# Book I

What is in Book I?

Book I is about basic basic geometry in two dimensions, that is on a plane. This involves lines, triangles, rectangles etc. However it doesn't include the study of circles to any depth, a topic that is reserved for Book III.

Many of the results from Book I are still taught in middle and high school. For example Book I includes the three well-known congruence theorems for triangles, that is SSS, SAS and SAA, and the theorem that the three angles inside a triangle add to 180 degrees.

Book I has 49 propositions, the second longest of all the books, with Pythagoras' theorem being one of the last proofs. It also has a relatively large number of definitions (23), postulates (5) and axioms (5), although these numbers can vary from edition to edition. The number of propositions generally don't change. The definitions, postulates and axioms have attracted a considerable amount of commentary from many authors over the last two thousand years.

The first part of the book are the definitions which describes the language used in the Books. For example, the definitions include the various types of angle such as obtuse and acute angles, and the main characteristic of a circle such as the center and diameter. Next come the postulates which describe self-evident truths related to geometry, for example, that one can draw a line between any two point. The third part are the axioms, also called common notions, these are non-geometric self-evident truths. The language can seem a bit strange but what they state are obvious when written using modern algebra and define basic arithmetic operations. For example, axiom two says "If equals be added to equals, the wholes are equal". Using algebra we'd say that if A = B and we add C to both A and B, then the axiom says that A + C = B + C.

After the axioms come the propositions which are the main results in each book. The most important aspect of all the books is that each proposition is built on previous ones. This is even the case between books where one book may reference propositions from a previous book. This page shows a dependency graph for Book I and illustrates how the results of Book I grow, culminating in Propositions 45 and 47.

To help navigate the contents of Book I, I have made the following groups. These are in order of the propositions as they are given in the book.

Group 1, Four Propositions:

Equilateral triangles, basic manipulation theorems, and SAS theorem.

Group 2, Two Propositions:

Isosceles Triangles

Group 3, One Proposition

SSS Theorem

Group 4, Six Propositions

Bisection and Perpendicular Theorems

Group 5, One Proposition

Opposite Angles

Group 6, Eleven Propositions

Properties of Triangles

Group 7, Five Propositions

Parallels Lines

Group 8, One Proposition

Angles of a Triangle

Group 9, Eight Propositions

Parallelogram and Triangle Theorems Related to Areas

Group 10, Six Propositions

Area Arithmetic Propositions

Group 11, Two Propositions

Pythagoras’ Theorem

DEFINITIONS, POSTULATES AND AXIOMS

The following definitions etc, will be based on a pre-Theon edition of Euclid. In this case, I am using the Heath text.

Euclid starts in Book I with a series of definitions, followed by postulates and then common notions or axioms.

DEFINITIONS

Let's start with the definitions.

Some of the definitions are quite obscure, and their meaning and interpretation have been argued literally for 1000s of years. Definition 4, in particular, is difficult to interpret as written. It doesn't take too much effort however to realize what Euclid's meaning was. For example, the first definition says "A point is that which has no part". I think its fairly obvious that what's being described is a position in space, even if the language is a bit odd. One has to understand that the ancient Greeks, as far as we know, had no concept of a coordinate system. I suppose they could have used "A point is a unique position in space", or 'A point is a location in space" but these don't specify the size of the point, is the point 1 cm across, 1 mile across? In some sense we are delving into realm of calculus where we might describe a point as an infinitesimal object that describes a unique location in space. In other words the definition we find in Euclid is not as bad as it first seems given the limitations the Greeks had.

Here are the definitions:

1. A point is that which has no part.

A point is a position in space. For example is we're considering 2D space then the coordinates (2,3) marks a position on a 2D surface at coordinates x=2, y=3. Obviously Euclid didn't have the concept of a coordinate so he (or who ever wrote the definition) had to do the best they could.

2. A line is a length without breadth.

Euclid here may not mean a line in our sense of the word but actually a curve. The reason being is that proposition 4, defines a particular kind of line, a straight-line. Hence a line in this definition could be straight or curved. Note he doesn't define length or breadth and it seems these are so obvious as to not require a definition.

3. The extremities of a line are points.

In simpler language he is just saying that the ends of a line (if it has an end, a circle line has no end) are points.

4. A straight line is a line which lies evenly with the points on itself.

This is where Euclid defines a particular type of line and is one of the more strangely worded definitions. Many people over the years have tried to rephrase this with limited success. Here are some examples from other editions of Euclid:

a) John Casey: A line which lies evenly between its extreme.

b) John Playfair: If two lines are such that they cannot coincide in any two points, with- out coinciding altogether, each of them is called a straight line.

c) John Keill: A right line is that which lieth evenly between its points.

b) D M'Curdy: A straight line is the path of a point, without a curve or angle.

e) Gracilis (1558) Translated from Latin by chatgpt: A straight line is one which equally intersects its points

5. A surface is that which has only length and breadth.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

9. When the lines containing the angle are straight, the angle is called rectilineal.

10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular

11. An obtuse angle is an angle which is greater than a right angle

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.

