Subject: PLANE AND SOLID GEOMETRY
At the end of the session/s, the students should be able to:
•Discuss theorems familiar from high school geometry the traditional viewpoint
•Discover any hidden assumptions that are made by Euclid in his axioms and proofs, or
appeals to intuition instead of logic
Introduction
The focus of geometry continues to evolve with time. The renewed emphasis on
geometry today is a response to the realization that visualization, problem-solving and
deductive reasoning must be a part of everyone’s education. Deductive reasoning has
long been an integral part of geometry, but the introduction in recent years of
inexpensive dynamic geometry software programs has added visualization and
individual exploration to the study of geometry. All the constructions underlying Euclidean
plane geometry can now be made accurately and conveniently. The dynamic nature
of the construction process means that many possibilities can be considered, thereby
encouraging exploration of a given problem or the formulation of conjectures. Thus,
geometry is ideally suited to the development of visualization and problem-solving skills
as well as deductive reasoning skills. Geometry itself hasn’t changed: technology has
simply added a powerful new tool for use while studying geometry.
So, what is geometry? Meaning literally “earth measure”, geometry began several
thousand years ago for strictly utilitarian purposes in agriculture and building construction.
The explicit 3-4-5 example of the Pythagorean Theorem, for instance, was used by the
Egyptians in determining a square corner for a field or the base of a pyramid long before
the theorem as we know it was established. But from the sixth through the fourth centuries
BC, Greek scholars transformed empirical and quantitative geometry into a logically
ordered body of knowledge. They sought irrefutable proof of abstract geometric truths,
culminating in Euclid’s Elements published around 300 BC. Euclid’s treatment of the
subject has had an enormous influence on mathematics ever since, so much so that
deductive reasoning is the method of mathematical inquiry today. In fact, this is often
interpreted as meaning “geometry is 2-column proofs”. In other words, geometry is a
formal axiomatic structure – typically the axioms of Euclidean plane geometry - and one
objective of this course is to develop the axiomatic approach to various geometries,
including plane geometry. This is a very important, though limited, interpretation of the
need to study geometry, as there is more to learn from geometry than formal axiomatic
structure. Successful problem solving requires a deep knowledge of a large body of
geometry and of different geometric techniques, whether these are acquired by
emphasizing the ‘proving’ of theorems.
Evidence of geometry is found in all cultures. Geometric patterns have always
been used to decorate buildings, utensils and weapons, reflecting the fact that
geometry underlies the creation of design and structures. Patterns are visually appealing
because they often contain some symmetry or sense of proportion. Symmetries are found
throughout history, from dinosaur tracks to tire tracks. Buildings remain standing due to
the rigidity of their triangular structures. Interest in the faithful representation of a three
dimensional scene as a flat two-dimensional picture has led artists to study perspective.
In turn perspective drawing led to the introduction of projective geometry, a different
geometry from the plane geometry of Euclid. The need for better navigation as trading
distances increased along with an ever more sophisticated understanding of astronomy
led to the study of spherical geometry. But it wasn’t until the 19th century, as a result of a
study examining the role of Euclid’s parallel postulate, that geometry came to represent
the study of the geometry of surfaces, whether flat or curved. Finally, in the 20th century
this view of geometry turned out to be a vital component of Einstein’s theory of relativity.
Thus through practical, artistic and theoretical demands, geometry evolved from the flat
geometry of Euclid describing one’s immediate neighborhood, to spherical geometry
describing the world, and finally to the geometry needed for an understanding of the
universe.
History of Geometry
Geometry began with a practical need to measure shapes. The word geometry means
to “measure the earth” and is the science of shape and size of things. It is believed that
geometry first became important when an Egyptian pharaoh wanted to tax farmers who
raised crops along the Nile River. To compute the correct amount of tax the pharaoh’s
agents had to be able to measure the amount of land being cultivated.
Around 2900 BC the first Egyptian pyramid was constructed. Knowledge of geometry was
essential for building pyramids, which consisted of a square base and triangular faces.
