Stage: 3, 4 and 5
1. Ancient and Classical Geometries
As an essential part of their daily lives, ancient cultures
knew a considerable amount of geometry as practical measurement
and as rules for dividing and combining shapes of different
kinds for building temples, palaces and for civil engineering.
For their everyday practical purposes, people lived on a 'flat'
Earth. A 'straight line' was a tightly stretched rope, and a
circle could be drawn by tracing round a fixed point.
Aristotle (384-322) BCE
Much of the knowledge of these peoples was well-known around
the Mediterranean, and when the Greek civilisation began to
assert itself in the 4th century BCE, philosophers like
Aristotle (384-322 BCE), developed a particular way of
thinking, and promoted a mode of discussion which required the
participants to state as clearly as possible the basis of their
argument. In this atmosphere, Greek Logic was born.
Euclid of Alexandria
(325-265) BCE
During this period, Alexandria became one of the important
centres of Greek learning and this is where Euclid's Elements
of Mathematics was written in about 300 BCE. Following
Aristotle's principles, Euclid based his mathematics on a
series of definitions of basic objects like points, straight
lines, surfaces, angles, circles and triangles, and axioms (or
postulates). These were the agreed starting points for his
development of mathematics.
The first three postulates are about what can be done, the next one about equality of right angles and the final statement uses the sum of two right angles to define whether two lines meet:
- Draw a straight line from any point to any other point.
- Produce (extend) a finite straight line continuously in a straight line.
- Describe a circle with any centre and distance.
- All right angles are equal to each other.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then if the two lines are produced indefinitely, they will meet on that side where the angles are less than the two right angles.
Almost as soon as Euclid put his pen down, mathematicians and
philosophers were having difficulty with the fifth postulate.
In contrast to the short statements of the first four, the
fifth looked as though it ought to be a theorem, not an axiom,
meaning that it ought to be deducible from the other axioms. We
know this from various logical analyses written by other
mathematicians. In the fifth century CE, Proclus (411-485 CE)
gave a simpler version of the fifth postulate:
- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
Playfair's Axiom
John Playfair (1748-1819)
Today, this is known as Playfair's axiom, after the English
mathematician John Playfair who wrote an important work on
Euclid in 1795, even though this axiom had been known for over
1200 years!
Arab mathematicians studied the Greek works, logically analysed
the relatively complex statement of the fifth postulate, and
produced their own versions.
Abul Wafa al-Buzjani (940-998)
Abul Wafa developed some important ideas in trigonometry and is
said to have devised a wall quadrant [See Note 1 below] for the
accurate measurement of the declination of stars. He also
introduced the tangent, secant and cosecant functions and
improved methods for calculating trigonometrical tables to 15'
intervals and accurate to 8 decimal places. All this was done
as part of an investigation into the Moon's orbit in his
Theories of the Moon .
The Abul Wafa crater is named after him.
As a result of his trigonometric investigations, he developed
ways of solving some problems of spherical triangles.
Greek astronomers had long since introduced a geometrical model
of the universe. Abul Wafa was the first Arab astronomer to use
the idea of a spherical triangle to develop ways of measuring
the distance between stars on the inside of a sphere. In the
accompanying diagram, the blue triangle with sides a, b, and c
represents the distances between stars on the inside of a
sphere. The apex where the three angles are marked is the
position of the observer.
Spherical Triangle
Omar Khayyam (1048-1131)
Famous for his poetry, Omar Khayyam was also an outstanding
astronomer and mathematician who wrote Commentaries on the difficult postulates
of Euclid's book . He tried to prove the fifth postulate
and found that he had discovered some non-Euclidean properties
of figures.
Omar Khayyam (1048-1131)
Omar Khayyam Quadrilateral
Omar Khayyam constructed the quadrilateral shown in the figure
in an effort to prove that Euclid's fifth postulate could be
deduced from the other four. He began by constructing equal
line segments AD and BC perpendicular to AB. He recognized that
if, by connecting C and D, he could prove that the internal
angles at the top of the quadrilateral are right angles, then
he would have shown that DC is parallel to AB. Although he
showed that the internal angles at the top are equal (try it
yourself) he could not prove that they were right angles.
Nasir al-Din al-Tusi (1201-1274)
Nasir al-Din al-Tusi (1201-1274)
Al-Tusi wrote commentaries on many Greek texts and his work on
Euclid's fifth postulate was translated into Latin and can be
found in John Wallis' work of 1693.
