*preliminary draft for student use only. not for citation...

Sci. Rev. Reader ('02/02/01) 2-P2_Euclid

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*Preliminary draft for student use only. Not for citation or circulation without permission of editor.

[Hobbes] was 40 years old before he looked on Geometry. . . . Being in a

Gentleman’s Library, Euclid’s Elements lay open, and ‘twas [the Pythagorean

theorem (Book I, Proposition 47)]. He read the Proposition. By God, sayd he, . .

. this is impossible! So he reads the Demonstration of it, which referred him back

to such a Proposition; which Proposition he read. That referred him back to

another, which he also read. Et sic deinceps [and so on] that at last he was

demonstratively convinced of the truth. This made him in love with Geometry.

- John Aubrey, Brief Lives

Euclid (fl. 300 B.C.E.), "Elements" (c. 300 B.C.E.)1

BOOK ONE

DEFINITIONS

1. A point is that which has no part.

2. A line is breadthless length.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on it-self.

8. A plane angle is the inclination to one another of two lines in a plane which

meet one another and do not lie in a straight line.

1 Euclid. “The Thirteen Books of Euclid’s Elements,” Great Books of the Western World ,vol. II, Sir Thomas Heath, trans. Chicago: Encyclopaedia Britannica, Inc., 1952. Pp. 1-2, 4, 9-10, 24, 27-28, 74-75, 123, 375, 395-396.


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9. And when the lines containing the angle are straight, the angle is called

rectilineal.

10. When a straight line set up on a straight line makes the adjacent angles

equal to one another, each of the equal angles is right and the straight line

standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle.

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.

14. A figure is that which is contained by any boundary or boundaries.

15. A circle is a plane figure contained by one line such that all the straight lines

falling upon it from one point among those lying within the figure are equal

to one another;

16. And the point is called the center of the circle.

17. A diameter of the circle is any straight line drawn through the center and

terminated in both directions by the circumference of the circle, and such a

straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference

cut off by it. And the center of the semicircle is the same as that of the

circle.

19. Rectilineal figures are those which are contained by straight lines, trilateral

figures being those contained by three, quadrilateral those contained by

four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides

equal, an isosceles triangle that which has two of its sides alone equal,

and a scalene triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right

angle, an obtuse-angled triangle that which has an obtuse angle, and an

acute-angled triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and right-

angled; an oblong that which is right-angled but not equilateral; a rhombus


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that which is equilateral but not right-angled; and a rhomboid that which

has its opposite sides and angles equal to one another hut is neither

equilateral nor right-angled. And let quadrilaterals other than these be

called trapezia.

23. Parallel straight lines are straight lines which, being in the same plane and

being produced indefinitely in both directions, do not meet one another in

either direction.

POSTULATES

Let the following be postulated:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and distance [radius].

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make the interior angles on

the same side less than two right angles, the two straight lines, if produced

indefinitely, meet on that side on which are the angles less than the two

right angles.

COMMON NOTIONS

1. Things which are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

(7) 4. Things which coincide with one another are equal to one another.

(8) 5. The whole is greater than the part.

BOOK I, Proposition 1 [N.B.: needs diagram]

On a given finite straight line to construct an equilateral triangle.

Let AB be the given finite straight line. Thus it is required to construct an

equilateral triangle on the straight line AB. With centre A and distance AB let the


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circle BCD be described; [Post. 3] again, with centre B and distance BA let the

circle ACE be described; [Post. 3] and from the point C, in which the circles cut

one another, to the points A, B let the straight lines CA, CR be joined. [Post. 1]

Now, since the point A is the centre of the circle CDB,

AC is equal to AB. [Def. 15]

Again, since the point B is the centre of the circle CAE,

BC is equal to BA. [Def. 15]

But CA was also proved equal to AB; therefore each of the straight lines

CA, CB is equal to AB. And things which are equal to the same thing are also

equal to one another; therefore CA is also equal to CR. [C. N. 1] Therefore the

three straight lines CA, AB, BC are equal to one another. Therefore the triangle

ABC is equilateral; and it has been constructed on the given finite straight line

AB. (Being) what it was required to do.

BOOK I, PROPOSITION 4

If two triangles have the two sides equal to two sides respectively, and

have the angles contained by the equal straight lines equal, they will also have

the base equal to the base, the triangle will be equal to the triangle, and the

remaining angles will be equal to the remaining angles respectively, namely

those which the equal sides subtend.

