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Bibliothèque de l'Institut de France, Paris

[I]

Any right angle joined to the center of a circle must delimit the fourth part of this circle1 with its two2 lines; and the angle of an equilateral triangle joined to this center delimits the sixth part of the circle with its two lines.

A complex geometrical drawing based on two overlapping circles, whose centers are labeled a and b; the circumference of each circle passes through the center of the other. The section common to both circles is divided into quarters by one vertical and one horizontal line; and the upper point where the vertical line intersects with the two circles is labeled f. A smaller circle is inscribed within the section common to both circles, tangent at points a and b, its center labeled S; a square is inscribed inside of this, its upper vertex labeled n. Another circle, the same size as the inscribed circle, is drawn with its center at n and intersecting with side a n of the square at point t. Points a, b, and f are joined, producing an equilateral triangle. Labels:

f - n - t - a S b

[II]

a b f is the sixth part of the larger circle; a n b is half of the square; t s is one-eighth of the smaller circle.

In the central column drawing of a large circle, with two perpendicular diameters intersecting at center b. Each of the two lower quarters is divided into twelve equal sectors, and the boundaries of these sectors, radiating from center b, are labeled, from right to left:

a e f - g h i K l m n o p d - 1 2 3 4 5 6 7 8 9 10 11 c

Right of the figure: [III]

e b d is the first degree of the obtuse angle.

b f first

b g 2nd

b h 3rd

b i 4

b K 5

b l 6

b m 7

b n 8

b o 9

b p 103

Left of the figure:

[IV]

d b 11 is the first degree of the acute angle

b 10 2nd

b 9 3rd

b 8 4th

b 7 5th

b 6 6

b 5 7

b 4 8

b 3 9

b 2 10

b 1 11

[V]

Here a certain rule is made, for designating names for angles according to their variety, because in the ancient writings,4 only two names are found for angles other than the right angle, namely, the "acute" and the "obtuse" angles. But they did not set5 the degrees, as the musicians did for human voices. So because of this, as I see it necessary to give names that are different, corresponding to the diversity of the angles that are greater or smaller than a right angle, I have designated, above here, degrees for the acute and the obtuse angles. From a to d6 will be all the lines arising from the obtuse angles positioned at point b, and likewise at b will be all the acute angles arising from the circle between d and c. And I will divide7 the obtuse angles into twelve degrees, and likewise for the acute angles, and the right angle will be in the middle.8

Geometry: naming and grading the degrees of angles

A protractor-like circle dividing obtuse and acute angles into twelve degrees each

Geometry

Leonardo proposes a system for naming angles by degree, noting that the ancients recorded only the 'acute' and 'obtuse' angles besides the right angle and set no degrees, 'as the musicians did for human voices.' A large circle centered at b has its two lower quadrants each divided into twelve sectors, the radiating lines lettered e f g h i K l m n o p on one side and numbered 1 to 11 on the other, grading the obtuse and acute angles with the right angle in the middle. An accompanying figure of two overlapping circles shows that a right angle at the center cuts off a quarter of the circle and an equilateral-triangle angle a sixth.

[I]

In the central column drawing of a large circle, with two perpendicular diameters intersecting at center b. Each of the two lower quarters is divided into twelve equal sectors, and the boundaries of these sectors, radiating from center b, are labeled, from right to left:

Right of the figure: [III]

Left of the figure:

[V]

Right angle and triangle-angle at a circle's center

Proportions in the overlapping-circles figure

The graduated circle's labels

Degrees of the obtuse angle (right of the figure)

Degrees of the acute angle (left of the figure)

A rule for naming angles by degree

Geometry: naming and grading the degrees of angles

On this page

A right angle cuts a quarter, a triangle's angle a sixth

Proportions in the overlapping-circles figure

The graduated circle divided into sectors

Degrees of the obtuse angle

Degrees of the acute angle

A new rule for naming angles by degree