Squaring of the Circle (1503) - Page 13

Third

Third conclusion.

To divide a straight line into four equal parts.

Let one circle be made; then, without closing or widening the compass original: "circino", let the foot of the compass be placed on the circumference and drawn around so that a second circle is formed, which intersects the first in two places and is intersected by the same, passing through the center of the first. Then, let a straight line be drawn through both centers from the outer edge to the outer edge of each circle. At the point where this line terminates on the circumference of the second circle, let the foot of the compass be placed (keeping the same setting used for the first original: "sub dispositione primi") and let it be drawn around so that a third circle is formed. This third circle will intersect the second in two places and be intersected by it, touching the first circle at the center of the second. Finally, let the aforementioned straight line be extended as far as the circumference of the third circle, as is clear in the accompanying figure.

A geometric diagram consisting of three overlapping circles of equal radius arranged horizontally. A single straight horizontal line passes through the centers of all three circles, extending from the far left edge of the first circle to the far right edge of the third circle. The line is labeled with points: 'A' at the left extremity, '1' at the first center, '2' at the intersection/second center, '3' at the third center, and '4' and 'B' at the right extremity.

Therefore, the aforementioned straight line passing through the three centers—from the edge of the first circle to the edge of the third—is divided into four equal parts. This is because any two adjacent parts of the said line share the same center a "center" here refers to the midpoint from which the circle was drawn and are drawn from that center to the circumference the outer boundary of the circle; they are therefore equal. This construction relies on the fact that all radii of circles drawn with the same compass opening are identical.