First treatise: triangles in semicircles and the quadrature of lunes

Right half, from the top.

First treatise. Account will be kept of the rectangle in its semicircle, in changing position with the angle, of how much it varies its height over the same base.

Four semicircles contained within five parallel lines, with the letters:

a - d - f - h - i p g o e n c m b

a b c is the greatest triangle that can fit in the semicircle.

Of triangles set upon the same base, the proportion of the quantity is as that of their axes (assis).

The axis f o, being double the axis h p, the triangle e f g is double the said triangle; and the axis d n, being double the axis f o, the triangle e f g is worth twice as much again.

The like follows in like order, and so on. There follows the order of the proportions of the increases and decreases of the lateral portions of the right angle (ortogonio) made within the circle. Always, of the 2 portions that within the semicircle put their right angle in the middle, the one that grows incorporates the right angle and the opposite portion.

The last increase of one of the portions that in the semicircle puts the right angle in the middle will be at the last destruction of the right angle and of the opposite portion.

The birth of one of the 2 portions in the semicircle will never be without the birth of the right angle.

Of the 2 portions lateral to the right angle set in the semicircle, that one will have the more acute angles which is of smaller figure; and the opposite portion will be made of greater angle as much as the smaller diminishes.

Never will the angle made of the whole diameter of the semicircle and touching the semicircle diminish its quantity, nor increase it, for any movement it may make along the whole length of the periphery of such semicircle.

The greatest space that can be included within the right angle will be at the last equality of the 2 portions that include the right angle between them. The greatest increase that can be made by one of the portions that include the right angle will be at the destruction of that right angle and of the opposite portion.

The last size of one of the portions that include the right angle between them will be equal to the semicircle that included it. The last increase that one of the hypotenuses of the right angle can acquire will be at the destruction of the opposite hypotenuse. It is impossible that the angle of the right angle made by the base of the whole diameter of a circle could ever go outside such semicircle.

The knowledge of the proportion the portions made in the same semicircle have among themselves will be had through the knowledge of the sides of the right angle divided by its cathetus, or, I mean, axis.

The 2 portions that put the right angle in the middle always consume the whole periphery of the semicircle.

The smallest right angle that can be made in the semicircle will be in the middle of the greatest difference of portions that can fit in such semicircle.

Right margin. Rectangle with two curved sides, divided into three lune-segments, with:

a a - c b - c

The triangles of 2 or 3 curved sides that can be made in the parallel of the same curvature will be equal to half of such parallel.

Figure similar to the preceding, with greater subdivisions and the letters:

c a b - n - m - g f

a is worth b c.

I take n from the half-portion n and take it again in m.

Left margin, at the top. Semicircle divided into sectors and rectangles:

b a - d c

Give me the third of the semicircle between parallel lines with a single portion, that is in the figure of a sector, and then give it to me between parallel lines; then remove from the parallel the 2 portions and from the sector one, and the value of the other lies in the angle of that sector.

Three semicircles and a quarter with the letters:

c - a - b

a b is worth half of b c. It is proven because the triangles are of double heights and of equal bases.

Two rectangles including a triangle, with:

c a b; e b f

The greatest rectilinear triangle that can be made in the parallel will always be equal to half of such parallel.

Left half. Right column. Three semicircles constructed on the sides of a right triangle, with:

b - d - a

If from the ends of a portion one removes part of the proportion of its whole, the remainder stays in its first proportion. Therefore, having here 3 semicircles of which the first, smaller, a has 2 of diameter and the greater d has 4 of diameter, which is the root of 16, subtracting the multiplication of the root 2, which is 4, the diameter of the smaller circle remains the root of 12, which is surd (irrational).

Such proportion is from circle to circle as from square to square, made from the multiplication in itself of their diameters.

Bound with a pen stroke on the left:

The semicircle a is worth a quarter of the semicircle d, to which it is quadruple, and the semicircle b is worth 3/4 of that semicircle d.

Three semicircles with the letters and numbers:

a 4; b 3; c 1

Three semicircles with a lune, with the letters:

a a; b b;

c c

Three semicircles constructed on the sides of a right triangle, with:

b - d 16 a 9 - 4 3 - c 25 - c 5 a

The lune, or semicircle a, is 9/25 of the semicircle c. The third semicircle d is 16/25 of that semicircle c. Therefore one is 9, the other is 16, the third 25.

Figure similar to the preceding, with:

b - a

The portion a is one-quarter (subquadruple) of the greater.

To the left of the preceding, a figure with the letters:

a - b - c

The rectilinear surface a b c is equal to the 2 lunes and likewise is worth the right angle, and the b is contained with the c.

Two opposed circular sectors, with:

b a

The surfaces a b, which are squarable except for their contact, which is the curve b that divides the concave contact of a from the convex contact of c: this contact being reserved, the remainder of their surfaces is squared into the pyramid a and the pyramid d.

Around a semicircle including a triangle, with:

b a

Here below one must describe the variety of the 2 portions a b in changing the rectangle of the right angle in the semicircle, and, true knowledge of it being made, one may proceed in the figures above.

