Reference · Book I
The Foundations.
The complete apparatus on which every proposition in Book I rests: twenty-three definitions, five postulates, and five common notions. Nothing else. Every proof in the Lab is licensed by something on this page; if a step is not, the step is unjustified.
Postulates · Five
What may be done.
A postulate is a permitted construction. Each names an action one may perform; together they are the only actions licensed in Book I. The Parallel Postulate is famous for being the one Euclid hesitated over.
Postulate 1
Postulate 2
To produce a finite straight line continuously in a straight line.
A line segment may be extended in either direction without limit. Distinct from drawing a new line.
USED IN I.5, I.16, I.20 …
Postulate 3
Postulate 4
That all right angles are equal to one another.
A claim of homogeneity: the right angle is the same angle everywhere in the plane. Less obvious than it sounds — it amounts to an assumption about the uniformity of space.
USED IN I.14, I.15, I.27 …
Postulate 5
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 the angles are less than the two right angles.
The Parallel Postulate. Famously contentious — many later mathematicians attempted to derive it from the others; the failure of those attempts produced non-Euclidean geometry. It is not invoked in Proposition I.1 or in any of the first 28 propositions.
USED FROM I.29 ONWARDS
Common notions · Five
What may be inferred.
A common notion is a logical principle: a permitted move about equality, addition, subtraction, coincidence, or part-and-whole. They license inference, not construction.
Common Notion 1
Common Notion 2
If equals be added to equals, the wholes are equal.
USED IN I.13, I.16 …
Common Notion 3
If equals be subtracted from equals, the remainders are equal.
USED IN I.15, I.34 …
Common Notion 4
Things which coincide with one another are equal to one another.
The basis of superposition arguments — laying one figure on top of another. Used cautiously by Euclid, who appeals to it sparingly.
USED IN I.4, I.8
Common Notion 5
The whole is greater than the part.
USED IN I.6, I.16, I.20 …
Definitions · Twenty-three
What things are.
Definitions fix the meaning of the terms used in the proofs. Some are descriptive (what a point or a line is); some are operative (the definition of equality of circles, of right angles, of triangles by side or by angle). Several invite philosophical scrutiny — what does it mean for a point to "have no part"? Euclid does not say.
Def. 1
A point is that which has no part.
Def. 2
A line is breadthless length.
Def. 3
The extremities of a line are points.
Def. 4
A straight line is a line which lies evenly with the points on itself.
Famously slippery. Modern axiomatisations replace this with an explicit primitive notion.
Def. 5
A surface is that which has length and breadth only.
Def. 6
The extremities of a surface are lines.
Def. 7
A plane surface is a surface which lies evenly with the straight lines on itself.
Def. 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.
Def. 9
When the lines containing the angle are straight, the angle is called rectilinear.
Def. 10
When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
Def. 11
An obtuse angle is an angle greater than a right angle.
Def. 12
An acute angle is an angle less than a right angle.
Def. 13
A boundary is that which is an extremity of anything.
Def. 14
A figure is that which is contained by any boundary or boundaries.
Def. 15
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
The operative definition: all radii of a circle are equal. The reason Proposition I.1 closes — every side of the constructed triangle is a radius of one of the two circles.
Def. 16
And the point is called the centre of the circle.
Def. 17
A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference; such a straight line also bisects the circle.
Def. 18
A semicircle is the figure contained by the diameter and the circumference cut off by it. The centre of the semicircle is the same as that of the circle.
Def. 19
Rectilinear figures are those contained by straight lines: trilateral figures by three, quadrilateral by four, multilateral by more than four.
Def. 20
Of trilateral figures: an equilateral triangle has three equal sides; an isosceles triangle has two equal sides; a scalene triangle has three unequal sides.
The target of Proposition I.1 is the first of these.
Def. 21
Of trilateral figures: a right-angled triangle has a right angle; an obtuse-angled triangle an obtuse angle; an acute-angled triangle three acute angles.