Let ABC, DEF be two triangles having the two sides AB, AC equal to the

two sides DE, DF respectively, namely AB to DE and AC to DF, and the angle

BAC equal to the angle EDF.


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Fig. 1.2.1) Book I, Proposition 4

I say that the base BC is also equal to the base EF, the triangle ABC will

be equal to the triangle DEF, and the remaining angles will be equal to the re-

maining angles respectively, namely those which the equal sides subtend, that is,

the angle ABC to the angle DEF, and the angle ACB to the angle DFE. For, if

the triangle ABC be applied to the triangle DEF, and if the point A be placed on

the point D and the straight line AB on DE, then the point B will also coincide with

E, because AB is equal to DE. Again, AB coinciding with DE, the straight line AC

will also coincide with DF, because the angle BAC is equal to the angle EDF;

hence the point C will also coincide with the point F, because AC is again equal

to DF. But B also coincided with E; hence the base BC will coincide with the base

EF. For if, when B coincides with E and C with F, the base BC does not coincide

with the base EF, two straight lines will enclose a space: which is impossible.

Therefore the base BC will coincide wnd will be equal to it. (Common Notion 4)

Thus the whole triangle ABC will coincide with the whole triangle DEF, and will

be equal to it. And the remaining angles will also coincide with the remaining

angles and will be equal to them, the angle ABC to the angle DEF, and the angle

ACB to the angle DFE. Therefore etc. (Being) what it was required to prove.


If with any straight line, and at a point on it, two straight lines not lying on

the same side make the adjacent angles equal to two right angles, the two

straight lines will be in a straight line with one another.


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For with any straight line AB, and at the point B on it, let the two straight

lines BC, BD not lying on the same side make the adjacent angles ABC, ABD

equal to two right angles;


I say that BD is in a straight line with CB. For, if BD is not in a straight line

with BC, let BE be in a straight line with CB. Then, since the straight line AB

stands on the straight line CBE, the angles ABC, ABE are equal to two right

angles. (Bk. I, Prop. 13) But the angles ABC, ABD are also equal to two right

angles; therefore the angles CBA, ABE are equal to the angles CBA, ABD.

(Postulate 4 and Common Notion 1) Let the angle CBA be subtracted from each;

therefore the remaining angle ABE is equal to the remaining angle ABD,

(Common Notion 3) the less to the greater: which is impossible. Therefore BE is

not in a straight line with CB. Similarly we can prove that neither is any other

straight line except BD. Therefore CB is in a straight line with BD. Therefore etc.

Q. E. D. [Quod erat demonstrandum, Latin for ‘which was to be demonstrated’.]

BOOK I, PROPOSITION 32 [N.B.: needs diagram]

In any triangle, if one of the sides be produced, the exterior angle is equal

to the two interior and opposite angles, and the three interior angles of the

triangle ABC equal to two right angles.

Let ABC be a triangle, and let one side of it BC be produced to D; I say


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that the exterior angle ACD is equal to the two interior and opposite angles CAB,

ABC, and the three interior angles of the triangle ABC, RCA, CAB are equal to

two right angles. For let CE be drawn through the point C parallel to the straight

line AB. [I. 31]

Then, since AB is parallel to CE, and AC has fallen upon them, AE the

alternate angles BAC, ACE are equal to one another. [I. 29] Again, since AB is

parallel to CE

and the straight line BD has fallen upon them, the exterior angle ECD is

equal to the interior B C D and opposite angle ABC. [I. 29] But the

angle ACE was also proved equal to the angle BAC; therefore the whole angle

ACD is equal to the two interior and opposite angles BAC, ABC. Let the angle

ACB be added to each; therefore the angles ACD, ACB are equal to the three

angles ABC, BCA, CAB. But the angles ACD, ACB are equal to two right angles;

[I. 13] therefore the angles ABC, BCA, CAB are also equal to two right angles.

Therefore etc. Q. E. D.


If a parallelogram have the same base with a triangle and be in the same

parallels, the parallelogram is double of the triangle.

For let the parallelogram ABCD have the same base BC with the triangle

EBC, and let it be in the same parallels BC, AE;


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I say that the parallelogram ABCD is double of the triangle BEC. For let

AC be joined. Then the triangle ABC is equal to the triangle EBC; for it is on the

same base with it and in the same parallels BC, AE. (Bk I, Prop. 37) But the

parallelogram ABCD is double of the triangle ABC; for the diameter AC bisects it;

(Bk I. Prop. 34) so that the parallelogram ABCD is also double of the triangle

EBC. Therefore etc. Q. E. D.


