Euclid. # Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

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Then each of the figures GBCA, DEFHis a parallelogram;

and they are equal to one another, (l. 36.)

because they are upon equal bases BC, EF,

and between the same parallels BF, GH.

And because the diameter AB bisects the parallelogram GBCA,

therefore the triangle ABC is. the half of the parallelogram GBCA ;

(I. 34.)

also, because the diameter Di^ bisects the parallelogram DEFH,

therefore the triangle DEF is the half of the parallelogram DEFH',

but the halves of equal things are equal ; (ax. 7.)

therefore the triangle ABC\& equal to the triangle DEF.

"Wherefore, triangles upon equal bases, &c. Q. E. D.

PROPOSITION XXXIX. THEOREM.

Equal triangles zcpon the same base and upon the same side of it, are

between the same parallels.

Let the equal triangles ABC, DBC be upon the same base BC,

and upon the same side of it.

Then the triangles ABC, Z)jBC shall be between the same parallels.

:v

Join AD ; then AD shall be parallel to BC.

For if ^Z) be not parallel to BC,

if possible, through the point A, draw AE parallel to BC, (l. 31.)

meeting BD, or BD produced, in E, and join EC.

Then the triangle ABC is equal to the triangle EBC, (l. 37.)

because they are upon the same base BC,

and between the same parallels BC, AE:

but the triangle ABC is equal to the triangle DBC; (h}T5.)

therefore the triangle DBC is equal to the triangle EBC,

I

BOOK I. PROP. XL, XLI.

the greater triangle equal to the less, which is impossible :

therefore AJ3 is not parallel to JBC.

In the same manner it can be demonstrated,

that no other line drawn from A but AD is parallel to JBC;

AD is therefore parallel to BC.

Wherefore, equal triangles upon, &c. Q. E. D.

PROPOSITION XL. THEOREM.

Equal triangles upon equal bases in the same straight line, and towards

the same parts, are between the same parallels.

Let the equal triangles ABC, DJEFhe upon equal bases BC, EF,

in the same straight line BF, and towards the same parts.

Then they shall be between the same parallels.

A D

Join AD ; then AD shall be parallel to BF.

For if ^Z) be not parallel to BF,

if possible, through A draw AG parallel to BF, (l. 31.)

meeting ED, or ED produced in G, and join GF.

Then the triangle ABCis equal to the triangle GEF, (i. 38.)

because they are upon equal bases BC, EF,

and between the same parallels BF, A G ;

but the triangle ^^Cis equal to the triangle DEF; (hyp.)

therefore the triangle DEFh equal to the triangle GEF, (ax. 1.)

the greater triangle equal to the less, which is impossible :

therefore AG is not parallel to BF.

And in the same manner it can be demonstrated,

that there is no other line drawn from A parallel to it but AD ;

AD is therefore parallel to BF.

Wherefore, equal triangles upon, &c. Q. E. D.

PROPOSITION XLI. THEOREM.

If a parallelogram, and a triangle be upon the same base, and beizceen

the same parallels ; the parallelogram shall be double of the triangle.

Let the parallelogram ABCD, and the triangle EBC be upon the

same base BC, and between the same parallels BC, AE.

Then the parallelogram AB CD shall be double of the triangle EB C.

D E

I

B

Join A C.

Then the triangle ABCis equal to the triangle EBC, (l. 37.)

ELEMENTS.

because they are upon the same base BC, and between the same

parallels BC, AJS.

But the parallelogram ABCD is double of the triangle ABC,

because the diameter ^C bisects it; (l. 34.)

wherefore ABCD is also double of the triangle EBC.

Therefore, if a parallelogram and a triangle, &c. Q.E.D.

PROPOSITION XLII. PROBLEM.

To describe a parallelogram that shall be equal to a given triangle^ and

have one of its angles equal to a given rectilineal angle.

Let ABC he the given triangle, and D the given rectilineal angle.

It is required to describe a parallelogram that sliall be equal to the

given triangle ABC, and have one of its angles equal to D,

A F G

B E c

Bisect ^Cin E, (l. 10.) and join AE-,

at the point E in the straight line EC,

make the angle C^i^ equal to the angle D; (I. 23.)

through C draw CG parallel to EF, and through A draw AF

parallel to BC, (l. 3L) meeting ^i^in F, and CG in G.

Then the figure CEFG is a parallelogram, (def. A.)

And because the triangles ABE, AEC are on the equal bases Bj

EC, and between the same parallels BC, AG;

they are therefore equal to one another ; (l. 38.)

and the triangle ABC is double of the triangle AEC;

but the parallelogram FECG is double of the triangle AEC, (l. 41

because they are upon the same base EC, and between the sam

parallels ^C,^(?;

therefore the parallelogram FECG is equal to the triangle ABC, (ax. 6

and it has one of its angles CEF equal to the given angle Z).

"Wherefore, a parallelogram FECG has been described equal toth

given triangle ABC, and having one of its angles CEF equal to th

given angle E. Q. E. f.

PROPOSITION XLIII. THEOREM.

The complements of the parallelograms, which are about the diamet

of any parallelogram, are equal to one another.

Let ABCD be a parallelogram, of which the diameter is AC', a:

EII,GF the parallelograms about ^ C, that is, throuf/h which A Cjjassa

also BK, KD the other parallelograms which make up the who!

figure ABCD, which are therefore called the complements.

Then the complwnent BK shall be equal to the complement KD

BOOK T. PROP. XLni, XLIV. SI

B G C

Because ABCD is a parallelogram, and ^Cits diameter,

therefore the triangle ABCi& equal to the triangle ADC. (l. 34.)

