Euclid's Elements

Euclidean geometry is one of humanity's greatest intellectual achievements — a complete logical system built from just a handful of definitions and self-evident truths. This chapter introduces the foundations: Euclid's definitions, axioms, and postulates, and the geometry that flows from them. Understanding these principles is essential before exploring any other area of geometry.

Introduction

Geometry can be defined as a quantitative and qualitative study of space and forms. The most fundamental attributes about space and form include shape and size. Geometry is a deductive system. This means there is a logical development and a few simple, fundamental principles. But before moving on to the content of geometry, it is important to understand a little about the nature of geometry and something about its course of historical development.

Geometry, in a fundamental sense, is the evolution of elementary ideas about shapes and forms that arise out of our exposure to the physical and natural world. In our interactions with our environment, we encounter physical shapes — rocks, mountains, trees, sticks — things which are different and diverse but can still be organised by patterns into groups and classes. Some are 'straight-ish', others are 'round-ish'. Developing from such primitive ideas, geometry grew to become the "Art of measuring" out of practical necessities. Geometry may have its roots in our curiosity and attempt to better understand these rudimentary ideas, but the geometry that we know today didn't start developing until someone separated the physical world from the mental.

History

Geometry started taking its modern form when the human mind started "abstraction" of perfect mental objects out of the imperfect physical objects. In the mental world, we can think of a perfectly straight line instead of having to deal with an imperfect, approximately straight-ish physical stick. In the same way, a random mountain terrain may provide the idea about a perfect triangular abstract form; the sun or moon's shape results in the idea of a perfect abstract circle and so on.

The key thing to understand is that in geometry we don't deal with worldly imperfections, but instead our objects of study are perfect mental constructs which have evolved from a process of abstraction out of physical and natural forms. The next important stage in the evolution of geometry was when the Greeks started learning geometry from the Egyptians. Where under the Egyptians, geometry remained more like a collection of random results, it developed into what we know today as the "axiomatic system" under the Greeks.

Thales and Pythagoras were among the first prominent figures who learnt geometry from the Egyptians and then went on starting their own schools in Greece. One of the most important features of the schools of both Pythagoras and Thales was their insistence that mathematical results be justified or proved true. This focus on proofs required a method of reasoning and argument that was precise and logical. This method culminated with The Elements by Euclid in about 300 BC, known as the Axiomatic Method.

Principles

The axiomatic method is based on a system of deductive reasoning. In a deductive system, statements used in an argument must be derived from prior statements, which must themselves be derived from even earlier statements, and so on. A postulate or axiom is a logical statement accepted as true without proof. Starting from a base of defined terms and agreed-upon axioms, we can define other terms and use our axioms to argue the truth of other statements.

Greek geometer Proclus (411–485 AD) believed that by training our minds in the most careful and rigorous forms of reasoning abstracted from the real world, we are preparing our minds for the harder task of reasoning about things we cannot perceive. For Proclus, this type of reasoning arouses our innate knowledge, awakens our intellect, purges our understanding, brings to light the concepts that belong essentially to us, takes away the forgetfulness and ignorance that we have from birth, and sets us free from the bonds of unreason.

The system of deductive reasoning was put into definitive form by Euclid around 300 BC in his 13-volume work Elements. Euclid was a scholar at the Museum of Alexandria, noted for his lucid exposition and great teaching ability. The Elements superseded all previous textbooks in geometry and persisted as the standard textbook for the next two millennia.

Euclid's Definitions

Euclid provides clear and concise definitions for the terms that he uses:

• A point is that which has no part.
• A line is breadthless length.
• The ends of a line are points.
• A straight line is a line which lies evenly with the points on itself.
• A surface is that which has length and breadth only.
• Rectilinear figures are those contained by straight lines — trilateral (three), quadrilateral (four), multilateral (more than four).
• An equilateral triangle has its three sides equal; an isosceles triangle has two sides equal; a scalene triangle has all three sides unequal.
• When a straight line standing on a straight line makes the adjacent angles equal to one another, each is a right angle.
• Parallel straight lines are straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction.

Common Notions

Common notions are universal statements about general logical systems:

1. Things which are equal to the same thing are also equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are 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.

Postulates

Postulates are assumptions specific to the objects under study:

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 centre and radius.
4. That all right angles equal one another.
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 are the angles less than two right angles.

The first three postulates provide the theoretical foundation for constructing figures based on a perfect straightedge and compass. The fourth follows directly from the definition of a right angle. The fifth — the Parallel Postulate — is the most significant.

The 5th Postulate

Euclid's fifth postulate does not look like the other four. It is considerably longer and more complicated. For that reason, generations of geometers hoped it might be provable from the first four — that it could be taken as a theorem rather than a postulate.

Many attempts were made to prove the fifth postulate. Invariably the mistake was assuming some 'obvious' property that turned out to be equivalent to the fifth postulate itself. We now know that it is impossible to derive the Parallel Postulate from the first four.

The best-known equivalent is Playfair's Axiom: "In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point."

Other equivalent statements include:

• The sum of the angles in every triangle is 180°.
• There exists a pair of similar non-congruent triangles.
• There exists a pair of straight lines everywhere equidistant from one another.
• For any three non-collinear points, there exists a circle passing through them.
• Two straight lines that intersect one another cannot be parallel to a third line.

Non-Euclidean Geometries

Non-Euclidean geometry arises by replacing the parallel postulate with an alternative:

• Euclidean geometry — parallel lines remain at a constant distance from each other.
• Hyperbolic geometry — parallel lines "curve away" from each other, increasing in distance.
• Elliptic geometry — lines "curve toward" each other and intersect.

Neutral geometry (or absolute geometry) is the study of results that hold regardless of which parallel postulate is assumed — the common ground between Euclidean and non-Euclidean geometry.

Projective Geometry

Projective geometry is a non-metric system that studies geometric properties invariant under projection. In projective space, parallel lines meet at a point on a horizon (the point at infinity) — unlike Euclidean space where they never meet. Projective geometry excludes compass constructions: there are no circles, no angles, no measurements, and no parallels.

Conclusion

A plethora of geometries exist in modern times that deviate from classical Euclidean geometry in many different ways. But Euclid's geometry still forms the core of geometric education and is the indispensable stepping stone for anyone interested in learning geometry.

The next chapter begins building on these foundations, examining the most elementary elements of Euclidean geometry in detail — Lines and Angles.