Water Pressure
Table of Contents
You are travelling along toward those vacua advocated by a certain ancient philosopher.
Galileo’s antagonist said Galileo “denied Divine Providence”
I hated the rancor of this ill-natured opponent.
It is unpleasant to the good tempered and well ordered mind of one so religious and pious, so orthodox and God-fearing as you.
The circumferences of the 2 circles are equal to the 2 straight lines, CE and BF
- BF is a continuum
- CE is interrupted with an infinity of empty points
How is the line AD drawn by the center and made up of an infinity of points equal to this center which is a single point?
This building up of lines out of points, divisibles out of indivisibles, and finites out of infinites, offers me an obstacle difficult to avoid.
The necessity of introducing a void was conclusively refuted by Aristotle.
These difficulties are real. We are dealing with infinities and indivisibles.
Both of these transcend our finite understanding.
- Infinities have magnitude.
- Indivisibles have smallness.
How can a single point be equal to a line?
Since I cannot do more at present I shall attempt to remove, or at least diminish, one improbability by introducing a similar or a greater one, just as sometimes a wonder is diminished by a miracle. *
I show you 2 equal surfaces with 2 equal solids located on these same surfaces as bases.
All 4 diminish continuously and uniformly.
Their remainders always preserve equality among themselves.
Both the surfaces and the solids terminate their previous constant equality by degenerating, the one solid and the one surface into a very long line, the other solid and the other surface into a single point; that is, the latter to one point, the former to an infinite number of points.
This proposition appears to me wonderful, indeed; but let us hear the explanation and demonstration.
Let AFB be a semicircle with center C.
Around it is the rectangle ADEB.
From the center draw the straight lines CD and CE to the points D and E.
Imagine the radius CF to be drawn perpendicular to either of the lines AB or DE.
The entire figure rotates around this radius as an axis.
The rectangle ADEB will thus describe a cylinder.
The semicircle AFB will describe a hemisphere
The triangle CDE will describe a cone.
Next let us remove the hemisphere but leave the cone and the rest of the cylinder, which we will call a “bowl”
First, we shall prove that the bowl and the cone are equal.
Then we shall show that a plane drawn parallel to the circle which forms the base of the bowl and which has the line DE for diameter and F for a center – a plane whose trace is GN – cuts the bowl in the points G, I, O, N, and the cone in the points H, L, so that the part of the cone indicated by CHL is always equal to (28) the part of the bowl whose profile is represented by the triangles GAI and BON.
Besides this we shall prove that the base of the cone, i. e. , the circle whose diameter is HL, is equal to the circular surface which forms the base of this portion of the bowl, or as one might say, equal to a ribbon whose width is GI. (Note by the way the nature of mathematical definitions which consist merely in the imposition of names or, if you prefer, abbreviations of speech established and introduced in order to avoid the tedious drudgery which you and I now experience simply because we have not agreed to call this surface a “circular band” and that sharp solid portion of the bowl a “round razor. “)
Call them by Fig 6 what name you please, it suffices to understand that the plane, drawn at any height whatever, so long as it is parallel to the base, i. e. , to the circle whose diameter is DE, always cuts the two solids so that the portion CHL of the cone is equal to the upper portion of the bowl; likewise the two areas which are the bases of these solids Y namely the band and the circle HL, are also equal. Here we have the miracle mentioned above; as the cutting plane approaches the line AB the portions of the solids cut off are always equal, so also the areas of their bases. And as the cutting plane comes near the top, the two solids (always equal) as well as their bases (areas which are also equal) finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely, the upper edge of the bowl and the apex of the cone. Now, since as these solids diminish equality is maintained between them up to the very last, we are justified in saying that, at the extreme and final end of this diminution, they are still equal and that one is not infinitely greater than the other. It appears therefore that we may equate the circumference of a large circle to a single point. And this which is true of the solids is true also of the surfaces which (29) form their bases; for these also preserve equality between themselves throughout their diminution and in the end vanish, the one into the circumference of a circle, the other into a single point. Shall we not then call them equal seeing that they are the last traces and remnants of equal magnitudes? Note also that, even if these vessels were large enough to contain immense celestial hemispheres, both their upper edges and the apexes of the cones therein contained would always remain equal and would vanish, the former into circles having the dimensions of the largest celestial orbits, the latter into single points. Hence in conformity with the preceding we may say that all circumferences of circles, however different, are equal to each other, and are each equal to a single point.