Dividing lines and circles into equal parts; squaring the circle

From the top, near the right margin:

Let an aliquot part of any line be given, and let there be given ... Let a known part of any line be given, and with it ...

Beside the previous note, a line divided into parts:

From three here I wish to make 5.

A line divided into three parts, one of them divided into 5.

7

A line divided into three and into nine parts.

3. If you wish to divide into 9 the line divided by 3, divide one of those 3 parts into 9, and take the 3/9 of that divided line, and with that measure of 3/9 you will divide the line exactly.

A line divided into parts:

If from 5 you wish to make 3, multiply 3 by 5, and you will have 15, which you will divide by 3, and you will have made 3 of 5 parts of a whole, without diminishing that whole.

Beneath the previous note:

To divide a circle from one division into another, into as many parts as you wish, with one span of the compass: first multiply one of the sides of the given first division by all the parts into which the circle is first divided, and from the sum take one, and take the remainder with the compass, and you will cover the circumference entirely with the intended numbers. It is proven: let the wheel first be divided into three equal parts, and you wish to divide it into 4. Multiply then 3 by itself, which gives 9, and from that result take one, and divide the remainder by the 4 that you assigned for the division of that circle, which gives 2 per part; and thus you have divided the wheel from 3 to 4. And if from the 4 you wish to make 5 parts, say: 4 times 4, 16; take away one, remains 15; divide it by 5, gives 3. And so continue to infinity.

Going up in the second column toward the center:

Divide one of the parts into which the circle is divided into as many parts as you wish to increase it, and add one of those small parts to one of the larger parts, and open the compass so that it spans one of the first parts with the said addition, and this will divide the said circle entirely into equal parts. It is proven: let the circle first be divided into 3 equal parts, and you wish to divide it into 4, dividing one of those parts into 4 and adding one ...

Beside two circles divided with straight and curved lines:

Suppose it were divided by three and you wish to make 4. Say: 3 times 4 makes 12; divide it by that 4; it gives three per part, and this 3 is the divisor of the wheel into 4 equal parts. 3 - 2. Let us say that one of those 3 parts into which the circle was divided were one braccio. Divide that braccio into as many numbers as you want the sides into which the circle is to be divided, and from that number take one, and divide the remainder into the number of sides. Therefore, if at first there were three sides with one braccio per side, you will divide this braccio by 3, and from the number take one, and divide this into the 4 sides, so that two divisions per side will fall in the same circle.

To the left of the previous passage:

If it is a third division of a circle, divide one of those parts into 4, and throw away one, and open the compass according to the remainder of the 4, which is 3, and with that division you will cover the whole circumference in equal parts, and so always continue in dividing. And if from the 4 parts you wish to make 5, divide one of those parts of 4 into 5, and throw away one, and continue.

To the left of the previous, a series of circles inscribed with polygons of three, four, five and six sides:

60 - 6. Divide the circle into 60 and you will say: the portion that occupies a third of the circumference is 20, the one that occupies a quarter is 15, the fifth is 12, the sixth is 10, and ... a half 30, a third 20, [a quarter] 15, a fifth [12], a sixth 10. 30 20 15 12 10. If you divide into 4 a triangular surface made in a circle: if the circle had three divisions, divide one of those parts into 4 and throw away one, and this will be the quarter of that circle. And if you wish to divide it into 5 with one opening of the compass, divide one of those parts into 5 and throw away one, and the remainder will be the fifth of that circumference. And this rule goes on to infinity.

Lower left corner:

And if the circumference of the wheel is 3 and you wished to divide it into 5, you will say: 3 times 5 makes 15. Now make the multiplier become the divisor by the number into which you wish to divide it (that is, the wheel), and so divide 15 by that 3. It gives the same, that is, 3; and opening the compass as much as the space of that 3 requires, you will measure it exactly 5 times upon the circumference of that wheel. And if you wish to divide the wheel of 3 braccia into seven equal parts with a single opening of the compass, say: 3 times 7 makes 21. Divide 21 by 3, it gives 3 for that seventh part, which you will enclose between the points of the compass, and with that space you will measure exactly seven times the whole circumference of the said wheel. And this division and invention will succeed for us upon every given number, with every number.

At the center of the upper margin, portions in a quadrant of a circle:

One portion is worth 3.

A semicircle with the letters:

e d - a - g with b f. The semicircle is worth two circles, and therefore that which exceeds that circle is worth a circle.

To the right of the previous, two overlapping circle sectors:

Quarter - 1/2 - Quadruple.

Two more overlapping sectors:

Quarter - Eighth - Double.

Below the semicircle, more overlapping sectors:

Third.

Above a semicircle with the letters:

a b c o. a, c, o are equal.

Below the same, a small figure with the letter:

a. a is worth the semicircle. a b is divided equally; c o is divided equally. And because a b is worth c o, a is worth o, and b is worth c. And if you give a square equal to c, you also give it equal to the other parts a b o; and giving that square equal to o, you have given the quadrature of the circle.

To the right of the previous, a sector with the letters:

b d - e a - f g c. The semicircle a b c is worth the lune d e f g. I will remove b of the circle and d of the lune; there will remain a c equal to g and f. Then I will remove c of the circle and e of the lune, and there will remain a equal to g f.

Higher up on the left, a figure with the letters:

b a - c m n. Quadruple proportion. Remove ... a n e b m are equal. Remove a from n and remove b from m. There remains n equal to m. From which m remove the portion c and give back 4 below. And because the greater circle is worth 4 of the smaller, the portion c of the greater circle restores the 4 smaller below, and the smaller portion retains in itself the sample, which stands in the corner of the lune.

Above a square enclosed in another square:

Such is the proportion from circle to circle, as ...

A little lower down:

The circle made from the half-diameter of a circle is worth 1/4 of the greater circle, and its diameter enters 6 [times] upon the circumference of that greater one.

On the right margin, circles and sectors:

Two various methods, one of removing, the other of covering.

Dall'alto, presso il margine destro:

Di fianco alla nota precedente, linea divisa in parti:

Linea divisa in tre parti con una d'esse divisa in 5.

Linea divisa in tre e in nove parti.

Linea divisa in parti:

Sotto la nota precedente:

Risalendo nella seconda colonna verso il centro:

Accanto a due cerchi divisi con linee rette e curve:

A sinistra del brano precedente:

A sinistra del precedente, una serie di cerchi con iscritti poligoni di tre, quattro, cinque, sei lati:

Angolo inferiore sinistro:

Al centro del margine superiore, porzioni in un quadrante di cerchio:

Semicerchio colle lettere:

A destra del precedente, due settori di cerchio sovrapposti:

Altri due settori sovrapposti:

Sotto il semicerchio altri settori sovrapposti:

Sopra un semicerchio colle lettere:

Sotto lo stesso, piccola figura colla lettera:

A destra del precedente, settore colle lettere:

Più sopra a sinistra, figura colle lettere:

Sopra un quadrato incluso in altro quadrato:

Poco più in basso:

Sul margine destro, cerchi e settori: