Direct Vision, Rationality, Realism and Common Sense.

Occamania wrote:This thread and its contents are absolutely AMAZING, vortexpuppy. WOW. This confirms the intuition I've had for YEARS regarding "being taught how to see reality wrong". This is explained wonderfully. I'm looking more into those resources you've mentioned above.

Thank you for this incredibly invaluable resource.

Thanks, am going to make some videos soon explaining more.

GoV - Geometry of Vision / Visibles – Part 1

This post seeks to show where GoV is situated within modern geometry and where the mathematics has been occulted. References are included for the mathematically inclined.

First some terminology and definitions.

An intrinsic view is the eye’s point of view in 2 dimensions (left/right and up/down). It is the direct visual appearance of length, distance or diameter of two dimensional lines, triangles and polygons that are visible figures. The eye is assumed stationery and is not rotating (i.e. you are not turning your body around). Note: If the eye did rotate, its ray(dial)s would generate a surface of revolution, but more of that in another post.

An extrinsic view is from above, where the eye is looking down, having “zoomed out”, so as to view longer segments, wider spreads and greater finite areas of the 2D figures below it. It necessitates the existence of a third dimension into which the eye can physically remove, or distance itself, from the surface below. It is in this point of view that a geometrical model is usually described.

A change of eye position (18 times for motion) changes the appearance of figures. When the eye is moved farther/nearer, higher/lower or wider/closer from the objects of vision, then lengths, spreads and areas of figures appear different.

The differences seen, are “in common sense” with the tangible 3D geometry learned in childhood, when we instinctively measured distance in the 3rd dimension when we started to move in all directions on our hands and feet.

We think that our eyes see in three dimensions, but they don’t. The Eye can only discriminate points having different positions with regards to the Eye. It cannot discriminate points which have the same position with regard to the Eye, but are at different distances from the Eye.

Irrespective of how we interpret, judge or perceive images in our minds, the momentary visual field only ever has two dimensions (Left/Right and Up/Down). The Geometry of Tangibles (touch) is experienced in three dimensions. We are conditioned and schooled to ignore GoV, to only see with the mind and to directly perceive the things which they signify. Signs and symbols rule the world.

The GoV is independent of the mind. It must be so, because despite constant brain washing of the mind, the optical instrument never changes and a visible exists whether seen by an eye or not. Visible figures create a kind of two-dimensional data map at the eye, using only the two up/down and left/right dimensions.

If the eye is moved to another position or set in motion, then this may assist our depth perception, but this depth perception does not add any dimension to the geometry of vision. Similarly, successive changes of eye position will create the illusion of motion (e.g. the platform moves, not the train), but the momentary visual snapshots remain two-dimensional.

Change (of eye position and ratios of angles and magnitudes of objects to each other), appears to be all there is.

Let’s look at some excerpts from RB Angells work.

“There is a geometry which fits precisely and naturally the configurations of the pure visual field, and that geometry is not a Euclidean geometry, but a two-dimensional elliptic, or Riemannian geometry.

By the visual field is meant a domain of objects which we can all attend to with one eye (monocular vision). It is 2-dimensional in the following sense:

Disregard any consideration of distance between the Eye and Objects in the visual field. Relations such as “in front of” or “behind”, or of “bulging towards” or “indented away” are not considered. Objects of the pure visual field are two-dimensional coloured regions, figures or shapes, as a painter might attend to in getting a “good likeness” onto a two-dimensional canvas.

We do distinguish points, lines and regions or areas, but the crucial determinations which make the field subject to geometrical laws are those which measure pure visual distance.

By visual distance we do not mean the psychologists “depth perception”, which involves assessing (or judging) a distance between the Eye and a visual object. We mean the visual width (length, or distance) between two points in the visual field. It is somewhat like the angular distance the astronomer measures between two stars.

We think of another person when we measure the angular distance. We think of observing them from above and noting the angle of the line which travels from object A to the Eye and then to object B. This angle we identify with the visual angle and its measure is the measure of angular distance."

See the diagrams in previous posts.

"This way of representing the concept of visual distance, although geometrically correct, can be somewhat misleading. What we see, when we see a visual distance is not a bent or v-shaped line connecting the Eye with each of the two objects. What we see is simply a one-directional visual extension (which can be called width, length, distance, diameter) between two visual objects.

These pure visual lengths may be measured rigorously by instruments (and measured roughly without instruments), and on the basis of such measurements, metrical properties can be assigned to visual lines and areas. Fixed units of visual distances are easily established.

So starting with this metrical concept of visual distances, suitably associated with distinct operations and instruments, we can give effective and rigorous definitions of circles and straight lines in the visual field.

