Common Questions about Geometry
Examples of Common Geometric Abbreviations
The segment AC is equal to the segments BD and FG
AC = BD,FG
Triangle ABC is equal to the square LKJH
^ABC = sq LKJH (or sq KH)
Angle ABC is equal to a right angle
<ABC = |_
< <> parallelogram ABCD
Circumference ABC is equal to segment FGH in circle ADG
(ABC = (|FGH in oADG
Line AC is parallel to line BF
AC || BF
As A is to B so is C to D
A is less than, equal to or great than B
A <=> B
The rectangle contained by A,BC is equal to the parallelogram ABCD.
 A,BC = <> ABCD
A alike exceeds, falls short of, or equals B as C does of D A<=>B as C<=>D
Whatever multiple A is of B, that multiple will C be of D. A m B = C m D
The ratio of A to B is equal to that of C to D. A:B::C:D
Sample proposition summary
Book 1, Proposition 5
|Euclid||Steps||Reasons- Just for your notes|
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
|This is the enunciation of the proposition. It tells you what the proposition will demonstrate and is not included in your summary.||
Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC.
|This is construction which you need to be able to explain, but is also not included in your summary.||
I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE.
|The “I say that” tells you once again what the proposition will demonstration and is not included in your proposition.||
Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB.
|Here include includes more construction. Once again, be prepared to explain this when you present the proposition, but do not include it in your summary.||
Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG.
Here begins the actual proof so you have the first step of your summary.
1. AF = AG
2. AB = AC
3. FA,AC = GA,AB
4. <FAG = <FAG (common)
3.Repeats steps 1 and 2
4. Common angle
Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB.
5. FC = GB
6. ^AFC = ^AGB
7. <ACF = <ABG
8. <AFC = <AGB
5. Prop 4- SAS
6. Prop 4- SAS
7. Prop 4- SAS
8. Prop 4- SAS
Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG.
9. AF = AG
10. AB = AC
11. BF = CG
9. Repeat of step 1
10. Repeat of step 2
11. Common notion 3
But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them.
12. FC = GB
13. BF,FC = CG,GB
14. <BFC = <CGB
12. Repeats 5
13. Repeats 5 and 11
14. repeats step 8
15. Common side
Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.
|16. ^BFC = ^CGB||
16. Prop 4- SAS
Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG.
17. <FBC = <GCB
18. <BCF = <CBG
17. Prop 4- SAS
18. Prop 4- SAS
Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC.
19. <ABG = <ACF
20. <CBG = <BCF
21. <ABC = <ACB
19. Repeats step 7
20. Repeats step 18
But the angle FBC was also proved equal to the angle GCB, and they are under the base.
|22. <FBC = <GCB||
22. repeats step 17
Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
|Here Euclid states what the proposition has proven.|
The steps you would send
into me would be the following- repeated steps may be excluded if you like.
Explain the steps of your proposition-
1. From memory (You cannot miss more than 2 steps)
2.From written notes
3. From the text
4. Did not have steps prepared
In order to guarantee a pass in Geometry, at the end of the year, students must have maintained an average of 2.0 or better for their score and have correctly identified at least 60 steps. Of students who have not maintained this standard, the lowest 15% will need to take Geometry again in order to continue with GBT. Whether a student fails or not is ultimately at the discretion of the tutor, however, the numerical system will be held outside of extraordinary circumstances. If you feel your students is falling behind, it would be good to encourage them to deliver their propositions by memory as often as they can and to volunteer for the large proofs. Please keep a record of your performance on all propositions that you demonstrated so that you can find your semester average.
The fundamentals of geometry: theories of triangles, parallels, and area.
Theory of circles.
Constructions for inscribed and circumscribed figures.
Theory of abstract proportions.
Similar figures and proportions in geometry.
Fundamentals of number theory.
Continued proportions in number theory.
Classification of incommensurables.
Measurement of figures.
The aim of the Classical Greek Tutorial is to provide students with the vocabulary and grammatical skills necessary to read ancient Greek authors. The translation exercises are chosen from the texts read in GBT I and II and also from writings prior to the GBT authors that provide historical and philosophical context to our GBT readings. The course will cover the paradigms and vocabulary for a reading knowledge of Greek, but each tutorial will go at a pace suited to the majority of the students in the class.
From ABRAHAM LINCOLN by G. Frederick Owen
"He ( Lincoln) read in the fields of politics, literature, philosophy, and science. But most of all he ‘read law’. In the course of that law-reading, he says:
‘ I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, ‘What do I mean when I demonstrate more than when I reason or prove?’. I consulted Webster’s Dictionary. That told of ‘certain proof’, ‘proof beyond the possibility of doubt’ ; but I could form no idea what of sort of proof that was. I thought that a great many things were proved beyond a possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood ‘demonstration’ to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined ‘blue’ to a blind man. At last I said, ‘Lincoln, you can never make a lawyer if you do not understand what ‘demonstrate’ means ; and I left my situation in Springfield, went home to my father’s house, and stayed there until I could give any proposition in the six books of Euclid at sight. I then found out what ‘demonstrate’ means, and went back to my law studies.’
"At noon we went home for dinner and then back again for history in the afternoon. The history was a pretty hard paper and I got dreadfully mixed up in the dates. Still, I think I did fairly well today. But oh, Diana, tomorrow the geometry exam comes off and when I think of it it takes every bit of determination I possess to keep from opening my Euclid . If I thought the multiplication table would help me any I would recite it from now till tomorrow morning."
