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Starvard University · Geometry
Learn geometry the way Euclid built it: start with a point, grant three simple tools, and prove your way to Pythagoras one careful step at a time.
Hello, my name is Finn, and I'm going to help you understand this course.
Here it is, explained like you are 10:
Imagine you are allowed only two tools, a ruler with no marks and a compass, and almost no rules at all. From that tiny start, Euclid builds all of geometry, one proved fact at a time. A point has no size. A line has no width. Put two circles together and you get a perfect triangle. Keep going, carefully, and hundreds of pages later you can prove the famous rule about right angled triangles that builders and sailors still use. This course walks that same road slowly, so you can see exactly why each step is true and not just believe it.
Euclid does not ask you to trust him. That is the whole point of this book. He begins by saying what his words mean, then he lists the few things he asks you to grant him, and after that he grants himself nothing else. Every new fact has to be earned from facts already earned. If a proof uses something, the proof says where it came from, right down to the number of an earlier proposition.
This course follows the first six books in John Casey's edition of 1885, with Casey's notes and exercises along the way. You will meet the famous hard places honestly: the fifth assumption about parallel lines, which Casey prints among the axioms, and the long proof about isosceles triangles that beginners have stumbled over for centuries. Take it slowly, draw every figure yourself, and by the end you will own a way of thinking, not just a list of facts. Bring a pencil to every lesson. Redraw each figure at the size of a playing card, label it in Euclid's own letters, and cover the printed proof while you try to say the next step aloud. Geometry was a hand skill before it was a school subject, and your hand will remember what your eyes only skim.
The whole course in brief
What you will be able to do after this course
The eight lessons
Lesson 1
Euclid starts by saying exactly what his words mean. In Casey's edition, Book I definition one says a point is that which has position but not dimensions: it is not a solid, not a surface, and not a line, so it has no length, breadth, or thickness. A line is length without breadth. A surface has length and breadth, and a surface where the straight line joining any two points lies wholly in it is a plane. His circle, definition thirty two, is a figure where all straight lines from one inside point, the centre, to the circumference are equal.
Next come the things he asks you to grant. The three postulates let you draw a straight line from any point to any point, produce a finished line onward in a straight line, and describe a circle from any centre with any radius. The axioms, often called common notions, cover all magnitudes: things equal to the same thing are equal to each other, equals added to equals give equal sums, and the whole is greater than its part.
Lesson 2
A construction in Euclid is a problem: something to be done, with a solution that builds it and a demonstration that the building worked. The postulates are his toolbox. Postulate one is the straightedge move, joining two points. Postulate three is the compass move, swinging a circle around a centre at a chosen radius. Casey remarks that Euclid never takes for granted the doing of anything for which a construction can be given from the postulates or from earlier problems.
Proposition one shows the method at its cleanest. On a given line AB, describe a circle with centre A and radius AB, then a second circle with centre B and radius BA. The circles cut each other at C. Join CA and CB. Because A is the centre of the first circle, AC equals AB. Because B is the centre of the second, BC equals BA. Things equal to the same thing are equal to each other, so all three sides match, and ABC is equilateral. The same building blocks do more work later: Proposition nine bisects a given angle by building an equilateral triangle on a line cut off equal on both sides, then proving the two halves equal.
Lesson 3
Two figures are congruent when one can be placed on the other and made to coincide in every part. Casey explains that this placing, called superposition, is done in the mind, and it rests on axiom eight: magnitudes that can be made to coincide are equal. Proposition four is the first great congruence test. If two triangles have two sides of one respectively equal to two sides of the other, and the angles included by those sides equal, then the triangles are equal in every respect: the third sides match, and the remaining angles match in pairs.
Proposition eight gives the companion test, three equal sides, and Proposition twenty six handles two angles and a side. With those tools Euclid attacks Proposition five: the angles at the base of an isosceles triangle are equal. Casey warns beginners that this proof feels hard because the two triangles inside it overlap each other, and he advises the teacher to draw those triangles separately and line up their corresponding parts. Notice the method: Euclid does not measure the base angles and declare them close. He builds extra lines, applies Proposition four twice, subtracts equals from equals, and the equality is forced.
Lesson 4
Parallel lines are straight lines in the same plane that never meet, however far they are produced. Euclid proves easy direction first. In Proposition twenty seven, if a line crossing two lines makes the alternate angles equal, the two lines are parallel, because if they met they would form a triangle whose exterior angle both equals and beats the same interior angle, which is absurd. Proposition twenty eight gives two more tests, using corresponding angles and interior angles that add to two right angles.
