| Name: Student Name | Classical Liberal Arts Academy |
| Student ID: Student ID | Classical Geometry I, Lesson 1 By: Dr. Daniele Arcara |
| Date: 5/20/2013 | Introduction to Classical Geometry |
Learn the Basic Definitions of Planar Geometry.Print this section | Hide this section
In this lesson, you must complete the following tasks:
Study the Lesson.
Complete the Memory Work.
Complete the Lesson Exam.
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Welcome to Classical Geometry I! In this course, we begin our study of one of the most important and influential books ever written, the Elements of Euclid. This is one of the oldest Mathematical books ever written, and it was so well-done that it is still in use today. We are going to navigate through most of it in our Classical Geometry courses, learning Geometry from one of the best Mathematicians of all time.
Euclid lived in Alexandria around 300 B.C., and he wrote the Elements in Greek, compiling all of his knowledge and all known notions of the time into one beautiful uniform treatise on Geometry.
Geometry comes from the Greek word Geometria, with “geo-” meaning “earth,” and “-metri” meaning “measurement.” As we learned in our Classical Arithmetic I course, Geometry is the science which has for its object the measurement of Magnitudes. Magnitudes can be considered under three dimensions: length, breadth, and height or thickness. In this first course, we will only deal with the first two dimensions, and we will study Planar Geometry.
Euclid organized his text neatly and coherently. He has a series of Definitions, which give us the meaning of the words he is going to use, then a series of Postulates, which are reasonable assumptions he is going to make (like “it is possible to draw a straight line” or similar statements), then a series of Axioms, which are self-evident truths. Finally, he uses all of the above to build Geometry from them, via a series of Propositions. Each Proposition is either a problem to be solved (usually a construction of a geometric figure) or a Theorem, i.e., a property which needs to be proved using the Postulates, the Axioms, and previous Propositions.
The beauty of Euclid's work is that, from just a few postulates and axioms, he is able to develop hundreds of propositions. And, out of those few statements, he constructs geometry as we know it.
In this first lesson, we are going to learn Euclid's definitions from the first Book of the Elements. In the next lesson, we will study his five postulates and his five axioms.
BASIC DEFINITIONS OF PLANAR GEOMETRY
| In this course, we are going to consider magnitudes in two dimensions: length and breadth. We shall call a Point something with no length and breadth, a Line something with length but no breadth, and a Surface something with length and breadth. |
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| The extremities of a Line are Points, and the extremities of a Surface are Lines. We will denote points by uppercase letters (A, B, C, and so on), lines by either lowercase letters (a, b, c, and so on) or by the two points at the extremities of the line, if the line has extremities (AB, AC, BC, and so on). |
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| A Line is called a Straight-Line if it lies evenly with points on itself. A Plane Surface is one which lies evenly with points on itself. This definitions are of course very intuitive, indicating concepts that we already know. |
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When two lines in a plane meet one another, we call Angle the inclination of the lines to one another.
| We denote angles either by Greek letters (α, β, γ, and so on) or by the three point indicating the lines meeting (writing the point where they meet in the middle). For example, in the picture on the right, the angle would be called ABC, because it is the inclination between AB and BC, which meet at B. |
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An angle could also be formed by two lines which are not straight, and we call Rectilinear Angle one which is formed by two straight-lines. However, all angles in this course will be assumed to be rectilinear unless otherwise specified.
If a straight-line meets another straight-line at a point in the middle of it, and the two angles it makes are equal to each other, we call those angles Right-Angles, and we call the two straight-lines Perpendicular. For example, in the picture below, the lines AB and CD are perpendicular, and the angles ADC and BDC are right-angles.
