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http://www.math.ru/l...euclid48-1.djvu

 

http://www.math.ru/l...euclid49-2.djvu

 

http://www.math.ru/l...euclid50-3.djvu

 

"Начала" ("Στοιχεῖα (Γεωμετρίας)", буквальный перевод чего, это также "Элементы (геометрии)") Евклида служили основой обучения в школах по крайней мере до 17 века. И поныне преподаваемая в школах геометрия зиждется на его "Началах". Считается, что эта книга пережила больше всех переизданий и публикаций, уступая в этом только Библии...

 

Древнегреческий текст:

 

http://users.ntua.gr...exelements.html

 

http://www.foundalis.../c/EuclidGr.htm

 

Древнегреческо-новогреческий словарь употребляемых в "Началах" слов:

 

http://users.ntua.gr.../lexiko_geo.pdf

THE ENCYCLOPEDIA OF

ANCIENT NATURAL
SCIENTISTS
The Greek tradition and its many heirs
Edited by Paul T. Keyser and Georgia L. Irby-Massie

First published 2008
by Routledge

 

Euclid of Alexandria (300 – 260 BCE)
We have remarkably little personal information about Euclid (Eukleide¯s), arguably the most
influential mathematician who ever lived. Pappos (Collection 7.35, p. 678.10–12 H.) says that
Apollonios of Perge studied with Euclid’s students in Alexandria, suggesting a floruit in
the middle of the 3rd c. BCE. Proclos (In Eucl. p. 68.10–11 Fr.) makes Euclid a contemporary
of the first Ptolemy (d. 282), but his evidence does not inspire confidence. The standard
edition of Euclid’s works (Heiberg and Menge) includes the following complete texts in
Greek: Elements, 13 books (vv. 1–4) plus a 14th book written by Hupsikles and a 15th book
at least in part due to a pupil of the elder Isidoros of Miletos (v. 5); Data (v. 6); Optics (in
two recensions) and Catoptrics (v. 7); Phenomena, Sectio Canonis, and Introductio Harmonica (v. 8).
Volume 8 also contains textual evidence relating to non-extant works ascribed in ancient
sources to Euclid: On Divisions, Fallacies, Porisms, Conics, and Surface Loci. Arabic evidence
indicates that Euclid also wrote on mechanics.
Mathematical texts are especially vulnerable to “improvements,” inserted “explanations,”
and recasting, as is shown, for example, by the 14th and 15th books of the Elements and the
two recensions of the Optics. Most Greek MSS of the Elements and all early printed versions
derive from an edition by Theon of Alexandria, whereas the standard printed edition
purports to be pre-Theonine. There are considerable variations between our Greek text and
Arabic translations and also among the Greek MSS themselves. Because the texts are so
subject to tampering, it is really not possible to speak about exactly what Euclid wrote, but
only about whether a work is based on something Euclid could have written. Of the complete
works published in the standard edition, only the Introductio Harmonica is universally
rejected as non-Euclidean in this sense. Older scholars tended to consider the Catoptrics and
Sectio Canonis spurious, but both works have been defended as Euclidean in more recent
years (the Sectio is treated in a separate entry). Only the other surviving works which can be 

considered Euclidean, all characterized by apparently rigorous, stylized deduction from first
principles, are discussed here.
Data (“Things Given”) is mentioned first by Pappos (Collection 7.3, p. 636.18–19 H.) in his
list of works useful for analysis, that is, the finding of solutions to problems and of proofs of
propositions, by supposing that what has to be done is accomplished or that what is to be
proved is true, and asking what else must be accomplished or true as a result: the idea is that
when one reaches things one knows how to accomplish or prove, one will be able to reverse
the steps and produce a solution or proof for what is sought.
Optics is essentially a treatise on monocular perspective. It is assumed that vision is a
matter of the emission of rectilinear rays from the eye which strike an object and form a
cone with vertex in the eye and base a plane figure determined by the shape of the object
seen, and that the relative apparent size of an object is determined by the size of the angle
“under which” it is seen and its relative apparent position by the relative position of the rays
under which it is seen; the rays are treated as discrete straight lines, so that an object will not
be seen if it falls between rays.
Catoptrics takes the same approach to mirror vision, treating plane, convex, and concave
mirrors.
Phenomena is an essay in very elementary geometric astronomy, the main point of which
seems to be showing that certain astronomical appearances can be represented and understood
geometrically. In the prologue simple astronomical data are invoked to justify the
claim that the sphere of the fixed stars rotates uniformly about a fixed axis and that the eye
of an observer is at the center of the sphere, and geometrical definitions are given of such
astronomical terms as “horizon,” and “meridian.” Among the theorems proved are the
assertion that if two stars lie on a great circle which has no point in common with the arctic
circle (the circle including all stars that are never seen to set), the one which rises earlier sets
earlier (prop. 4).

 

Euclid’s geometric algebra
(1. prop. 2) © Mueller

 

The name of “Euclid” is associated first and foremost with

the Elements, apparently a single treatise in which propositions
are derived from principles labeled as “definitions,” “postulates,”
and “common notions” (the last frequently called
axioms). Careful scholarship of the last century has made
clear that the work is a compilation based on several sources.
The subject of book 1 is the geometry of plane rectilinear
figures. The book is noteworthy for avoiding the use of proportions
and for postponing the use of the parallel postulate
until it is required. Book 2 introduces what is now frequently
called geometric algebra in a series of geometric propositions
corresponding to what we know as algebraic equations; for
example, proposition 2, which corresponds to “(x+y)2 = x2 + y2
+ 2xy,” says that if AGB is a straight line, the square with side
equal to AB [SQ(AB)] is equal to SQ(AG) plus SQ(BG) plus two times the rectangle with
sides equal to AG and BG.
Book 3 treats circles and their relations to straight lines and angles, Book 4 the inscription
in circles and circumscription about circles of rectilinear figures. Book 5 brings in proportionality,
developing a theory based on a definition which says of four magnitudes A, B,
C, D that A:B :: C:D if and only if for any multiples m·A, n·B, m·C, n·D of those magnitudes,
if m·A is greater than, equal to, or less than n·B, m·C is accordingly greater than, equal to, or 

