# Geometry: A History From Practice to Abstraction

### 1. Ancient and Classical Geometries

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Aristotle (384-322) BCE

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Euclid of Alexandria

(325-265)

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.

- 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)

#### Abul Wafa al-Buzjani (940-998)

*Theories of the Moon*. The Abul Wafa crater is named after him.

###### Spherical Triangle

#### Omar Khayyam (1048-1131)

*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

#### Nasir al-Din al-Tusi (1201-1274)

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Nasir al-Din al-Tusi (1201-1274)

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Al-Tusi's diagram

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Al-Tusi's original diagram

*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:

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Plane triangle sine rule

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Spherical triangle sine rule

###### Great Circles Triangle

*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

**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

**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

Piero della Francesca (1412-1492)

###### Piero Euclid VI, 21 diagram

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?).

*any rigid Euclidean shape can be transformed into another 'similar' shape by a perspective transformation*.

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Durer's cone picture

*invariance*and

*duality*.

*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 $\frac{AC}{BC}=\frac{AD}{BD}$

*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

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.

**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, $CD$> $AB$)

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Saccheri Hypotheses Diagram

###### 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 *Éléments de géométrie*. 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.

- 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 Friedrich Gauss (1777-1855)

###### Carl Friedrich Gauss (1777-1855)

###### 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.

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

**János 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.

###### János Bolyai (1802-1860) )

**János**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 János two more years before it was completed and his work was published as an appendix to his father's text-book. János had shown that a consistent geometry using the acute angle hypothesis case was possible.

János 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.

###### Eugenio Beltrami (1835-1900)

###### 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.

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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/

### 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

### Articles

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