He criticised Euclid's proposition I, 28
"If a straight line falling on two straight lines makes the
exterior angle equal to the interior and opposite angle on the
same side, or the sum of the interior angles on the same side
equal to two right angles, then the straight lines are parallel
to one another."
Al-Tusi's diagram
Al-Tusi's original diagram
Al-Tusi's argument looked at the second part of the statement.
Given two lines, AB and CD in the plane and a series of
perpendiculars to CD drawn from PQ to XY so that they meet AB.
On each side of these perpendiculars, one angle is acute
(towards A), and the other obtuse (towards B). Clearly the
perpendicular PQ is longer than each of the others and finally
longer than XY. The opposite is also true; perpendicular XY is
shorter than all those up to and including EF. So, if any pair
of these perpendiculars is chosen to make a rectangle, the
rectangle will contain an acute angle (on the A side) and an
obtuse angle (on the B side). So how can we ensure that the
perpendiculars are the same length, or show that both angles
are right angles?
One of al-Tusi's most important mathematical contributions was
to show that the whole system
of plane and spherical trigonometry was an independent branch
of mathematics . In setting up the system, he discussed
the comparison of curved lines and straight lines. The 'sine
formula' for plane triangles had been known for some time, and
Al-Tusi established an analogous formula for spherical
triangles:
Plane triangle sine rule
Spherical triangle sine rule
Great Circles Triangle
The important idea here is that Abul Wafa and al-Tusi were
dealing with the real problems of astronomy and between them
they produced the first
real-world non-Euclidean geometry which required
calculation for its justification as well as logical argument.
It was the ' Geometry of the
Inside of a Sphere '.
2. Renaissance and Early Modern Developments
The Painters' Perspective
In the Middle Ages the function of Christian Art was largely
hierarchical. Important people were made larger than others in
the picture, and sometimes to give the impression of depth,
groups of saints or angels were lined up in rows one behind the
other like on a football terrace. Euclid's Optics provided a
theoretical geometry of vision, but when the optical work of
Al-Haytham (965-1039) became known, artists began to develop
new techniques. Pictures in correct perspective appear in the
fourteenth century, and methods of constructing the 'pavement'
were no doubt handed down from master to apprentice.
Leone Battista Alberti
(1404-1472) published the first description of the method in
1435, and dedicated his book to Fillipo Brunelleschi
(1377-1446) who is the person who gave the first correct method
for constructing linear perspective and was clearly using this
method by 1413.
Leone Battista Alberti (1404-1472)
Alberti Perspective Construction
Alberti's method here is called distance point construction. In
the centre of the picture plane, mark a line H (the horizon)
and on it mark V (the vanishing point). Draw a series of
equally spaced lines from V to the bottom of the picture. Then
mark any point Z on the horizon line and draw a line from Z to
the corner of the frame underneath H. This line will intersect
all the lines from V. The points of intersection give the
correct spaces for drawing the horizontal lines of the
'pavement' on which the painting will be based.
Piero della Francesca (1412-1492) was a highly
competent mathematician who wrote treatises on arithmetic and
algebra and a classic work on perspective in which he
demonstrates the important converse of proposition 21 in Euclid
Book VI:b
"Figures which are similar to the same rectilinear figure are
also similar to one another".
Euclid uses this proposition to establish that similarity is a
transitive relation.
Piero's converse showed that if a pair of unequal parallel
segments are divided into equal parts, the lines joining
corresponding points converge to the vanishing point.
Piero della Francesca (1412-1492)
Piero Euclid VI, 21 diagram
Piero's argument was based on the fact that each of the pairs
of triangles $ABD$ and $AHK, ADE$ and $AKL$, etc. are similar,
because $HK$ is parallel to $BC$, and that the ratio $AB$ to
$BC$ is the same as $AH$ to $HI$. This implies that all the
converging lines meet at A, the vanishing point (at infinity).
Other famous artists improved on these methods, and in 1525 Albrecht Durer (1471-1528) produced a book demonstrating a number of mechanical aids for perspective drawing.
Durer "Reclining woman" perspective picture
Albrecht Durer (1471-1528)
Desargues and Projective Geometry
In 1639, Girard Desargues (1591-1661) wrote his ground-breaking
treatise on projective geometry. He had earlier produced a
manual of practical perspective for Architects and another on
stone cutting for Masons, but his approach was theoretical and
difficult to understand. In his 1639 treatise he introduced
many new fundamental concepts. The term 'point at infinity'
(the vanishing point) appears for the first time. He also uses
the ideas of a 'cone of vision' and talks about 'pencils of
lines', like the lines emanating from the vanishing point, (and
if you can have a point at infinity, why not more, to make
lines at infinity?).