1. A circle is a closed line such that all points on it are equidistant from a point.

2. A line segment AB is straight if and only if the distance from end-point A to end-point B is equal to the distance from A to any point C on the line plus the distance from C to B. (AB = AC+CB).

From these definitions we may proceed to define angles, specifically right angles, then triangles, quadrilaterals (closed figures with four straight lines as sides), rectangles (equiangular quadrilaterals), squares (equilateral rectangles) and so on.

The definitions thus given are suitable just as well for plane or solid Euclidean geometry, as long as the word “distance” is left ambiguous.

Now the question arises whether the geometrical propositions which hold for the objects of the visual field, belong to Euclidean geometry or not. The answer is plainly that they do not.

All of the *counter-Euclidean propositions are theorems in the two-dimensional non-Euclidean bipolar elliptic geometry of Riemann. “

What RB Angell is saying is that the GoV (Geometry of Visibles) that describes how the world APPEARS to us and how our eyes experience light and colour when viewed intrinsically (i.e. seeing what the eye would see), are consistent with the axioms and definitions underlying Riemannian geometry and whose theorems are therefore applicable.

On a side-note, Riemann was the protégé of Gauss, the master of curvature. Gauss was the son of a freemason and is revered as “a titan of science” and “prince of mathematicians”, on a par with Newton.
http://cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/c/Carl_Friedrich_Gauss.htm

The above given definitions of a line and circle are mathematically equivalent to corresponding Euclidean postulates. We have already given examples of visual geometry in a previous post on GoV vs Euclidean* and some of them are repeated below. These basic propositions of intrinsic visual geometry can be evidenced by anybody with eyes to see.

1. Railway tracks appearing to converge in both directions (parallel lines meeting at two points),

2. The corners of a room (in reality three mutually perpendicular planes, all at right angles to each other) appearing as a flat “Y” figure with angles of more than a right angle (obtuse angles), in the 2D visual field of the eye.

3. Corners appearing more and more like spherical triangles when we bring the eye closer to the vertex. To see this yourself, cut out the corner of a cardboard box and look at its vertex point with one eye. Move the vertex in a straight line closer to or farther from the eye. At arm’s length, the corner will appear like a flat Y and expand into a spherical triangle or bell shape when it is directly in front of the eye.

4. The sky/earth or light/dark horizon appearing as a “straight circle”, like the lune of a sphere with diminishing spread, where area and interval shrink, so as to no longer be visible, and we only see a straight line.

5. Turning full circle, to see a straight line return on itself. Any two points can be traversed in two ways, clockwise (right) or anti-clockwise (left). The length in one direction is usually shorter than the other. The greatest distance is finite and half the length of the line, where the circumference becomes the diameter.

This is the mathematics of two-dimensional spherical geometry and planar curves. This is not the geometry of the Earth or of fake Spaceballs, but of our direct vision, the GoV.

People like Riemann, Hilbert or Klein undertook pretty major surgery in re-modelling the prevalent geometry systems, making it difficult to be precise on the exact axioms and definitions. Before the re-writing in abstract algebra and projective geometry, mathematicians were basically removing or replacing Euclids 5th postulate, that parallel lines exist, but keeping the other four.

Many theorems of Euclid do not require the 5th postulate in their proof, and so these theorems are also valid in any new geometries created when redefining the meaning of parallelism.

One way of re-expressing the 5th postulate and creating new geometries as consistent as that of Euclid, is by means of the Three Hypothesis, which goes like this:

“The angles at the extremities of two equal perpendiculars are either right angles, acute angles, or obtuse angles, at least for restricted figures. We distinguish the three cases by speaking of them as the hypothesis of the right angle (Euclidean), the hypothesis of the acute angle (Hyperbolic), and the hypothesis of the obtuse angle (Elliptic), respectively. “

This change of the 5th postulate creates three consistent geometries, the original Euclidean one and two new ones called Hyperbolic and Elliptic. To confuse matters Euclidean geometry is sometimes then named Parabolic geometry.

GoV - Geometry of Vision / Visibles – Part 2

Now let’s look at the work of other mathematicians. For example, Henry Manning and his paper “Non-Euclidean Geometry”. Below are relevant excerpts from the paper that can be found here. http://www.gutenberg.org/ebooks/13702

“It has been shown, that it is possible to take a set of axioms, wholly or in part contradicting those of Euclid, and build up a Geometry as consistent as his.

We shall give the two most important Non-Euclidean Geometries. In these, the axioms and definitions are taken as in Euclid, with the exception of those relating to parallel lines.