less than n·D, and that A:B > C:D if and only if for some m and n, m·A > n·B and m·C ≤ n·D.
Euclid’s theory deals only with proportionalities among magnitudes, not with ratios between
pairs of magnitudes, but it is a simple matter to reformulate the theory of Book 5, by
treating ratios A:B as “cuts” in the system of positive fractions m/n. In Book 6 Euclid applies
the theory of proportion to geometric entities and develops the notion of similarity. Books
7–9 introduce numbers as objects of study using a separately developed theory of proportion.
The major topic of the very difficult Book 10 is a classification of straight lines A which
are called irrational relative to a given straight line R if both A and R and SQ(A) and SQ®
are incommensurable.
Book 11 develops basic ideas of solid geometry. Book 12 uses a method, which
is called the “method of exhaustion,” to prove a series of sophisticated results, the
simplest of which is prop. 2: if C and C’ are circles with diameters d and d’, then
C:C’ :: SQ(d ):SQ(d’ ).
In Book 13 Euclid constructs the five regular solids, triangular pyramid, octahedron,
cube, icosahedron, and dodecahedron, circumscribes spheres around them, and characterizes
their edges relative to the diameters of the circumscribing spheres using in the last three
cases the classification of Book 10.
It is clear from Proklos (In Eucl. pp. 65–68 Fr.) that Euclid’s Elements had more than
one predecessor, starting with a work of Hippokrates of Khios. It is also clear that
much of the contents of the Elements is based on the work of others, most clearly
Eudoxos (Books 5 and 12) and Theaitetos (Books 10 and 13). Nevertheless, Euclid’s
Elements is an outstanding achievement which replaced all of its predecessors and sources,
and became both an inspiration and a foil for much of the subsequent history of Western
mathematics.
Ed.: J.L. Heiberg, and H. Menge, Euclidis Opera Omnia, 9 vv. (1883–1916);
P. Ver Eecke, trans., Euclide, L’optique et al catoptrique (1938); DSB 4.414–437, I. Bulmer-Thomas;
B. Vitrac, trans., Euclide, Les Elements 4 vv. (1990–2001); J.L. Berggren and R.S.D. Thomas,
trans., Euclid’s Phaenomena (1996); DPA 3 (2000) 252–272, B. Vitrac; C.M. Taisbak, trans., Dedomena
(2003).
автор статьи: Ian Mueller
 

Euclidean Sectio Canonis (300 – 260 BCE?)
“Division of the Monochord” (= kanonos katatome¯ = Sectio Canonis), a short text on mathematical
harmonics ascribed in most MSS to Euclid. Fragments are quoted by Porphurios
(title and authorship: In Ptolemaei Harmonica Commentarium 98.19 Düring; preface: 90.7–22;
props.1–16: 99.1–103.25) and Boethius (De Institutione Musica iv). Its authorship, date and
unity of composition have been long debated: a logical error in prop. 11 has been used as
evidence against Euclid’s authorship, but arguments for dating it substantially later than
Euclid, and for excising the preface and two (or four) final propositions as late accretions,
have not met with consensus.
The text as we now have it is comprised of five types of material: (1) a discursive preface
attempting to derive mathematical harmonics from a physical acoustics which can account
for the behavior of strings (an essential connection in order for the monochord to be used
demonstratively in props.19–20); (2) nine purely mathematical propositions demonstrating

 

the properties of the simplest multiple (mn:n) and epimoric ((n+1):n) ratios; (3) seven subsequent
propositions (10–16) wherein the properties of simple musical intervals are shown
to be analogous to those of the ratios of props.1–9; (4) two propositions (17–18) locating the
“movable” notes in the scale by the method of concordance; (5) a final two propositions
(19–20) introducing the monochord and marking on it the bridge-positions corresponding
to the notes of a two-octave scale-system.
The Sectio owes as much to 4th c. developments in acoustics and harmonics as it does to
Euclidean mathematics. In the preface, the basic assumptions of Arkhutan acoustics (e.g.,
that sound is caused by impact, ple¯ge¯ ) are adopted and modified, apparently with the aim of
allowing mathematical propositions to be demonstrated on strings, in ways that suggest the
influence of theories akin to those expressed in the Aristotelian On Sounds and the
Aristotelian Problems (11.6, 19.12, 19.23, 19.39). The Sectio is also a polemical text;
certain propositions (e.g. 16, 18) are clearly intended to refute not only the conclusions but
also the basic assumptions of Aristoxenian harmonics.
Ed.: MSG; H. Menge, Euclides Phaenomena et scripta musica (1916).
Düring (1932); A.D. Barker, “Methods and aims in the Euclidean Sectio Canonis,” JHS 101 (1981) 1–16;
A. Barbera, “Placing Sectio Canonis in historical and philosophical contexts,” JHS 104 (1984)
157–161; Barker (1989); A. Barbera, The Euclidean Division of the Canon (1991); A.D. Barker, “Three
approaches to canonic division,” Apeiron 24 (1991) 49–83; A.C. Bowen, “Euclid’s Sectio canonis and
the history of Pythagoreanism,” in Bowen (1991); O. Busch, Logos Syntheseos (1998); Mathiesen
(1999); S. Hagel, “Zur physikalischen Begründung der pythagoreischen Musikbetrachtung,” WS
114 (2001) 85–93; Barker (2007) ch. 14, 364–410.
автор статьи: David Creese

 

A

History
of
Mathematics

 

T H I RD EDI T ION
Uta C.Merzbach and Carl B. Boyer

 

Copyrightr 1968, 1989, 1991, 2011 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada

 

5
Euclid of Alexandria

 

Ptolemy once asked Euclid whether there was any shorter way to a
knowledge of geometry than by a study of the Elements, whereupon
Euclid answered that there was no royal road to geometry.

