there are two directions for acquiring knowledge of the world, each beginning from a different...

There are two directions for acquiring knowledge of the world, each beginning from a different starting point.

First, knowledge can start with the complicated sense object; such a starting point is certain, but it is confused and vague.

There are two places that knowledge can begin.

Second, knowledge can start with the extremely simple; such a beginning is clear and precise, but also partial and incomplete.

When we start with a complicated sensible object, our knowledge is holistic, but it is necessarily vague and confused. Nevertheless, we can then proceed to break the whole down into its simpler parts and principles. This process is called analysis.

When we start with a simple element, our knowledge is precise, but it is necessarily partial and incomplete. Nevertheless, we can then proceed to build the simple up into clear, precise knowledge of the complex whole. This process is called synthesis.

These two methods must be combined to get perfect knowledge: we must start with common sense, then analyze down into the principles….…and end with

synthesizing back up to a more perfect knowledge of the ultimate whole. Through this process, we can never deny our starting common sense. We only improve it.

Geometry is a science. That means it must be certain about everything it says. In order to be certain, it must start out saying things that are sure and true.

What can we be sure about? Some people would say, “I think, therefore I am.” This is making things way too complicated. There are many things that are a lot more certain. Even before we think about thinking, we know that we exist, and that the world around us exists. We know we have a body and that the world is full of other bodies. When we move our body around, we take note of those other bodies and behave accordingly. No sane person tries to walk through walls. We open doors before passing through a door way. Even the most skeptical people do this.

It is obvious that bodies exist.


What can we be sure about? Some people would say, “I think, therefore I am.” This argument requires us to think and talk about thinking, and so it is making things way too complicated. There are many things that are a lot more certain.

Even before we think about thinking, we know that we exist, and that the world around us exists. We know we have a body and that the world is full of other bodies. When we move our body around, we take note of those other bodies and behave accordingly. No sane person tries to walk through walls. We open doors before passing through a door way. Even the most skeptical people do this.



What can we be sure about? Some people would say, “I think, therefore I am.” This argument requires us to think and talk about thinking, and so it is making things way too complicated. There are many things that are a lot more certain.

Even before we think about thinking, we know that we exist, and that the world around us exists. We know we have a body and that the world is full of other bodies. When we move our body around, we take note of those other bodies and behave accordingly. No sane person tries to walk through walls. We open doors before passing through a door way. Even the most skeptical people do this.


The Foundations of Geometry• By body, we do not mean just

living things, but anything that has volume, including inorganic things such as rocks, and minerals.

• Nor do we mean just massive bodies, but even the seemingly empty places we see about us are filled with bodies. Bodies include the tiny gaps between molecules, to the gaps around us filled with invisible air, to the enormous gaps between the stars.

• Bodies are everywhere and fill the entirety of space. In fact, the universe can be viewed as one enormous body made of innumerable parts in immediate contact with one another.

The Foundations of Geometry• We come into the world

surrounded by bodies. The crib around us is a body. A mobile that hangs above us is a body. We are separated from the mobile by another body, the invisible air. A mattress lays beneath us. We are separated from the mattress by another body, the sheet. We are separated from the sheet by our pajamas.

• What separates one body from another? It is just another body. But when two bodies are divided from each other by another body, that third body creates a gap between the two bodies that pushes them apart.

• But we shall see that it is not possible for every body to be divided from its surroundings by another body. Something else is required.

The Foundations of Geometry• What separates Body A from Body

F? Let’s say it is a third body, E.• What separates Body A from

Body E? Say, another body, D.

• What separates Body A from Body D? Say, another body, C.

• What separates Body A from Body C? Say, another body, B.

• If we need to go through an infinity of bodies to account for the separation of bodies, then bodies could never be separated. Eventually, we must say that we come to something like B, the body next to A, with no bodies between. Since A and B are not separated by a body, then what divides A and B? It cannot be, it must not be, another body.

• But something has to divide them from each other or they would be the same body… It just can’t be a body…

• Another way to look at this problem is this: what divides A from all other bodies put together?

• Do you see where this is going? We can ask what separates A and B, but if we keep asking the same kind of question and giving the same kind of answer, we will never get an explanation. A

F

E

BC

D

The Foundations of Geometry• What is the first thing that divides two bodies?

• Is the first divider the two bodies’ color? Is it their temperature? Is it any quality? After all, qualities are not bodies; they have no bulk. This cannot be the solution, however, because before A and B can have different qualities, they have got to be already different bodies.

• What do we know about this thing that separates Body A from Body B? First, we know it cannot be a body. It cannot have any bulk. It must divide the bodies without separating them by a gap.

• The divider must “spread” out so it can wrap around the body and divide it from the rest of the world, but since it is not a body, it cannot have any the bulkiness that comes with the thickness of body, otherwise, it would just be another body. Such a magnitude we call “a surface.”

• What else do we know about what divides A from B? We also know that the divider has got to spread all around A, spreading from the right side to the left, in order to divide the right and left sides of A and B. It even has to spread from the front of A to the back of A to divide the front and back sides of A and B. Therefore, the divider, to reach all these places, must be extended.