Left column, from the top. Two semicircles constructed on a right triangle, with:

a - b

The semicircle a is double the semicircle b, and so do their similar portions, which will always be double one to the other.

Figure similar to the preceding, with:

d c

The semicircle d is quadruple the semicircle c. Therefore the quarter of the greater is worth the whole of the smaller, c, and with such a rule one proceeds toward infinity.

Semicircle with crossed-out portions and the letters:

m - n - a - b

Because the quadrature of this lune is made, as much as it can be, with the surface a b, I will keep the pointed angle and draw out the rest of the right angle.

Below the preceding figure, a small triangle with:

c b a

Figure of two semicircles constructed on a right triangle, with the letters:

d e f - S a - c b

Here it happens that the smaller semicircle is 3/4 of the greater semicircle. The like do the 3 portions of the smaller with the 3 of the greater, and so each of the smaller with each of the greater. Therefore, to square by approximation such a lune, I will give it the 2 inner portions, which are worth 8/3 of the 3 portions above, c d f e, from which I will remove f, which is 1/3 of the portion f e. There remains e, which is 2/3 of that portion; there remains to me e d c exactly equal to the a b because, as said, a b are worth 8 and the d c are worth 6; adding the 2/3 of f e, that is e, there will be made in e d c a quantity of 8/3 equal to the 8/3 of a b. It follows that, removing e d c from the lune and giving back a b below, we have f a S b equal to the whole lune.

There remains to us f, which is like 1/8 of a b, because the 8 parts of c d e are worth the 8 of a b; for if the 2 portions a e and f e are 3/4 one of the other, the third of the smaller is worth 1/4 of the greater. The example by numbers: see 9, which is 3/4 of 12, so that the third part of 9, which is 3, is worth as much as the fourth part of 12, which is likewise 3.

Therefore it is concluded that the periphery, or portion of portion, f is worth the fourth part of the portion b n m, that is, the periphery n m is the curved side of that fourth part of b n m. Therefore I will remove f from its surface and join the same remaining surface with the surface a, and the part n will restore to the S b joined to it the part f that was removed from it, which on the far side is placed in b between the a c surface.

At the top, arithmetic operations:

[Columns of figures and multiplications:] 72 / 200 / 300 / 200 / 72 / 9 / 600 / 2100 / 25 / 18 / 21600 / 50 / 36 / 016 / 20016 / 100 / 21600 / 20000 / 200 / 2 / 108 / 72 / 200

Metà destra, dall'alto.

Quattro semicerchi contenuti entro cinque linee parallele, colle lettere:

Margine destro. Rettangolo con due lati curvi, diviso in tre falcate, con:

Figura analoga alla precedente, con maggiori suddivisioni e le lettere:

Margine sinistro, in alto. Semicerchio diviso in settori e rettangoli:

Tre semicerchi e un quarto colle lettere:

Due rettangoli includenti un triangolo, con:

Metà sinistra. Colonna destra. Tre semicerchi costruiti sui lati d'un triangolo rettangolo, con:

Legato con tratto di penna sulla sinistra:

Tre semicerchi colle lettere e i numeri:

Tre semicerchi con lunola, colle lettere:

Tre semicerchi costruiti sui lati d'un triangolo rettangolo, con:

Figura simile alla precedente, con:

A sinistra della precedente, figura colle lettere:

Due settori di cerchio contrapposti, con:

Intorno a un semicerchio includente un triangolo, con:

Colonna sinistra, dall'alto. Due semicerchi costruiti su un triangolo rettangolo, con:

Figura simile alla precedente, con:

Semicerchio con porzioni depennate e le lettere:

Sotto la figura precedente, piccolo triangolo con:

Figura di due semicerchi costruiti su un triangolo rettangolo, colle lettere:

In alto, operazioni aritmetiche:

Right half, from the top.

Four semicircles contained within five parallel lines, with the letters:

Right margin. Rectangle with two curved sides, divided into three lune-segments, with:

Figure similar to the preceding, with greater subdivisions and the letters:

Left margin, at the top. Semicircle divided into sectors and rectangles:

Three semicircles and a quarter with the letters:

Two rectangles including a triangle, with:

Left half. Right column. Three semicircles constructed on the sides of a right triangle, with:

Bound with a pen stroke on the left:

Three semicircles with the letters and numbers:

Three semicircles with a lune, with the letters:

Three semicircles constructed on the sides of a right triangle, with:

Figure similar to the preceding, with:

To the left of the preceding, a figure with the letters:

Two opposed circular sectors, with:

Around a semicircle including a triangle, with:

Left column, from the top. Two semicircles constructed on a right triangle, with:

Figure similar to the preceding, with:

Semicircle with crossed-out portions and the letters:

Below the preceding figure, a small triangle with:

Figure of two semicircles constructed on a right triangle, with the letters:

At the top, arithmetic operations:

On this page

Largest triangle in a semicircle; triangles proportional to their axes

Circle is to circle as square to square of diameters

Lunes on a right triangle in the ratio 9 : 16 : 25

Approximate quadrature of a lune by the 8/3 method

The number 9 as three-quarters of 12

Columns of arithmetic