc. i. 30.

But one of the least objectionable of all the definitions which have

been proposed on this subject, appears to be that which simply expresses

the conception of equidistance. It may be formally stated thus :

" Parallel lines are such as lie in the same plane, and which neither

recede from, nor approach to, each other." This includes the con-

50 Euclid's elements.

ception stated by Euclid, that parallel lines never meet. Dr. Wallis

observes on this subject, '* Parallelismus et aequidistantia vel idem sunt,

vel certe se rautuo comitantur."

As an additional reason for this definition being preferred, it may-

be remarked that the meaning of the terms ypamxal Trapd\kr]\oi, suggests

the exact idea of such lines.

An account of thirty methods which have been proposed at different

times for avoiding the difficulty in the twelfth axiom, will be

found in the appendix to Colonel Thompson's " Geometry without

Axioms."

Prop. XXX. In the diagram, the two lines AB and CD are placed

one on each side of the line EF : the proposition may also be proved

when both AB and GD are on the same side of EF.

Prop. XXXII. From this proposition, it is obvious that if one angle

of a triangle be equal to the sum of the other two angles, that angle

is a right angle, as is shewn in Euc. iii. 31, and that each of the angles

of an equilateral triangle, is equal to two thirds of a right angle, as

it is shewn in Euc. iv. 15. Also, if one angle of an isosceles triangle

be a right angle, then each of the equal angles is half a right angle, as

in Euc. II. 9.

The three angles of a triangle may be shewn to be equal to two

right angles without producing a side of the triangle, by drawing through

any angle of the triangle a line parallel to the opposite side, as Proclus

has remarked in his Commentary on this proposition. It is manifest

from this proposition, that the third angle of a triangle is not inde-

pendent of the sum of the other two ; but is known if the sum of any

two is known. Cor. 1 may be also proved by drawing lines from any

one of the angles of the figure to the other angles. If any of the

sides of the figure bend inwards and form what are called re-entering

angles, the enunciation of these two corollaries will require some

modification. As Euclid gives no definition of re-entering angles, it

may fairly be concluded, he did not intend to enter into the proofs

of the properties of figures which contain such angles.

Prop. XXXIII. The words 'â€¢ towards the same parts" are a necessary

restriction : for if they were omitted, it would be doubtful whether

the extremities A, C, and B, D were to be joined by the lines AC and

BD ; or the extremities A, D, and B, C, by the lines AD and BC.

Prop, xxxiv. If the other diameter be drawn, it may be shewn

that the diameters of a parallelogram bisect each other, as well as bisect

the area of the parallelogram. If the parallelogram be right angled,

the diagonals are equal ; if the parallelogram be a square or a rhombus,

the diagonals bisect each other at right angles. The converse of this

Prop., namely, " If the opposite sides or opposite angles of a quadrilateral

figure be equal, the opposite sides shall also be parallel ; that is, the

figure shall be a parallelogram," is not proved by Euclid,

Prop. XXXV. The latter part of the demonstration is not expressed

very intelligibly. Simson, who altered the demonstration, seems in fact

to consider two trapeziums of the same form and magnitude, and from

one of them, to take the triangle ABE\ and from the other, the tri-

angle /)CF; and then the remainders are equal by the third axiom:

that is, the parallelogram ABCD is equal to the parallelogram EBCF.

Otherwise, the triangle, whose base is DE, (fig. 2.) is taken twice from

the trapezium, which would appear to bo impossible, if the sense m

which Euclid applies the third axiom, is to be retained here.

I

NOTES TO BOOK I. 57

It may be observed, that the two parallelograms exhibited in fig. 2

partially lie on one another, and that the triangle whose base is ^C is a

common part of them, but that the triangle whose base is DE is entirely

without both the parallelograms. After having proved the triangle JBE

equal to the triangle DCF, if we take from these equals (fig. 2.) the

triangle whose base is I)E, and to each of the remainders add the

triangle whose base is BC, then the parallelogram ABCD is equal to

the parallelogram EBCF. In fig. 3, the equality of the parallelograms

ABCD, EBCF, is shewn by adding the figure EBCD to each of the

triangles ABE, DCF.