On a given straight line to describe a square.

Let AB be the given straight line; thus it is required to describe a square

on the straight line AB.


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Let AC be drawn at right angles to the straight line AB from the point A on

it (Bk I, Prop. 11), and let AD be made equal to AB; through the point D let DE be

drawn parallel to AB, and through the point E let BE be drawn parallel to AD. (Bk

I, Prop. 31) Therefore ADEB is a parallelogram; therefore AB is equal to DE and

AD to BE. (Bk I, Prop. 34) But AB is equal to AD; therefore the four straight lines

BA, AD, DE, EB are equal to one another; therefore the parallelogram ADEB is

equilateral. I say next that it is also right-angled. For, since the straight line AD

falls upon the parallels AB, DE, the angles BAD, ADE are equal to two right

angles. (Bk I, Prop. 29)


In right-angled triangles the square on the side subtending the right angle

is equal to the squares on the sides containing the right angle.

Let ABC be a right-angled triangle having the angle BAC right;


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I say that the square on BC is equal to the squares on BA, AC. For let

there be described on BC the square BDEC, and on BA, AC the squares GB,HC;

(Bk 1, Prop. 46) through A let AL be drawn parallel to either BD or CE, and let

AD, FC be joined. Then, since each of the angles BAC, BAG is right, it follows

that with a straight line BA, and at the point A on it, the two straight lines AC, AG

not lying on the same side make the adjacent angles equal to two right angles;

therefore CA is in a straight line with AG (Bk I, Prop. 14) For the same

reason BA is also in a straight line with AH. And, since the angle DBC is equal to

the angle FBA: for each is right: let the angle ABC be added to each; therefore

the whole angle DBA is equal to the whole angle FBC. (Common Notion 2) And,

since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the

two sides FB BC respectively; and the angle ABD is equal to the angle FBC;

therefore the base AD is equal to the base FC, and the triangle ABD is equal to

the triangle FBC. (Bk I, Prop. 41) Now the parallelogram BL is double of the

triangle ABD for they same base BD and are in the same parallels BD, AL. (Bk I,


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Prop. 4) And the square GB is double of the triangle FBC, for they again have

the same base FB and are in the same parallels have the FB, CC. (Bk 1, Prop.

41) (But the doubles of equals are equal to one another.) Therefore the

parallelogram BL is also equal to the square GB. Similarly, if AE, BK be joined,

the parallelogram CL can also be proved equal to the square HC; therefore the

whole square BDEC is equal to the two squares GB, HC. (Common Notion 2)

And the square BDEC is described on BC, and the squares GB, HC on BA, AC.

Therefore the square on the side BC is equal to the squares on the sides

BA, AC. Therefore etc. Q. E. D.

BOOK IV, PROPOSITION 11

In a given circle to inscribe an equilateral and equiangular pentagon.

Let ABCDE be the given circle; thus it is required to inscribe in the circle

ABCDE an equilateral and equiangular pentagon.

Fig. 1.2.6) Book IV, Proposition 11

Let the isosceles triangle FGH be set out having each of the angles at G,

H double of the angle at F; (Bk IV, Prop. 10) let there be inscribed in the circle

ABCDE the triangle ACD equiangular with the triangle FGH, so that the angle

CAD is equal to the angle at F and the angles at G, H respectively equal to the

angles ACD, CDA; (Bk IV, Prop. 2) therefore each of the angles ACD, CDA is

also double of the angle CAD. Now let the angles ACD, CDA be bisected


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respectively by the straight lines CE, DB (Bk I, Prop. 9), and let AB, BC, DE, EA

be joined. Then, since each of the angles ACD, CDA is double of the angle

CAD, and they have been bisected by the straight lines CE, DB, therefore the

five angles DAC, ACE, ECD, CDB, BDA are equal to one another.

But equal angles stand on equal circumferences; (Bk III, Prop. 26)

therefore the five circumferences AB, BC, CD, DE, EA are equal to one another.

