Euclid's Elements: Reference Sheet for Book 1

At times, Euclid makes some mistakes or omissions. I sometimes point these out, but other times I do not. If the omission is relatively minor, I often correct it without comment.

The purpose of this project is NOT to write an axiomatic geometry textbook (at least not yet, he said threateningly). That has been done many times to date. Rather, the purpose of this project is to look at the historical work of Euclid from a modern perspective. I'm personally interested in axiomatics, so this project approaches the topic with a view to the type of axiomatic questions I like to ask. For example, when Euclid needs an axiom/postulate/assumption he has not stated, I will point this out. Also, I'm interested in questions of logical dependence, so I'm hoping to (slowly) include more information on that here.

See my list of planned improvements for a list of plans I have to expand this project over the next few(?) decades.

The following terms are undefined from a modern perspective. While Euclid gives "definitions", they should be read as "notations" instead, in the sense that Euclid's descriptions are how one depicts these objects in a cartoon.

Point

depicted by a dot, and an indivisible element.

Line

depicted by a single, straight stroke on the canvas.

Plane

the canvas itself.

Some of these definitions assume things which Euclid either does not prove or does not postulate. In addition, because this is intended as a dictionary of sorts, Euclid does not worry about the logical order of the statements. In particular, some of these definitions need later postulates or propositions before they are well-defined axiomatically.

Rigid Motion

A rigid motion is a sequence of translations, rotations, and reflections.

Line

A line is a collection of points. The points are said to be on the line. We often write \( A \in \ell \) to denote that \( A \) is a point on the line \( \ell \). The line \( AB \) is the line containing both points \( A \) and \( B \). Points \( A \), \( B \), and \( C \) are collinear when there is a line \( \ell \) with \( A, B, C \in \ell \).

Plane

The collection of points is called the plane.

NB: This works for Book 1, but much more care is necessary if we venture into later books!.

Segment

A segment between points \( A \) and \( B \) is the collection of all points lying on the line containing \( A \) and \( B \) and between them. The points \( A \) and \( B \) are the ends or endpoints of the segment, denoted \( \segment{AB} \). We write \( C \in \segment{AB} \) to denote that \( C \) is on the segment, i.e., between \( A \) and \( B \).

Segment Comparisons

Segments \( \segment{AB} \) and \( \segment{CD} \) are equal in measure when there is a rigid motion taking \( A \) to \( C \) and \( B \) to \( D \) respectively. We write \( \mSegment{AB} = \mSegment{CD} \) in this case. Segments satisfy \( \mSegment{AB} < \mSegment{CD} \) when there is a rigid motion sending \( A \) to \( C \) and \( B \) to \( E \neq D \) such that \( E \in \segment{CD} \).

Sides of a Line

Points \( A \) and \( B \) lie on the same side of line \( \ell \) when \( \segment{AB} \cap \ell = \emptyset \). Otherwise, \( A \) and \( B \) lie on opposite sides \( \ell \).

Angle

The angle \( \angle{BAC} \) is formed by the line segments \( \segment{AB} \) and \( \segment{AC} \).

Interior of an Angle

A point \( D \) is on the interior of \( \angle{BAC} \) when the segments forming the angle can be extended so that \( B \in \segment{AE} \), \( C \in \segment{AF} \), and there exist points \( G \in \segment{AE} \) and \( H \in \segment{AF} \) such that \( D \in \segment{GH} \). In other words, the point \( D \) lies on a segment with ends in the rays \( \ray{AB} \) and \( \ray{AC} \).