Proclus Diadochus

Alexandria

The death of Alexander the Great had led to internecine strife among the
generals in the Greek army, but after 300 BCE, control of the Egyptian
portion of the empire was firmly in the hands of the Ptolemies, the
Macedonian rulers of Egypt. Ptolemy I laid the foundations for two
institutions at Alexandria that would make it the leading center of
scholarship for generations. They were the Museum and the Library,
both amply endowed by him and his son, Ptolemy II, who brought to this
great research center men of outstanding scholarship in a variety of
fields. Among these was Euclid, the author of the most successful
mathematics textbook ever written—the Elements (Stoichia). Considering
the fame of the author and of his best-seller, remarkably little is
known of Euclid’s life. He was so obscure that no birthplace is

 

the identification of the author as Euclid of Megara and a portrait of
Euclid of Megara appears in histories of mathematics, this is a case of
mistaken identity.
From the nature of his work, it is presumed that Euclid of Alexandria
had studied with students of Plato, if not at the Academy itself. There is a
tale told of him that when one of his students asked of what use was the
study of geometry, Euclid asked his slave to give the student three pence,
“since he must needs make gain of what he learns.”
Lost Works
Of what Euclid wrote, more than half has been lost, including some of
his more important compositions, such as a treatise on conics in four
books. Both this work and an earlier lost treatise on Solid Loci (the
Greek name for the conic sections) by the somewhat older geometer
Aristaeus were soon superseded by the more extensive work on conics
by Apollonius. Among Euclid’s lost works are also one on Surface Loci,
another on Pseudaria (or fallacies), and three books on Porisms. It is not
even clear from ancient references what material these contained. As far
as we know, the Greeks did not study any surface other than that of a
solid of revolution.
The loss of the Euclidean Porisms is particularly tantalizing. Pappus
later reported that a porism is intermediate between a theorem, in which
something is proposed for demonstration, and a problem, in which
something is proposed for construction. Others have described a porism
as a proposition in which one determines a relationship between known
and variable or undetermined quantities, perhaps the closest approach
in antiquity to the concept of function.
Extant Works
Five works by Euclid have survived to our day: the Elements, the Data,
the Division of Figures, the Phaenomena, and the Optics. The lastmentioned
is of interest as an early work on perspective, or the geometry
of direct vision. The ancients had divided the study of optical phenomena
into three parts: (1) optics (the geometry of direct vision), (2) catoptrics
(the geometry of reflected rays), and (3) dioptrics (the geometry of
refracted rays). A Catoptrica sometimes ascribed to Euclid is of doubtful
authenticity, being perhaps by Theon of Alexandria, who lived some six
centuries later. Euclid’s Optics is noteworthy for its espousal of an
“emission” theory of vision, according to which the eye sends out rays that
travel to the object, in contrast to a rival Aristotelian doctrine in which an

 

activity in a medium travels in a straight line from the object to the eye. It
should be noted that the mathematics of perspective (as opposed to the
physical description) is the same, no matter which of the two theories is
adopted. Among the theorems found in Euclid’s Optics is one widely used
in antiquity—tan α / tan β ,α / β if 0,α,β ,π / 2. One object of the
Optics was to combat an Epicurean insistence that an object was just as
large as it looked, with no allowance to be made for the foreshortening
suggested by perspective.
The Euclidean Division of Figures is a work that would have been lost
had it not been for the learning of Arabic scholars. It has not survived in
the original Greek, but before the disappearance of the Greek versions,
an Arabic translation had been made (omitting some of the original
proofs “because the demonstrations are easy”), which in turn was later
translated into Latin and ultimately into current modern languages. This
is not atypical of other ancient works. The Division of Figures includes a
collection of thirty-six propositions concerning the division of plane
configurations. For example, Proposition 1 calls for the construction of
a straight line that shall be parallel to the base of a triangle and shall
divide the triangle into two equal areas. Proposition 4 requires a bisection
of a trapezoid abqd (Fig. 5.1) by a line parallel to the bases; the
required line zi is found by determining z such that ze2 51
2ðeb
2 1ea2Þ.
Other propositions call for the division of a parallelogram into two equal
parts by a line drawn through a given point on one of the sides (Proposition
6) or through a given point outside the parallelogram (Proposition
10). The final proposition asks for the division of a quadrilateral in
a given ratio by a line through a point on one of the sides of the
quadrilateral.
Somewhat similar in nature and purpose to the Division of Figures is
Euclid’s Data, a work that has come down to us through both the Greek
and the Arabic. It seems to have been composed for use at the Museum
of Alexandria, serving as a companion volume to the first six books of
the Elements in much the way that a manual of tables supplements a
textbook. It opens with fifteen definitions concerning magnitudes and
loci. The body of the text comprises ninety-five statements concerning
the implications of conditions and magnitudes that may be given in a
problem. The first two state that if two magnitudes a and b are given,