Anne of Green Gables
"if God exists and if He really did create the world, then, as we all know, He created it according to the geometry of Euclidand the human mind with the conception of only three dimensions in space. Yet there have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in Euclid’s geometry ; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. [Lobechevski] I have come to the conclusion that, since I can’t understand even that, I can’t expect to understand about God."
"On my return to Geneva, I passed two or three years at my uncle’s, expecting the determination of my friends respecting my future establishment. His own son being devoted to engineering, was taught drawing, and instructed by his father in the elements of Euclid : I partook of these instructions, but was principally fond of drawing."
The Confessions of Jean-Jacques Rousseau
Then when a little more I raised my brow,
I spied the master of the sapient throng,
Seated amid the philosophic train.
Him all admire, all pay him reverence due.
There Socrates and Plato both I mark’d
Nearest to him in rank, Democritus,
Who sets the world at chance, Diogenes,
With Heraclitus, and Empedocles,
And Anaxagoras, and Thales sage,
Zeno, and Dioscorides well read
In nature’s secret lore. Orpheus I mark’d
And Linus, Tully and moral Seneca,
Euclid and Ptolemy, Hippocrates,
Galenus, Avicen, and him who made
The commentary vast, Averroes.
The Divine Comedy (Inferno)
How is it, then, with the whale? True, both his eyes, in themselves, must simultaneously act; but is his brain so much more comprehensive, combining, and subtle than man’s, that he can at the same moment of time attentively examine two distinct prospects, one on one side of him, and the other in
an exactly opposite direction? If he can, then is it as marvellous a thing in him, as if a man were able simultaneously to go through the demonstrations of two distinct problems in Euclid . Nor, strictly investigated, is there any incongruity in this comparison.
I WILL give no more of the details of my hero’s earlier years. Enough that he struggled through them, and at twelve years old knew every page of his Latin and Greek Grammars by heart. He had read the greater part of Virgil, Horace, and Livy, and I do not know how many Greek plays: he was proficient in arithmetic, knew the first four books of Euclid thoroughly, and had a fair knowledge of French. It was now time he went to school, and to school he was accordingly to go, under the famous Dr. Skinner of Roughborough.
Way of All Flesh- Samuel Butler
Euclid alone has looked on Beauty bare.
Edna St. Vincent Millay
When I am violently beset with temptations, or cannot rid myself of evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some other study, which necessarily engages all my thoughts, and unavoidably keeps them from wandering.
I picked up the dignified-looking book called Euclid’s Elements for the first time as a 15-year-old sophomore with a passionate dislike of math. Although the black hardcover book with its silky white pages had a striking visage unlike any math textbook I had previously encountered, I was not going to let appearances deceive me. I knew math was math, and no pretty disguise would change that. My evaluation of Euclid was dramatically altered from the first page. Euclidimpressed me with three primary items that had a significant impact on my thinking and greatly prepared me for future studies.
First, his method of intellectual organization surprised and delighted me. He did not confuse his point with pointless palaver or confounding circumlocutions. He knew exactly what he wanted to prove, why he wanted to prove it, how he wanted to prove it, and he proved it! I had always valued organization in real life, but I had never thought about intellectual organization before. Euclid presented a picture of what mental organization looked like, and how beautifully intricate it could be. “Perhaps,” I began to consider, “this structure and clarity could be applied to other intellectual pursuits, like writing, speaking, and heavens, maybe even Algebra.” Euclid made me aware of the beauty of orderly thought and inspired me to seek and create that structure elsewhere.
Secondly, the Elements introduced a cast of abstract ideas that forced me to exercise my “mind’s eye” and greatly aided my comprehension of other abstract ideas. For instance, discovering that every “line” (based on Euclid ’s definition) was merely a representation of a real line, which could not be reproduced physically, helped clarify what Plato meant by his theory of the forms. I was fascinated by the almost mystic quality of “points,” “triangles” and the rest, and contemplating them created a new “cabinet” in my mind where I could file away related information of an abstract nature.
The third thing I loved about Euclid was his method of building irrefutable arguments. By constructing simple proofs from undeniable definitions and “common notions” and then using his simple proofs to prove complex ones, he assembles a venerable army of unassailable arguments. Often I would read the thesis of a complex proof and think, “Oh wow! He’s never going to be able to convince me of this ,” but in the end, I was always forced to concede that what he said was true. As a somewhat over-confident youth, these intellectual defeats surprised me at first, then humbled me, and finally thrilled me.
I fell in love with geometric syllogism, because of its organization, abstract content, and dazzling certainty. Studying Euclid encouraged me to grasp these principles for my own, and enriched my reading of philosophers who have striven for clarity in these areas as well. Euclid was also vital stepping stone to reading Kant, who was the greaquiz academic challenge I have faced.
From Kate Peacock's (former Great Books V student) admissions essay
“If Euclid failed to kindle your youthful imagination, then you were not born to be a scientific thinker”
How does Geometry apply to cathedral construction?
Mathematicians of the Seventeenth and Eighteenth Centuries
Web Resources for the History of Mathematics
Fibonacci sequence and the Golden Mean
Recommended Supplemental reading
Mathematics- Is God Silent?- James Nickel