Right-angled triangles and the doubling of a square

Doubling a triangle: half right angles make a whole right angle

An isosceles right triangle is doubled, and the note at n reads 'half right angles: therefore n is a whole right angle.' A smaller triangle inserted gives a 'small right angle at m.' The construction shows how right angles are built up and subdivided.

Squares on the leg and hypotenuse, 'double' and 'sub-double'

A right triangle carries squares raised on one leg and on the hypotenuse, marked 'double' and 'sub-double' to compare their areas. The two preceding figures are then combined at the left under the same labels. The pair studies the ratio of the squares on a right triangle's sides.

A line divided and a perpendicular half its length

A line is divided into equal parts (a, b, c) and into equal and unequal parts (a, b, d, c; 12 then 6, 4, 2). On it a perpendicular equal to half the line is raised to form an isosceles right ('ortogonio') triangle. A parallel drawn to the base produces a further right angle.

Arithmetical sums of the squared sides

Two additions are set at the left: 100 plus 4 giving 104, and 36 plus 16 giving 52. They tally the numerical areas belonging to the drawn right triangles. The calculation runs alongside the geometric doubling.

Linea divisa in parti uguali, con:

Linea divisa in parti uguali e disuguali, con:

A sinistra, idem, con:

Perpendicolare sulla linea predetta, uguale alla sua metà.

Formazione di un triangolo isoscele:

Raddoppio del medesimo, con:

Inserimento di un triangolo minore, con:

Con una parallela alla base si forma un altro ortogonio.

Triangolo rettangolo coi quadrati su un cateto e l'ipotenusa, con:

A sinistra. Sintesi delle due figure precedenti, con:

Tre figure senza lettere.

Operazioni aritmetiche:

Line divided into equal parts, with:

Line divided into equal and unequal parts, with:

On the left, the same, with:

Perpendicular on the aforesaid line, equal to its half.

Formation of an isosceles triangle:

Doubling of the same, with:

Insertion of a smaller triangle, with:

With a parallel to the base another right angle is formed.

Right triangle with the squares on a leg and the hypotenuse, with:

On the left. Synthesis of the two preceding figures, with:

Three figures without letters.

Arithmetical operations:

Right-angled triangles and the doubling of a square

On this page

Doubling a triangle: half right angles make a whole right angle

Squares on the leg and hypotenuse, 'double' and 'sub-double'

A line divided and a perpendicular half its length

Arithmetical sums of the squared sides