Def. 22
Of quadrilateral figures: a square is equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; a rhomboid has opposite sides and angles equal but is neither; other quadrilaterals are called trapezia.
Def. 23
Parallel straight lines are straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction.
Conventions · Closing markers
Q.E.F. vs Q.E.D.
Euclid's propositions divide into two kinds, and each kind closes with its own Latin tag in Heath's translation. The distinction is not decorative — it tells you what the proposition was for.
Q.E.F.
Quod erat faciendum — "which was to be done." Closes a problem: a proposition that constructs or produces something. The Greek original is ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai, "the very thing it was required to do"). In Book I, the problems are I.1, I.2, I.3, I.9, I.10, I.11, I.12, I.22, I.23, I.31, I.42, I.44, I.45, and I.46 — each ends in Q.E.F.
Q.E.D.
Quod erat demonstrandum — "which was to be demonstrated." Closes a theorem: a proposition that proves a statement. The Greek is ὅπερ ἔδει δεῖξαι (hoper edei deixai, "the very thing it was required to show"). Everything in Book I that is not a problem is a theorem — including I.4 (SAS), I.5 (the bridge of asses), and I.47 (Pythagoras) — each ends in Q.E.D.
Why it matters
Reading Euclid, the closing tag is the cleanest signal of what the proposition delivered. A problem hands you an object (a triangle, a perpendicular, a square equal in area to a given figure) that the rest of the book may now use as a primitive. A theorem hands you a fact (an equality, a congruence, a relation) that subsequent reasoning may invoke. In the Lab, each proposition's closing line carries the appropriate marker: I.1–I.3 close on Q.E.F.; I.4 and I.5 close on Q.E.D.
Letter mappings · Lab vs Heath
Lab letters and Heath letters.
The Lab labels points in strict Latin alphabetical order from D, in the order they appear on the canvas. Heath's English translation uses different letters in several propositions — following Euclid's Greek by position in each alphabet. The Greek Γ (gamma, 3rd letter) becomes C (3rd Latin letter); Ζ (zeta, 6th) becomes F; Η (eta, 7th) becomes G; Λ (lambda, 11th) becomes L. Heath's letters jump around the Latin alphabet because Euclid's letters proceed straight through the Greek one. The tables below map each Lab letter to its Heath and original-Greek equivalents and to the role the point plays in the proof.
Note: the Lab is being migrated to display these doctrine-compliant letters. Some propositions still show Heath letters on the canvas pending the tutor-copy sweep; this table is the target state.
I.1
On a given finite straight line to construct an equilateral triangle.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Given endpoint of the segment. |
| B | B | Β | Given endpoint of the segment. |
| C | C | Γ | Apex of the equilateral triangle; intersection of the two circles. |
Identity mapping — Heath and the Lab agree throughout I.1. Heath's C is the positional Romanisation of Euclid's Γ (gamma, 3rd letter).
I.2
To place at a given point a straight line equal to a given segment.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | The given point. |
| B | B | Β | Endpoint of the given segment BC. |
| C | C | Γ | Endpoint of the given segment BC. |
| D | D | Δ | Apex of the equilateral triangle on AB. |
| — | E | Ε | Heath names the extension of DA past A; the Lab leaves the extension unlabelled. |
| — | F | Ζ | Heath names the extension of DB past B; the Lab leaves the extension unlabelled. |
| E | G | Η | Intersection of the circle centred at B radius BC with the extension of DB. |
| F | L | Λ | Intersection of the circle centred at D radius DE with the extension of DA. AF is the constructed segment of length BC. |
Heath gives a letter to each extension as a named ray (Ε→E, Ζ→F); the Lab treats extensions as unbounded rays without endpoint labels. The two intersection points consequently take the next Lab letters (E, F) rather than Heath's G and L. Note Heath's F for Ζ and G for Η — both positional Romanisations.