Again, because EKHA is a parallelogram, and AK its diameter,

therefore the triangle AJEK is equal to the triangle AHK; (l. 34.)

and for the same reason, the triangle KG Cis equal to the triangle KFC,

, Wherefore the two triangles AEK, KGC are equal to the two

triangles AHK, KFC, (ax. 2.)

but the whole triangle ABCis equal to the whole triangle ADC]

therefore the remaining complement JBK is equal to the remaining

complement KD. (ax. 3.)

Wherefore the complements, &c. Q.e.d.

PROPOSITION XLIV. PROBLEM.

To a given straight line to apply a parallelogram^ which shall he equal

to a given triangle^ and have owe of its angles equal to a given rectilineal

angle.

Let ABhe the given straight line, and Cthe given triangle, and D

the given rectilineal angle.

It is required to apply to the straight line AB, a, parallelogram

equal to the triangle C, and having an angle equal to the angle D.

F E K

k

Make the parallelogram BEFG equal to the triangle C,

and having the angle EBG equal to the angle D, (i. 42.)

so that BE be in the same straight line with AB ;

produce FG to II,

through A draw ^ J/ parallel to BG or EF, (l. 31.) and join HB.

Then because the straight line ^jP falls upon the parallels AH, EF,

therefore the angles AHF, HFE are together equal to two right

angles ; (l. 29.)

wherefore the angles BIIF, HFE are less than two right angles :

but straight lines which with another straight line, make the two

interior angles upon the same side less than two right angles, do meet

if produced far enough : (ax. 12.)

therefore HB, FE shall meet if produced ;

let them be produced and meet in K,

through K di'aw KL parallel to EA or FH,

and produce HA, GB to meet KL in the points L, 31.

Then HLKFh a parallelogram, of which the diameter is HK)

and AG, ME, are the parallelograms about UK;

also LB, BF are the complements ;

therefore the complement LB is equal to the complement BF; (l. 43.)

but the complement BF is equal to the triangle C; (constr.)

wherefore LB is equal to the triangle C.

And because the angle GBE is equal to the angle AB3I, (l. 15.)

and likewise to the angle D ; (constr.)

therefore the angle AB 31 is equal to the angle D. (ax. 1.)

Therefore to the given straight line AB, the parallelogram LB has

been applied, equal to the triangle C, and having the angle AB3I

equal to the given angle D. q.e.f.

PROPOSITION XLV. PROBLEM.

To describe a parallelogram equal to a given rectilineal figure^ and

having an angle equal to a given rectilineal angle.

Let ABCD be the given rectilineal figure, and F the given recti-

lineal angle.

It is required to describe a parallelogram that shall be equal to the

figure ABCD, and having an angle equal to the given angle F.

D F G L

H M

Join DB.

Describe the parallelogram FH equal to the triangle ABB, and

having the angle i^^JET equal to the angle F; (l. 42.)

to the straight line GIT, apply the parallelogram G3I equal to the

triangle DB C, having the angle GHM equal to the angle E.

(I. 44.)

Then the figure FK3IL shall be the parallelogram required.

Because each of the angles FKH, GHM, is equal to the angle E,

therefore the angle FKH\& equal to the angle GII3I;

add to each of these equals the angle KHG ;

therefore the angles FKH, KHG are equal to the angles KHG, GHM;

but FKH, KHG are equal to two right angles ; (i. 29.)

therefore also KHG, GH31 are equal to two right angles ;

and because at the point H, in the straight line GH, the two

straight lines KH, H3I, upon the opposite sides of it, make the ad-

jacent angles KHG, GHM equal to two right angles,

therefore HK is in the same straight line with H3L (l. 14.)

And because the line HG meets the parallels KM, FG,

therefore the angle 3IHG is equal to the alternate angle J^G^jP; (l. 29.)

add to each of these equals the angle HGL ;

therefore the angles 3IHG, HGL are equal to the angles HGF, HGL;

but the angles 3IHG, HGL are equal to two right angles ; (l. 29.)

therefore also the angles HGF, HGL are equal to two lignt angles,

and therefore FG is in the same straight line with GL, (l. 14.)

I^I^^HP BOOK I. PROF. XLV; XLVI. 39

|H| And because KFh parallel to HG, and HG to ML,

VK therefore KFh parallel to ML ; (I. 30.)

I^H and FL has been proved parallel to K3Â£,

|H| wherefore the figure FKML is a parallelogram ;

I^E and since the parallelogram HF is equal to the triangle ABD,

and the parallelogram GM to the triangle BDC;

therefore the whole parallelogram KFLM is equal to the whole

rectilineal figui-e A BCD.

Therefore the parallelogram KFLM has been described equal to

the given rectilineal figure A BCD, having the angle FKM equal to

the given angle E. q.e.f.

Cor. From this it is manifest how, to a given straight line, to apply

a parallelogram, which shall have an angle equal to a given rectilineal

angle, and shall be equal to a given rectilineal figure ; viz. by applying

to the given straight line a parallelogram equal to the first triangle

ABD, (l. 44.) and having an angle equal to the given angle.

PROPOSITION XLVI. PROBLEM.

To describe a square upon a given straight line.

Let AB be the given straight line.

I

It is required to describe a square upon AB.

From the point A di-aw ^Cat right angles to AB; (l. 11.)

make AD equal to AB-, (l. 3.)

through the point D draw DE parallel to AB; (i. 31.)

and through B, draw BE parallel to AD, meeting DE in E;

therefore ABED is a parallelogram ;

whence AB is equal to DE, and AD to BE; (l. 34.)

but AD is equal to AB,

therefore the four lines AB, BE, ED, DA are equal to one another,

and the parallelogram ABED is equilateral.

It has likewise all its angles right angles ;

since AD meets the parallels AB, DE,

therefore the angles BAD, ADEaie equal to two right angles ; (l.29.)

but BAD is a right angle ; (constr.)

therefore also ADE is a right angle.