In this proposition, the word equal assumes a new meaning, and is no

longer restricted to mean coincidence in all the parts of two figures.

Prop. XXXVIII. In this proposition, it is to be understood that the

bases of the two triangles are in the same straight line. If in the

diagram the point E coincide with C, and D with A, then the angle

of one triangle is supplemental to the other. Hence the following

property : â€” If two triangles have two sides of the one respectively equal

to two sides of the other, and the contained angles supplemental, the

two triangles are equal.

A distinction ought to be made between equal triangles and equivalent

triangles, the former including those whose sides and angles mutually

coincide, the latter those whose areas only are equivalent.

Prop. XXXIX. If the vertices of all the equal triangles which can be

described upon the same base, or upon the equal bases as in Prop. 40,

be joined, the line thus formed will be a straight line, and is called the

locus of the vertices of equal triangles upon the same base, or upon

equal bases.

A locus in plane Geometry is a straight line or a plane curve, every

point of which and none else satisfies a certain condition. With the

exception of the straight line and the circle, the two most simple loci ;

all other loci, perhaps including also the Conic Sections, may be more

readily and effectually investigated algebraically by means of their

rectangular or polar equations.

Prop. xLi. The converse of this proposition is not proved by Euclid ;

viz. If a parallelogram is double of a triangle, and they have the same base,

or equal bases upon the same straight line, and towards the same parts,

they shall be between the same parallels. Also, it may easily be shewn

that if two equal triangles are between the same parallels ; they are either

upon the same base, or upon equal bases.

Prop. XLiv. A parallelogram described on a straight line is said to

be applied to that line.

Prop. XLv. The problem is solved only for a rectilineal figure of four

sides. If the given rectilineal figure have more than four sides, it may

be divided into triangles by drawing straight lines from any angle of the

figure to the opposite angles, and then a parallelogram equal to the third

triangle can be applied to LM, and having an angle equal to E: and

so on for all the triangles of which the rectilineal figure is composed.

Prop. XLvi. The square being considered as an equilateral rectangle,

its area or surface may be expressed numerically if the number of lineal

units in a side of the square be given, as is shewn in the note on Prop, i.,

Book II.

The student will not fail to remark the analogy which exists between

the area of a square and the product of two equal numbers ; and between

the side of a square and the square root of a number. There is, however,

t

d5

5S Euclid's elements.

this distinction to be observed ; it is always possible to find the product

of two equal numbers, (or to find the square of a number, as it is usually

called,) and to describe a square on a given line ; but conversely, though,

the side of a given square is known from the figure itself, the exact

number of units in the side of a square of given area, can only be found

exactly, in such cases where the given number is a square number. For

example, if the area of a square contain 9 square units, then the square

root of 9 or 3, indicates the number of lineal units in the side of that

square. Again, if the area of a square contain 12 square units, the side

of the square is greater than 3, but less than 4 lineal units, and there is

no number which will exactly express the side of that square: an approxi-

mation to the true length, however, may be obtained to any assigned

degree of accuracy.

Prop. xLvii. In a right-angled triangle, the side opposite to the right

angle is called the hypotenuse, and the other two sides, the base and

perpendicular, according to their position.

In the diagram the three squares are described on the outer sides of

the triangle -4^ C. The Proposition may also be demonstrated (1) when

the three squares are described upon the inner sides of the triangle : (2)

when one square is described on the outer side and the other two squares

on the inner sides of the triangle : (3) when one square is described on the

inner side and the other two squares on the outer sides of the triangle.

As one instance of the third case. If the square BE on the hypote-

nuse be described on the inner side of BC and the squares BG, EC on

the outer sides of ^^, AC\ the point D falls on the side FG (Euclid's

fig.) of the square BG, and iiC// produced meets CE in E. Let LA meet

BC in M. Join DA ; then the square GB and the oblong LB are each

double of the triangle DAB, (Euc. i. 41.); and similarly by joining EA^

the square HC and oblong LC are each double of the triangle EAC.

Whence it follows that the squares on the sides AB, AC are together

equal to the square on the hypotenuse BC.

By this proposition may be found a square equal to the sum of any given

next, with center D and radius DO, describing the circle GKL^ and then

producing DA to meet the circumference in L.