Omitting the axiom on parallels, we are led to three hypotheses; one of these establishes the Geometry of Euclid, while each of the other two (hyperbolic, elliptic) gives us a series of propositions both interesting and useful. Indeed, as long as we can examine but a limited portion of the universe, it is not possible to prove that the system of Euclid is true, rather than one of the two Non-Euclidean Geometries which we are about to describe.

We shall adopt an arrangement which enables us to prove first the propositions common to the three Geometries, then to produce a series of propositions and the trigonometrical formula for each of the two Geometries which differ from that of Euclid, and by analytical methods to derive some of their most striking properties.

We do not propose to investigate directly the foundations of Geometry, nor even to point out all of the assumptions which have been made, consciously or unconsciously, in this study. Leaving undisturbed that which these Geometries have in common, we are free to fix our attention upon their differences. By a concrete exposition it may be possible to learn more of the nature of Geometry than from abstract theory alone.

Thus we shall employ most of the terms of Geometry without repeating the definitions given in our text-books, and assume that the figures defined by these terms exist. In particular, we assume:

I. The existence of straight lines determined by any two points, and that the shortest path between two points is a straight line.
II. The existence of planes determined by any three points not in a straight line, and that a straight line joining any two points of a plane lies wholly in the plane.
III. That geometrical figures can be moved about without changing their shape or size.
IV. That a point moving along a line from one position to another passes through every point of the line between, and that a geometrical magnitude, for example, an angle, or the length of a portion of a line, varying from one value to another, passes through all intermediate values.”

His introduction shows that the foundation is Euclidean (first four postulates) whilst the 5th is replaced by The Three Hypothesis as stated above and described in more detail in his paper. In Chapter 2, he first introduces the two dimensional “Pangeometry”, a collection of fundamental propositions and theorems, only two of which are highlighted here:

"6. Theorem.
In a circle the radius bisecting an angle at the centre is perpendicular to the chord which subtends the angle and bisects this chord. "

A chord is a straight line connecting any two points lying on a circle. If the EYE is at the centre of the circle, then the chord subtends that angle at the EYE. The chord measures a length between two points. A line bisecting the angle at the EYE, is perpendicular to the chord and also bisects the chord.

"7. Theorem.
Angles at the centre of a circle are proportional to the intercepted arcs and may be measured by them. "

In the GoV this means that the angles subtended at the EYE, if it were situated at the centre of the circle, can be measured by the arc lengths of the segments created by the intercepted arcs. This is just direct vision, looking or focussing on the centre of a linear magnitude, where ray(dial)s are imagined to connect the EYE with the extremes of the chord.

Manning then continues with more theorems that are at least true for restricted figures whose lines do not exceed a certain length. This is especially important with respect to the elliptic (spherical) geometry.

“The following propositions are true at least for figures whose lines do not exceed a certain length. That is, if there is any exception, it is in a case where we cannot apply the theorem or some step of the proof on account of the length of some of the lines. For convenience we shall use the word restricted in this sense and say that a theorem is true for restricted figures or in any restricted portion of the plane.”

Remember, this is 2D geometry, the geometry of a surface, usually taken to be a plane. So we have the ability to ZOOM IN and OUT, and by doing so, distancing the eye from the 2D figure(s) that are the object(s) of view. Only when we zoom out sufficiently far, can we see whole object(s). When we are close, we only see a restricted portion of the object or of the plane in which it lies, like reading this paragraph on a sheet of paper.

Visible 2D figures are unique, mind-independent objects (or collection of objects or parts of objects) that appear to an eye at a specific location with respect to the objects of view. A perpendicular line extending from the eye to the plane (or surface), meeting it at the point which is the centre of the 2D figure, is the eye-line, the direct line of sight.

“Distancing” yourself from an object, means going backwards, keeping the same point of view by not changing the position or rake of your head. It is moving the eye backwards on the line that is perpendicular to the finite surface on which the figure lies, and which connects the eye to the centre of the figure.

“Removing” yourself from an object, means the eye is moved OBLIQUELY to the original perpendicular eye-line. Moving obliquely is measured in the SAME two dimensions as the object / figure itself. The eye is moving to the left/right or up/down, with respect to its original position.

We do not move in these two dimensions when “distancing” ourselves from an object, since this necessarily entails a change of position in the direction of a 3rd dimension, namely forward / backward. The different positions of the eye can be a combination of “distancing” and “removing”.

Restricted figures can be thought as having ZOOMED in, so as to consider finite portions of the surface and objects thereupon. This is what we are told is the case on a sphere, where we cannot see nor measure the curve and it looks flat due to its immense size. When we zoom into a circle or sphere (e.g. using geometry software), it appears as a straight line rather than a curve, when the portion viewed is sufficiently small.