But the opposite angles of parallelograms are equal ; (l. 34.)

therefore each of the opposite angles ABE, BED is a right angle ;

wherefore the figure ABED is rectangular,

and it has been proved to be equilateral ;

therefore the figure ABED is a square, (def. 30.)

and it is described upon the given straight line AB. Q.E.F,

40

EUCLID S ELEMENTS.

Cor. Hence, every parallelogram that has one ofits angles a right

angle, has all its angles right angles.

PROPOSITION XLVII. THEOREM.

In any right-angled triangle, the square which is described vpon the side

subtending the right angle, is equal to the squares described upon the sides

which contain the right angle.

Let ABC he a right-angled triangle, having the right angle BAC.

Then the square described upon the side B C, shall be equal to the

squares described upon BA, AC.

G

On ^C describe the square BDEC, (l. 46.)

and on BA, ^ C the squares GB, HC;

through A di-aw AL parallel to BJD or CH; (l. 31.)

and join AD, FC.

Then because the angle ^^Cis a right angle, (h)^.)

and that the angle BA 6^ is a right angle, (def. 30.)

the two straight lines AC, AG upon the opposite sides of AB, make

with it at the point A, the adjacent angles equal to two right angles;

therefore CA is in the same straight line with A G. (l. 14.)

For the same reason, BA and AH are in the same straight line.

And because the angle DBCis equal to the angle FBA,

each of them being a right angle,

add to each of these equals the angle ABC,

therefore the whole angle ABD is equal to the whole angle FBC. (ax. 2.)

And because the two sides AB, BD, are equal to the two sides FB,

BC, each to each, and the included angle -4^ J9 is equal to the included

angle FBC,

therefore the base AD is equal to the base FC, (l. 4.)

and the triangle ABD to the triangle FBC.

Now the parallelogram BL is double of the triangle ABD, {l. 41.)

because they are upon the same base BD, and between tne same

parallels BD, AL-, ^

also the square GB is double of the triangle FBC,

because these also are upon the same base FB, and between the

same parallels FB, GC.

But the doubles of equals are equal to one another ; (ax. 6.)

therefore the parallelogram BL is equal to the square GB.

Similarly, by joining AE, BK, it can be proved,

that the j)arallelogram CL is equal to the square JIC.

BOOK 1.

PROP. XLVUl.

41

Therefore the whole square BDEC is equal to the two squares GB,

IIC; (ax. 2.)

and the square BDBCis described upon the straight line BC,

and the squares GB, HC, upon AB, AC:

therefore the square upon the side BC, is equal to the squares upon

the sides AB, AC.

Therefore, in any right-angled triangle, &c. q.e.d.

PROPOSITION XLVIII. THEOREM.

Jf the square described upon one of the sides of a triangle, be equal to

the squares described upon the other two sides of it; the angle contained by

these two sides is a right angle.

Let the square described upon BC, one of the sides of the triangle

ABC, be equal to the squares upon the other two sides, AB, AC.

Then the angle BA C shall be a right angle.

D

From the point A draw AD at right angles to A C, (l. 11.)

make AD equal to AB, and join DC.

Then, because AD is equal to AB,

the square on ^Z) is equal to the square on AB',

to each of these equals add the square on ^C;

therefore the squares on ^Z>, ^ Care equal to the squares onAB,A C:

but the squares on AD, A C are equal to the square on DC, (i. 47.)

because the angle D AC is a right angle ;

and the square on B C, by hypothesis, is equal to the squares on BA,A C;

therefore the square on DC is equal to the square on J5C;

and therefore the side DC is equal to the side B C.

And because the side AD is equal to the side AB,

and ^Cis common to the two triangles DAC, BAC;

the two sides DA, A C, are equal to the two BA, AC, each to each ;

and the base DC has been proved to be equal to the base BC;

therefore the angle DA C is equal to the angle BA Cj (i. 8.)

but DA C is a right angle ;

therefore also BA C is a right angle.

Therefore, if the square described upon, &c. Q.E.D.

NOTES TO BOOK I.

ON THE DEFINITIONS.

Geometry is one of the most perfect of the deductive Sciences, and

seems to rest on the simplest inductions from experience and observation.

The first principles of Geometry are therefore in this view consistent

hypotheses founded on facts cognizable by the senses, and it is a subject

of primary importance to draw a distinction between the conception of

things and the things themselves. These hypotheses do not involve any

property contrary to the real nature of the things, and consequently cannot

be regarded as arbitrary, but in certain respects, agree with the concep-

tions which the things themselves suggest to the mind through the

medium of the senses. The essential definitions of Geometry therefore

being inductions from observation and experience, rest ultimately on the

evidence of the senses.

It is by experience we become acquainted with the existence of indi-

vidual forms of magnitudes ; but by the mental process of abstraction,

which begins with a particular instance, and proceeds to the general

idea of all objects of the same kind, we attain to the general conception

of those forms which come under the same general idea.

The essential definitions of Geometry express generalized conceptions

of real existences in their most perfect ideal forms : the laws and appear-

ances of nature, and the operations of the human intellect being sup-

posed uniform and consistent.

But in cases where the subject falls under the class of simple ideas,

the terms of the definitions so called, are no more than merely equivalent

expressions. The simple idea described by a proper term or terms, does

not in fact admit of definition properly so called. The definitions in

Euclid's Elements may be divided into two classes, those which merely

explain the meaning of the terms employed, and those, which, besides

explaining the meaning of the terms, suppose the existence of the things

described in the definitions.

Definitions in Geometry cannot be of such a form as to explain the

nature and properties of the figures defined : it is sufficient that they give

marks whereby the thing defined may be distinguished from every other

of the same kind. It will at once be obvious, that the definitions of

Geometry, one of the pure sciences, being abstractions of space, are not

like the definitions in anj' one of the physical sciences. The discovery

of any new physical facts may render necessary some alteration or modi-

fication in the definitions of the latter.