 

a second, the second magnitude is given. There are about two dozen
similar statements, serving as algebraic rules or formulas. After this, the
work lays out simple geometric rules concerning parallel lines and
proportional magnitudes, while reminding the student of the implications
of the data given in a problem, such as the advice that when two line
segments have a given ratio then one knows the ratio of the areas of
similar rectilinear figures constructed on these segments. Some of the
statements are geometric equivalents of the solution of quadratic equations.
For example, we are told that if a given (rectangular) area AB is
laid off along a line segment of given length AC (Fig. 5.2) and if the area
BC by which the area AB falls short of the entire rectangle AD is given,
the dimensions of the rectangle BC are known. The truth of this
statement is easily demonstrated by modern algebra. Let the length of
AC be a, the area of AB be b2, and the ratio of FC to CD be c:d. Then, if
FC5x and CD5y, we have x / y5c / d and (a x)y5b2. Eliminating
y, we have (a2x)dx5b2c or dx22adx1b2c50, from which
x5a=26 ða=2Þ2 2b2c=d:
p
The geometric solution given by Euclid is
equivalent to this, except that the negative sign before the radical is used.
Statements 84 and 85 in the Data are geometric replacements of the
familiar Babylonian algebraic solutions of the systems xy5a2, x 6 y
5b, which again are the equivalents of solutions of simultaneous
equations. The last few statements in the Data concern relationships
between linear and angular measures in a given circle.
The Elements
The Elements was a textbook and by no means the first one. We know
of at least three earlier such Elements, including that by Hippocrates of
Chios, but there is no trace of these or of other potential rivals from
ancient times. The Elements of Euclid so far outdistanced competitors
that it alone survived. The Elements was not, as is sometimes thought, a
compendium of all geometric knowledge; it was instead an introductory
textbook covering all elementary mathematics—that is, arithmetic (in
the sense of the English “higher arithmetic” or the American “theory
of numbers”), synthetic geometry (of points, lines, planes, circles, and

 

spheres), and algebra (not in the modern symbolic sense, but an
equivalent in geometric garb). It will be noted that the art of calculation
is not included, for this was not a part of mathematical instruction; nor
was the study of the conics or higher plane curves part of the book, for
these formed a part of more advanced mathematics. Proclus described
the Elements as bearing to the rest of mathematics the same sort of
relation as that which the letters of the alphabet have in relation to
language. Were the Elements intended as an exhaustive store of information,
the author might have included references to other authors,
statements of recent research, and informal explanations. As it is, the
Elements is austerely limited to the business in hand—the exposition in
logical order of the fundamentals of elementary mathematics. Occasionally,
however, later writers interpolated into the text explanatory
scholia, and such additions were copied by later scribes as part of the
original text. Some of these appear in every one of the manuscripts now
extant. Euclid himself made no claim to originality, and it is clear that
he drew heavily from the works of his predecessors. It is believed
that the arrangement is his own, and, presumably, some of the proofs
were supplied by him, but beyond that, it is difficult to estimate the
degree of originality that is to be found in this, the most renowned
mathematical work in history.
Definitions and Postulates
The Elements is divided into thirteen books or chapters, of which the first
half-dozen are on elementary plane geometry, the next three on the
theory of numbers, the tenth on incommensurables, and the last three
chiefly on solid geometry. There is no introduction or preamble to
the work, and the first book opens abruptly with a list of twenty-three
definitions. The weakness here is that some of the definitions do not
define, inasmuch as there is no prior set of undefined elements in terms
of which to define the others. Thus, to say, as does Euclid, that “a point is
that which has no part,” or that “a line is breadthless length,” or that “a
surface is that which has length and breadth only,” is scarcely to define
these entities, for a definition must be expressed in terms of things that
precede and are better known than the things defined. Objections can
easily be raised on the score of logical circularity to other so-called
definitions of Euclid, such as “The extremities of a line are points,”
or “A straight line is a line which lies evenly with the points on itself,” or
“The extremities of a surface are lines,” all of which may have been due
to Plato.
Following the definitions, Euclid lists five postulates and five common
notions. Aristotle had made a sharp distinction between axioms (or
common notions) and postulates; the former, he said, must be convincing

 

in themselves—truths common to all studies—but the latter are less
obvious and do not presuppose the assent of the learner, for they pertain
only to the subject at hand. We do not know whether Euclid distinguished
between two types of assumptions. Surviving manuscripts are not in
agreement here, and in some cases, the ten assumptions appear together
in a single category. Modern mathematicians see no essential difference
between an axiom and a postulate. In most manuscripts of the Elements,
we find the following ten assumptions:
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 radius.
4. That all right angles are equal.
5. That, if a straight line falling on two straight lines makes 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 the angles
are 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.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Aristotle had written that “other things being equal, that proof is the
better which proceeds from the fewer postulates,” and Euclid evidently
subscribed to this principle. For example, Postulate 3 is interpreted in the
very limited literal sense, sometimes described as the use of the Euclidean
(collapsible) compass, whose legs maintain a constant opening so
long as the point stands on the paper, but fall back on each other when
they are lifted. That is, the postulate is not interpreted to permit the use
of a pair of dividers to lay off a distance equal to one line segment on a
noncontiguous longer line segment, starting from an end point. It is
proved in the first three propositions of Book I that the latter construction
is always possible, even under the strict interpretation of Postulate 3. The
first proposition justifies the construction of an equilateral triangle ABC
on a given line segment AB by constructing through B a circle with a
center at A and another circle through A with a center at B, and letting C
be the point of intersection of the two circles. (That they do intersect is
tacitly assumed.) Proposition 2 then builds on Proposition 1 by showing
that from any point A as extremity (Fig. 5.3), one can lay off a straight
line segment equal to a given line segment BC. First, Euclid drew AB,

 

and on this he constructed the equilateral triangle ABD, extending the
sides DA and DB to E and F, respectively. With B as center, describe
the circle through C, intersecting BF in G; then, with D as center, draw a
circle through G, intersecting DE in H. Line AH is then easily shown
to be the line required. Finally, in Proposition 3 Euclid made use of
Proposition 2 to show that given any two unequal straight lines, one can
cut off from the greater a segment equal to the smaller.
Scope of Book I
In the first three propositions, Euclid went to great pains to show that a very
restricted interpretation of Postulate 3 nevertheless implies the free use of a
compass as is usually done in laying off distances in elementary geometry.
Nevertheless, by modern standards of rigor, the Euclidean assumptions are
woefully inadequate, and in his proofs Euclid often makes use of tacit
postulates. In the first proposition of the Elements, for example, he assumes
without proof that the two circles will intersect in a point. For this and
similar situations, it is necessary to add to the postulates one equivalent
to a principle of continuity. Moreover, Postulates 1 and 2, as they were
expressed by Euclid, guarantee neither the uniqueness of the straight line
through two noncoincident points nor even its infinitude; they simply assert
that there is at least one and that it has no termini.
Most of the propositions in Book I of the Elements are well known to
anyone who has had a high school course in geometry. Included are
the familiar theorems on the congruence of triangles (but without an
axiom justifying the method of superposition), on simple constructions
by straightedge and compass, on inequalities concerning angles and