By a similar construction the less of two given straight lines may be

produced, so that the less together with the part produced may be equal

to the greater.

Prop. III. This problem admits of two solutions, and it is left unde-

termined from which end of the greater line the part is to be cut off.

By means of this problem, a straight line may be found equal to the

um or the difierence of two given lines.

Prop. IV. This forms the first case of equal triangles, two other cases

re proved in Prop. viii. and Prop. xxvi.

The term base is obviously taken from the idea of a building, and the

ame may be said of the term altitude. In Geometry, however, these

erms are not restricted to one particular position of a figure, as in the

ase of a building, but may be in any position whatever.

Prop. V. Proclus has given, in his commentary, a proof for the

quality of the angles at the base, without producing the equal sides.

The construction follows the same order, taking in AB one side of

he isosceles triangle ABC, a point D and cutting off from AC o. part

VE equal to AD, and then joining CD and BE.

A corollary is a theorem which results from the demonstration of

proposition.

Prop. VI. is the converse of one part of Prop. v. One proposition

D 2

52 Euclid's elements.

is defined to be the cotiverse of another when the hypothesis of the

former becomes the predicate of the latter ; and vice versa.

There is besides this, another kind of conversion, when a theorem

has several hypotheses and one predicate ; by assuming the predicate

and one, or more than one of the hypotheses, some one of the hypotheses

may be inferred as the predicate of the converse. In this manner,

Prop. VIII. is the converse of Prop. iv. It may here be observed,

that converse theorems are not universally true : as for instance, the

following direct proposition is universally true; **If two triangles have

their three sides respectively equal, the three angles of each shall be

respectively equal." But the converse is not universally true ; namely,

"If two triangles have the three angles in each respectively equal,

the three sides are respectively equal." Converse theorems require,

in some instances, the consideration of other conditions than those

which enter into the proof of the direct theorem. Converse and contrary

propositions are by no means to be confounded ; the contrnry proposition

denies what is asserted, or asserts what is denied, in the direct pro-

position, but the subject and predicate in each are the same. A contrary

-proposition is a completely contradictory proposition, and the distinction

consists in this â€” that two contrary propositions may both be false, but

of two contradictory propositions, one of them must be true, and the

other false. It may here be remarked, that one of the most common

intellectual mistakes of learners, is to imagine that the denial of a

proposition is a legitimate ground for affirming the contrary as true :

whereas the rules of sound reasoning allow that the affirmation of a

proposition as true, only affords a ground for the denial of the contrary

as false.

Prop. VI. is the first instance of indirect demonstrations, and they

are more suited for the proof of converse propositions. All those pro-

positions which are demonstrated ex absurdo, are properly analytical

demonstrations, according to the Greek notion of analysis, which first

supposed the thing required, to be done, or to be true, and then shewed

the consistency or inconsistency of this construction or hypothesis

with truths admitted or already demonstrated.

In indirect demonstrations, where hypotheses are made which are

not true and contrary to the truth stated in the proposition, it seems

desirable that a form of expression should be employed diff'erent from

that in which the hypotheses are true. In all cases therefore, whether

noted by Euclid or not, the words if possible have been introduced,

or some such qualifying expression, as in Euc. i. 6, so as not to leave

upon the mind of the learner, the impression that the hypothesis

which contradicts the proposition, is really true.

Prop. VIII. When the three sides of one triangle are shewn to

coincide with the three sides of any other, the equality of the triangles

is at once obvious. This, however, is not stated at the conclusion ot

Prop. VIII. or of Prop. xxvi. For the equality of the areas of two

coincident triangles, reference is always made by Euclid to Prop. iv.

A direct demonstration may be given of this proposition, and Prop.

VII. may be dispensed with altogether.

Let the triangles ABC, DEF be so placed that the base BC may

coincide with the base EF, and the vertices A, D may be on opposite

sides of EF. Join AD. Then because EAD is an isosceles triangle,

the angle EAD is equal to the angle EDA; and because CD A is an

isosceles triangle, the angle CAD is equal to the angle CDA. Hence

NOTES TO BOOK I.

53

the angle EAF is equal to the angle EDF, (ax. 2 or 3) : or the angle

BDC is equal'to the angle EDF.