In Chapter 2, Manning explores the Three Hypothesis and the excesses and defects of triangles. Excesses and defects of triangles are the amount that the sum of their angles deviates, from that of two right angles. (Note: They play an integral role in advanced mathematics of surface areas and curvature. A particular theorem involving them, is said to be the favourite theorem of Gauss).

Important excepts from Chapter 2 are:

"10. Theorem.
Given a right triangle with a fixed angle; if the sides of the triangle diminish indefinitely, the ratio of the opposite side to the hypotenuse and the ratio of the adjacent side to the hypotenuse approach as limits the sine and cosine of this angle.”

“Corollary. When any plane triangle diminishes indefinitely, the relations of the sides and angles approach those of the sides and angles of plane triangles in the ordinary geometry and trigonometry with which we are familiar. “

"11. Theorem.
Spherical geometry is the same in all three hypotheses, and the formula of spherical trigonometry are exactly those of the ordinary spherical trigonometry. “

“The three hypotheses give rise to three systems of Geometry, which are called the Parabolic, the Hyperbolic, and the Elliptic Geometries. They are also called the Geometries of Euclid, of Lobachevsky, and of Riemann.”

“In the first and second hypotheses we prove that a straight line must be of infinite length.”

“The third hypothesis cannot be true unless the straight line is of finite length returning into itself, and these two points are one and the same point, its distance in either direction being one-half the entire length of the line. In this way, however, we can build up a consistent Geometry on the third hypothesis, and this Geometry it is which is called the Elliptic Geometry.”

“The restrictions which we have placed upon some of the propositions of this chapter is necessary in the third hypothesis. “



“If two angles of a triangle are equal and the side between them is just an entire straight line, it does not follow necessarily that the opposite sides are equal. It may be said, however, that the opposite sides form one continuous line, and, therefore, this figure is not strictly a triangle, but a figure somewhat like a lune. The points A and B are the same point, and the angles A and B are vertical angles. “

“Finally, though we assume that the shortest path between two points is a straight line, it is not always true that a straight line drawn between two points is the shortest path between them. We can pass from one point to another in two ways on a straight line; namely, over each of the two parts into which the two points divide the line determined by them. One of these parts will usually be shorter than the other, and the longer part will be longer than some paths along broken lines or curved lines. “

“When, however, the straight line is of infinite* length, that is, in the hypothesis of the right angle (Euclidean) and in the hypothesis of the acute angle (Hyperbolic), all the propositions of this chapter hold without restriction. “

* Euclid wrote “extended indefinitely” instead of “infinite”.

So all theorems of two dimensional Pangeometry hold true in all three geometries as long as we define a straight line in the Elliptic geometry (obtuse angle hypothesis) to be finite, continuous and returning on itself.

The axioms and definitions that underlie these three geometries produce many theorems valid across all of them. This two dimensional geometry is the basis for three dimensional solid geometries and the fantasies in higher dimensions of curved space/time. All of them share the same geometric foundation, rooted in two dimensions.

The systemization of a foundational geometry is mostly associated with Riemann, but before him it was known by different names. It was called

• Astralgeometry or Astral-lehre by Lambert, Taurinus and Gauss.
  
• Imaginary geometry by Lobachevsky and  Boylai.
  
• Pangeometry by Manning and Halsted.
  
• Neutral geometry is a modern naming.
  

The name given by the authors depends on the exact axioms and definitions used. The mathematical geometrical descriptions are created by re-expressing the parallel postulate and exploring the properties of its basic elements and figures.

They perfectly describe the visual geometry of the eye and perspective.

GoV - Geometry of Vision / Visibles – Part 3

Before continuing with Manning’s paper, here are some more excerpts and references from others.

1. Theory of parallels, by Nicholas Lobachevsky
http://www.stmarys-ca.edu/sites/default/files/attachments/files/Theory_of_Parallels.pdf

This short paper is the basis of the hyperbolic geometry of Lobachevsky or what he also called “Imaginary geometry”, mentioned briefly in Part 2. The paper might appear intimidating at first, but ignoring the trigonometry towards the end, it assists greatly in understanding the logical and philosophical underpinnings of parallelism and the Three Hypothesis.

2. From the book Double-Elliptic-Geometry, by JR Kline.
https://archive.org/details/jstor-2007148

“In his Rational Geometry, Halsted built up two-dimensional double elliptic geometry, in terms of the undefined symbols point, order, association and congruence. “

3. Here is the book Rational Geometry, by Halsted
https://archive.org/details/cu31924001509557

4. From the book Geometry of four dimensions, by Henry Manning
https://archive.org/details/geometryoffourdi033495mbp

“In the double Elliptic Geometry, two lines in the same plane intersect in two points, and a line meets any plane in which it does not lie, in two points, the distance between the points in each of these cases being one-half of the entire length of the line. The length of the line is most conveniently taken as 2 PI.