This was a completely new kind of geometry. The fundamental
relationships were based on ideas of 'projection and section'
which means that any rigid
Euclidean shape can be transformed into another 'similar' shape
by a perspective transformation .
A square can be transformed into a parallelogram (think of
shadow play) and while the number and order of the sides remain
the same, their length varies.
Durer's cone picture
The new geometry was not recognized at the time, because
Desargues' technical language was difficult, and also because
Rene Descartes' coordinate geometry published three years
earlier was so popular. In the late 18th century Desargues'
work was rediscovered, and developed both theoretically and
practically into a coherent system, with central concepts of
invariance and
duality .
In Projective geometry lengths, and ratios of lengths, angles
and the shapes of figures, can all change under projection.
Parallel lines do not exist because any pair of distinct lines
intersect in a point.
Properties that are invariant
under projection are the order of three or more points
on a line and the 'cross ratio, among four points, $A, B, C,
D,$ so that $\frac{AC}{BC}=\frac{AD}{BD}$
Another important concept in projective geometry is
duality . In the
plane, the terms 'point' and 'line' are dual and can be
interchanged in any valid statement to yield another valid
statement.
See Leo's
articles on Proof
(for
the Invariants and Projection and Section) and on
the Four Colour
Theorem (for
Duality).
3. Modern Geometries
In spite of the practical inventions of Spherical Trigonometry
by Arab Astronomers, of Perspective Geometry by Renaissance
Painters, and Projective Geometry by Desargues and later 18th
century mathematicians, Euclidean Geometry was still held to be
the true geometry of the real world. Nevertheless,
mathematicians still worried about the validity of the parallel
postulate.
In 1663 the English mathematician John Wallis had translated
the work of al-Tusi and followed his line of reasoning. To
prove the fifth postulate he assumed that for every figure
there is a similar one of arbitrary size. However, Wallis
realized that his proof was based on an assumption equivalent
to the parallel postulate.
Saccheri's title page
Girolamo Saccheri
(1667-1733) entered the Jesuit Order in 1685. He went to Milan,
studied philosophy and theology and mathematics. He became a
priest and taught at various Jesuit Colleges, finally teaching
philosophy and theology at Pavia, and holding the chair of
mathematics there until his death. Saccheri knew about the work
of the Arab mathematicians and followed the reasoning of
al-Tusi in his investigation of the parallel postulate, and in
1733 he published his famous book, Euclid Freed from Every
Flaw.
In his first proposition at the beginning of his book, Saccheri
constructed a quadrilateral in a similar manner to that of Omar
Khayyam (above) and proved that the angles $ADB$ and $BCA$ are
equal. He then considered the length of the upper side of the
quadrilateral $CD$, and in Proposition III set up the three
possibilities, depending on whether $CD$ is equal to, or less,
or greater than the base $AB$.
These possibilities are equivalent to:
Hypothesis I : There is
exactly one parallel (the right angle case, $CD=AB$)
Hypothesis II: There
are no parallels (the obtuse angle case, $CD$< $AB$)
Hypothesis III : There
are more than one parallel (the acute angle case, $CD$>
$AB$)
Saccheri Hypotheses Diagram
Saccheri assumes that (i) a straight line divides the plane
into two separate regions and (ii) that straight line can be
infinite in extent. These assumptions are incompatible with the
obtuse angle case, and so this is rejected. However, they are
compatible with the acute angle case, and we can see from his
diagram (fig. 33) and Proposition XXXII below that he is
treating the intersection at infinity as a finite point, and
this is where his contradiction lies.
Proposition XXXII
"Now I say there is (in the hypothesis of acute angle) a
certain determinate acute angle $BAX$ drawn under which $AX$
(fig. 33) only at an infinite distance meets $BX$, and thus is
a limit in part from within, in part from without; on the one
hand of all those which under lesser acute angles meet the
aforesaid $BX$ at a finite distance; on the other hand also of
the others which under greater acute angles, even to a right
angle inclusive, have a common perpendicular in two distinct
points with $BX$."
To us now, the curved line AX looks like an asymptote, but he
says that $AX$ meets $BX$ "at an infinite distance" so that in
the next Proposition XXXIII he states:
"The hypothesis of acute angle is absolutely false; because it
is repugnant to the nature of the straight line".