But equal circumferences are subtended by equal straight lines; (Bk III,

Prop. 29) therefore the five straight lines AB, BC, CD, DE, EA are equal to one

another; therefore the pentagon ABCDE is equilateral. I say next that it is also

equiangular. For, since the circumference AB is equal to the circumference DE,

let BCD be added to each; therefore the whole circumference ABCD is equal to

the whole circumference EDCB. And the angle AED stands on the

circumference ABCD, and the angle BAE on the circumference EDCB; therefore

the angle BAE is also equal to the angle AED. (Bk III, Prop. 27) For the same

reason each of the angles ABC, BCD, CDE is also equal to each of the angles

BAE, AED; therefore the pentagon ABCDE is equiangular. But it was also

proved equilateral; therefore in the given circle an equilateral and equiangular

pentagon has been inscribed. Q. E. F. [Quod erat faciendum, Latin for ‘which

was to be done’.]

BOOK VI, PROPOSITION 30

To cut a given finite straight line in extreme and mean ratio.2

2 Euclid defines the mean extreme ratio thus: “A straight line is said to have been cut inextreme and mean ratio when, as the whole line is to the greater segment, so thegreater is to the lesser.” (Bk VI, Definition 2). That is,

a b ______________________________

A C BAB is to AC as CB is to AC. The mean and extreme ratio is also the only ratio that isalso a proportion; i.e., AB:AC::AC:CB. Algebraically, it is expressed as a/b = b/(a + b).Numerically, it is 0.618… (for AB = 1). Plato devoted considerable attention to ‘the


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Let AB be the given finite straight line; thus it is required to cut AB in

extreme and mean ratio.

Fig. 1.2.7) Book VI, Proposition 30

On AB let the square BC be described; and let there be applied to AC the

parallelogram CD equal to BC and exceeding by the figure AD similar to BC. (Bk

VI, Prop. 29) Now BC is a Square; therefore AD is also a square.

And, since BC is equal to CD, let CE be subtracted from each; therefore

the remainder BF is equal to the remainder AD. But it is also equiangular with it;

therefore in BF, AD the sides about the equal angles are reciprocally

proportional; (Bk VI, Prop. 14) therefore, as FE is to ED, so is AE to EB.

But FB is equal to AB, and ED to AE. Therefore as BA is to AE, so is AE

Section’ (as he called he), and it appears repeatedly in the composition of regular solids(also called the ‘Platonic solids’). It held a particular fascination for artists of theRenaissance, who saw in the ratio the secret to harmonious proportions in botharchitecture and the human form. In 1509 Luca Pacioli (c. 1445-c. 1514), a member ofthe Franciscan Order and leading mathematician of his day, published his study of theratio, Divina Proportione, the illustrations for which were executed by Leonardo da Vinci.Pacioli also included in his work a translation of the treatise on the five regular solids bythe Italian artist, Piero della Francesco (c. 1410-1492).


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to EB.

And AB is greater than AE; therefore AE is also greater than EB.

Therefore the straight line AB has been cut in extreme and mean ratio at and the

greater segment of it is AE. Q. E. F.

BOOK XIII, PROPOSITION 8

If in an equilateral and equiangular pentagon straight lines subtend two

angles taken in order, they cut one another in extreme and mean ratio, and their

greater segments are equal to the side of the pentagon.

For in the equilateral and equiangular pentagon ABCDE let the straight

lines AC, BE, cutting one another at the point H, subtend two angles taken in

order, the angles at A, B;

Fig. 1.2.8) Book XIII, Proposition 8

I say that each of them has been cut in extreme and mean ratio at the

point H, and their greater segments are equal to the side of the pentagon. For let

the circle ABCDE be circumscribed about the pentagon ABCDE. (Bk IV, Prop.


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14)

Then, since the two straight lines EA, AB are equal to the two AB, BC, and

they contain equal angles, therefore the base BE is equal to the base AC the

triangle ABE is equal to the triangle ABC, and the remaining angles will be equal

to the remaining angles respectively, namely those which the equal sides

subtend. Bk 1, Prop. 4) Therefore the angle BAC is equal to the angle ABE;

therefore the angle AHE is double of the angle BA H. (Bk I, Prop. 32)

But the angle EAC is also double of the angle BAC, inasmuch as the

circumference EDC is also double of the circumference CB; (Bk III Prop. 28; Bk

VI, Prop. 33) therefore the angle HAE is equal to the angle AHE; hence the

straight line HE is also equal to EA, that is, to AB. (Bk I, Prop. 6) And, since the

straight line BA is equal to AE, the angle ABE is also equal to the angle AEB. (Bk

I, Prop. 5) But the angle ABE was proved equal to the angle BAH; therefore the

angle BEA is also equal to the angle BAH. And the angle ABE is common to the

two triangles ABE and ABH; therefore the remaining angle BAE is equal to the

remaining angle AHB; (Bk I, Prop. 32) therefore the triangle ABE is equiangular

with the triangle ABH; therefore, proportionally, as EB is to BA, so is AB to BH.