 

sides of a triangle, on properties of parallel lines (leading to the fact that
the sum of the angles of a triangle is equal to two right angles), and on
parallelograms (including the construction of a parallelogram having
given angles and equal in area to a given triangle or a given rectilinear
figure). The book closes (in Propositions 47 and 48) with the proof of the
Pythagorean theorem and its converse. The proof of the theorem as given
by Euclid was not that usually given in textbooks of today, in which
simple proportions are applied to the sides of similar triangles formed by
dropping an altitude on the hypotenuse. For the Pythagorean theorem,
Euclid used instead the beautiful proof with a figure sometimes described
as a windmill or the peacock’s tail or the bride’s chair (Fig. 5.4). The
proof is accomplished by showing that the square on AC is equal to twice
the triangle FAB or to twice the triangle CAD or to the rectangle AL, and
that the square on BC is equal to twice the triangle ABK or to twice the
triangle BCE or to the rectangle BL. Hence, the sum of the squares is
equal to the sum of the rectangles—that is, to the square on AB. It has
been assumed that this proof was original with Euclid, and many conjectures
have been made as to the possible form of earlier proofs. Since
the days of Euclid, many alternative proofs have been proposed.
It is to Euclid’s credit that the Pythagorean theorem is immediately
followed by a proof of the converse: If in a triangle the square on one of
the sides is equal to the sum of the squares on the other two sides, the
angle between these other two sides is a right angle. Not infrequently in
modern textbooks, the exercises following the proof of the Pythagorean
theorem are such that they require not the theorem itself but the still
unproved converse. There may be many a minor flaw in the Elements,
but the book had all of the major logical virtues.

 

Geometric Algebra
Book II of the Elements is a short one, containing only fourteen propositions,
not one of which plays any role in modern textbooks, yet in
Euclid’s day this book was of great significance. This sharp discrepancy
between ancient and modern views is easily explained—today we have
symbolic algebra and trigonometry, which have replaced the geometric
equivalents from Greece. For instance, Proposition 1 of Book II states,
“If there be two straight lines, and one of them be cut into any number of
segments whatever, the rectangle contained by the two straight lines
is equal to the rectangles contained by the uncut straight line and each
of the segments.” This theorem, which asserts (Fig. 5.5) that AD
(AP1PR1RB)5AD AP1AD PR1AD RB, is nothing more than a
geometric statement of one of the fundamental laws of arithmetic known
today as the distributive law: a(b1c1d)5ab1ac1ad. In later books
of the Elements (V and VII), we find demonstrations of the commutative
and associative laws for multiplication. In Euclid’s day magnitudes were
pictured as line segments satisfying the axioms and the theorems of
geometry.
Book II of the Elements, which is a geometric algebra, served much the
same purpose as does our symbolic algebra. There can be little doubt that
modern algebra greatly facilitates the manipulation of relationships among
magnitudes. Yet, it is undoubtedly also true that a Greek geometer versed
in the fourteen theorems of Euclid’s “algebra” was far more adept in
applying these theorems to practical mensuration than is an experienced
geometer of today. Ancient geometric algebra was not an ideal tool, but it
was far from ineffective and its visual appeal to an Alexandrian schoolboy
must have been far more vivid than its modern algebraic counterpart can
ever be. For example, Elements II.5 contains what we should regard as an
impractical circumlocution for a22b25(a1b)(a2b):
If a straight line be cut into equal and unequal segments, the rectangle
contained by the unequal segments of the whole, together with the square
on the straight line between the points of section, is equal to the square on
the half.

 

The diagram that Euclid uses in this connection played a key role
in Greek algebra; hence, we reproduce it with further explanation.
(Throughout this chapter, the translations and most of the diagrams
are based on the Thirteen Books of Euclid’s Elements, as edited by
T. L. Heath.) If in the diagram (Fig. 5.6) we let AC5CB5a,
and CD5b, the theorem asserts that (a1b)(a2b)1b25a2. The geometric
verification of this statement is not difficult; however,
the significance of the diagram lies not so much in the proof of the
theorem as in the use to which similar diagrams were put by Greek
geometric algebraists. If the Greek scholar were required to construct a
line x having the property expressed by ax x25 b2, where a and b are
line segments with a.2b, he would draw line AB5a and bisect it at C.
Then, at C he would erect a perpendicular CP equal in length to b; with
P as center and radius a / 2, he would draw a circle cutting AB in point D.
Then, on AB he would construct rectangle ABMK of width BM5BD and
complete the square BDHM. This square is the area x2, having the property
specified in the quadratic equation. As the Greeks expressed it, we have
applied to the segment AB (5a) a rectangle AH (5ax2x2), which is
equal to a given square (b2) and falls short (of AM) by a square DM. The
demonstration of this is provided by the proposition cited earlier (II.5), in
which it is clear that the rectangle ADHK equals the concave polygon
CBFGHL—that is, it differs from (a / 2)2 by the square LHGE, the side of
which by construction is CD5 ða=2Þ2 2b2
p
:
The figure used by Euclid in Elements II.11 and again in VI.30 (our
Fig. 5.7) is the basis for a diagram that appears today in many geometry
books to illustrate the iterative property of the golden section. To
the gnomon BCDFGH (Fig. 5.7), we add point L to complete the rectangle
CDFL (Fig. 5.8), and within the smaller rectangle LBGH, which is
similar to the larger rectangle LCDF, we construct, by making GO5GL,
the gnomon LBMNOG similar to gnomon BCDFGH. Now within the