The irony is that in the next twenty or so pages, in order to
show that the acute angle case is impossible, he demonstrates a
number of elegant theorems of non-Euclidean geometry! It was
clear that Saccheri could not cope with a perfectly logical
conclusion that appeared to him to be against common sense.
Saccheri's work was virtually unknown until 1899 when it was
discovered and republished by the Italian mathematician,
Eugenio Beltrami (1835-1900). As far as we know it had no
influence on Lambert, Legendre or Gauss.
Johan Heinrich Lambert
(1728-1777)
Johan Heinrich Lambert
(1728-1777) followed a similar plan to Saccheri. He
investigated the hypothesis of the acute angle without
obtaining a contradiction. Lambert noticed the curious fact
that, in this new geometry, the angle sum of a triangle
increased as the area of the triangle decreased.
Adrien-Marie Legendre
(1752-1833) spent many years working on the parallel postulate
and his efforts appear in different editions of his
Elements de Geometrie
. Legendre proved that the fifth postulate is equivalent to the
statement that the sum of the
angles of a triangle is equal to two right angles .
Legendre also obtained a number of consistent but
counter-intuitive results in his investigations, but was unable
to bring these ideas together into a consistent system.
Many of the consequences of the Parallel Postulate, taken with
the other four axioms for plane geometry, can be shown
logically to imply the Parallel Postulate. For example, these
statements can also be regarded as equivalent to the Parallel
Postulate.
- In any triangle, the three angles sum to two right angles.
- In any triangle, each exterior angle equals the sum of the two internally opposite angles.
- If two parallel lines are cut by a transversal, the alternate interior angles are equal, and the corresponding angles are equal.
Carl Freidrich Gauss (1777-1855)
Carl Freidrich Gauss (1777-1855)
Gauss was the first person to truly understand the problem of
parallels. He began work on the fifth postulate by attempting
to prove it from the other four. But by 1817 he was convinced
that the fifth postulate was independent of the other four, and
then began to work on a geometry where more than one line can
be drawn through a given point parallel to a given line. He
told one or two close friends about his work, though he never
published it and in a private letter of 1824 he wrote:
"The assumption that (in a triangle) the sum of the three
angles is less than 180o leads to a curious geometry, quite
different from ours, but thoroughly consistent, which I have
developed to my entire satisfaction".
The final breakthrough was made quite independently by two men,
and it is clear that both Bolyai and Lobachevski were
completely unaware of each other's work.
Nikolai Ivanovich Lobachevski
(1792-1856)
Nikolai Ivanovich
Lobachevski (1792-1856) did not try to prove the fifth
postulate but worked on a geometry where the fifth postulate
does not necessarily hold. Lobachevski thought of Euclidean
geometry as a special case of this more general geometry, and
so was more open to strange and unusual possibilities. In 1829
he published the first account of his investigations in Russian
in a journal of the university of Kazan but it was not noticed.
His original work, Geometriya had already been
completed in 1823, but not published until 1909.
Lobachevski explained how his geometry works, "All straight
lines which in a plane go out from a point can, with reference
to a given straight line in the same plane, be divided into two
classes - into cutting and non-cutting. The boundary lines of
the one and the other class of those lines will be called
parallel to the given line."
The red line is the boundary, the 'parallel' to the line BC.
The red line is the boundary, the 'parallel' to the line BC.
Lobachevski tried to get his work Geometrical investigations on the theory
of parallels recognized, and an account in French in
1837 brought his work on non-Euclidean geometry to a wide
audience but the mathematical community was not yet ready to
accept these revolutionary ideas.
Lobachevski Diagram
Janos Bolyai
(1802-1860) was the son of the mathematician Farkas Bolyai, a
friend of Gauss. Farkas had worked on the problem of the fifth
postulate, but had not been able to make any headway.
Janos Bolyai (1802-1860) )
In 1823 young Janos wrote to his father saying, "I have discovered things so wonderful
that I was astounded ... out of nothing I have created a
strange new world." However it took Janos two more years
before it was completed and his work was published as an
appendix to his father's text-book. Janos had shown that a
consistent geometry using the acute angle hypothesis case was
possible.
Janos Bolyai set out to investigate the three basic hypotheses of the right, obtuse, and acute angles by separating the case where the fifth postulate was true (the right angle case) from the cases where it was not true. On this basis he set up two systems of geometry, and searched for theorems that could be valid in both.