If we start at an intersection of two lines and follow one of them until we come to the other, we shall come, not to the same intersection point, but to an “opposite” point. We have traversed only one-half of the line, and we arrive at the starting point only when we have gone the same distance further. “

But back to Manning and his paper Non-Euclidean Geometry. He does not delve into Euclidean geometry (right angle hypothesis) since it is trivially provable to be the same as the original Euclidean. He continues in Chapter 3 to detail Hyperbolic geometry using the acute angle hypothesis.

“We have now the hypothesis of the acute angle. Two lines in a plane perpendicular to a third diverge on either side of their common perpendicular. The sum of the angles of a triangle is less than two right angles, and the propositions of the last chapter hold without restriction. “

Two of the hyperbolic theorems are mentioned here.

“4. Theorem.
Parallel lines continually approach each other.

5. Theorem.
As the perpendicular distance varies, starting from zero and increasing indefinitely, the angle of parallelism decreases from a right angle to zero. “

Manning then addresses the topics of (equidistant) boundary curves and surfaces of which some excepts are given below.

“From their definition it follows that all boundary-curves are equal, and the boundary-curve is symmetrical with respect to its axis; if revolved through two right angles about its axis, it will coincide with itself.

1 Theorem.
Any line parallel to the axis of a boundary-curve may be taken for axis.

Corollary. The boundary-curve may be slid along on itself without altering its shape; that is, it has a constant curvature.

2 Theorem.
Two boundary-curves having a common set of axes cut off the same distance on each of the axes, and the ratio of corresponding arcs depends only on this distance.

3 Theorem.
The area enclosed by two boundary-curves having the same axes and by two of their common axes, is proportional to the difference of the intercepted arcs.

4. Theorem.
The boundary-curve is a limiting curve between the circle and the equidistant-curve; it may be regarded as a circle with infinitely large radius, or as an equidistant-curve whose base line is infinitely distant.

6. Theorem.
Geometry on the boundary-surface is the same as the ordinary Euclidean Plane Geometry.

If we assume that figures on the boundary-surface become more and more like plane figures when we diminish indefinitely their size, it follows that a triangle on this surface approaches more and more the form of an infinitesimal plane triangle, for which the sum of the angles is two right angles, and the angles and sides have the same relations as in the Euclidean Plane Geometry.

All the formula of Plane Trigonometry with which we are familiar hold, then, for triangles on the boundary-surface.”

He then continues in like manner to look at Elliptical Geometry, which includes what we call spherical geometry.

“In the hypothesis of the obtuse angle a straight line is of finite length and returns into itself. This length is the same for all lines, since any two lines can be made to coincide.

Two straight lines always intersect, and two lines perpendicular to a third intersect at a point whose distance from the third on either line is half the entire length of a straight line.”

“A straight line does not divide the plane. Starting from the point of intersection of two lines and passing along one of them a certain finite distance, we come to the intersection point again without having crossed the other line. Thus, we can pass from one side of the line to the other without having crossed it. “

“There is one point through which pass all the perpendiculars to a given line. It is called the pole of that line, and the line is its polar. Its distance from the line is half the entire length of a straight line, and the line is the locus of points at this distance from its pole. “

“The locus of points at a given distance from a given line is a circle having its centre at the pole of the line. The straight line is a limiting form of a circle when the radius becomes equal to half the entire length of a line. “

“All the perpendiculars to a plane in space meet at a point which is the pole of the plane. It is the centre of a system of spheres of which the plane is a limiting form when the radius becomes equal to half the entire length of a straight line. “

“Figures on a plane can be projected from similar figures on any sphere which has the pole of the plane for centre. That is, they have equal angles and corresponding sides in a constant ratio that depends only on the radius of the sphere.”

“Geometry on a plane is, therefore, like Spherical Geometry, but the plane corresponds to only half a sphere, just as the diameters of a sphere correspond to the points of half the surface. “

“Indeed, the points and straight lines of a plane correspond exactly to the lines and planes through a point, but we can realize the correspondence better that compares the plane with the surface of a sphere. If we can imagine that the points on the boundary of a hemisphere at opposite extremities of diameters are coincident, the hemisphere will correspond to the elliptic plane. There is no particular line of the plane that plays the part of boundary. All lines of the plane are alike; the plane is unbounded, but not infinite in extent. “

“The entire straight line corresponds to a semicircle. We will take such a unit for measuring length that the entire length of a line shall be PI; the formula of Spherical Trigonometry will then apply without change to our plane. “

“Distances on a line will then have the same measure as the angles which they subtend at the pole of the line, and the angle between two lines will be equal to the distance between their poles. The distance from any point to its polar, half the entire length of a straight line, may then be called a quadrant. “

Remember: In GoV we take the “pole” (or origin) to be the position of the eye. Manning then continues with Elliptic Analytic Geometry, but now deriving the theorems using only the relations between poles and lines.