I.3
From the greater of two given lines to cut off a line equal to the less.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Endpoint of the greater segment AB. |
| B | B | Β | Endpoint of the greater segment AB. |
| C | C | Γ | Endpoint of the lesser given. (Heath represents the lesser as a single labelled length C; the Lab displays it as a segment CD for visual clarity.) |
| D | — | — | Other endpoint of the Lab's CD representation; no Heath or Euclid equivalent. |
| E | D | Δ | Transport scaffold — point at distance CD from A, produced by applying Prop. I.2 at A. |
| F | E | Ε | Cut-off point on AB; AF = CD by construction. |
The Lab adds D as the second endpoint of the lesser-segment visualisation; Heath treats the lesser as a free-floating length. The Lab's transport scaffold and cut-off point therefore shift one letter later than Heath's D and E (Δ and Ε in the original).
I.4
SAS: triangles with two equal sides and the included angle equal are congruent.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Apex of △ABC. |
| B | B | Β | Vertex of △ABC. |
| C | C | Γ | Vertex of △ABC. |
| D | D | Δ | Apex of △DEF (the target of superposition). |
| E | E | Ε | Vertex of △DEF. |
| F | F | Ζ | Vertex of △DEF. |
No constructed points — I.4 is a proof by superposition. The two triangles are given; the construction is the motion that maps one onto the other. Heath's F here is Romanised Ζ (zeta, the 6th Greek letter) — same positional logic as elsewhere.
I.5
Pons asinorum: in an isosceles triangle, the angles at the base are equal.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Apex of the isosceles triangle (AB = AC). |
| B | B | Β | Base vertex of △ABC. |
| C | C | Γ | Base vertex of △ABC. |
| — | D | Δ | Heath names the extension of AB past B; the Lab leaves the extension unlabelled. |
| — | E | Ε | Heath names the extension of AC past C; the Lab leaves the extension unlabelled. |
| D | F | Ζ | Any point on the extension of AB beyond B. |
| E | G | Η | Point on the extension of AC beyond C with AE = AD; produced by cutting off an equal length (Prop. I.3). |
As in I.2, Heath labels the extensions themselves (Δ→D and Ε→E); the Lab treats extensions as unbounded rays. The two cut-off points consequently take Lab letters D and E in place of Heath's F and G — which are themselves the Romanisations of Euclid's Ζ and Η, the 6th and 7th Greek letters.
I.6
Converse of pons asinorum: if two angles of a triangle are equal, the sides opposite them are equal. Proved by contradiction.
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Apex of △ABC (the vertex opposite the base whose angles are equal). |
| B | B | Β | Base vertex; the figure is drawn with AB > AC, the suppose-not position of the reductio. |
| C | C | Γ | Base vertex; AC is the shorter side in the supposed-wrong figure. |
| D | — | — | Transport scaffold — point at distance CA from B, produced by applying Prop. I.2 at B. No Heath or Euclid equivalent; Heath packages the I.2 step inline. |
| E | D | Δ | Cut-off point on segment BA with BE = CA; the meaningful new point that the reductio inspects. |
Heath's D on segment BA is one letter behind the Lab's because the Lab makes Prop. I.2's transport step visible — its scaffold point consumes the letter D before the cut-off intersection lands as E.
I.7
Apex uniqueness: no second apex distinct from C can satisfy the same pair of equalities on the same side of AB. The second reductio in Book I; a uniqueness lemma whose load-bearing use is I.8 (SSS).
| Lab | Heath | Greek | Role |
|---|---|---|---|
| A | A | Α | Left endpoint of the base segment AB. |
| B | B | Β | Right endpoint of the base segment AB. |
| C | C | Γ | First apex above AB; AC and CB are the “given two straight lines constructed on AB”. |
| D | D | Δ | Supposed second apex on the same side of AB; the supposition the reductio refutes. |
A clean alignment: Heath's letters for I.7 (A, B, C, D) match the Lab's alphabetical convention exactly. Heath does not name any extensions or auxiliary rays in this proof — the figure is just four points and five segments, all named identically across the two systems.
TRANSLATION · T. L. Heath, The Thirteen Books of Euclid's Elements (Cambridge, 1908). Public domain. Glosses and use-notes are editorial.