Janos Bolyai's work was read by Gauss who recognized and gave
credit to the young genius. However, when Gauss later explained
to Janos that he himself had made these discoveries some years
before, Janos was devastated. Later, Janos learned that
Lobachevski had anticipated his work which disappointed him
even more. He continued to work in mathematics, presenting some
original ideas, but his enthusiasm and health deteriorated and
he never published again.
Lobachevski and Bolyai had discovered what we now call
Hyperbolic Geometry. This is the geometry of the acute angle
hypothesis where a 'line' is no longer a straight line and
there are many possible lines through a given point which do
not intersect another line. This is very difficult to
visualize, and for people brought up to believe Euclidean
geometry was 'true' this was counter-intuitive and
unacceptable.
Eugenio Beltrami (1835-1900)
It was not until Beltrami produced the first model for
hyperbolic geometry on the surface of a pseudo-sphere in 1868
that many mathematicians began to accept this strange new
geometry.
Imagine a circular polar grid (like a dart board) pulled up from the origin. It forms a trumpet-like surface. Any triangle drawn on this grid will become even more distorted when an apex is near the origin. All the lines going up the surface are asymptotes to a single central line rising vertically from the origin. These lines are all 'parallel' lines passing through a single limit point at infinity.
If the Tractrix is rotated about its vertical axis, the surface formed will be a complete Pseudo-sphere.
In the Poincare Model, all 'lines' are arcs of circles, except
for the diameter (the arc of a circle with infinite radius).
'Parallel' lines are thought of as asymptotes where the limit
point is on the circumference. With this model many 'parallels'
can pass through the same point. This disc has a basic four-fold
symmetry. The Yellow Poincare Disc has symmetry order seven.
Maurits Escher used a six-fold symmetry for his "Circle Limit IV"
engraving - the picture with the interlocking angels and devils.
For more on Escher see: http://www.mcescher.com/
Gradually other models helped to make the new ideas more secure
and in 1872 the famous German mathematician Felix Klein
(1849-1925) produced his general view of geometry by unifying
the different Spherical, Perspective Projective and Hyperbolic
geometries with others as sets of axioms and properties
invariant under the action of certain transformations. In this
way, mathematicians at last became free to think of geometry in
the abstract as a set of axioms, operations and logical rules
that were not tied to the physical world.
For pedagogical notes: Use the notes tab at the top of this article or click here .
Notes
1. Wall quadrants were invented and used for many years by astronomers for measuring the altitude of heavenly bodies. They have been specially built as part of ancient observatories, and as they became larger had to be supported by solid walls to keep them steady. It was believed that the larger the instrument was, the more accurate were the results obtained. It is true that the larger the instrument is, the easier it is to divide the scale of the quadrant into degrees, minutes and seconds. However, the accuracy can also depend on other things like the sighting instrument. For example, telescopes were not developed well enough to be reliable until the early 18th century, and because the mounting was fixed, it had limited use. In spite of the problems, Arab astronomers were able to achieve an accuracy of about 20 seconds of arc.
Weblinks
These are the most reliable and accurate links. A quick
search in Wikipedia often gives basic information, but be
careful. It is always best to cross-check details with other
sites.
For all biographical details and special pages on
non-Euclidean geometries and Mathematics and Art the MacTutor
site at St Andrews University go to:
For more detail on Mathematical techniques in Astronomy go to
the 'Starry Messenger' site of the History of Science at
Cambridge University:
The Cut-the-Knot site has a good set of pages on
non-Euclidean Geometry:
For excellent exposition and explanations of Euclid with Java
applets go to David Clark's site at:
'The Origins of Perspective' is section 11 of a more
extensive course on Art and Architecture based at Dartmouth
college:
And, if you have on-line access to Encyclopaedia Britannica
or the Dictionary of Scientific Biography, then of course
these give you much greater detail if you need it.
Articles
Some books that open us to the range and fascination of
cultural links are:
Michele Emmer, (1993) The Visual Mind; Art and Mathematics MIT Press
J.L. Heilbron, (1998) Geometry Civilised; History, Culture and Technique . Clarendon Press, Oxford.
AND a book to look out for:
Eleanor Robson and Jackie Stedall (Editors) (December 2008), The Oxford Handbook of the History of Mathematics . Oxford University Press
History of mathematics. Making and testing hypotheses. Patterned numbers. History of number systems. Mathematical reasoning & proof. History of measurement. Mixed trig ratios. physics. Generalising. Famous mathematicians.
Published November 2008.
Copyright © 2007. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project, which also includes the Plus and Motivate sites.
email: nrich@damtp.cam.ac.uk