Angle Comparisons

Two angles are equal in measure when one can be transformed into the other via a rigid motion. We write \( \mAngle{BAC}} = \mAngle{EDF} \) to denote that these angles are equal in measure. Angles satisfy \( \mAngle{BAC}} < \mAngle{EDF}} \) when there is a rigid motion transforming the ray \( \ray{AB} \) into the ray \( \ray{DE} \) so that \( C \) lies in the interior of \( \angle{EDF} \). The sum of two angles \( \angle{ABC} \) and \( \angle{DEF} \) is any angle equal in measure to the angle obtained by making a rigid motion to transform the ray \( \ray{DE} \) into the ray \( \ray{AB} \) so that \( \)

Angle Types

A right angle is an angle \( \angle{BAC} \) so that, given \( D \) with \( A \in \segment{CD} \) we have \( \mAngle{DAC} = \mAngle{BAC} \). An acute angle is one which is smaller in measure than a right angle. An obtuse angle is one which is greater in measure than a right angle.

Perpendicularity

Two lines \( \ell_1 \) and \( \ell_2 \) are perpendicular when they intersect in a single point \( A \) and form a right angle at \( A \). We denote this by \( \ell_1 \perp \ell_2 \).

Figure

A figure is a set of points with nonempty interior and boundary. No point on the interior of the figure may belong to the boundary of the figure. Moreover, if \( A \) is in the interior of the figure and \( B \) is not, then there must be a point on \( \segment{AB} \) which lies in the boundary.

Triangle

A triangle \( \triangle{ABC} \) is a figure which has a boundary made of the three segments \( \segment{AB} \), \( \segment{BC} \), and \( \segment{CA} \), and interior all those points which lie on the interior of a segment with ends on one of the boundary segments. The boundary segments of the triangle are called sides thereof, and the points \( A \), \( B \), and \( C \) are its vertices.

Congruence

Two triangles are congruent when there is a rigid motion taking one triangle to the other.

Triangle Types

Consider a triangle \( \triangle{ABC} \). This triangle is…

• equilateral when its three boundary segments are of equal measure.
• isosceles when it has (at least) two sides of equal measure.
• scalene when it is not isosceles.
• right when it has a right angle.
• obtuse when it has an obtuse angle.
• acute when it is neither right nor obtuse.

Polygon

A polygon is a figure whose boundary is a union of finitely many line segments and so that if the segments \( \segment{AB} \), \( \segment{BC} \), and \( \segment{CA} \) are all contained in the union of the boundary and the interior of the figure, then the triangle \( \triangle{ABC} \) is also contained therein. The sides of a polygon are the maximal segments contained in the boundary. The angles of a polygon are the angles formed by the sides sharing a common point.

Quadrilateral

A quadrilateral is a polygon with four sides. A quadrilateral is a…

• rectangle when its angles are all right angles.
• rhombus when its sides are all equal in measure.
• square when it is both a rhombus and a rectangle.
• parallelogram when opposing the sides may be labelled \( \alpha \), \( \beta \), \( \gamma \), and \( \delta \) with \( \alpha \parallel \gamma \) and \( \beta \parallel \delta \).

Circle

The circle centred at \( C \) with radius \( \segment{CA} \) is a convex figure whose boundary is the set of those points \( B \) with \( \mSegment{CB} = \mSegment{CA} \). A diameter of the circle is a segment \( \segment{AB} \) with \( \segment{CA} \) and \( \segment{CB} \) both radii of the circle and \( C \in \segment{AB} \). We also say this is the circle with centre \( C \) through \( A \).

Parallelism

Two lines \( \ell_1 \) and \( \ell_2 \) are parallel when either \( \ell_1 = \ell_2 \) or \( \ell_1 \cap \ell_2 = \emptyset \). We denote this by \( \ell_1 \parallel \ell_2 \).

For every pair of points \( A \) and \( B \), there is a line \( AB \) containing both \( A \) and \( B \).

Every line segment can be extended at either end. In particular, given points \( A \) and \( B \), there are points \( C \neq A \) and \( D \neq B \) for which \( A \in \segment{BC} \) and \( B \in \segment{AD} \).

If \( A \) and \( C \) are points, then there is a circle centred at \( C \) and through \( A \).