Def. I. Simson has adopted Theon's definition of a point. Euclid's

definition is, o-jj/xcToi/ e'o-tji/ ou /ut'pos oCosv, â™¦' A point is that, of which there

is no part," or which cannot be parted or divided, as it is explained by

Proclus. The Greek term o-jj/xtloi/, literally means, a visible siff7i or mark

on a surface, in other words, a physical point. The English term point,

means the sharp end of any thing, or a mark made by it. The word

point comes from the Latin punctiim, through the French word point.

Neither of these terms, in its literal sense, appears to give a very exact

notion of what is to be understood by a point in Geometry. Euclid's

definition of a point merely expresses a negative property, which excludes

the proper and literal meaning of the Greek term, as applied to denote a

physical point, or a mark which is visible to the senses.

Pythagoras defined a point to be novd<i dsa-iv Ixovaa, '* a monad having

position." By uniting the positive idea of position, with the negative

idea of defect of magnitude, the conception of a point in Geometry may

NOTES TO BOOK I. 4o

be rendered perhaps more intelligible- A point is defined to be that

which has no magnitude, but position only.

Def. II. Every visible line has both length and breadth, and it is im-

possible to draw any line whatever which shall have no breadth. The

definition requires the conception of the length only of the line to be

considered, abstracted from, and independently of, all idea of its breadth.

Def. III. This definition renders more intelligible the exact meaning

of the definition of a point : and we may add, that, in the Elements,

Euclid supposes that the intersection of two lines is a point, and that two

lines can intersect each other in one point only.

Def. IV. The straight line or right line is a term so clear and intel-

ligible as to be incapable of becoming more so by formal definition.

Euclid's definition is Eudala ypafxixn fo-nv, T/Vts e'^ ta-ov ToTs Â£^' iavTij^

arrifxtLOLi KEiTat, wherein he states it to lie evenli/, or equally/, or upon an

equality (t^ "i-o-ov) between its extremities, and which Proclus explains as

being stretched between its extremities, tj tV' dKpwv TtTafxivt],

If the line be conceived to be drawn on a plane surface, the words

f 5 io'ou may mean, that no part of the line which is called a straight line

deviates either from one side or the other of the direction which is fixed

by the extremities of the line ; and thus it may be distinguished from a

curved line, which does not lie, in this sense, evenly between its extreme

points. If the line be conceived to be drawn in space, the words i^ taov,

must be understood to apply to every direction on every side of the line

between its extremities.

Every straight line situated in a plane, is considered to have two sides ;

and when the direction of a line is known, the line is said to be given in

position ; also, when the length is known or can be found, it is said to be

given in magnitude.

From the definition of a straight line, it follows, that two points fix a

straight line in position, which is the foundation of the first and second

postulates. Hence straight lines which are proved tocoincideintwoormore

points, are called, "one and the same straight line," Prop. 14, Book i,

or, which is the same thing, that " two straight lines cannot have a

common segment," as Simson shews in his Corollary to Prop. 11, Book i.

The following definition of straight lines has also been proposed.

*' Straight lines are those which, if they coincide in any two points, coin-

cide as far as they are produced." But this is rather a criterion of straight

lines, and analogous to the eleventh axiom, which states that, *' all right

angles are equal to one another," and suggests that all straight lines may

be made to coincide wholly, if the lines be equal ; or partially, if the lines

be of unequal lengths. A definition should properly be restricted to the

description of the thing defined, as it exists, independently of any com-

parison of its properties or of tacitly assuming the existence of axioms.

Def. VII, Euclid's definition of a plane surface is 'E-TrtTrsSos sTrKpa-

VEid icrriv ffxis e'J 1(tov tol^ i(p' gauTt)? evdtiai^ KfiTai, *' A plane surface is

that which lies evenly or equally with the straight lines in it ;" instead

of which Simson has given the definition which was originally proposed

by Hero the Elder. A plane superficies may be supposed to be situated

in any position, and to be continued in every direction to any extent.

Def. viii. Simson remarks that this definition seems to include the

angles formed by two curved lines, or a curve and a straight line, as well

as that formed by two straight lines.

Angles made by straight lines only, are treated of in Elementary

Geometry.

44

ELEMENTS.

Def. IX. It is of the highest importance to attain a clear conceptioi

of an angle, and of the sum and difference of two angles. The litera

meaning of the term angtilus suggests the Geometrical conception of ai

angle, which may be regarded as formed by the divergence of two straigh

lines from a point. In the definition of an angle, the magnitude of thi

angle is independent of the lengths of the two lines by which it ii

included ; their mutual divergence from the point at which they meet, ii

the criterion of the magnitude of an angle, as it is pointed out in tin

succeeding definitions. The point at which the two lines meet is callec

the angular point or the vertex of the angle, and must not be confounde(

with the magnitude of the angle itself. The right angle is fixed in mag

nitude, and, on this account, it is made the standard with which a.'

other angles are compared.

Two straight lines which actually intersect one another, or whic

when produced would intersect, are said to be inclined to one another)

and the inclination of the two lines is determined by the angle whicl

they make with one another.

Def. X. It may be here observed that in the Elements, Euclid alwayi

assumes that when one line is perpendicular to another line, the latter ii

also perpendicular to the former ; and always calls a right angle, opQx

yuivia ; but a straight line, tvQila ypafxfjLt].

Def. XIX. This has been restored from Proclus, as it seems to have i

meaning in the construction of Prop. 14, Book ii ; the first case of Prop

33, Book III, and Prop. 13, Book vi. The definition of the segment of i

circle is not once alluded to in Book i, and is not required before the dis-

cussion of the properties of the circle in Book iii. Proclus remarks oi

this definition : *' Hence you may collect that the center has three places

and they are equal to one another, (l. 36.)

because they are upon equal bases BC, EF,

and between the same parallels BF, GH.