A

B

The Foundations of Geometry

• Bodies exist. “This is obvious” - though a habitual skeptic may yet deny it.

• But bodies are also divided from the entirety of all surrounding bodies by something between them which is not a body, which does not leave a gap.

• So if bodies exist, this something must also exist. We call it “a surface”.• Since surface cannot be just another body, it must have no thickness. (It is

not enough for it to be just really thin!)• So even though a surface has absolutely no thickness – that is hard to

imagine! – we know it must exist in order for the universe to have parts.

A

B

The Foundations of Geometry• Consider any surface, like the one shown below. In

order to divide bodies, it must have parts.• It has two parts, such as the two ends, call them surface A and

surface F• What separates Part A from Part

F? Say, another surface, E.

• What separates Part A from Part E? Say, another surface, D. • Another way to look at the question is this:

what divides A from all other parts of the entire surface?

• Clearly, what divides one part of a surface from all other parts cannot be another surface. It has got to lack the spread of a surface and the bulk of a body so that it can come between the touching parts of the surface without imposing a gap between them.

• And what separated Part A from Part D? Does this sound familiar?

• Eventually, we will go get to C, then B, until we get to surfaces A and B which are right beside each other, which are divided but have no surface gap between them?

FEDCBA

Even though it might seem impossible for something to exist which has no bulkiness like a body, and no spread like a surface, we know that such a thing must exist in order that surfaces be divided into parts. We call such a divider a “line”.

Since a line cannot spread out over a body -- it has no width -- we cannot see it. Nevertheless, we know it is there. We can see the blue part and yellow part of this surface.

We know that between these parts, there must be a line dividing them. This line is neither yellow nor blue; it can’t have a color because it is not spread over the body. As a result, we cannot directly see it. Yet we see the surface it divides, and so we know it is there. It must be.

We also know that this line must extend out continuously across the surface from top to bottom so that it can divide the surface at the top, the bottom, and the middle. So a line must have parts. This particular line has straight parts at top, and curved parts at the bottom. A line must be a magnitude lacking depth and width.

Summarizing what we have discussed so far: (1)We are surrounded by bulky things, called bodies, which extend

continuously through the space around us, and our own body, which extends through us. It is self-evident that bodies exist - nothing can be more obvious than this.

(2)We next argued that a “divider” of bodies must exist. This divider must be between a body and all other bodies; it cannot be a body, yet it must have parts that spread out continuously over and around the body.

(3)We called this divider of bodies “surface” and noted that since it’s parts spread out continuously, it is a magnitude, but since it cannot be a body, it must lack the bulk of a body. We understand it as a magnitude without depth. We know surfaces exist because we know bodies’ parts must have some divider between them, namely a magnitude which has no bodily bulk.

(4)Because a surface is a magnitude, a “divider” of surfaces must exist. This divider must be between a surface and all other parts; it cannot be a surface, yet it must have parts that extend out continuously beside and around the surface.

(5)We called this divider of surfaces “line” and noted that since a line’s parts spread out continuously, it is a magnitude, but since it cannot be a body or surface, it must lack the bulk of a body and the spread of a surface. We understand it as a magnitude without depth or thickness. We know lines exist because we know a surface’s parts must have some divider between them, namely a magnitude which has no superficial spread.

Twice we have considered a magnitude, and been led to conclude that it must have a divider. First we were led from body to surface, then from surface to line. But a line is also a magnitude, and so a “divider” of lines must exist.

We can see the green, and red parts of this surface. We know that there is a line dividing the green and red parts. The line has parts. There is a curved part between the light red and green surfaces, and a bent part between the dark red and green surfaces. But these two parts meet and are divided at an invisible point. We cannot see it, but we know its there.

This divider must be between the parts of a line; it cannot be a body, or a surface, or a line. It must be lacking in all these ways. We call this divider of lines “point”.

This divider is different from previous dividers, however. Other dividers needed to have parts to divide, but the line has nothing but length, so its divider must lack everything but its position. We can define a point as “that principle of magnitude which has no parts.”

Do we need to go on? Do we need to find a “divider” of points? If we did keep going on, we would end up with an infinite regress problem. But there is no reason to go on because a point has no parts – it cannot be divided.

It is amazing to think that something can exist when it has absolutely no parts. And because it has no parts, it cannot have color, temperature, texture, or shape. All one can do is to point at it, and say, “It’s there.”

Once we have understood what a point is, and seen that it must exist, we are now in a position to start geometry. Here we have something that is both certain, true, and simple.

We know points exist because we know a line’s parts must have some divider between them, namely something with no linear extension.

HAVE YOU UNDERSTOOD?

(1) Where does analysis start and where does it go?(2) Where does synthesis start and where does it go?(3) How do we know that bodies exist?(4) What is a surface and how can we know that it exists?(5) What is a line and how can we know that it exists?(6) What is a point and how can we know that it exists?

WHAT’S ITS JOB?SURFACE: What’s its job? What does the job tell us it can’t be? What’s the job tell us it must be?

LINE: What’s its job? What does the job tell us it can’t be? What’s the job tell us it must be?

POINT: What’s its job? What does the job tell us it can’t be? What’s the job tell us it must be?