All right angles are equal in measure to all other right angles.

Let \( AB \) and \( CD \) be lines with points \( E \in \segment{AB} \) and \( F \in \segment{CD} \) so that \( A \) and \( C \) lie on the same side of \( EF \). If \( \mAngle{AEF} \) and \( \mAngle{CEF} \) sum to an angle less than two right angles, then the lines \( AB \) and \( CD \) intersect at a point on the same side of \( EF \) as \( A \).

There are some three points \( A \), \( B \), and \( C \) which are non-collinear.

The line \( AB \) is the unique line passing through points \( A \) and \( B \).

If \( A \), \( B \), and \( C \) are distinct collinear points, then exactly one of \( A \in \segment{BC} \)

If \( \ell_1 \) and \( \ell_2 \) are lines, then \( \ell_1 \cap \ell_2 \) is at most one point.

Given a pair of triangles \( \triangle{ABC} \) and \( \triangle{DEF} \), there is a rigid motion taking the ray \( \ray{AB} \) to the ray \( \ray{DE} \) and carrying \( C \) to the same side of line \( DE \) as \( F \).

If \( \ell \) is a line passing through the interior of a circle, then \( \ell \) intersects the boundary of the circle at exactly two points.

Let \( A \) and \( B \) be distinct points. There is a point \( C \) for which \( \triangle{ABC} \) is an equilateral triangle.

Let \( A \), \( B \), and \( C \) be points. There is a point \( D \) for which \( \mSegment{AD} = \mSegment{BC} \).

Let \( A \), \( B \), \( C \), and \( D \) be points. If \( \mSegment{AB} \leq \mSegment{CD} \), then there is a point \( E \in \segment{CD} \) with \( \mSegment{CE} = \mSegment{AB} \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. If \( \mSegment{AB} = \mSegment{DE} \), \( \mSegment{AC} = \mSegment{DF} \), and \( \mAngle{BAC} = \mAngle{EDF} \), then \( \triangle{ABC} \cong \triangle{DEF} \).

Let \( A \), \( B \), and \( C \) be non-collinear points. If \( \mSegment{AB} = \mSegment{AC} \), then \( \mAngle{ABC} = \mAngle{ACB} \).

Let \( A \), \( B \), and \( C \) be non-collinear points. If \( \mAngle{ABC} = \mAngle{ACB} \), then \( \mSegment{AB} = \mSegment{AC} \).

Let \( A \), \( B \), \( C \), and \( D \) be points. Assume \( \mSegment{AC} = \mSegment{AD} \) and \( \mSegment{BC} = \mSegment{BD} \). If \( C \) and \( D \) are on the same side of \( AB \), then \( C = D \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. If \( \mSegment{AB} = \mSegment{DE} \), \( \mSegment{BC} = \mSegment{EF} \), and \( \mSegment{CA} = \mSegment{FD} \), then \( \mAngle{ABC} = \mAngle{DEF} \).

Let \( A \), \( B \), and \( C \) be non-collinear points. There is a point \( D \) in the interior of \( \angle{BAC} \) with \( \mAngle{BAD} = \mAngle{CAD} \).

Let \( A \) and \( B \) be points. There is a point \( C \in \segment{AB} \) with \( \mSegment{AC} = \mSegment{BC} \).

Let \( A \), \( B \), and \( C \) be distinct points. If \( C \in \segment{AB} \), then there is a line \( \gamma \) with \( \gamma \perp AB \) and \( C \in \gamma \).

Let \( A \) be a point not on line \( \beta \). There is a line \( \alpha \) with \( \alpha \perp \beta \) and \( A \in \alpha \).

Let \( A \), \( B \), \( C \), and \( D \) be distinct points. If \( B \in \segment{CD} \), then the sum of the angles \( \angle{CBA} \) and \( \angle{ABD} \) is the same in measure as the sum of two right angles.