Prop. IX. If BA, ^C be in the same straight line. This problem

then becomes the same as Prob. xi, which may be regarded as drawing

a line which bisects an angle equal to two right angles.

If FA be produced in the fig. Prop. 9, it bisects the angle which

is the defect of the angle B AC from four right angles.

By means of this problem, any angle may be divided into four,

eight, sixteen, &c. equal angles.

Prop. X. A finite straight line may, by this problem, be divided

into four, eight, sixteen, &c. equal parts.

Prop. XI. When the point is at the extremity of the line ; by

the second postulate the line may be produced, and then the construction

applies. See note on Euc. III. 31.

The distance between two points is the straight line which joins

the points ; but the distance between a point and a straight line, is

the shortest line which can be drawn from the point to the line.

From this Prop, it follows that only one perpendicular can be drawn

from a given point to a given line ; and this perpendicular may be

shewn to be less than any other line which can be drawn from the

given point to the given line : and of the rest, the line which is nearer

to the perpendicular is less than one more remote from it : also only

two equal straight lines can be drawn from the same point to the line,

one on each side of the perpendicular or the least. This property

is analogous to Euc. iii. 7, 8.

The corollary to this proposition is not in the Greek text, but

was added by Simson, who states that it "is necessary to Prop. 1,

Book XI., and otherwise."

Prop. XII. The third postulate requires that the line CD should

be drawn before the circle can be described with the center C, and

radius CD.

Prop. XIV. is the converse of Prop. xiii. " Upon the opposite sides

of it." If these words were omitted, it is possible for two lines to make

with a third, two angles, which together are equal to two right angles, in

such a manner that the two lines shall not be in the same straight line.

The line BE may be supposed to fall above, as in Euclid's figure,

or below the line BD^ and the demonstration is the same in form.

Prop. XV. is the development of the definition of an angle. If the lines

at the angular point be produced, the produced lines have the same incli-

nation to one another as the original lines, but in a different position,

The converse of this Proposition is not proved by Euclid, namely : â€”

If the vertical angles made by four straight lines at the sajne point

be respectively equal to each other, each pair of opposite lines shall

be in the same straight line.

Prop. XVII. appears to be only a corollary to the preceding pro-

position, and it seems to be introduced to explain Axiom xii, of which

it is the converse. The exact truth respecting the angles of a triangle

is proved in Prop, xxxii.

Prop, xviii. It may here be remarked, for the purpose of guarding

the student against a very common mistake, that in this proposition

and in the converse of it, the hypothesis is stated before the predicate.

Prop. XIX. is the converse of Prop, xviii. It may be remarked,

that Prop. XIX. bears the same relation to Prop, xviii., as Prop. vi.

does to Prop. v.

64 Euclid's elements.

Prop. XX. The following corollary arises from this proposition: â€”

A straight line is the shortest distance between two points. For

the straight line J5C is always less than BA and AC, however near

the point J may be to the line BC.

It may be easily shewn from this proposition, that the difference

of any two sides of a triangle is less than the third side.

Prop. XXII. When the sum of two of the lines is equal to, and

when it is less than, the third line ; let the diagrams be described,

and they will exhibit the impossibility implied by the restriction laid

down in the Proposition.

The same remark may be made here, as was made under the first

Proposition, namely: â€” if one circle lies partly within and partly without

another circle, the circumferences of the circles intersect each other

in two points.

Prop. XXIII. CD might be taken equal to CE, and the construction

effected by means of an isosceles triangle. It would, however, be less

general than Euclid's, but is more convenient in practice.

Prop. XXIV. Simson makes the angle EBG at D in the line ED,

the side which is not the greater of the two ED, DF ; otherwise, three

different cases would arise, as may be seen by forming the different

figures. The point G might fall below or upon the base EF produced

as well as above it. Prop. xxiv. and Prop. xxv. bear to each other

the same relation as Prop. iv. and Prop. viii.

Prop. xxvi. This forms the third case of the equality of two tri-

angles. Every triangle has three sides and three angles, and when

any three of one triangle are given equal to any three of another, the

triangles may be proved to be equal to one another, whenever the

three magnitudes given in the hypothesis are independent of one another.