The earliest record of a formula for calculating the area of a triangle dates back to 2000
BC. The Egyptians (5000–500 BC) and the Babylonians (4000–500 BC) developed practical
geometry to solve everyday problems, but there is no evidence that they logically
deduced geometric facts from basic principles.
It was the early Greeks (600 BC–400 AD) that developed the principles of modern
geometry beginning with Thales of Miletus (624–547 BC). Thales is credited with bringing
the science of geometry from Egypt to Greece. Thales studied similar triangles and wrote
the proof that corresponding sides of similar triangles are in proportion.
The next great Greek geometer was Pythagoras (569–475 BC). Pythagoras is regarded as
the first pure mathematician to logically deduce geometric facts from basic principles.
Pythagoras founded a brotherhood called the Pythagoreans, who pursued knowledge
in mathematics, science, and philosophy. Some people regard the Pythagorean School
as the birthplace of reason and logical thought. The most famous and useful contribution
of the Pythagoreans was the Pythagorean Theorem. The theory states that the sum of the
squares of the legs of a right triangle equals the square of the hypotenuse.
Euclid of Alexandria (325–265 BC) was one of the greatest of all the Greek geometers
and is considered by many to be the “father of modern geometry”. Euclid is best known
for his 13-book treatise The Elements. The Elements is one of the most important works in
history and had a profound impact on the development of Western civilization.
Euclid began The Elements with just a few basics, 23 definitions, 5 postulates, and 5
common notions or general axioms. An axiom is a statement that is accepted as true.
From these basics, he proved his first proposition. Once proof was established for his first
proposition, it could then be used as part of the proof of a second proposition, then a
third, and on it went. This process is known as the axiomatic approach. Euclid’s Elements
form the basis of the modern geometry that is still taught in schools today.
Archimedes of Syracuse (287–212 BC) is regarded as the greatest of the Greek
mathematicians and was also the inventor of many mechanical devices including the
screw, the pulley, and the lever. The Archimedean screw – a device for raising water from
a low level to a higher one – is an invention that is still in use today. Archimedes works
include his treatise Measurement of a Circle, which was an analysis of circular area, and
his masterpiece On the Sphere and the Cylinder in which he determined the volumes
and surface areas of spheres and cylinders.
There were no major developments in geometry until the appearance of Rene Descartes
(1596–1650). In his famous treatise Discourse on the Method of Rightly Conducting the
Reason in the Search for Truth in the Sciences, Descartes combined algebra and
geometry to create analytic geometry. Analytic geometry, also known as coordinate
geometry, involves placing a geometric figure into a coordinate system to illustrate proofs
and to obtain information using algebraic equations.
The next great development in geometry came with the development of non-Euclidean
geometry. Carl Friedrich Gauss (1777–1855) who along with Archimedes and Newton is
considered to be one of the three greatest mathematicians of all time, invented nonEuclidian geometry prior to the independent work of Janos Bolyai (1802–1860) and Nikolai
Lobachevsky (1792-1856). Non-Euclidian geometry generally refers to any geometry not
based on the postulates of Euclid, including geometries for which the parallel postulate
is not satisfied. The parallel postulate states that through a given point not on a line, there
is one and only one line parallel to that line. Non-Euclidian geometry provides the
mathematical foundation for Einstein’s Theory of Relativity.
The most recent development in geometry is fractal geometry. Fractal geometry was
developed and popularized by Benoit Mandelbrot in his 1982 book The Fractal Geometry
of Nature. A fractal is a geometric shape, which is self-similar (invariance under a change
of scale) and has fractional (fractal) dimensions. Similar to chaos theory, which is the
study of non-linear systems; fractals are highly sensitive to initial conditions where a small
change in the initial conditions of a system can lead to dramatically different outputs for
that system.
General Terms
A proof is a course of reasoning by which the truth or falsity of any statement is logically
established.
An axiom is a statement admitted to be true without proof.