14. A figure is that which is contained by any boundary or boundaries.

15. A circle is a plane figure contained by one line such that all the straight lines failing upon it from one point among those lying within the figure are equal to one another.

16. And the point is called the centre of the circle.

17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by Jour, and roultilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides egual, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle ‘hat which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle ¢hat which has an obtuse angle, and an acute-angled triangle ¢hat which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and rightangled; an oblong that which ts right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction,

POSTULATES

That a straight line may be drawn from any one point to any other point

That a terminated straight line may be produced to any length in a straight line

To describe a circle with any center and radius.

That all right angles equal one another.

This seems to be an odd postulate since it seems to be stating the obvious. Even ancient commentaries discuss this postulate whether it should be here or not. Heath spends 2 whole pages trying to explain its meaning.

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

COMMON NOTIONS (AXIONS)

Things equal to the same thing are also equal to one another.

In other words, if A = B and B = C, then it must be the case that A = C

If equal things are added to equal things then the wholes are equal.

This means is A = B, and we add C to both A and B, then A + C = B + C

If equal things are subtracted from equals, the then the remainders are equal.

This means if A = B, and we subtract C from both A and B, then A - C = B - C

Things which coincide with one another are equal to one another.

This is saying that 5 = 5, which isn't terribly useful but nevertheless is a truth. It's real application is in geometry where it can be used to justify superposition of lines, angles, shapes etc. For example, let’s say we have two squares, A and B, and we move one of the squares onto the other one. If both squares match up exactly, then we can say both squares are the same.

The whole is greater than the part.

This is the pie axiom. A piece of a pie is smaller than the pie itself. This axiom recognizes the observation that the parts of any object will be smaller than the object itself. If we break the number 10 into 2, 5 and 3, then each of these parts is smaller than the original 10.

PROPOSITION 1

The first preposition is a nice gentle introduction to a Euclidian proof. It one of the few construction proofs in Euclid which in this case is to construct an equilateral triangle. If you remember your school geometry you'll recall that an equilateral triangle is a triangle that has three equal length sides. The construction proceeds as follows. I'm not using the same language used in Euclid, but it's the same construction and proof.

1, Draw a straight line of any convenient length., label the line ends A and B.

2. Using a compass, place the compass point on A and open the compass so that the pencil touches B.

3. Draw circle, C1, using the compass. This circle will have a center A and a radius given by the length of the AB.

3. Next, place the compass on point B, and open the compass so that the pencil end touches A and draw a circle, C2.

4. You should now have two circles drawn. You should also notice that the circles cross over at two points.

5. Pick the upper point where the circles cross and mark it point C.

6. Draw two lines, from C to A and from C to B.

7. We claim that the triangle drawn on the points A, B and C is an equilateral triangle.

These steps are shown below in graphical form.

We now need to give the proof that triangle ABC, is an equilateral triangle.

The radius of circle C1, has a length of AB.

The length of line AC is also the radius of C1.

Therefore AB = AC.

The radius of circle C2, has a length of AB.

The length of line BC is also the radius of C2.

Therefore AB = BC.

Since AB = AC and AB = BC, therefore AC = BC.

Since all sides have equal length, ABC is an equilateral tringle.

Hopefully you agree that proposition 1 is reasonably straight forward. It seems like a typical problem in geometry.

What readers will find puzzling are the next two propositions. After proposition 3, the book returns to more familiar territory. So what are propositions 2 and 3 about?

Proposition 2 is about duplicating a line segment and 3 is about cutting off segments. The reader is probably expecting to dive straight into proofs related to angles, triangles, and rectangles, much like a modern textbook on geometry. Why does Proposition 2 go to tall the lengths to copy a line using a series of constructions? It would seem more convenient to just copy a line using a compass to physically measure a length, then move the compass to the new location and draw the copied line. However, the Greeks didn’t like to use the compass-distance measure as part of a proof. Instead they preferred a purely geometric method to copy a line.

Proposition 2 and 3 are a pair. Proposition 2 is only referenced once in the whole of Book I where we find it used to prove Proposition 3. Proposition 3 is used in twelve propositions in Book I as well as propositions in the other books on geometry in the series.

Proposition 2 copies a line to a given point, and Proposition 3 allows the line to be rotated in any orientation. Proposition 3 is relatively simple, but Proposition 2 is a very clever use of geometry and is the first hint of the sophistication of the Greek geometry. These two propositions are existence proofs that the given operation can be done.