Two Systems
Table of Contents
Let us draw a few shapes on this paper.
First we shall mark these 2 points, A and B, and draw from one to the other the curved lines ACB and ADE, and the straight line P3.
(Fig. 1)
Which of them determines the distance between the ends A and B, and why?
The straight line does because it is shorter and unique, distinct, and determinate.
The infinite others are indefinite, unequal, and longer. The choice should depend on that which is unique and definite.
We have the straight line, then, as determining the distance between the two points.
We now add another straight line parallel to AB – let it be CD – so that between them there lies a surface of which I want you to show the breadth.
(Fig. 2)
Therefore starting from point A, tell me how and which way you will go, stopping on the line CD, so as to show me the breadth included between those lines.
Would you determine it according to the measure of the curve AF, or the straight line AF, or. . . ?
According to the straight line AF, and not according to the curve, such being already excluded for such a use.
But I should take neither of them, seeing that the straight line AF runs obliquely.
I should draw a line perpendicular to CD, for this would seem to me to be the shortest, as well as being unique among the infinite number of longer and unequal ones which may be drawn from the point A to every other point of the opposite line CD.
Your choice and the reason you adduce for it seem to me most excellent.
So now we have it that the first dimension is determined by a straight line; the second (namely, breadth) by another straight line, and not only straight, but at right angles to that which determines the length.
Thus we have defined the two dimensions of a surface; that is, length and breadth. But suppose you had to determine a height – for example, how high this platform is from the pavement down below there. Seeing that from any point in the platform we may draw infinite lines, curved or straight, and all of different lengths, to the infinite points of the pavement below, which of all these lines would you make use of?
I would fasten a string to the platform and, by hanging a plummet from it, would let it freely stretch till it reached very near to the pavement; the length of such a string being the straightest and shortest of all the lines that could possibly be drawn from the same point to the pavement, I should say that it was the true height in this case.
If, from the point on the pavement indicated by this hanging string (taking the pavement to be level and not inclined), you should produce two other straight lines, one for the length and the other for the breadth of the surface of the pavement, what angles would they make with the thread?
They would surely meet at right angles, since the string falls perpendicularly and the pavement is quite flat and level.
Therefore if you assign any point for the point of origin of your measurements, and from that produce a straight line as the determinant of the first measurement (that is, of the length) it will necessarily follow that the one which is to define the breadth leaves the first at a right angle.
That which is to denote the altitude, which is the third dimension, going out from the same point, also forms right angles and not oblique angles with the other two.
Thus by three perpendiculars you will have determined the three dimensions AB length, AC breadth, and AD height, by three unique, definite, and shortest lines.
(Fig. 3)
Since clearly no more lines can meet in the said point to make right angles with them, and the dimensions must be determined by the only straight lines which make right angles with each other, then the dimensions are no more than three.
Whatever has the three has all of them, and that which has all of them is divisible in every way, and that which is so, is perfect, etc.
Who says that I cannot draw other lines? Why may I not bring another line from beneath to the point A, which will be perpendicular to the rest?
Surely you cannot make more than three straight lines meet in the same point and form right angles with each other!
Yes, because it seems to me that what Simphcio means would be the same DA prolonged downward. In that way there might also be drawn two others; but they would be the same as the first 3, differing only in that whereas now they merely touch, they would then intersect. But this would not produce any new dimensions.
I shall not say that this argument of yours cannot be conclusive. But I still say, with Aristotle, that in physical matters one need not always require a mathematical demonstration.
Granted, where none is to be had; but when there is one at hand, why do you not wish to use it?
But it would be good to spend no more words on this point, for I think that Salviati will have conceded both to Aristotle and to you, without further demonstration, that the world is a body, and perfect; yea, most perfect, being the chief work of God.
Exactly.
Aristotle in his first division separates the whole into 2 differing contrary parts:
- Celestial
This is ingenerable, incorruptible, inalterable, impenetrable, etc.
- Elemental
This is exposed to continual alteration, mutation, etc.
He takes this difference from the diversity of local motions as his original principle.
He leaves the sensible world and goes into the ideal world.
He begins architectonically to consider that, nature being the principle of motion, it is appropriate that natural bodies should be endowed with local motion.
He then declares local motions to be of three kinds: namely, circular, straight, and mixed straight-and-circular.
The first two he calls simple, because of all lines only the circular and the straight are simple.
Hereupon, restricting himself somewhat, he newly defines among the simple motions one, the circular, to be that which is made around the center; and the other, the straight, to be upward and downward – upward, that which goes from the center; and downward, whatever goes toward the center.
From this he infers it to be necessary and proper that all simple motions are confined to these three kinds; namely, toward the center, away from the center, and around the center.
This answers, he says, with a certain beautiful harmony to what has been said previously about the body; it is perfect in three things, and its motion is likewise.
These motions being established, he goes on to say that some natural bodies being simple, and others composites of those (and he calls those bodies simple which have a natural principle of motion, such as fire and earth), it is proper that simple motions should be those of simple bodies, and that mixed motions should belong to compound bodies; in such a way, moreover, that compounds take the motion of that part which predominates in their composition.