Let \( A \), \( B \), \( C \), and \( D \) be distinct points. If \( C \) and \( D \) are on opposite sides of \( AB \) and the sum of \( \angle{ABC} \) and \( \angle{ABD} \) is the same in measure as the sum of two right angles, then \( B \in \segment{CD} \).

Let \( A \), \( B \), \( C \), and \( D \) be distinct points. If \( E \in \mSegment{AB} \cap \mSegment{CD} \), then \( \mAngle{AEC} = \mAngle{BED} \).

Let \( \triangle{ABC} \) be given and \( D \neq C \). If \( C \in BD \), then \( \mAngle{ACD} > \mAngle{CBA} \) and \( \mAngle{ACD} > \mAngle{BAC} \).

Let \( \triangle{ABC} \) be given. The sum of \( \angle{ABC} \) and \( \angle{ACB} \) is less in measure than the sum of two right angles.

Let \( \triangle{ABC} \) be given. If \( \mSegment{AB} < \mSegment{BC} \), then \( \mAngle{ACB} < \mAngle{BAC} \).

Let \( \triangle{ABC} \) be given. If \( \mAngle{ACB} < \mAngle{BAC} \), then \( \mSegment{AB} < \mSegment{BC} \).

Let \( \triangle{ABC} \) be given. Then the sum of the segments \( \segment{AB} \) and \( \segment{BC} \) is greater in measure than the segment \( \segment{BC} \).

Let \( \triangle{ABC} \) be given. If \( D \) is in the interior of \( \triangle{ABC} \), then \( \mSegment{BD} < \mSegment{AB} \) and \( \mAngle{BAC} < \mAngle{BDC} \).

Let \( \alpha \), \( \beta \), and \( \gamma \) be segments. If the measure of the pairwise sums of these segments always exceed the measure of the remaining segment, then there is a triangle with sides equal in measure to \( \alpha \), \( \beta \), and \( \gamma \).

In particular, if \( \mSegment{\alpha} + \mSegment{\beta} \geq \mSegment{\gamma} \), \( \mSegment{\beta} + \mSegment{\gamma} \geq \mSegment{\alpha} \), and \( \mSegment{\gamma} + \mSegment{\alpha} \geq \mSegment{\beta} \), then there is a triangle \( \triangle{ABC} \) with \( \mSegment{\alpha} = \mSegment{BC} \), \( \mSegment{\beta} = \mSegment{CA} \), and \( \mSegment{\gamma} = \mSegment{AB} \).

Let \( A \), \( B \), \( C \), \( D \), and \( E \) be points with \( C \), \( D \), and \( E \) non-collinear. There is a point \( F \) with \( \mAngle{FAB} = \mAngle{DCE} \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. If \( \mSegment{AB} = \mSegment{DE} \), \( \mSegment{AC} = \mSegment{DF} \), and \( \mAngle{BAC} > \mAngle{EDF} \), then \( \mSegment{BC} > \mSegment{EF} \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. If \( \mSegment{AB} = \mSegment{DE} \), \( \mSegment{AC} = \mSegment{DF} \), and \( \mSegment{BC} > \mSegment{EF} \), then \( \mAngle{BAC} > \mAngle{EDF} \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. Assume \( \mAngle{ABC} = \mAngle{DEF} \) and \( \mAngle{BCA} = \mAngle{EFD} \). If either \( \mSegment{BC} = \mSegment{EF} \) or \( \mSegment{AB} = \mSegment{DE} \), then \( \triangle{ABC} \cong \triangle{DEF} \).

Let lines \( AB \), \( CD \), and \( EF \) be given with \( E \in \segment{AB} \) and \( F \in \segment{CD} \). If \( \mAngle{AEF} = \mAngle{EFD} \), then \( AB \parallel CD \).