 

rectangle BHOP, which is similar to the larger rectangles CDFL and
LBHG, we construct the gnomon PBHQRN similar to the gnomons
BCDFGH and LBMNOG. Continuing indefinitely in this manner, we
have an unending sequence of nested similar rectangles tending toward a
limiting point Z. It turns out that Z, which is easily seen to be the point of
intersection of lines FB and DL, is also the pole of a logarithmic spiral
tangent to the sides of the rectangles at points C, A, G, P, M, Q, . . . Other
striking properties can be found in this fascinating diagram.
Propositions 12 and 13 of Book II are of interest because they adumbrate
the concern with trigonometry that was shortly to blossom in Greece.
These propositions will be recognized by the reader as geometric
formulations—first for the obtuse angle and then for the acute angle—of
what later became known as the law of cosines for plane triangles:
Proposition 12. In obtuse-angled triangles, the square on the side subtending
the obtuse angle is greater than the squares on the sides containing

 

the obtuse angle by twice the rectangle contained by one of the sides
about the obtuse angle, namely, that on which the perpendicular falls, and the
straight line cut off outside by the perpendicular toward the obtuse angle.
Proposition 13. In acute-angled triangles, the square on the side subtending
the acute angle is less than the squares on the sides containing
the acute angle by twice the rectangle contained by one of the sides of the
acute angle, namely, that on which the perpendicular falls, and the straight
line cut off within by the perpendicular toward the acute angle.
The proofs of Propositions 12 and 13 are analogous to those used today
in trigonometry through double application of the Pythagorean theorem.
Books III and IV
It has generally been supposed that the contents of the first two books of
the Elements are largely the work of the Pythagoreans. Books III and IV,
on the other hand, deal with the geometry of the circle, and here the
material is presumed to have been drawn primarily from Hippocrates of
Chios. The two books are not unlike the theorems on circles contained in
textbooks of today. The first proposition of Book III, for example, calls for
the construction of the center of a circle, and the last, Proposition 37, is the
familiar statement that if from a point outside a circle a tangent and a
secant are drawn, the square on the tangent is equal to the rectangle on the
whole secant and the external segment. Book IV contains sixteen propositions,
largely familiar to modern students, concerning figures inscribed
in, or circumscribed about, a circle. Theorems on the measure of
angles are reserved until after a theory of proportions has been established.
Theory of Proportion
Of the thirteen books of the Elements, those most admired have been the
fifth and the tenth—the one on the general theory of proportion and
the other on the classification of incommensurables. The discovery of the
incommensurable had threatened a logical crisis that cast doubt on
proofs appealing to proportionality, but the crisis had been successfully
averted through the principles enunciated by Eudoxus. Nevertheless,
Greek mathematicians tended to avoid proportions. We have seen that
Euclid put off their use as long as possible, and such a relationship
among lengths as x:a5b:c would be thought of as an equality of the
areas cx5ab. Sooner or later, however, proportions are needed, and so
Euclid tackled the problem in Book V of the Elements. Some commentators
have gone so far as to suggest that the whole book, consisting
of twenty-five propositions, was the work of Eudoxus, but this seems to
be unlikely. Some of the definitions—such as that of a ratio—are so
vague as to be useless. Definition 4, however, is essentially the axiom of

 

Eudoxus and Archimedes: “Magnitudes are said to have a ratio to one
another which are capable, when multiplied, of exceeding one another.”
Definition 5, the equality of ratios, is precisely that given earlier in
connection with Eudoxus’s definition of proportionality.
Book V deals with topics of fundamental importance in all mathematics.
It opens with propositions that are equivalent to such things as
the left-hand and right-hand distributive laws for multiplication over
addition, the left-hand distributive law for multiplication over subtraction,
and the associative law for multiplication (ab)c5a(bc). Then the
book lays out rules for “greater than” and “less than” and the wellknown
properties of proportions. It is often asserted that Greek geometric
algebra could not rise above the second degree in plane geometry
or above the third degree in solid geometry, but this is not really the case.
The general theory of proportions would permit work with products of
any number of dimensions, for an equation of the form x45abcd is
equivalent to one involving products of ratios of lines such as x / a x / b
5c / x d / x.
Having developed the theory of proportions in Book V, Euclid
exploited it in Book VI by proving theorems concerning ratios and
proportions related to similar triangles, parallelograms, and other polygons.
Noteworthy is Proposition 31, a generalization of the Pythagorean
theorem: “In right-angled triangles the figure on the side subtending
the right angle is equal to the similar and similarly described figures
on the sides containing the right angle.” Proclus credits this extension to
Euclid himself. Book VI also contains (in Propositions 28 and 29) a
generalization of the method of application of areas, for the sound basis
for proportion given in Book V now enabled the author to make free
use of the concept of similarity. The rectangles of Book II are now
replaced by parallelograms, and it is required to apply to a given straight
line a parallelogram equal to a given rectilinear figure and deficient
(or exceeding) by a parallelogram similar to a given parallelogram.
These constructions, like those of II.5 6, are in reality solutions of the
quadratic equations bx5ac6x2, subject to the restriction (implied in
IX.27) that the discriminant is not negative.
Theory of Numbers
The Elements of Euclid is often mistakenly thought of as restricted to
geometry. We already have described two books (II and V) that are
almost exclusively algebraic; three books (VII, VIII, and IX) are devoted
to the theory of numbers. The word “number,” to the Greeks, always
referred to what we call the natural numbers—the positive whole
numbers or integers. Book VII opens with a list of twenty-two definitions