“The formula of Elliptic Plane Analytic Geometry may then be applied to a sphere in any of our three Geometries. “

“On a sphere, if we take as origin the pole of the equator, ρ and θ are colatitude and longitude. x and y, one with its sign changed, are the “bearings” of the point from two points 90◦ apart on the equator. “

“An equation of the first degree in α and β represents a curve which enjoys on this surface all the properties of the straight line in the plane of the Euclidean Geometry. Through any two points one, and only one, such line can be drawn, because two sets of coordinates are just sufficient to determine the coefficients of an equation of the first degree. Such a line on a surface is called a geodesic line, or, so far as the surface is concerned, a straight line."

"Triangles formed of these lines have all the properties of plane triangles in the Euclidean Geometry: the sum of the angles is π, etc. In fact, this surface has the same relation to elliptic space that the boundary-surface has to hyperbolic space. "

Manning concludes with historical notes which are worth quoting here in full. I have made some passages bold or underlined them.

“The history of Non-Euclidean Geometry has been so well and so often written that we will give only a brief outline.

There is one axiom (the parallel postulate) of Euclid that is somewhat complicated in its expression and does not seem to be, like the rest, a simple elementary fact. It is this:

If two lines are cut by a third, and the sum of the interior angles on the same side of the cutting line is less than two right angles, the lines will meet on that side when sufficiently produced.

Attempts were made by many mathematicians, notably by Legendre, to give a proof of this proposition; that is, to show that it is a necessary consequence of the simpler axioms preceding it. Legendre proved that the sum of the angles of a triangle can never exceed two right angles, and that if there is a single triangle in which this sum is equal to two right angles, the same is true of all triangles. This was, of course, on the supposition that a line is of infinite length. He could not, however, prove that there exists a triangle the sum of whose angles is two right angles.

At last some mathematicians began to believe that this statement was not capable of proof, that an equally consistent Geometry could be built up if we suppose it not always true, and, finally, that all of the postulates of Euclid were only hypotheses which our experience had led us to accept as true, but which could be replaced by contrary statements in the development of a logical Geometry.

The beginnings of this theory have sometimes been ascribed to Gauss, but it is known now that a paper was written by Lambert, in 1766, in which he maintains that the parallel axiom needs proof, and gives some of the characteristics of Geometries in which this axiom does not hold. Even as long ago as 1733 a book was published by an Italian, Saccheri, in which he gives a complete system of Non-Euclidean Geometry, and then saves himself and his book by asserting dogmatically that these other hypotheses are false. It is his method of treatment that has been taken as the basis of the first chapter of this book.

Gauss was seeking to prove the axiom of parallels for many years, and he may have discovered some of the theorems which are consequences of the denial of this axiom, but he never published anything on the subject.

(Note: We will return in another post to the question of what did Gauss know and when did he know it.)

Lobachevsky, in Russia, and Johann Bolyai, in Hungary, first asserted and proved that the axiom of parallels is not necessarily true. They were entirely independent of each other in their work, and each is entitled to the full credit of this discovery. Their results were published about 1830.

It was a long time before these discoveries attracted much notice. Meanwhile, other lines of investigation were carried on which were afterwards to throw much light on our subject, not, indeed, as explanations, but by their striking analogies.

Thus, within a year or two of each other, in the same journal (Crelle) appeared an article by Lobachevsky giving the results of his investigations, and a memoir by Minding on surfaces on which he found that the formula of Spherical Trigonometry hold if we put “ia” (imaginary) for a, etc. Yet these two papers had been published thirty years before their connection was noticed by Beltrami.

Again, Cayley, in 1859, in the Philosophical Transactions, published his Sixth Memoir on Quantics, in which he developed a projective theory of measurement and showed how metrical properties can be treated as projective by considering the anharmonic relations of any figures with a certain special figure that he called the absolute. In 1872 Klein took up this theory and showed that it gave a perfect image of the Non-Euclidean Geometry.

It has also been shown that we can get our Non-Euclidean Geometries if we think of a unit of measure varying according to a certain law as it moves about in a plane or in space.

The older workers in these fields discovered only the Geometry in which the hypothesis of the acute angle is assumed. It did not occur to them to investigate the assumption that a line is of finite length.

