Conclusion

And since those things which are equal to one and the same thing are also equal among themselves original: "inter esse," likely a scribal variant for "inter se." This refers to Euclid’s First Common Notion: "Things which are equal to the same thing are also equal to one another.", it follows that any part of the line contained within one of the aforementioned circles is equal to any part of the line circumscribed in another circle. This same point is proven thus: Let one circle be made; then, by means of a compass compass circini: a tool for drawing circles and measuring distances that has not been adjusted original: "non diuersificati," meaning the distance between the two legs of the compass remains fixed, let its foot be placed on the circumference of that circle, and let the compass, still unvaried, be extended outside the aforementioned circle. There, with the center fixed, let it be drawn so that a second circle is established, touching the first at the aforementioned point. Then, with the same foot of the compass unvaried or unchanged, let the other foot of the compass be moved so that a third circle is established. A straight line straight line linea recta should then be drawn through their three centers; this line is cut into four equal parts, as is evident in the figure mentioned above.

This third theorem theorem theorema: a statement that has been proven based on previously established statements is proven by the author in two ways—though they are not entirely different—in which he briefly seems to hold this view: If you wish to cut a straight line into four equal parts, he says, you should establish one circle. Then, let one foot of an unvaried compass be placed on the circumference circumference circunferentia of that first circle, and let the other, mobile foot be led around so that a second circle is formed. This second circle, passing through the center center centrum of the first, will intersect the first circle in two places and be intersected by it. Again, another circle should be established in the same manner. Then, let a straight line be drawn from one end to the other, according to the definition of a straight line, passing through the three centers—for example, from point a to point b. In this way, it will happen that the straight line is cut into four equal portions, as is expressly shown in the author’s figure above. And just as it can be cut into four, so you will be able to divide a straight line into however many equal portions you please by cutting it with any line... The text ends mid-sentence, suggesting that this method of overlapping circles can be extended indefinitely to divide a line into any number of equal segments.