A theorem is a statement to be proved.
A construction is the representation of a required figure by means of points and lines.
A postulate is a construction admitted to be possible.
A problem is a construction to be made so that it shall satisfy certain given conditions.
A proposition is an axiom, a theorem, a postulate, or a problem.
A corollary is a truth that is easily deduced from known truths.
A scholium is a remark upon some particular feature of a proposition.
The solution of a problem consists of four parts:
1. The analysis, or course of thought by which the construction of the
required figure is discovered.
2. The construction of the figure with the aid of ruler and compasses.
3. The proof that the figure satisfies all the conditions.
4. The discussion of the limitations, if any, within which the solution is possible.
A theorem consists of two parts: the hypothesis, or that which is assumed; and the
conclusion, or that which is asserted to follow from the hypothesis.
The contradictory of a theorem is a theorem which must be true if the given theorem is
false, and must be false if the given theorem is true.
Thus, A theorem: If A is B, then C is D.
Its contradictory: If A is B, then C is not D.
The opposite of a theorem is obtained by making both the hypothesis and the conclusion
negative.
Thus, A theorem: If A is B, then C is D. Its opposite: If A is not B, then C is not D.
The converse of a theorem is obtained by interchanging the hypothesis
and conclusion. Thus,
A theorem: If A is B, then C is D. Its converse: If C is D, then A is B.
The converse of a truth is not necessarily true.
Thus, Every horse is a quadruped is true, but the converse, Every quadruped is a horse, is
not true.
If a direct proposition and its opposite are true, the converse proposition is true; and if a
direct proposition and its converse are true, the opposite proposition is true.
Thus, if it were true that
1. If an animal is a horse, the animal is a quadruped;
2. If an animal is not a horse, the animal is not a quadruped;
it would follow that
3. If an animal is a quadruped, the animal is a horse.
Moreover, if 1 and 3 were true, then 2 would be true.
Undefined Terms
POINT
• a point indicates a location (or position) in space.
• a point has no dimension (actual size).
• a point has no length, no width, and no height (thickness).
• a point is usually named with a capital letter.
• in the coordinate plane, a point is named by an ordered pair, (x,y).
While we represent a point with a dot, the dot can be very tiny or very large. Remember,
a point has no size.
LINE (straight line)
• a line has no thickness.
• a line's length extends in one dimension.
• a line goes on forever in both directions.
• a line has infinite length, zero width, and zero height.
• a line is assumed to be straight.
• a line is drawn with arrowheads on both ends.
• a line is named by a single lowercase script letter, or by any two (or more) points which
lie on the line.
PLANE
• a plane has two dimensions.
• a plane forms a flat surface extending indefinitely in all directions.
• a plane has infinite length, infinite width and zero height (thickness).
• a plane is drawn as a four-sided figure resembling a tabletop or a parallelogram.
• a plane is named by a single letter (plane m) or by three coplanar, but non-collinear,*
points (plane ABC).
Definition of some Geometric terms (from Elements Book 1):
1.
2.
3.
4.
5.
6.
7.
A point is that of which there is no part.
And a line is a length without breadth
And the extremities of a line are points.
A straight-line is (any) one which lies evenly with points on itself.
And a surface is that which has length and breadth only.
And the extremities of a surface are lines.
A plane surface is (any) one which lies evenly with the straight-lines on itself.
8.
And a plane angle is the inclination of the lines to one another, when two lines in
a plane meet one another,and are not lying in a straight-line.
9. And when the lines containing the angle are straight then the angle is called
rectilinear.
10. And when a straight-line stood upon (another) straight-line makes adjacent angles
(which are) equal to one another, each of the equal angles is a right-angle, and
the former straight-line is called a perpendicular to that upon which it stands.
11. An obtuse angle is one greater than a right-angle.
12. And an acute angle (is) one less than a right-angle.
13. A boundary is that which is the extremity of some
14. A figure is that which is contained by some boundary or boundaries
15. A circle is a plane figure contained by a single line [which is called a
circumference], (such that) all of the straight-lines radiating towards [the
circumference] from one point amongst those lying inside the figure are equal to
one another.