Its job is to divide bodies, so it can’t be a body but it must be a magnitude that can spread out over a body or between the parts of a body.

Its job is to divide surfaces, so it can’t be a surface but it must be a magnitude that can extend out between the parts of a surface.

Its job is to divide lines, so it can’t be a line but it must be something with a position between the parts of a line.

WHAT’S DOES ITS JOB REVEAL ABOUT IT?

What is it?

BODY: a magnitude with bulk

b/c it has parts in continuity with each other

SURFACE: a magnitude without bodily bulk

b/c it must spread out continuously over and between the bodies it divides

b/c it immediately divides bodies, it can’t be a body

LINE: a magnitude without superficial spread

b/c it must extend continuously around and between the surfaces it divides

b/c it immediately divides surfaces, it can’t be a surface

POINT: that w/ position but w/o linear extension

b/c it immediately divides lines, it cannot be a line

b/c it must fall between the parts of the line it divides

b/c it fills up its space

Points and Dots

One cannot draw a point because a point has no parts. So in order to show where a point is, we draw a dot around it. The point is not a dot, but the dot is a sign of the point that it is around. This sign is a similitude of a point because from a distance, a dot appears to have no parts. Even though we may draw or imagine a dot when doing geometry, we are thinking of a point.

Lines and StripesOne cannot draw a line because a line has no thickness. So in order to show where a line goes, we draw a thin stripe. The line is not a stripe, but the stripe is a sign, and makes us think of the line that runs along it. This sign is a similitude of a line because from a distance, a stripe appears to have no thickness. Even though we may draw or imagine a stripe when doing geometry, we are thinking of a line.

It is now necessary to synthesize our way back up to the complex.

Geometry begins with the point….

…and analyzed our way down to the simple reality of the point.

Because the point is so very simple, there is nothing to confuse, no features to mix together. We can know it perfectly and completely.

We started with the complicated world of bodies….

The next question is: how do we go from a point, to a line.Can a line be synthesized from points?It is sometimes said, “A line is composed of an infinity of points.”

Is that the solution then, we just need to add an infinity of points together, and we will thereby generate a line?

Although it is said, “A line is composed of an infinity of points,” we shall see that this statement is absurd.

SYNTHESIS

Let us try to generate a line from points. We will start with two points, represented by the green and red dots, and these points will represent the beginning and end of the line that we hope to generate.

We will now add an infinity of points to the beginning of the line and see how far we get toward reaching the end.Since a point has no part, when we add two points, we still have no part. As soon as the points touch, they coincide in the very same position. Adding an infinity of points gets us no closer to a line.

SYNTHESIS

Is there any way we can generate a line from points?Certainly not by adding points.

SYNTHESIS

The only things points have is position, so the only way two points can be different, is by having a different position. Let the green dot and the red dot represent two points with different position.

One could also have a third point with a changing position.

It would then be possible to connect these two points with another point.

The trick is to have the third point moving….

A moving point generates a magnitude with length and shape, but no breadth.

That line has an awfully complicated shape.

The simpler the movement of the generating point, the simpler the shape of the line produced.

But what makes one movement of shape simpler that another?

As it turns out, a simpler line can slide along itself and say within its place. Such a line, one whose parts

are all the same, is called “regular”. Regular is simpler than irregular.

Name some lines with regular shapes? How many can you name?

A regularly shaped line has a repeating pattern.

This means one part of the line can fit exactly on the next part… …and the next part, etc.

But to be perfectly regular, it must be able to fit on any part.

Since it is impossible to check every part, the best way to show whether a line is perfectly regular or not

is to slide one part over the whole line, and see whether the part stays in place. Since this part does not stay even with every other

part of the line, the line is not perfectly regular.

Can you think of a line which is perfectly regular?

There are only three kinds of perfectly regular lines.

The first kind is called straight. The whole line is evenly situated with the points on itself.

The second kind is called circular. The sliding test shows that any part of this line is just

like any other part.

The sliding test shows that any part of this line is just like any other part.

Can you think of a line which is perfectly regular?

There are only three kinds of perfectly regular lines.

The third kind is called a helical line. A helix looks like a spring. It is generated by a point revolving in a circular motion around a second point

moving in a straight line perpendicular to the circle.

Since a helix does not fit on a flat surface, we cannot show it.

If it sounds complicated, it is, because it involves straight lines, circular lines, and perpendicular angles! The helix cannot known with the same precision as the straight or

circular line because it results from the mixing up (confundere), that is, the confusion, of two motions, the

straight and the circular.

After we have seen how lines are synthesized, we next see how surfaces are synthesized.

Do you think we could make a surface by laying a multitude of lines side by side,

like threads in a clothe?

Since lines have absolutely no width, they could never touch each other this way. Adding no width together can never make

some width.Gaps would always exist between the lines where more lines

could be inserted.

So can you guess how we can synthesize a surface from a line?The generating line shows the continuity of surface along

its length, and the movement shows the continuity of surface along its width.

The shape comes from both the generating line and its movement.

One could take an irregular line and move it in a straight line….

…or take a straight line and move it in an irregular line.

But what if you took a straight line and moved it in a straight line?