 

distinguishing various types of numbers—odd and even, prime and
composite, plane and solid (that is, those that are products of two or three
integers)—and finally defining a perfect number as “that which is equal
to its own parts.” The theorems in Books VII, VIII, and IX are likely to
be familiar to the reader who has had an elementary course in the theory
of numbers, but the language of the proofs will certainly be unfamiliar.
Throughout these books, each number is represented by a line segment,
so that Euclid will speak of a number as AB. (The discovery of the
incommensurable had shown that not all line segments could be associated
with whole numbers, but the converse statement—that numbers
can always be represented by line segments—obviously remains true.)
Hence, Euclid does not use the phrases “is a multiple of” or “is a factor
of,” for he replaces these by “is measured by” and “measures,”
respectively. That is, a number n is measured by another number m if
there is a third number k such that n5km.
Book VII opens with two propositions that constitute a celebrated
rule in the theory of numbers, which today is known as “Euclid’s
algorithm” for finding the greatest common divisor (measure) of two
numbers. It is a scheme suggestive of a repeated inverse application of
the axiom of Eudoxus. Given two unequal numbers, one subtracts the
smaller a from the larger b repeatedly until a remainder r1 less than
the smaller is obtained; then, one repeatedly subtracts this remainder r1
from a until a remainder r2,r1 results; then, one repeatedly subtracts r2
from r1; and so on. Ultimately, the process will lead to a remainder rn,
which will measure rn 1, hence all preceding remainders, as well as a and
b; this number rn will be the greatest common divisor of a and b. Among
succeeding propositions, we find equivalents of familiar theorems in
arithmetic. Thus, Proposition 8 states that if an5bm and cn5dm, then
(a2c)n5(b2d)m; Proposition 24 states that if a and b are prime to c,
then ab is prime to c. The book closes with a rule (Proposition 39) for
finding the least common multiple of several numbers.
Book VIII is one of the less rewarding of the thirteen books of the
Elements. It opens with propositions on numbers in continued proportion
(geometric progression) and then turns to some simple properties of
squares and cubes, closing with Proposition 27: “Similar solid numbers
have to one another the ratio which a cube number has to a cube
number.” This statement simply means that if we have a “solid number”
ma mb mc and a “similar solid number” na nb nc, then their ratio will
be m3:n3—that is, as a cube is to a cube.
Book IX, the last of the three books on the theory of numbers, contains
several theorems that are of special interest. Of these, the most celebrated
is Proposition 20: “Prime numbers are more than any assigned
multitude of prime numbers.” Euclid here gives the well-known elementary
proof that the number of primes is infinite. The proof is indirect,

 

for one shows that the assumption of a finite number of primes leads to a
contradiction. Let P be the product of all of the primes, assumed to be
finite in number, and consider the number N5P11. Now, N cannot
be prime, for this would contradict the assumption that P was the product
of all primes. Hence, N is composite and must be measured by some
prime p. But p cannot be any of the prime factors in P, for then it would
have to be a factor of 1. Hence, p must be a prime different from all of
those in the product P; therefore, the assumption that P was the product
of all of the primes must be false.
Proposition 35 of this book contains a formula for the sum of numbers
in geometric progression, expressed in elegant but unusual terms:
If as many numbers as we please be in continued proportion, and there be
subtracted from the second and the last numbers equal to the first, then as
the excess of the second is to the first, so will the excess of the last be to
all those before it.
This statement is, of course, equivalent to the formula

 

which in turn is equivalent to

 

The following and last proposition in Book IX is the well-known formula
for perfect numbers: “If as many numbers as we please, beginning
from unity, be set out continuously in double proportion until the sum of
all becomes prime, and if the sum is multiplied by the last, the product
will be perfect.” That is, in modern notation, if Sn51121221?1
2n 152n21 is prime, then 2n 1(2n21) is a perfect number. The proof
is easily established in terms of the definition of a perfect number given
in Book VII. The ancient Greeks knew the first four perfect numbers:
6, 28, 496, and 8128. Euclid did not answer the converse question—
whether his formula provides all perfect numbers. It is now known that
all even perfect numbers are of Euclid’s type, but the question of the
existence of odd perfect numbers remains an unsolved problem. Of
the two dozen perfect numbers now known, all are even, but to conclude
by induction that all must be even would be hazardous.
In Propositions 21 through 36 of Book IX, there is a unity that
suggests that these theorems were at one time a self-contained mathematical
system, possibly the oldest in the history of mathematics and
stemming presumably from the middle or early fifth century BCE. It has

 

even been suggested that Propositions 1 through 36 of Book IX were
taken over by Euclid, without essential changes, from a Pythagorean
textbook.
Incommensurability
Book X of the Elements was, before the advent of early modern algebra,
the most admired—and the most feared. It is concerned with a systematic
classification of incommensurable line segments of the forms
a6 b;
p
a
p
6 b;
p
a6 b
p p
and a
p
6 b
p p
, where a and b, when of
the same dimension, are commensurable. Today, we would be inclined
to think of this as a book on irrational numbers of the types above, where
a and b are rational numbers, but Euclid regarded this book as a part of
geometry, rather than of arithmetic. In fact, Propositions 2 and 3 of the
book duplicate for geometric magnitudes the first two propositions of
Book VII, where the author had dealt with whole numbers. Here he
proves that if to two unequal line segments one applies the process
described previously as Euclid’s algorithm, and if the remainder never
measures the one before it, the magnitudes are incommensurable. Proposition
3 shows that the algorithm, when applied to two commensurable
magnitudes, will provide the greatest common measure of the segments.
Book X contains 115 propositions—more than any other—most of
which contain geometric equivalents of what we now know arithmetically
as surds. Among the theorems are counterparts of rationalizing
denominators of fractions of the form a=ðb6 c
p
Þ and a=ð b
p
6 c
p
Þ:
Line segments given by square roots, or by square roots of sums of
square roots, are about as easily constructed by straightedge and compasses
as are rational combinations. One reason that the Greeks turned to
a geometric, rather than an arithmetic, algebra was that in view of the
lack of the real-number concept, the former appeared to be more general
than the latter. The roots of ax2x25b2, for example, can always be
constructed (provided that a.2b). Why, then, should Euclid have gone
to great lengths to demonstrate, in Propositions 17 and 18 of Book X, the
conditions under which the roots of this equation are commensurable
with a? He showed that the roots are commensurable or incommensurable,
with respect to a, according as a2 24b2
p
and a are commensurable
or incommensurable. It has been suggested that such considerations
indicate that the Greeks also used their solutions of quadratic equations
for numerical problems, much as the Babylonians had in their system of
equations x1y5a, xy5b2. In such cases, it would be advantageous to
know whether the roots will or will not be expressible as quotients of
integers. A close study of Greek mathematics seems to give evidence
that beneath the geometric veneer, there was more concern for logistic