The Elliptic Geometry was left to be discovered by Riemann, who, in 1854, took up a study of the foundations of Geometry. He studied it from a very different point of view, an abstract algebraic point of view, considering not our space and geometrical figures, except by way of illustration, but a system of variables.

He investigated the question: What is the nature of a function of these variables which can be called element of length or distance? He found that in the simplest cases it must be the square root of a quadratic function of the differentials of the variables whose coefficients may themselves be functions of the variables.

By taking different forms of the quadratic expressions we get an infinite number of these different kinds of Geometry, but in most of them we lose the axiom that bodies may be moved about without changing their size or shape.

Two more names should be included in this sketch, Helmholtz and Clifford. These did much to make the subject popular by articles in scientific journals. To Clifford we owe the theory of parallels in elliptic space, as explained on page 55. He showed that we can have in this Geometry a finite surface on which the Euclidean Geometry holds true.

The chief lesson of Non-Euclidean Geometry is that the axioms of Geometry are only deductions from our experience, like the theories of physical science. For the mathematician, they are hypotheses whose truth or falsity does not concern him, but only the philosopher.

He may take them in any form he pleases and on them build his Geometry, and the Geometries so obtained have their applications in other branches of mathematics.

The “axiom”, so far as this word is applied to these geometrical propositions, is not “self-evident”, and is not necessarily true. If a certain statement can be proved, that is, if it is a necessary consequence of axioms already adopted, then it should not be called an axiom.

When two or more mutually contradictory statements are equally consistent with all the axioms that have already been accepted, then we are at liberty to take either of them, and the statement which we choose becomes for our Geometry an axiom. Our Geometry is a study of the consequences of this axiom.

The assumptions which distinguish the three kinds of Geometry that we have been studying may be expressed in different forms.

We may say that one or two or no parallels can be drawn through a point; or, that the sum of the angles of a triangle is equal to, less than, or greater than two right angles; or, that a straight line has two real points, one real point, or no real point at infinity; or, that in a plane we can have similar figures or we cannot have similar figures, and a straight line is of finite or infinite length, etc.

But any of these forms determines the nature of the Geometry, and the others are deducible from it. “

GoV - Geometry of Vision / Visibles – Part 4 - Summary

I realize that a lot of the information being given is perhaps too abstract, contains too much text, might appear disjointed, or that I am repeating myself, but I have tried to summarize in a non-mathematical way and use different words, so as to reach the widest possible audience. I do intend on showing what this means in practice in more detail as soon as time permits.

To summarize thus far:

It is clear that the geometry of the sphere is the double (bi-polar) elliptical geometry of two dimensions. The theorems developed therein are the same ones used in navigation, astronomy, optics, surveying, architecture, etc.

All the different aspects and theorems of “Pangeometry” as described by Manning, can be modeled by projecting a sphere onto a plane as was done by Riemann. This is the main reason why all the geometries have so much in common with each other.  Models such as Bertrami-Klein, Poincare Model, Minkowski can be mapped to each other, because they all use the concept of a sphere that is then projected back onto a plane surface.

But wait. Is this not putting the cart before the horse? Surely 2D comes before 3D? Why model as a sphere on the surface of a ball, only to project it back onto a plane, to get back to where we started, a two dimensional geometry?

When we know the earth is flat and space is fantasy, then we can see how this could be abused, in order to deceive. This is especially clear when we understand that all measurements of moderate to extremely large distances, can only be done via optical instruments. A spherical ruler big enough to measure the earths curvature has never been constructed and a straight ruler is useless on a curve. So we have to rely on optics to measure Earth or Celestial objects.

I have hopefully also helped show that this same basic geometry correctly describes how the eye sees or performs vision. The geometry of vision can be explained by the very same 2D geometry that underlies Euclidean, Hyperbolic and Elliptic geometry.

Personally, I think it is readily seen that all three geometric descriptions are integral to our visual experience and that it depends on the distance and position of the eye to the objects and the relations of angles and lengths of objects to each other, that determines which geometry (Euclidean, hyperbolic or elliptic) manifests itself in which situation.

Why is this never discussed? Why is it conveniently ignored and seemingly of no interest for mathematics or the natural sciences? Pure coincidence?

As Thomas Reid said, “the geometry of the objects of vision has entirely escaped the notice of mathematicians” and “while that figure and that extension which are objects of touch, have been tortured ten thousand ways for twenty centuries, [...] not a single proposition do we find with regard to the figure and extension which are the immediate objects of sight”.

Does it need the realization of a flat earth and not just the associated inquiry into the perspective science of parallel lines, to bring the geometry of vision to the attention of the scientific community? Due to the deliberate and enduring mockery of the flat earth theory, most scientist would avoid closer examination like the plague.