Prop. IV. contains the first case, when the hypothesis consists of two

sides and the included angle of each triangle. Prop. viii. contains

the second, when the hypothesis consists of the three sides of each

triangle. Prop. xxvi. contains the third, when the hypothesis consists

of two angles, and one side either adjacent to the equal angles, or

opposite to one of the equal angles in each triangle. There is another

case, not proved by Euclid, when the hypothesis consists of two sides

and one angle in each triangle, but these not the angles included by

the two given sides in each triangle. This case however is only true

under a certain restriction, thus :

If two triangles have two sides of one of them equal to tico sides of the

other, each to each, and have also the angles opposite to one of the equal sides

in each triangle, equal to one another, and if the angles opposite to the other

equal sides he both acute, or both obtuse angles ; then shall the third sides

he equal in each triangle, as also the remaining angles of the one to the

remaijiing angles of the other.

Let ABC, DEF be two triangles which have the sides AB, AC equal

to the two sides DE, DF, each to each, and the angle ABC equal to the

angle DEF: then, if the angles ACB, DEF, be both acute, or both obtuse

angles, the third side BC shall be equal to the third side EF, and also

the angle BCA to the angle EFD, and the angle BJC to the angle EDF.

First. Let the angles ACB, DFE opposite to the equal sides AB,

DE, be both acute angles.

if BC be not equal to EF, let BC be the greater, and from BC, cut off

BG equal to EF, and join AG.

Then in the triangles ABG, DEF, Euc. i. 4. AG is equal to DF,

NOTES TO BOOK I. 55

and the angle AGE to DFE. But since AC i^ equal to DF, AG is, equal

to AC: and therefore the angle ACG is equal to the angle AGC^ which

is also an acute angle. But because AGC^ AGB are together equal

to two right angles, and that AGC is an acute angle, AGB must be

an obtuse angle ; which is absurd. Wherefore, BC is not unequal

to EF, that is, BC is equal to EF, and also the remaining angles of

one triangle to the remaining angles of the other.

Secondly. Let the angles ACB, DFE, be both obtuse angles. By

proceeding in a similar way, it may be shewn that BC cannot be

otherwise than equal to EF.

If ACB, DFE be both right angles: the case falls under Euc. i. 26.

Prop. xxviT. Alternate angles are defined to be the two angles

which two straight lines make with another at its extremities, but upon

opposite sides of it.

When a straight line intersects two other straight lines, two pairs of

alternate angles are formed by the lines at their intersections, as in the

figure, BEF, EEC are alternate angles as well as the angles AEF^ EFD.

Prop. XXVIII. One angle is called " the exterior angle," and another

"the interior and opposite angle," when they are formed on the same

side of a straight line which falls upon or intersects two other straight

lines. It is also obvious that on each side of the line, there will be two

exterior and two interior and opposite angles. The exterior angle EGB

has the angle GHD for its corresponding interior and opposite angle :

also the exterior angle FHD has the angle HGB for its interior and

opposite angle.

Prop. XXIX is the converse of Prop, xxvii and Prop, xxviii. â€¢

As the definition of parallel straight lines simply describes them

by a statement of the negative property, that they never meet ; it is

necessary that some positive property of parallel lines should be assumed

as an axiom, on which reasonings on such lines may be founded.

Euclid has assumed the statement in the twelfth axiom, which has

been objected to, as not being self-evident. A stronger objection

appears to be, that the converse of it forms Euc. i. 17; for both the

assumed axiom and its converse, should be so obvious as not to require

formal demonstration.

Simson has attempted to overcome the objection, not by any improved

definition and axiom respecting parallel lines ; but, by considering Euclid's

twelfth axiom to be a theorem, and for its proof, assuming two definitions

and one axiom, and then demonstrating five subsidiary Propositions.

Instead of Euclid's twelfth axiom, the following has been proposed

as a more simple property for the foundation of reasonings on parallel

lines ; namely, " If a straight line fall on two parallel straight lines,

the alternate angles are equal to one another." In whatever this may

exceed Euclid's definition in simplicity, it is liable to a similar objection,

being the converse of Euc. i. 27.