(Bk VI, Prop. 4) But BA is equal to EH; therefore, as BE is to EH, so is EH to HB.

And BE is greater than EH; therefore EH is also greater than HB. (Bk V, Prop.

14) Therefore BE has been cut in extreme and mean ratio at H, and the greater

segment HE is equal to the side of the pentagon. Similarly we can prove that AC

has also been cut in extreme and mean ratio at H, and its greater segment CH is

equal to the side of the pentagon. Q. E. D.


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Fig. 1.2.9) Iterated Pentagrams & Pentagonal Pyramid

The Pythagoreans adopted the pentagram as the secret insignia of their

community, a sign to which they gave the meaning ‘Health’ and by which

members could recognize one another while traveling. Surely their fascination

with this construction had much to do with the unexpected frequency with which

one encounters the mean extreme ratio. Consider the following three

(undemonstrated) propositions: 1) The base of each isosceles triangle in the

pentagram forms the mean and extreme ratio with the longer sides. 2)

Connecting each vertex of a pentagon with the two opposite it yields a

pentagram which in turn reproduces a pentagon at its center, a process that can

be iterated ad infinitum and thereby the mean and extreme ratio is replicated in

an infinity of successively smaller pentagrams. Folding the five triangles of the

pentagram until they meet produces a five-sided pyramid the base AO and height

OH of which give the mean and extreme ratio.

BOOK XIII, [FINAL PROPOSITION]


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I say next that no other figure, besides the said five figures, can be

constructed which is contained by equilateral and equiangular figures equal to

one another.3

For a solid angle cannot be constructed with two triangles, or indeed

planes. With three triangles the angle of the pyramid is constructed, with four the

angle of the octahedron, and with five the angle of the icosahedron; but a solid

angle cannot be formed by six equilateral and equiangular triangles placed

together at one point, for, the angle of the equilateral triangle being two-thirds of

a right angle, the six will be equal to four right angles: which is impossible, for

any solid angle is contained by angles less than four right angles. (Bk XI, Prop.

21)

For the same reason, neither can a solid angle be constructed by more

than six plane angles. By three squares the angle of the cube is contained, but

by four it is impossible for a solid angle to be contained, for they will again be four

right angles. By three equilateral and equiangular pentagons the angle of the

dodecahedron is contained; but by four such it is impossible for any solid angle to

be contained, for, the angle of the equilateral pentagon being a right angle and a

fifth, the four angles will be greater than four right angles: which is impossible.

Neither again will a solid angle be contained by other polygonal figures by reason

of the same absurdity. Therefore etc. Q. E. D.

3 Several of the previous propositions develop the properties of the five regular solids:tetrahedron, cube (or hexahedron), octahedron, icosahedron, and dodecahedron. SeeFigure 1.2.10 below.


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Fig. 1.2.10) Book XIII, Final Proposition

The regular or ‘Platonic’ solids were among the most fascinating objects from

ancient Greek geometry, and they continued to fascinate through the centuries.

One of the simplest relations found among the solids was not discovered until the

seventeenth century: for each solid the number of vertices (or corners) added to

the number of faces equals the number of edges plus 2 (that is, V + F = E + 2).

There also exist reciprocal relations between certain pairs of solids. The six


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vertices of an octahedron placed inside a cube touch the geometric centers of

the six faces of the cube. Conversely, the eight vertices of a cube placed inside

an octahedron touch the centers of its eight faces. Similar reciprocal

relationships exist for the icosahedron and dodecahedron and the tetrahedron

with itself. Plato used the tetra-, hexa-, octa-, and dodecahedron as the

geometrical basis of his four-element matter theory; and Kepler believed that the

concentric nesting of the regular solids solved two great ‘mysteries of the

cosmos’, i.e., the number and spacing of the Copernican orbits. See ?? Kepler,

“Harmony of the World,” pp. 000 below).

*preliminary draft for student use only. not for citation...

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