Dialogue Concerning the Two Chief World Systems

...assigned the width; however, starting from point A, tell me where and how you wish to end at line C.D. to assign me the width contained between these lines; I ask if you will determine it according to the length of the curve A.E., or perhaps the straight line A.F., or even—

SIMP. According to the straight line A.F. and not the curve, curves having already been excluded from such use. Simplicio follows the Aristotelian preference for straight lines in measuring distances between points.

SAGR. But I would use neither the one nor the other, seeing that the straight line A.F. goes obliquely; instead, I would want to draw a line that was square original: "a squadra"—meaning perpendicular or at a 90-degree angle. to C.D., because this seems to me that it would be the shortest, and the only one among the infinite longer and unequal lines that can be produced from point A to various other points on the opposite line C.D.

SALV. Your choice and the reason you give for it seem most perfect to me; so that thus far we have established that the first dimension is determined by a straight line; the second, that is, the width, by another straight line—and not only straight, but furthermore at right angles to the other which determined the length. Thus we have defined the two dimensions of a surface, namely length and width. But when you had to determine a height—as, for example, how high this ceiling ceiling: original "palco," referring here to the structural ceiling of the room. is from the floor floor: original "pavimento." that we have under our feet—since from any point on the ceiling one can draw infinite lines, both curved and straight, and all of different lengths, to infinite points on the floor below, which of these lines would you use?

SAGR. I would attach a string to the ceiling, and with a small lead weight lead weight: original "piombino," a plumb bob used by builders to find a true vertical line. hanging from it, I would let it hang freely until it reached near the floor; and the length of such a string being the straightest and shortest of all lines that could be drawn from that same point to the floor, I would say that it was the true height of this room.

SALV. Very well. And when, from the point marked on the floor by this hanging string (assuming the floor is level and not inclined), you were to draw two other straight lines, one for the length and the other for the width of the surface of that floor, what angles would they form with that string?

SAGR. They would surely form right angles, since the string falls plumb original: "a piombo"—perfectly vertical. and the floor is quite flat and well-leveled.

SALV. Therefore, if you establish any point as the head and termi—