Professor Playfair has adopted in his Elements of Geometry, that

â€¢* Two straight lines which intersect one another cannot be both parallel

to the same straight line." This apparently more simple axiom follows

as a direct inference from Euc. i. 30.

But one of the least objectionable of all the definitions which have

been proposed on this subject, appears to be that which simply expresses

the conception of equidistance. It may be formally stated thus :

" Parallel lines are such as lie in the same plane, and which neither

recede from, nor approach to, each other." This includes the con-

50 Euclid's elements.

ception stated by Euclid, that parallel lines never meet. Dr. Wallis

observes on this subject, '* Parallelismus et aequidistantia vel idem sunt,

vel certe se rautuo comitantur."

As an additional reason for this definition being preferred, it may-

be remarked that the meaning of the terms ypamxal Trapd\kr]\oi, suggests

the exact idea of such lines.

An account of thirty methods which have been proposed at different

times for avoiding the difficulty in the twelfth axiom, will be

found in the appendix to Colonel Thompson's " Geometry without

Axioms."

Prop. XXX. In the diagram, the two lines AB and CD are placed

one on each side of the line EF : the proposition may also be proved

when both AB and GD are on the same side of EF.

Prop. XXXII. From this proposition, it is obvious that if one angle

of a triangle be equal to the sum of the other two angles, that angle

is a right angle, as is shewn in Euc. iii. 31, and that each of the angles

of an equilateral triangle, is equal to two thirds of a right angle, as

it is shewn in Euc. iv. 15. Also, if one angle of an isosceles triangle

be a right angle, then each of the equal angles is half a right angle, as

in Euc. II. 9.

The three angles of a triangle may be shewn to be equal to two

right angles without producing a side of the triangle, by drawing through

any angle of the triangle a line parallel to the opposite side, as Proclus

has remarked in his Commentary on this proposition. It is manifest

from this proposition, that the third angle of a triangle is not inde-

pendent of the sum of the other two ; but is known if the sum of any

two is known. Cor. 1 may be also proved by drawing lines from any

one of the angles of the figure to the other angles. If any of the

sides of the figure bend inwards and form what are called re-entering

angles, the enunciation of these two corollaries will require some

modification. As Euclid gives no definition of re-entering angles, it

may fairly be concluded, he did not intend to enter into the proofs

of the properties of figures which contain such angles.

Prop. XXXIII. The words 'â€¢ towards the same parts" are a necessary

restriction : for if they were omitted, it would be doubtful whether

the extremities A, C, and B, D were to be joined by the lines AC and

BD ; or the extremities A, D, and B, C, by the lines AD and BC.

Prop, xxxiv. If the other diameter be drawn, it may be shewn

that the diameters of a parallelogram bisect each other, as well as bisect

the area of the parallelogram. If the parallelogram be right angled,

the diagonals are equal ; if the parallelogram be a square or a rhombus,

the diagonals bisect each other at right angles. The converse of this

Prop., namely, " If the opposite sides or opposite angles of a quadrilateral

figure be equal, the opposite sides shall also be parallel ; that is, the

figure shall be a parallelogram," is not proved by Euclid,

Prop. XXXV. The latter part of the demonstration is not expressed

very intelligibly. Simson, who altered the demonstration, seems in fact

to consider two trapeziums of the same form and magnitude, and from

one of them, to take the triangle ABE\ and from the other, the tri-

angle /)CF; and then the remainders are equal by the third axiom:

that is, the parallelogram ABCD is equal to the parallelogram EBCF.

Otherwise, the triangle, whose base is DE, (fig. 2.) is taken twice from

the trapezium, which would appear to bo impossible, if the sense m

which Euclid applies the third axiom, is to be retained here.

I

NOTES TO BOOK I. 57

It may be observed, that the two parallelograms exhibited in fig. 2

partially lie on one another, and that the triangle whose base is ^C is a

common part of them, but that the triangle whose base is DE is entirely

without both the parallelograms. After having proved the triangle JBE

equal to the triangle DCF, if we take from these equals (fig. 2.) the

triangle whose base is I)E, and to each of the remainders add the

triangle whose base is BC, then the parallelogram ABCD is equal to

the parallelogram EBCF. In fig. 3, the equality of the parallelograms

ABCD, EBCF, is shewn by adding the figure EBCD to each of the

triangles ABE, DCF.