And because the diameter AB bisects the parallelogram GBCA,

therefore the triangle ABC is. the half of the parallelogram GBCA ;

(I. 34.)

also, because the diameter Di^ bisects the parallelogram DEFH,

therefore the triangle DEF is the half of the parallelogram DEFH',

but the halves of equal things are equal ; (ax. 7.)

therefore the triangle ABC\& equal to the triangle DEF.

"Wherefore, triangles upon equal bases, &c. Q. E. D.

PROPOSITION XXXIX. THEOREM.

Equal triangles zcpon the same base and upon the same side of it, are

between the same parallels.

Let the equal triangles ABC, DBC be upon the same base BC,

and upon the same side of it.

Then the triangles ABC, Z)jBC shall be between the same parallels.

:v

Join AD ; then AD shall be parallel to BC.

For if ^Z) be not parallel to BC,

if possible, through the point A, draw AE parallel to BC, (l. 31.)

meeting BD, or BD produced, in E, and join EC.

Then the triangle ABC is equal to the triangle EBC, (l. 37.)

because they are upon the same base BC,

and between the same parallels BC, AE:

but the triangle ABC is equal to the triangle DBC; (h}T5.)

therefore the triangle DBC is equal to the triangle EBC,

I

BOOK I. PROP. XL, XLI.

the greater triangle equal to the less, which is impossible :

therefore AJ3 is not parallel to JBC.

In the same manner it can be demonstrated,

that no other line drawn from A but AD is parallel to JBC;

AD is therefore parallel to BC.

Wherefore, equal triangles upon, &c. Q. E. D.

PROPOSITION XL. THEOREM.

Equal triangles upon equal bases in the same straight line, and towards

the same parts, are between the same parallels.

Let the equal triangles ABC, DJEFhe upon equal bases BC, EF,

in the same straight line BF, and towards the same parts.

Then they shall be between the same parallels.

A D

Join AD ; then AD shall be parallel to BF.

For if ^Z) be not parallel to BF,

if possible, through A draw AG parallel to BF, (l. 31.)

meeting ED, or ED produced in G, and join GF.

Then the triangle ABCis equal to the triangle GEF, (i. 38.)

because they are upon equal bases BC, EF,

and between the same parallels BF, A G ;

but the triangle ^^Cis equal to the triangle DEF; (hyp.)

therefore the triangle DEFh equal to the triangle GEF, (ax. 1.)

the greater triangle equal to the less, which is impossible :

therefore AG is not parallel to BF.

And in the same manner it can be demonstrated,

that there is no other line drawn from A parallel to it but AD ;

AD is therefore parallel to BF.

Wherefore, equal triangles upon, &c. Q. E. D.

PROPOSITION XLI. THEOREM.

If a parallelogram, and a triangle be upon the same base, and beizceen

the same parallels ; the parallelogram shall be double of the triangle.

Let the parallelogram ABCD, and the triangle EBC be upon the

same base BC, and between the same parallels BC, AE.

Then the parallelogram AB CD shall be double of the triangle EB C.

D E

I

B

Join A C.

Then the triangle ABCis equal to the triangle EBC, (l. 37.)

ELEMENTS.

because they are upon the same base BC, and between the same

parallels BC, AJS.

But the parallelogram ABCD is double of the triangle ABC,

because the diameter ^C bisects it; (l. 34.)

wherefore ABCD is also double of the triangle EBC.

Therefore, if a parallelogram and a triangle, &c. Q.E.D.

PROPOSITION XLII. PROBLEM.

To describe a parallelogram that shall be equal to a given triangle^ and

have one of its angles equal to a given rectilineal angle.

Let ABC he the given triangle, and D the given rectilineal angle.

It is required to describe a parallelogram that sliall be equal to the

given triangle ABC, and have one of its angles equal to D,

A F G

B E c

Bisect ^Cin E, (l. 10.) and join AE-,

at the point E in the straight line EC,

make the angle C^i^ equal to the angle D; (I. 23.)

through C draw CG parallel to EF, and through A draw AF

parallel to BC, (l. 3L) meeting ^i^in F, and CG in G.

Then the figure CEFG is a parallelogram, (def. A.)

And because the triangles ABE, AEC are on the equal bases Bj

EC, and between the same parallels BC, AG;

they are therefore equal to one another ; (l. 38.)

and the triangle ABC is double of the triangle AEC;

but the parallelogram FECG is double of the triangle AEC, (l. 41

because they are upon the same base EC, and between the sam

parallels ^C,^(?;

therefore the parallelogram FECG is equal to the triangle ABC, (ax. 6

and it has one of its angles CEF equal to the given angle Z).

"Wherefore, a parallelogram FECG has been described equal toth

given triangle ABC, and having one of its angles CEF equal to th

given angle E. Q. E. f.

PROPOSITION XLIII. THEOREM.

The complements of the parallelograms, which are about the diamet

of any parallelogram, are equal to one another.

Let ABCD be a parallelogram, of which the diameter is AC', a:

EII,GF the parallelograms about ^ C, that is, throuf/h which A Cjjassa

also BK, KD the other parallelograms which make up the who!

figure ABCD, which are therefore called the complements.

Then the complwnent BK shall be equal to the complement KD

BOOK T. PROP. XLni, XLIV. SI

B G C

Because ABCD is a parallelogram, and ^Cits diameter,

therefore the triangle ABCi& equal to the triangle ADC. (l. 34.)