 

and numerical approximations than the surviving classical treatises
portray.
Solid Geometry
The material in Book XI, containing thirty-nine propositions on the
geometry of three dimensions, will be largely familiar to one who has
taken a course in the elements of solid geometry. Again, the definitions
are easily criticized, for Euclid defines a solid as “that which has length,
breadth, and depth” and then tells us that “an extremity of a solid is a
surface.” The last four definitions are of four of the regular solids. The
tetrahedron is not included, presumably because of an earlier definition
of a pyramid as “a solid figure, contained by planes, which is constructed
from one plane to any point.” The eighteen propositions of
Book XII are all related to the measurement of figures, using the
method of exhaustion. The book opens with a careful proof of the
theorem that areas of circles are to each other as squares on the diameters.
Similar applications of the typical double reductio ad absurdum
method are then applied to the volumetric mensuration of pyramids,
cones, cylinders, and spheres. Archimedes ascribed the rigorous proofs
of these theorems to Eudoxus, from whom Euclid probably adapted
much of this material.
The last book is devoted entirely to properties of the five regular
solids. The closing theorems are a fitting climax to a remarkable treatise.
Their object is to “comprehend” each of the regular solids in a sphere—
that is, to find the ratio of an edge of the solid to the radius of the circumscribed
sphere. Such computations are ascribed by Greek commentators
to Theaetetus, to whom much of Book XIII is probably due. In
preliminaries to these computations, Euclid referred once more to the
division of a line in mean and extreme ratio, showing that “the square on
the greater segment added to half the whole is five times the square
on the half”—as is easily verified by solving a / x5x / (a2x)—and
citing other properties of the diagonals of a regular pentagon. Then, in
Proposition 10, Euclid proved the well-known theorem that a triangle
whose sides are respectively sides of an equilateral pentagon, hexagon,
and decagon inscribed in the same circle is a right triangle. Propositions
13 through 17 express the ratio of edge to diameter for each of the
inscribed regular solids in turn: e/d is 2
3
p
for the tetrahedron, 1
2
p
for
the octahedron, 1
3
p
for the cube or hexahedron, (5 þ 5
p
) 10
p
for the
icosahedron, and ð 5
p
21Þ=2 3
p
for the dodecahedron. Finally, in Proposition
18, the last in the Elements, it is easily proved that there can be
no regular polyhedron beyond these five. About 1,900 years later, the
astronomer Kepler was so struck by this fact that he built a cosmology

 

on the five regular solids, believing that they must have been the
Creator’s key to the structure of the heavens.
Apocrypha
In ancient times, it was not uncommon to attribute to a celebrated author
works that were not by him; thus, some versions of Euclid’s Elements
include a fourteenth and even a fifteenth book, both shown by later
scholars to be apocryphal. The so-called Book XIV continues Euclid’s
comparison of the regular solids inscribed in a sphere, the chief results
being that the ratio of the surfaces of the dodecahedron and the icosahedron
inscribed in the same sphere is the same as the ratio of their
volumes, the ratio being that of the edge of the cube to the edge of
the icosahedron, that is, 10 [3(5 5
p
)]
p
. This book may have been
composed by Hypsicles (fl. ca. 150 BCE) on the basis of a treatise (now
lost) by Apollonius comparing the dodecahedron and the icosahedron.
Hypsicles is also the author of an astronomical work, De ascensionibus,
an adaptation for the latitude of Alexandria of a Babylonian technique
for computing the rise times of the signs of the zodiac; this work also
contains the division of the ecliptic into 360 degrees.
The spurious Book XV, which is inferior, is thought to have been
(at least in part) the work of a student of Isidore of Miletus’s (fl. ca.
532 CE), the architect of the Hagia Sophia at Constantinople. This
book also deals with the regular solids, showing how to inscribe
certain of them within others, counting the number of edges and solid
angles in the solids, and finding the measures of the dihedral angles of
faces meeting at an edge. It is of interest to note that despite such
enumerations, all of the ancients apparently missed the so-called
polyhedral formula, known to Rene´ Descartes and later enunciated by
Leonhard Euler.
Influence of the Elements
The Elements of Euclid was composed in about 300 BCE and was copied
and recopied repeatedly after that. Errors and variations inevitably crept
in, and some later editors, notably Theon of Alexandria in the late fourth
century, sought to improve on the original. Later accretions, generally
appearing as scholia, add supplementary information, often of a historical
nature, and in most cases they are readily distinguished from the
original. The transmission of translations from Greek to Latin, starting
with Boethius, has been traced in some detail. Numerous copies of the
Elements have also come down to us through Arabic translations, later

 

turned into Latin, largely in the twelfth century, and finally, in the sixteenth
century, into the vernacular. The study of transmission of these
variants presents continuing challenges.
The first printed version of the Elements appeared at Venice in 1482,
one of the very earliest of mathematical books to be set in type. It has
been estimated that since then, at least a thousand editions have been
published. Perhaps no book other than the Bible can boast so many
editions, and, certainly, no mathematical work has had an influence
comparable to that of Euclid’s Elements.