Historical evidence
A proposition to make a triangle
In the cited English edition, Proposition I.1 asks for an equilateral triangle on a given finite straight line. Its text names the segment AB, two circles, an intersection C, and the joins CA and CB.
The proposition is constructive: it tells the reader what to draw and then justifies why the result has the required property. The English wording and marginal references belong to a modern edition and teaching tradition.[EUCLID]
Historical evidence
Construction and justification
A circle centred at A with radius AB makes every point on that circle as far from A as B is. The second circle repeats the relation from B. Their shared point carries both radius equalities.
The proof then uses the common notion that things equal to the same thing are equal to one another. In the Atlas figure, colour, step numbers and live lengths are modern reading aids.[EUCLID]
Reconstruction
Construct the equilateral triangle
Advance through the segment, two circles and final joins. Choose either circle intersection: the construction works above or below AB, and the live proof reports the same three lengths.
Changing AB rescales the entire figure without changing its logical structure. Buttons and select controls provide keyboard and touch access to every step.[EUCLID]
Reconstruction · compass and straightedge
Build the triangle
Modern coordinate drawing, uniformly scaled to fit. Circles are clipped by the frame; their exact computed radii remain AB.
- Draw the given segment AB.
- Draw the circle with centre A and radius AB.
- Draw the circle with centre B and radius BA.
- Choose one intersection C and join CA and CB.
AC and AB are radii of the first circle. BC and BA are radii of the second. Therefore all three segments are equal.
Modern translation
Coordinates behind the drawing
Modern coordinates make the two intersections explicit. The distance formula gives AC = BC = AB = s, so the computed reconstruction can check the same equalities numerically.
Coordinates are not attributed to the ancient text. They are a present-day verification of the diagram used on this page.[EUCLID]
Open question
Where the circles meet
The proposition names the point where the circles cut one another. Joyce’s guide highlights that the postulates stated earlier in the Elements do not by themselves establish that such a point exists.
The gap is a precise question about foundations. It should not be turned into a general claim that the construction or the Elements is invalid.[EUCLID]
The cited guide notes that Proposition I.1 assumes the circles intersect. Modern axiom systems need an appropriate existence or continuity assumption. This foundations question does not erase the historical construction.
Historical evidence
A proof travels through texts
The page links to one English edition rather than claiming direct access to an authorial original. Copies, commentaries and translations are part of how the Elements reached later readers.
Elsewhere in the Atlas, the transmission timeline names particular Arabic translators. A documented translation is more specific than a broad arrow between civilizations.[EUCLID]
Open question
Why the first proposition matters
Proposition I.1 joins an executable procedure to a chain of reasons. That pairing remains useful when reading algorithms and computer experiments: producing an object and proving a property of it are related tasks, but they are not identical.
Continue to the Geometry exhibit to compare triangles in different geometries, or follow proof and transmission in the historical Atlas.[EUCLID]