Again, because EKHA is a parallelogram, and AK its diameter,

therefore the triangle AJEK is equal to the triangle AHK; (l. 34.)

and for the same reason, the triangle KG Cis equal to the triangle KFC,

, Wherefore the two triangles AEK, KGC are equal to the two

triangles AHK, KFC, (ax. 2.)

but the whole triangle ABCis equal to the whole triangle ADC]

therefore the remaining complement JBK is equal to the remaining

complement KD. (ax. 3.)

Wherefore the complements, &c. Q.e.d.

PROPOSITION XLIV. PROBLEM.

To a given straight line to apply a parallelogram^ which shall he equal

to a given triangle^ and have owe of its angles equal to a given rectilineal

angle.

Let ABhe the given straight line, and Cthe given triangle, and D

the given rectilineal angle.

It is required to apply to the straight line AB, a, parallelogram

equal to the triangle C, and having an angle equal to the angle D.

F E K

k

Make the parallelogram BEFG equal to the triangle C,

and having the angle EBG equal to the angle D, (i. 42.)

so that BE be in the same straight line with AB ;

produce FG to II,

through A draw ^ J/ parallel to BG or EF, (l. 31.) and join HB.

Then because the straight line ^jP falls upon the parallels AH, EF,

therefore the angles AHF, HFE are together equal to two right

angles ; (l. 29.)

wherefore the angles BIIF, HFE are less than two right angles :

but straight lines which with another straight line, make the two

interior angles upon the same side less than two right angles, do meet

if produced far enough : (ax. 12.)

therefore HB, FE shall meet if produced ;

let them be produced and meet in K,

through K di'aw KL parallel to EA or FH,

and produce HA, GB to meet KL in the points L, 31.

Then HLKFh a parallelogram, of which the diameter is HK)

and AG, ME, are the parallelograms about UK;

also LB, BF are the complements ;

therefore the complement LB is equal to the complement BF; (l. 43.)

but the complement BF is equal to the triangle C; (constr.)

wherefore LB is equal to the triangle C.

And because the angle GBE is equal to the angle AB3I, (l. 15.)

and likewise to the angle D ; (constr.)

therefore the angle AB 31 is equal to the angle D. (ax. 1.)

Therefore to the given straight line AB, the parallelogram LB has

been applied, equal to the triangle C, and having the angle AB3I

equal to the given angle D. q.e.f.

PROPOSITION XLV. PROBLEM.

To describe a parallelogram equal to a given rectilineal figure^ and

having an angle equal to a given rectilineal angle.

Let ABCD be the given rectilineal figure, and F the given recti-

lineal angle.

It is required to describe a parallelogram that shall be equal to the

figure ABCD, and having an angle equal to the given angle F.

D F G L

H M

Join DB.

Describe the parallelogram FH equal to the triangle ABB, and

having the angle i^^JET equal to the angle F; (l. 42.)

to the straight line GIT, apply the parallelogram G3I equal to the

triangle DB C, having the angle GHM equal to the angle E.

(I. 44.)

Then the figure FK3IL shall be the parallelogram required.

Because each of the angles FKH, GHM, is equal to the angle E,

therefore the angle FKH\& equal to the angle GII3I;

add to each of these equals the angle KHG ;

therefore the angles FKH, KHG are equal to the angles KHG, GHM;

but FKH, KHG are equal to two right angles ; (i. 29.)

therefore also KHG, GH31 are equal to two right angles ;

and because at the point H, in the straight line GH, the two

straight lines KH, H3I, upon the opposite sides of it, make the ad-

jacent angles KHG, GHM equal to two right angles,

therefore HK is in the same straight line with H3L (l. 14.)

And because the line HG meets the parallels KM, FG,

therefore the angle 3IHG is equal to the alternate angle J^G^jP; (l. 29.)

add to each of these equals the angle HGL ;

therefore the angles 3IHG, HGL are equal to the angles HGF, HGL;

but the angles 3IHG, HGL are equal to two right angles ; (l. 29.)

therefore also the angles HGF, HGL are equal to two lignt angles,

and therefore FG is in the same straight line with GL, (l. 14.)

I^I^^HP BOOK I. PROF. XLV; XLVI. 39

|H| And because KFh parallel to HG, and HG to ML,

VK therefore KFh parallel to ML ; (I. 30.)

I^H and FL has been proved parallel to K3Â£,

|H| wherefore the figure FKML is a parallelogram ;

I^E and since the parallelogram HF is equal to the triangle ABD,

and the parallelogram GM to the triangle BDC;

therefore the whole parallelogram KFLM is equal to the whole

rectilineal figui-e A BCD.

Therefore the parallelogram KFLM has been described equal to

the given rectilineal figure A BCD, having the angle FKM equal to

the given angle E. q.e.f.

Cor. From this it is manifest how, to a given straight line, to apply

a parallelogram, which shall have an angle equal to a given rectilineal

angle, and shall be equal to a given rectilineal figure ; viz. by applying

to the given straight line a parallelogram equal to the first triangle

ABD, (l. 44.) and having an angle equal to the given angle.

PROPOSITION XLVI. PROBLEM.

To describe a square upon a given straight line.

Let AB be the given straight line.

I

It is required to describe a square upon AB.

From the point A di-aw ^Cat right angles to AB; (l. 11.)

make AD equal to AB-, (l. 3.)

through the point D draw DE parallel to AB; (i. 31.)

and through B, draw BE parallel to AD, meeting DE in E;

therefore ABED is a parallelogram ;

whence AB is equal to DE, and AD to BE; (l. 34.)

but AD is equal to AB,

therefore the four lines AB, BE, ED, DA are equal to one another,

and the parallelogram ABED is equilateral.

It has likewise all its angles right angles ;

since AD meets the parallels AB, DE,

therefore the angles BAD, ADEaie equal to two right angles ; (l.29.)

but BAD is a right angle ; (constr.)

therefore also ADE is a right angle.

