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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
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
- All right angles are equal to each
- 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
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.
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
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
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
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 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
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
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
are similar to the same rectilinear figure are also similar to one
Euclid uses this proposition to establish that similarity is a
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
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
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
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
Invariants and Projection and Section) and on the Four Colour
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
Saccheri's title page
(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.
"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:
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
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
(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:
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
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
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.
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
(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
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
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
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
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
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 .
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.
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
The Cut-the-Knot site has a good set of pages on non-Euclidean
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.
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,
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