In this proposition, the word equal assumes a new meaning, and is no

longer restricted to mean coincidence in all the parts of two figures.

Prop. XXXVIII. In this proposition, it is to be understood that the

bases of the two triangles are in the same straight line. If in the

diagram the point E coincide with C, and D with A, then the angle

of one triangle is supplemental to the other. Hence the following

property : â€” If two triangles have two sides of the one respectively equal

to two sides of the other, and the contained angles supplemental, the

two triangles are equal.

A distinction ought to be made between equal triangles and equivalent

triangles, the former including those whose sides and angles mutually

coincide, the latter those whose areas only are equivalent.

Prop. XXXIX. If the vertices of all the equal triangles which can be

described upon the same base, or upon the equal bases as in Prop. 40,

be joined, the line thus formed will be a straight line, and is called the

locus of the vertices of equal triangles upon the same base, or upon

equal bases.

A locus in plane Geometry is a straight line or a plane curve, every

point of which and none else satisfies a certain condition. With the

exception of the straight line and the circle, the two most simple loci ;

all other loci, perhaps including also the Conic Sections, may be more

readily and effectually investigated algebraically by means of their

rectangular or polar equations.

Prop. xLi. The converse of this proposition is not proved by Euclid ;

viz. If a parallelogram is double of a triangle, and they have the same base,

or equal bases upon the same straight line, and towards the same parts,

they shall be between the same parallels. Also, it may easily be shewn

that if two equal triangles are between the same parallels ; they are either

upon the same base, or upon equal bases.

Prop. XLiv. A parallelogram described on a straight line is said to

be applied to that line.

Prop. XLv. The problem is solved only for a rectilineal figure of four

sides. If the given rectilineal figure have more than four sides, it may

be divided into triangles by drawing straight lines from any angle of the

figure to the opposite angles, and then a parallelogram equal to the third

triangle can be applied to LM, and having an angle equal to E: and

so on for all the triangles of which the rectilineal figure is composed.

Prop. XLvi. The square being considered as an equilateral rectangle,

its area or surface may be expressed numerically if the number of lineal

units in a side of the square be given, as is shewn in the note on Prop, i.,

Book II.

The student will not fail to remark the analogy which exists between

the area of a square and the product of two equal numbers ; and between

the side of a square and the square root of a number. There is, however,

t

d5

5S Euclid's elements.

this distinction to be observed ; it is always possible to find the product

of two equal numbers, (or to find the square of a number, as it is usually

called,) and to describe a square on a given line ; but conversely, though,

the side of a given square is known from the figure itself, the exact

number of units in the side of a square of given area, can only be found

exactly, in such cases where the given number is a square number. For

example, if the area of a square contain 9 square units, then the square

root of 9 or 3, indicates the number of lineal units in the side of that

square. Again, if the area of a square contain 12 square units, the side

of the square is greater than 3, but less than 4 lineal units, and there is

no number which will exactly express the side of that square: an approxi-

mation to the true length, however, may be obtained to any assigned

degree of accuracy.

Prop. xLvii. In a right-angled triangle, the side opposite to the right

angle is called the hypotenuse, and the other two sides, the base and

perpendicular, according to their position.

In the diagram the three squares are described on the outer sides of

the triangle -4^ C. The Proposition may also be demonstrated (1) when

the three squares are described upon the inner sides of the triangle : (2)

when one square is described on the outer side and the other two squares

on the inner sides of the triangle : (3) when one square is described on the

inner side and the other two squares on the outer sides of the triangle.

As one instance of the third case. If the square BE on the hypote-

nuse be described on the inner side of BC and the squares BG, EC on

the outer sides of ^^, AC\ the point D falls on the side FG (Euclid's

fig.) of the square BG, and iiC// produced meets CE in E. Let LA meet

BC in M. Join DA ; then the square GB and the oblong LB are each

double of the triangle DAB, (Euc. i. 41.); and similarly by joining EA^

the square HC and oblong LC are each double of the triangle EAC.

Whence it follows that the squares on the sides AB, AC are together

equal to the square on the hypotenuse BC.

By this proposition may be found a square equal to the sum of any given

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