But the opposite angles of parallelograms are equal ; (l. 34.)

therefore each of the opposite angles ABE, BED is a right angle ;

wherefore the figure ABED is rectangular,

and it has been proved to be equilateral ;

therefore the figure ABED is a square, (def. 30.)

and it is described upon the given straight line AB. Q.E.F,

40

EUCLID S ELEMENTS.

Cor. Hence, every parallelogram that has one ofits angles a right

angle, has all its angles right angles.

PROPOSITION XLVII. THEOREM.

In any right-angled triangle, the square which is described vpon the side

subtending the right angle, is equal to the squares described upon the sides

which contain the right angle.

Let ABC he a right-angled triangle, having the right angle BAC.

Then the square described upon the side B C, shall be equal to the

squares described upon BA, AC.

G

On ^C describe the square BDEC, (l. 46.)

and on BA, ^ C the squares GB, HC;

through A di-aw AL parallel to BJD or CH; (l. 31.)

and join AD, FC.

Then because the angle ^^Cis a right angle, (h)^.)

and that the angle BA 6^ is a right angle, (def. 30.)

the two straight lines AC, AG upon the opposite sides of AB, make

with it at the point A, the adjacent angles equal to two right angles;

therefore CA is in the same straight line with A G. (l. 14.)

For the same reason, BA and AH are in the same straight line.

And because the angle DBCis equal to the angle FBA,

each of them being a right angle,

add to each of these equals the angle ABC,

therefore the whole angle ABD is equal to the whole angle FBC. (ax. 2.)

And because the two sides AB, BD, are equal to the two sides FB,

BC, each to each, and the included angle -4^ J9 is equal to the included

angle FBC,

therefore the base AD is equal to the base FC, (l. 4.)

and the triangle ABD to the triangle FBC.

Now the parallelogram BL is double of the triangle ABD, {l. 41.)

because they are upon the same base BD, and between tne same

parallels BD, AL-, ^

also the square GB is double of the triangle FBC,

because these also are upon the same base FB, and between the

same parallels FB, GC.

But the doubles of equals are equal to one another ; (ax. 6.)

therefore the parallelogram BL is equal to the square GB.

Similarly, by joining AE, BK, it can be proved,

that the j)arallelogram CL is equal to the square JIC.

BOOK 1.

PROP. XLVUl.

41

Therefore the whole square BDEC is equal to the two squares GB,

IIC; (ax. 2.)

and the square BDBCis described upon the straight line BC,

and the squares GB, HC, upon AB, AC:

therefore the square upon the side BC, is equal to the squares upon

the sides AB, AC.

Therefore, in any right-angled triangle, &c. q.e.d.

PROPOSITION XLVIII. THEOREM.

Jf the square described upon one of the sides of a triangle, be equal to

the squares described upon the other two sides of it; the angle contained by

these two sides is a right angle.

Let the square described upon BC, one of the sides of the triangle

ABC, be equal to the squares upon the other two sides, AB, AC.

Then the angle BA C shall be a right angle.

D

From the point A draw AD at right angles to A C, (l. 11.)

make AD equal to AB, and join DC.

Then, because AD is equal to AB,

the square on ^Z) is equal to the square on AB',

to each of these equals add the square on ^C;

therefore the squares on ^Z>, ^ Care equal to the squares onAB,A C:

but the squares on AD, A C are equal to the square on DC, (i. 47.)

because the angle D AC is a right angle ;

and the square on B C, by hypothesis, is equal to the squares on BA,A C;

therefore the square on DC is equal to the square on J5C;

and therefore the side DC is equal to the side B C.

And because the side AD is equal to the side AB,

and ^Cis common to the two triangles DAC, BAC;

the two sides DA, A C, are equal to the two BA, AC, each to each ;

and the base DC has been proved to be equal to the base BC;

therefore the angle DA C is equal to the angle BA Cj (i. 8.)

but DA C is a right angle ;

therefore also BA C is a right angle.

Therefore, if the square described upon, &c. Q.E.D.

NOTES TO BOOK I.

ON THE DEFINITIONS.

Geometry is one of the most perfect of the deductive Sciences, and

seems to rest on the simplest inductions from experience and observation.

The first principles of Geometry are therefore in this view consistent

hypotheses founded on facts cognizable by the senses, and it is a subject

of primary importance to draw a distinction between the conception of

things and the things themselves. These hypotheses do not involve any

property contrary to the real nature of the things, and consequently cannot

be regarded as arbitrary, but in certain respects, agree with the concep-

tions which the things themselves suggest to the mind through the

medium of the senses. The essential definitions of Geometry therefore

being inductions from observation and experience, rest ultimately on the

evidence of the senses.

It is by experience we become acquainted with the existence of indi-

vidual forms of magnitudes ; but by the mental process of abstraction,

which begins with a particular instance, and proceeds to the general

idea of all objects of the same kind, we attain to the general conception

of those forms which come under the same general idea.

The essential definitions of Geometry express generalized conceptions

of real existences in their most perfect ideal forms : the laws and appear-

ances of nature, and the operations of the human intellect being sup-

posed uniform and consistent.

But in cases where the subject falls under the class of simple ideas,

the terms of the definitions so called, are no more than merely equivalent

expressions. The simple idea described by a proper term or terms, does

not in fact admit of definition properly so called. The definitions in

Euclid's Elements may be divided into two classes, those which merely

explain the meaning of the terms employed, and those, which, besides

explaining the meaning of the terms, suppose the existence of the things

described in the definitions.

Definitions in Geometry cannot be of such a form as to explain the

nature and properties of the figures defined : it is sufficient that they give

marks whereby the thing defined may be distinguished from every other

of the same kind. It will at once be obvious, that the definitions of

Geometry, one of the pure sciences, being abstractions of space, are not

like the definitions in anj' one of the physical sciences. The discovery

of any new physical facts may render necessary some alteration or modi-

fication in the definitions of the latter.

