First Dialogue

...of the measures, and from it you will make a straight line depart, as the determinant of the first measure, that is, the length; it will be necessary by necessity that the one which must define the width departs at a right angle upon the first, and that the one which has to mark the height—which is the third dimension—departing from the same point, also forms with the other two angles that are not oblique, but right. And thus from the three perpendiculars you will have, as from three unique, certain, and shortest lines, the three dimensions determined: A.B. length, A.C. width, A.D. height. And because it is a clear thing that at the same point no other line can meet that makes right angles with those, and the dimensions must be determined by the only straight lines that among themselves make right angles, therefore the dimensions are no more than three; and he who has the three has them all, and he who has them all is divisible in all directions, and he who is such is perfect, etc.

A geometric diagram showing three lines meeting at a central point 'A'. Line 'AB' extends horizontally to the right. Line 'AD' extends vertically upwards. Line 'AC' extends diagonally upwards and to the right, illustrating the three dimensions of space intersecting at right angles.

SIMP. And who says that other lines cannot be drawn? And why can I not make another line come from below up to point A that is square original: "a squadra"—meaning perpendicular or forming a 90-degree angle. with the others?

SALV. You certainly cannot, at one and the same point, make anything else meet except three straight lines alone which constitute right angles among themselves.

SAGR. Yes, because the one that Mr. Simplicio mentions seems to me would be the same D.A. prolonged downwards; and in this way one could draw another two, but they would be the same first three, differing in nothing except that where they now only touch, they would then intersect; but they would not bring new dimensions.

In natural proofs, one must not seek geometric exactness.

SIMP. I will not say that this reason of yours cannot be conclusive, but I will say indeed with Aristotle original: "Aristotile" that in natural things natural things: refers to the study of the physical world or physics, as opposed to the abstract world of mathematics. one must not always seek the necessity of a mathematical demonstration.

SAGR. Yes, perhaps where it cannot be had; but if it is available here, why do you not want to use it? But it will be well not to spend any more words on this particular, because I believe that Mr. Salviati would have granted to Aristotle and to you, without other demonstrations, that the World is a body, and is perfect, and most perfect, as the greatest work of God.