The hard direction needs an extra grant. What most editions call the fifth postulate, Casey prints as axiom twelve: if a line meeting two lines makes the interior angles on the same side add to less than two right angles, those lines, produced, will meet on that side. Proposition twenty nine uses it, and only there: assuming the alternate angles unequal forces the interior sum below two right angles, so the supposedly parallel lines would meet, a contradiction. Proposition thirty one then constructs a parallel through a given point, and Proposition thirty two spends the result. The angle sum of a triangle depends on this one assumption, which is why generations of readers tried, and failed, to prove it from the rest.
Lesson 5
Proposition forty seven of Book I states the famous result exactly: in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Euclid does not check it with numbers. He builds a square outward on each of the three sides, and then shows that the big square is cut into two rectangles, each one equal to one of the smaller squares.
Here is the engine of the proof. Draw a line through the right angle parallel to a side of the big square, splitting it. Join two diagonal lines to form two triangles. Those two triangles have two pairs of equal sides, because sides of squares are equal, and their included angles are equal because each is a right angle plus the same triangle angle. By Proposition four, the triangles are congruent. Now a key area fact from Proposition forty one does the heavy lifting: a parallelogram on the same base and between the same parallels is double the triangle. So the rectangle is double one triangle, and a smaller square is double the congruent one. Doubles of equals are equal, says axiom six. Repeat on the other side, add the two rectangles, and the big square equals the two small squares.
Lesson 6
Book III turns to the figure the compass has been drawing all along. Everything flows from the definition: a circle is the figure whose centre sits at an equal distance, the radius, from every point on the circumference, and a diameter is a straight line through the centre ending both ways at the circumference. Because all radii are equal, the centre controls the whole figure, and many circle proofs are just clever ways of joining points to that centre and using isosceles triangles.
The jewel of the book is Proposition thirty one. The angle in a semicircle is a right angle. Take a diameter, pick any point on the circumference, and join it to both ends of the diameter: the angle you get is always right, wherever you picked the point. Casey adds that the angle in a segment bigger than a semicircle is acute, and in a smaller segment, obtuse. Proposition thirty two goes further: the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle. Book III also shows that two chords that are not both diameters cannot bisect each other, a neat proof because assuming they do would make a part equal to the whole.
Lesson 7
Books V and VI answer a different question: what stays true when a figure grows or shrinks? Book V defines the language. A less magnitude is a part of a greater when it measures it exactly, a multiple contains it an exact number of times, and ratio is the relation between two magnitudes of the same kind with respect to quantity. Casey is frank: he says Euclid's propositions on proportion are still quoted by every writer, but the easiest treatment is algebraic, so his notes translate the ideas into that language.
Book VI applies the theory to shape. Similar rectilineal figures, by definition one, are those whose angles are equal each to each and whose sides about the equal angles are proportional. Similar figures agree in shape; if they also agree in size, they are congruent. Proposition two shows that a line drawn parallel to one side of a triangle cuts the other sides proportionally, and the converse holds too. Propositions four and five then show something special about triangles: if two triangles are equiangular, their sides about the equal angles are proportional, and if their sides are proportional, they are equiangular. Casey notes triangles are unique in this: for other figures, equal angles and proportional sides can come apart.
Lesson 8
Casey calls Euclid's Elements the oldest mathematical book in existence, and his introduction calls geometry the science of figured space, built from the simplest lines that can be drawn on a plane, the right line and the circle. What made the book revolutionary was not any single result. Theorems about triangles were known in pieces before Euclid. His invention was the chain: a small set of definitions, postulates, and axioms at the front, and then every proposition locked to the ones before it by explicit references.
His own explanation of terms shows the discipline. An axiom is self evident and incapable of demonstration. A theorem is not: it has a hypothesis, what is assumed, and a conclusion, what is asserted to follow, and the demonstration must carry you from one to the other. A problem must be solved by a construction and then demonstrated. Nothing is true because it looks true in a tidy diagram, and nothing is borrowed from later in the book to rescue an earlier gap. That standard, that a claim owes you its reasons in order, is the tool this course has used in every lesson. It is also the tool you can carry into any subject: ask what is defined, what is merely granted, and what has actually been proved.
Sources
This course is taught from classic texts that are free for anyone to read. Every method in the lessons traces back to one of these books.
Graduation
You started with a point that has position but no size, and you finished with squares, circles, and similar figures, every step earned. Keep a compass and a plain ruler by your notebook, draw each figure before you read its proof, and the Elements will keep teaching you long after this course ends. Congratulations on thinking the Euclidean way.
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