In Classical Geometry, students must learn how to study Propositions. To do this, they must understand the parts of propositions and how propositions work.
Table of contents
THE ELEMENTS OF GEOMETRY
We see that the title of the classic text on the art of Geometry is Euclid’s “Elements”. When we begin this study, we find three sets of information provided by Euclid:
- Common Notions, or “Axioms”
These are the “elements” from which the art of Geometry is constructed. Every point we learn about the science of “magnitudes at rest”, will be proven from these definitions, postulates and axioms. It is recommended that these elements be memorized, but it is possible to get started in Geometry so long as they are constantly reviewed.
After learning the elements, we begin to take our first steps in Geometry. We move forward like a man taking steps in a dark place, moving slowly, feeling around, making sure we have something to step on, then moving to that place. We do this in Geometry through the study of “propositions”.
In a proposition, some simple action or idea is “proposed” to us. There are two kinds of propositions in Geometry: Problems and Theorems. In a problem, something is to be done. In a theorem, some truth is to be proven. The solutions and proofs for these propositions have already been provided by the ancient philosophers—we simply need to learn them. The study of these propositions is our work in classical Geometry.
THE PARTS OF A PROPOSITION
Propositions are composed of six parts:
For each proposition, we need to learn each of these parts. So, let’s learn what each of them is, and how to identify them.
The Proposition states what is given (known) and what is sought (unknown). The Exposition clarifies what is given in the proposition and adapts it for use in the investigation. The Determination makes clear what particular thing is sought in the proposition. The Construction adds what is lacking to what has been given for the purpose of finding what is sought. The Demonstration provides the proof for the solution of the proposition, showing how it is reached from what is known and certain. Lastly, the Conclusion repeats the Proposition, confirming what has been done and/or demonstrated.
It is important to note that not all propositions have all six parts. The most important parts, which are found in all propositions, are these three: the Proposition, the Proof and the Conclusion.
IDENTIFYING THE PARTS OF A PROPOSITION
In the text of Euclid’s Elements, the parts will not be labeled. Therefore, you need to be able to identify them by knowing what to look for. Here are a few tips.
- View Euclid’s Elements (pdf)
1. The Proposition is given first, in italics. You should be able to see a statement of what is given and what is sought—in general terms. Looking at the top of Proposition 1, we read,
On a given finite straight line to construct an equilateral triangle.
2. The Exposition comes next and states, in more specific terms, what is sought. In Proposition 1, we read,
Let AB be the given finite straight line.
3. Next, the Demonstration clarifies what is sought. In Proposition 1, the demonstration reads,
“Thus it is required to construct an equilateral triangle on the straight line AB.
4. Then, the Construction for Proposition 1 follows, in three steps:
With centre A and distance AB let the circle BCD be described; [Post. 3]
again, with centre B and distance BA let the circle ACB be described; [Post. 3]and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1]
You can see here that for each action taken in the Construction, the Postulate on which it is based is stated. Note that one clue to help you find the Construction in a Proposition is reference to the Postulates.
5. After the Construction, we find the Demonstration (or Proof). In Proposition I, we read:
(1) Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15]
(2) Again, since the point B is the centre of the circle CAB, BC is equal to BA. [Def. 15]
(3) But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [C. N. 1] Therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another.
Tthe first two lines of the demonstration are based on Definition 15. The third line is concluded from Common Notion 1—the first axiom. We prove here that the triangle produced in the Construction is composed of three equal lines.
Note that the demonstration is easy to recognize because it reasons from definitions, axioms and previous propositions.
6. Lastly, after the Demonstration is completed, the Conclusion confirms that we have obtained what was sought in the Proposition. We read,
Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. Being what it was required to do.
Note the last line, “Being what it was required to do.” This is a key that tells us that the proposition is a problem. When we study a proposition that is a theorem, we will read, “Being what it was required to prove.”. These phrases are abbreviated in Latin as Q.E.F. (quod erat faciendum) and Q.E.D. (quod erat demonstandum), respectively.
In this article, we have examined how Propositions are to be studied in Classical Geometry. We have learned how the “Elements” of Geometry are used to prove every “Proposition” presented in the study. We learned the parts of which Propositions are composed and how they may be identified. It’s important to remember that all propositions do not have all of these parts, but that three will be found in all. With the help of this guide, students should be prepared to begin the study of Classical Geometry–and enjoy success in doing so.
If you have any questions about the study of Classical Geometry, please contact me.
God bless your studies,
Mr. William C. Michael, Headmaster
Classical Liberal Arts Academy