16. And the point is called the center of the circle.
17. And a diameter of the circle is any straight-line, being drawn through the center,
and terminated in each direction by the circumference of the circle. (And) any
such (straight-line) also cuts the circle in half.
18. And a semi-circle is the figure contained by the diameter and the circumference
cuts off by it. And the center of the semi-circle is the same (point) as (the center
of) the circle.
19. Rectilinear figures are those (figures) contained by straight-lines: trilateral figures
being those contained by three straight-lines, quadrilateral by four, and multi.
lateral by more than four.
20. And of the trilateral figures: an equilateral is that having three equal sides, an
isosceles (triangle) that having only two equal sides, and a scalene (triangle) that
having three unequal sides.
21. And further of the trilateral figures: a right-angled triangle is that having a rightangle, an obtuse-angled (triangle) that having an obtuse angle, and an acute
angled (triangle) that having three acute angles.
22. And of the quadrilateral figures: a square is that which is right-angled and
equilateral, a rectangle that which is right-angled but not equilateral, a rhombus
that which is equilateral but not right-angled, and a rhomboid that having
opposite sides and angles equal to one other which is neither right-angled nor
equilateral. And let quadrilateral figures besides these be called trapezia.
23. Parallel lines are straight-lines which, being in the same plane, and being
produced to infinity in each direction, meet with one another in neither (of these
directions).
Euclid’s Postulates
Any statement that is assumed to be true on the basis of reasoning or discussion is a
postulate or axiom. The postulates stated by Euclid are the foundation of Geometry and
are rather simple observations in nature. ‘Euclid’ was a Greek mathematician regarded
as the ‘Father of Modern Geometry‘. He is credited with profound work in the fields of
algebra, geometry, science, and philosophy. Euclid introduced the fundamentals of
geometry in his book called “Elements”.
Postulate – I
A straight line segment can be formed by joining any two points in space.
In Geometry, a line segment is a part of a line that is bounded by 2 distinct points
on either end. It consists of a series of points bounded by the two endpoints. Thus a line
segment is measurable as the distance between the two endpoints. A line segment is
named after the two endpoints with an overbar on them.
Postulate – II
Any straight line can be extended indefinitely on both sides. Unlike a line segment, a line
is not bounded by any endpoint and so can be extended indefinitely in either direction.
A line is uniquely defined as passing through two points which are used to name it.
Postulate – III
A circle can be drawn with any centre and any radius. For any line segment, a
circle can be drawn with its centre at one endpoint and the radius of the circle as the
length of the line segment. Consider a line segment bounded by two points. If one of
these points is taken as the centre of a circle and the radius of the circle is taken as equal
to the length of the segment, a circle can be drawn with its diameter twice than the
length of the line segment.
Postulate – IV
All right angles are congruent or equal to one another. A right angle is an angle
measuring 90 degrees. So, irrespective of the length of a right angle or its orientation all
right angles are identical in form and coincide exactly when placed one on top of the
other.
Postulate – V
Two lines are parallel to each other if they intersect the third line and the interior angle
between them is 180 degrees.
‘Parallel lines’ are a set of 2 or more lines that never cross or intersect each other at any
point in space if they are extended indefinitely. As you can see in the above image, line
1 and line 2 are parallel if and only if the sum of angles ‘a’ and ‘b’ they make with the
transversal is 180 degrees.
Reference:
http://www.thegeodes.com/templates/geometryhistory.asp
https://mathbitsnotebook.com/Geometry/BasicTerms/BTundefined.html#:~:text=There%
20are%2C%20however%2C%20three%20words,three%20undefined%20terms%20of%20g
eometry%22.
The Project Gutenberg EBook of Plane Geometry, by George Albert Wentworth
https://www.toppr.com/guides/maths/introduction-to-euclids-geometry/euclidspostulates/