Let distinct lines \( AB \), \( CD \), and \( EF \) be given so that no three of these points are collinear. Assume there are points \( G \in \segment{AB} \cap \segment{EF} \) and \( H \in \segment{CD} \cap \segment{EF} \). If either \( \mAngle{EGB} = \mAngle{GHD} \) or the measure of the sum of \( \angle{BGH} \) and \( \angle{DHG} \) is equal to that of two right angles, then \( AB \parallel CD \).

Let distinct lines \( AB \), \( CD \), and \( EF \) be given so that no three of these points are collinear. Assume there are points \( G \in \segment{AB} \cap \segment{EF} \) and \( H \in \segment{CD} \cap \segment{EF} \). If \( AB \parallel CD \), then the sum of the angles \( \angle{BGH} \) and \( \angle{GHD} \) is equal in measure with the sum of two right angles.

If lines \( \alpha \parallel \beta \) and \( \beta \parallel \gamma \), then \( \alpha \parallel \gamma \). That is, parallelism is a transitive relation on lines.

Let \( A \), \( B \), and \( C \) be non-collinear points. There is a line \( \ell \) with \( A \in \ell \) and \( \ell \parallel BC \).

Let \( \triangle{ABC} \) be given. If side \( \segment{BC} \) is extended to \( \segment{BD} \) with \( C \neq D \), then the angle \( \angle{ACD} \) is equal in measure with the sum of the angles \( \angle{BAC} \) and \( \angle{ABC} \). Moreover, the sum of the three angles of the triangle is equal in measure with the sum of two right angles.

Let \( AB \) and \( CD \) be lines with \( A \) and \( D \) on opposite sides of \( BC \). If \( \mSegment{AB} = \mSegment{CD} \) and \( AB \parallel CD \), then \( \mSegment{AC} = \mSegment{BD} \) and \( AC \parallel BD \).

If \( ABCD \) is a parallelogram, then the angles satisfy \( \mAngle{ABD} = \mAngle{DCA} \). Moreover, the segment \( \segment{BC} \) bisects both of these angles.

If \( ABCD \) and \( EBCF \) are parallelograms and the lines \( AD = EF \), then the these parallelograms are equal in area.

Let \( A \), \( B \), \( C \), \( D \), \( E \), \( F \), \( G \), and \( H \) be points. If \( AD = EH \), \( BC = FG \), \( \mSegment{AD} = \mSegment{EH} \), and \( \mSegment{BC} = \mSegment{FG} \), then the parallelograms \( ABCD \) and \( EFGH \) are equal in area.

Let \( A \), \( B \), \( C \), and \( D \) be points. If \( AD \parallel BC \), then the triangles \( \triangle{ABC} \) and \( \triangle{DBC} \) are equal in area.

Let \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \) be points. If \( BC = EF \), \( AD \parallel BC \), and \( \mSegment{BC} = \mSegment{EF} \), then the triangles \( \triangle{ABC} \) and \( \triangle{DEF} \) are equal in area.

Let \( \triangle{ABC} \) and \( \triangle{DBC} \) be given. If these triangles are equal in area and \( A \) and \( D \) are on the same side of \( BC \), then \( AD \parallel BC \).

Let \( \triangle{ABC} \) and \( \triangle{DEF} \) be given. Assume \( BC = EF \), \( \mSegment{BC} = \mSegment{EF} \), and \( A \) and \( D \) lie on the same side of \( BC \). If these triangles are equal in area, then \( AD \parallel BC \).

Let parallelogram \( ABCD \) and triangle \( \triangle{BCE} \) be given. If \( E \in AD \), then the area of the triangle is half that of the parallelogram.

Let \( \triangle{ABC} \) and angle \( \angle{\alpha} \) be given. There is a parallelogram \( DEFG \) with \( \mAngle{DEF} = \mAngle{\alpha} \) and having the same area as \( \triangle{ABC} \).