Def. I. Simson has adopted Theon's definition of a point. Euclid's

definition is, o-jj/xcToi/ e'o-tji/ ou /ut'pos oCosv, â™¦' A point is that, of which there

is no part," or which cannot be parted or divided, as it is explained by

Proclus. The Greek term o-jj/xtloi/, literally means, a visible siff7i or mark

on a surface, in other words, a physical point. The English term point,

means the sharp end of any thing, or a mark made by it. The word

point comes from the Latin punctiim, through the French word point.

Neither of these terms, in its literal sense, appears to give a very exact

notion of what is to be understood by a point in Geometry. Euclid's

definition of a point merely expresses a negative property, which excludes

the proper and literal meaning of the Greek term, as applied to denote a

physical point, or a mark which is visible to the senses.

Pythagoras defined a point to be novd<i dsa-iv Ixovaa, '* a monad having

position." By uniting the positive idea of position, with the negative

idea of defect of magnitude, the conception of a point in Geometry may

NOTES TO BOOK I. 4o

be rendered perhaps more intelligible- A point is defined to be that

which has no magnitude, but position only.

Def. II. Every visible line has both length and breadth, and it is im-

possible to draw any line whatever which shall have no breadth. The

definition requires the conception of the length only of the line to be

considered, abstracted from, and independently of, all idea of its breadth.

Def. III. This definition renders more intelligible the exact meaning

of the definition of a point : and we may add, that, in the Elements,

Euclid supposes that the intersection of two lines is a point, and that two

lines can intersect each other in one point only.

Def. IV. The straight line or right line is a term so clear and intel-

ligible as to be incapable of becoming more so by formal definition.

Euclid's definition is Eudala ypafxixn fo-nv, T/Vts e'^ ta-ov ToTs Â£^' iavTij^

arrifxtLOLi KEiTat, wherein he states it to lie evenli/, or equally/, or upon an

equality (t^ "i-o-ov) between its extremities, and which Proclus explains as

being stretched between its extremities, tj tV' dKpwv TtTafxivt],

If the line be conceived to be drawn on a plane surface, the words

f 5 io'ou may mean, that no part of the line which is called a straight line

deviates either from one side or the other of the direction which is fixed

by the extremities of the line ; and thus it may be distinguished from a

curved line, which does not lie, in this sense, evenly between its extreme

points. If the line be conceived to be drawn in space, the words i^ taov,

must be understood to apply to every direction on every side of the line

between its extremities.

Every straight line situated in a plane, is considered to have two sides ;

and when the direction of a line is known, the line is said to be given in

position ; also, when the length is known or can be found, it is said to be

given in magnitude.

From the definition of a straight line, it follows, that two points fix a

straight line in position, which is the foundation of the first and second

postulates. Hence straight lines which are proved tocoincideintwoormore

points, are called, "one and the same straight line," Prop. 14, Book i,

or, which is the same thing, that " two straight lines cannot have a

common segment," as Simson shews in his Corollary to Prop. 11, Book i.

The following definition of straight lines has also been proposed.

*' Straight lines are those which, if they coincide in any two points, coin-

cide as far as they are produced." But this is rather a criterion of straight

lines, and analogous to the eleventh axiom, which states that, *' all right

angles are equal to one another," and suggests that all straight lines may

be made to coincide wholly, if the lines be equal ; or partially, if the lines

be of unequal lengths. A definition should properly be restricted to the

description of the thing defined, as it exists, independently of any com-

parison of its properties or of tacitly assuming the existence of axioms.

Def. VII, Euclid's definition of a plane surface is 'E-TrtTrsSos sTrKpa-

VEid icrriv ffxis e'J 1(tov tol^ i(p' gauTt)? evdtiai^ KfiTai, *' A plane surface is

that which lies evenly or equally with the straight lines in it ;" instead

of which Simson has given the definition which was originally proposed

by Hero the Elder. A plane superficies may be supposed to be situated

in any position, and to be continued in every direction to any extent.

Def. viii. Simson remarks that this definition seems to include the

angles formed by two curved lines, or a curve and a straight line, as well

as that formed by two straight lines.

Angles made by straight lines only, are treated of in Elementary

Geometry.

44

ELEMENTS.

Def. IX. It is of the highest importance to attain a clear conceptioi

of an angle, and of the sum and difference of two angles. The litera

meaning of the term angtilus suggests the Geometrical conception of ai

angle, which may be regarded as formed by the divergence of two straigh

lines from a point. In the definition of an angle, the magnitude of thi

angle is independent of the lengths of the two lines by which it ii

included ; their mutual divergence from the point at which they meet, ii

the criterion of the magnitude of an angle, as it is pointed out in tin

succeeding definitions. The point at which the two lines meet is callec

the angular point or the vertex of the angle, and must not be confounde(

with the magnitude of the angle itself. The right angle is fixed in mag

nitude, and, on this account, it is made the standard with which a.'

other angles are compared.

Two straight lines which actually intersect one another, or whic

when produced would intersect, are said to be inclined to one another)

and the inclination of the two lines is determined by the angle whicl

they make with one another.

Def. X. It may be here observed that in the Elements, Euclid alwayi

assumes that when one line is perpendicular to another line, the latter ii

also perpendicular to the former ; and always calls a right angle, opQx

yuivia ; but a straight line, tvQila ypafxfjLt].

Def. XIX. This has been restored from Proclus, as it seems to have i

meaning in the construction of Prop. 14, Book ii ; the first case of Prop

33, Book III, and Prop. 13, Book vi. The definition of the segment of i

circle is not once alluded to in Book i, and is not required before the dis-

cussion of the properties of the circle in Book iii. Proclus remarks oi

this definition : *' Hence you may collect that the center has three places

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38