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Bibliothèque de l'Institut de France, Paris

Proving Pythagoras: the 3-4-5 triangle and squares on the sides

Three columns of triangle diagrams citing Euclid I.4, I.41 and the penultimate of Book I

Geometry

The folio is filled with three columns of small triangle diagrams working through the Pythagorean theorem, cited as 'the penultimate of the first' (Euclid I.47). A right triangle with legs 3 and 4 and hypotenuse 5 is shown with the squares on its sides, together with the parallelogram-between-parallels lemma ('41,' Euclid I.41), congruent triangles ('the fourth'), and the converse ('it says the angle is not right'). Further citations to 'the eleventh,' 'the fourth and the eighth,' and 'the common notion' mark the logical steps. A few lines of annotation at the lower left are only briefly captured in the transcription, so further untranscribed writing is present on the sheet.

Colonna destra.1 Triangolo rettangolo.

Lo stesso col quadrato dell'ipotenusa.

Lo stesso coi quadrati dei cateti:

Lo stesso colla perpendicolare e una congiungente calate dal vertice:

Lo stesso con altra congiungente e:

Colonna centrale. Perpendicolare a una retta, con:

Triangolo col quadrato del cateto e:

Parte del triangolo e del quadrato, con:

Triangolo e parallelogrammo fra parallele, con:

Colonna sinistra. Due triangoli incrociati, con:

Tre figure analoghe. Angolo inferiore sinistro.2

Due triangoli.

Triangolo, con:

Right column. Right-angled triangle.

The same, with the square of the hypotenuse.

The same, with the squares of the legs:

The same, with the perpendicular and a connecting line dropped from the vertex:

The same, with another connecting line and:

Central column. Perpendicular to a straight line, with:

Triangle with the square of the leg and:

Part of the triangle and of the square, with:

Triangle and parallelogram between parallels, with:

Left column. Two crossed triangles, with:

Three analogous figures. Lower-left corner.

Two triangles.

Triangle, with:

Proving Pythagoras: the 3-4-5 triangle and squares on the sides

On this page

Right triangle with the squares on the legs (3, 4, 5)

Perpendicular dropped from the vertex (the penultimate of Book I)

Triangle and parallelogram between parallels ('41')

Congruent triangles by 'the fourth'

The converse: 'it says the angle is not right'

Further annotation at the lower left