Geometry: constructing a triangle, a square, and a pentagon

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Bibliothèque de l'Institut de France, Paris

Geometric construction of an equilateral triangle over a line whose endpoints are labeled a and b. With the compass point placed once at each of points a and b, two arcs are drawn. Their intersection at point c determines the third vertex of an equilateral triangle whose first side is a b. Labels:

c – a b

[I]

For making an equilateral triangle over a line.

When I want to make an equilateral triangle over line a b, I put the point of the compass at a and make circle c b, then I put it at b and make a c, and where the circles intersect at c, there is the third angle.

Geometric construction of a square over a line segment, whose endpoints are labeled a and e. With the compass point placed once at each of the points a and e, two arcs are drawn, intersecting at point c. An estimate of the midpoint of arc a c is marked, at point b, and arc a c is continued, for approximately that length, to point d, to determine one of the upper vertices of the square. Labels:

d – c – b – a e

[II]

For making a rectangular quadrangle over a line.

If I want to make a quadrangle with four right angles over a line, I will make equilateral triangle a c e, and I will make circle a c pass as far beyond intersection c as I believe it receives the half of a c into itself, and I will put that at c d, and the right angle will be at d.

Geometric construction of a regular pentagon, each side being equal to the length of line segment m n. Over line m n, an equilateral triangle is constructed, with upper vertex labeled c. The perpendicular bisector to segment m n passes through point c; the point of bisection is labeled r. Line c n is then bisected at point b, and the perpendicular to c n at point b is drawn, namely, m b. With the compass opened to length m b, point a is found along perpendicular c r, the same distance from point m as b is, which is the center of the circle that will circumscribe a regular pentagon having segment m n as one of its sides. Labels:

c r b m n

[III]

For making an equilateral and equiangular pentagon over a line.

I want to position an equilateral and equiangular pentagon over line m n, and I will do it in this way. I will make equilateral triangle1 m c n over line m n, and I will intersect the middle of face c n at point b with a line starting from angle m, and I will join its end, b, to line c r, and where it touches at point a,2 there is where you will fix one point3 of the compass, and with the other point, draw a circle that touches the ends of line m n, and this circle will contain five of these4 lines within itself.

Reader's Edition — described with AI from the e-Leo transcription.

Geometric construction of an equilateral triangle over a line whose endpoints are labeled a and b. With the compass point placed once at each of points a and b, two arcs are drawn. Their intersection at point c determines the third vertex of an equilateral triangle whose first side is a b. Labels:

For making an equilateral triangle over a line

For making a rectangular quadrangle over a line

For making an equilateral and equiangular pentagon over a line

Geometry: constructing a triangle, a square, and a pentagon

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Equilateral triangle over line a b

Square (right-angled quadrangle) over line a e

Regular pentagon on line m n