Tag: geometry definitions

Before We Begin … Definitions

The building blocks for classical mathematics are definitions and theorems (or propositions). The goal is to prove new theorems based solely on theorems that have already been proved and definitions. Examples and pictures can be used to gain an idea whether a statement is true, but are not sufficient when it comes to proving that statement.

Definitions – Book 1

The definitions for Book 1 can be found on pages 1 and 2 of the book by Green Lion’s Press (Euclid’s Elements). You can also find a free digital version here.

To work through Proposition 1 you should be familiar with the following definitions:

  • Finite straight line: A straight line is a line which lies evenly with the points on itself. A finite straight line can be thought of as a line segment.
  • Circle: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; and the point is called the center of the circle.
  • Equilateral triangle: Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three. An equilateral triangle is that which has its three sides equal.

You should also be familiar with these two postulates. Let the following be postulated:

  1. To draw a straight line from any point to any point. [We can draw a finite straight line connecting any two points.\
  2. To describe a circle with any center and distance. [We can draw a circle provided that we know its center and its distance (or radius).]

Finally, there is a common notion that we will use: Things which are equal to the same thing are also equal to one another.

 

Proposition 1

On a given finite straight line to construct an equilateral triangle.

In other words, if you are given a line segment, can you explain how to construct an equilateral triangle from it? All of the tools you will need are listed above. See if you can develop a plan. A construction will be presented in the next post.

If you have questions, or would like to share your construction, please use the comments section. Thanks – George