We call an angle Obtuse if it is greater than a right-angle, and we call it Acute if it is less than a right-angle.
| Besides straight-lines, the other most useful and basic geometric figure is the Circle. A circle is a plane figure enclosed by one single line, called the Circumference. It also has a point, called the Center, such that all straight-lines from the center to the points on the circumference are equal to each other, in the sense that they have the same shape and length. We call each of those straight-lines a Radius. |
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| A straight-line between two points on the circumference that passes through the center is called a Diameter. In the picture on the side, the straight-line AB is a diameter. Each diameter cuts the circle exactly in half, and each of the two halves is called a Semi-Circle. |
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A Figure that is bounded by straight-lines is called a Rectilinear Figure. If it is contained by three straight-lines, it is called a Triangle, if it is contained by four, it is called a Quadrilateral, and if it is contained by more than four sides, it is called a Multilateral. Here is a picture of several rectilinear figures:
A triangle is called Equilateral if it has three equal sides, Isosceles if it has only two equal sides, and Scalene if all three sides are unequal. Also, a triangle is called Right-angled if it has a right-angle, Obtuse-angled if it has an obtuse angle, and Acute-angled if all three angles are acute.
If a quadrilateral has four equal sides and four right-angles, we call it a Square. If it has four right-angles, but not all four sides are equal, we call it a Rectangle. If it has four equal sides, but the angles are not right-angles, we call it a Rhombus. Its opposite sides and opposite angles are equal, and it is not a square, a rectangle, or a rhombus, we call it a Rhomboid. All other quadrilaterals are called Trapezia.
The last basic definition we are going to learn is that of Parallel Lines, which are straight-lines in the same plane that never meet, even if produced to infinity in each direction:
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Note: Since these definitions are from the first book of the Elements, we shall denote them by a "I." followed by their number. We also broke some of the original definitions into parts to facilitate the memorization.
Definition I.1. A point is that of which there is no part: no length and no breadth.
Definition I.2. A line is a length without breadth.
Definition I.3. The extremities of a line are points.
Definition I.4. A straight line is one which lies evenly with points on itself.
Definition I.5. A surface is that which has length and breadth only.
Definition I.6. The extremities of a surface are lines.
Definition I.7. A plane surface is one which lies evenly with the straight-lines on itself.
Definition I.8. 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.
Definition I.9. An angle is called rectilinear when the lines containing it are straight.
Definition I.10a. 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.
Definition I.10b. When a straight-line stood upon another straight-line makes adjacent angles which are equal to one another, the two straight-lines are said to be perpendicular to one another.
Definition I.11. An obtuse angle is one which is greater than a right-angle.
Definition I.12. An acute angle is one which is less than a right-angle.
Definition I.13. A boundary is that which is the extremity of something.
Definition I.14. A figure is that which is contained by some boundary.
Definition I.15a. A circle is a plane figure contained by a single line, called 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.
Definition I.15b. The circumference of a circle is the boundary of the circle.
Definition I.16. The center of a circle is the point lying inside the circle such that all of the straight-lines from it to the circumference are equal to one another.
Definition I.17a. The radius of a circle is a straight-line from the center of the circle to a point on the circumference.
Definition I.17b. The diameter of a circle is a straight-line between two points on the circumference that passes through the center of the circle.
Definition I.18. A semi-circle is one of the two halves of a circle bounded by the circumference and a diameter.
Definition I.19a. A rectilinear figure is a figure contained by straight-lines.
Definition I.19b. A triangle is a rectilinear figure with three sides.
Definition I.19c. A quadrilateral is a rectilinear figure with four sides.
Definition I.20a. An equilateral triangle is a triangle having three equal sides.
Definition I.20b. An isosceles triangle is a triangle having only two equal sides.
Definition I.20c. A scalene triangle is a triangle having three unequal sides.
Definition I.21a. A right-angled triangle is a triangle having a right-angle.
Definition I.21b. A obtuse-angled triangle is a triangle having an obtuse angle.
Definition I.21c. A acute-angled triangle is a triangle having three acute angles.
Definition I.22a. A square is a quadrilateral having four right-angles and four equal sides.
Definition I.22b. A rectangle is a quadrilateral having four right-angles, but that is not a square.
Definition I.22c. A rhombus is a quadrilateral having four equal sides, but that is not a square.
Definition I.22d. A rhomboid is a quadrilateral having opposite sides and angles equal to each other, but that is not a square, a rectangle, or a rhombus.
Definition I.22e. A trapezia is a quadrilateral that is not a square, a rectangle, a rhombus, or a rhomboid.
Definition I.23. Parallel lines are straight-lines that, being in the same plane, never meet one another even if prolonged to infinity in each direction.
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