Let \( ABCD \) be a parallelogram and \( E \) a point on the diagonal \( \segment{AC} \). Let \( F \in \segment{BC} \) and \( G \in \segment{AD} \) have \( E \in \segment FG \) and \( FG \parallel AB \). Similarly, let \( H \in \segment{AB} \) and \( I \in \segment{CD} \) have \( E \in \segment HI \) and \( HI \parallel BC \). Then the figures \( BFEH \) and \( DGEI \) are parallelograms of equal area.

Let line \( AB \), triangle \( \triangle{\tau} \), and angle \( \angle{\alpha} \) be given. There is a parallelogram \( ABCD \) with \( \mAngle{ABC} = \mAngle{\alpha} \) and having the same area as \( \triangle{tau} \).

Let \( ABCD \) be a quadrilateral and \( \angle{\alpha} \) be an angle. There is a parallelogram \( EFGH \) with \( \mAngle{EFG} = \mAngle{\alpha} \) and equal in area with \( ABCD \).

Let \( \segment{AB} \) be a given segment. There are points \( C \) and \( D \) with \( ABCD \) a square.

Let \( \triangle{ABC} \) be given. If angle \( \angle{ACB} \) is a right angle, then the sum of the areas of the squares with sides measuring \( \mSegment{BC} \) and \( \mSegment{AC} \) is equal to the area of the square with side measuring \( \mSegment{AB} \).

Let \( \triangle{ABC} \) be given. If the sum of the areas of the squares with sides measuring \( \mSegment{BC} \) and \( \mSegment{AC} \) is equal to the area of the square with side measuring \( \mSegment{AB} \), then the angle \( \angle{ACB} \) is a right angle.

In a project like this—a low-priority vanity project I mostly work on when I'm too tired to do research—there are bound to be some errors, some possibly glaring. I'd like those to go away over time. If you, dear reader, find any errors or think I have misrepresented something, please email me! My email address for this purpose is eppolito (/dot/) math AT gmail [dawt] com. Sorry if my response takes some time.

The one exception to my rule "don't change what Euclid does", has to do with definitions. Most of Euclid's definitions are rather shaky from a modern perspective. I have clarified how Euclid thinks about certain terms in my list of definitions. However, this is not perfect, and I should rethink these very carefully at some point.

I hope to create a separate page for each proposition translating its proof into more modern language.

Each proof should have a picture depicting the case(s) resolved by Euclid in his proof. Implementing this will require some thought. At the time of writing, I think the best way is to…

1. Write some lisp or Haskell or Elm code to generate SVG from simple commands describing the constructions.
2. Add the generated SVG to the site on each proofs page an on a common page of figures.

I plan to open-source all the code I write for this purpose. Note that I am already aware of software that would allow me to generate the SVG files. That's not the point here; to my knowledge, there is no major framework for generating these images from human-readable code which would also allow me to effectively just write the proofs from Euclid as code. That is the point of this improvement…

In my humble opinion, such a module/framework would be a major boon for education in geometry. Ideally, this module will also be able to generate interactive versions of the canvas. But that is a very far off goal; it would also be cool if this could be used to generate a slideshow for each proof… I find this prospect exciting!

Each proof should be accompanied by a list of the postulates and propositions upon which it relies. After compiling such a list (e.g. as a JSON or YAML file), I plan to use that database to create an interactive dependence graph on its own page.

As it stands, I've preserved the way Euclid talks about certain things. For example, I leave statements of the form "the sum of the two angles is equal in measure with the sum of two right angles" or "equal in area" mostly unchanged. There are much better ways to handle this, e.g., by introducing some notations, but they will take some care to implement. One day I'll have the time…

I love axiomatics. In the future, I hope to expand this portion of the site with proofs that various historical Parallel Postulate replacements (e.g., Playfair's Axiom, Pi is constant, etc.) are logically equivalent to Postulate 5.

This comes dangerously close to me saying "I'd like to write a textbook on axiomatic geometry". That means I should not consider doing this one until other parts of my academic career are a bit more stable.