Is it only because they are blinded, like everybody else, or perhaps even more so by the very nature of their schooling, that it is overlooked? Is it because idols of authority like Gauss and Newton can never be questioned? Or that the same underlying 2D mathematics means it works in all practical situations, so nobody stops to consider it? Or have there been cunning and deliberate attempts to conceal and deceive? Maybe all of the above?

The association of two seemingly separate notions:

1) parallel lines exist, as per the original Euclidean postulate  
2) categorizing the measures of angles into three types (acute, right, obtuse) as in the Three Hypothesis,

also presents itself in other areas.

In the science of perspective art, a similar categorization exists, with respect to the distance and position of the Eye. The eye is the most natural choice with which to categorize projections and not as others would claim by the position of the object or the position of the intercepting plane on which the images are projected.

The only three possible positions of the eye with respect to the objects of view are:

a) immoderately far away – distant, or for example stars
b) in contact with - very near or touching,
c) at some moderate distance in-between.

In the domain of artists or draughtsmen, these are called ORTHOGRAPHICAL, STEREOGRAPHICAL and SCENOGRAPHICAL and they suspiciously correspond to the Three Hypothesis and three geometries we have investigated.

Enjoying your posts very much VP, I have a lot of reading to do, thanks!

csp wrote:Enjoying your posts very much VP, I have a lot of reading to do, thanks!

You took the words out of my mouth.
Ive been wanting to say that ever since I read VPs posts on geometry. Very eye-opening and something that makes perfect sense. We have to hand it to these Jews (Freemasons or whatever of their hydras head), they covered all angles (literally).

Thanks fellow IFERS, more coming soon ....

Perspective Appearances and Representations - Version 1

This post links to a paper I have written to understand perspective in the context of the flat earth and the geometry of vision.

The paper can be downloaded here:
Download Perspective Appearances and Representations-V1.pdf

I am grateful for any comments, feedback, suggestions for improvements, and reports of any errors.

Know the difference ...

The Geometry of Touch measures the proportions and intervals of tangible figure and extension of object(s) and/or part(s) of tangible objects WITH RESPECT TO EACH OTHER.

The Geometry of Vision measures the proportions and intervals of visible figure and extension of object(s) and/or part(s) of visible objects WITH RESPECT TO THE EYE, in a particular situation.

Here is a collection of passages from Kim H. Veltman, 2004_Sources_of_Perspective.pdf, that are relevant to this thread ...

Sorry for the long post ..


Geometry and the Distance Point Construction
In one of his propositions, Piero della Francesca mentioned the possibility of confirming these principles with physical demonstrations. In the course of the 1480's and 1490's, individuals such as Francesco di Giorgio Martini sought to carry this out by reconstructing the geometrical principles in terms of actual surveying situations. In all likelihood it was Francesco who first explored the principle of the distance-point.

In determining the distance point one begins by extending the converging sides of a foreshortened square (fig. 14) until they meet at a central vanishing point. Through this point a horizon line, parallel to the base is drawn. Through the foreshortened square one also draws a diagonal which is extended until it meets the horizon line at the distance point, so-called because the space from this point to the central vanishing point marks in scale the original viewer's distance from the picture plane which produced the foreshortening in question.

This ability to work backwards from the foreshortened square to reconstruct the original viewpoint which caused it has been termed the reversibility principle of perspective. This only functions in the case of regular squares (or cubes) positioned at right angles to the picture plane. That perspectival drawings tend to feature regular geometrical and idealized architectural shapes is therefore no coincidence.

1. Optics and the legitimate construction

The development of the second major method was closely tied with the history of optics. Euclid's Optics had dealt primarily with what would today be termed psychological optics, study of subjective aspects of vision. But the treatise also contained four propositions devoted to surveying problems and thereby the accurate perception and measurement of distance became part of the optical heritage.

By the ninth century thinkers in the Arabic tradition such as Al-Farabi could define optics in terms of measuring the heights of mountains and even distances of stars.

Through this tradition there evolved an overlap between the ideals of optics and those of surveying. In the optical treatises of Alhazen (early 11th c.) and Witelo (c. 1280) the concept of measured distance acquired new significance.

By the fourteenth century treatises on optics frequently appeared together with those on surveying or practical geometry. One important consequence of this interplay between optics and surveying was that theoretical propositions in optics were increasingly tested in terms of practical demonstrations. Euclid, for instance, had claimed that visual angles do not vary inversely with distance. Blasius of Parma, in the 1390's, tested this experimentally, just as he used candles to test experimentally the projections of armillary spheres. Brunelleschi's picture-plane or window (c. 1415-1425) was probably a direct outgrowth of this tradition: a practical demonstration of the visual pyramids and other principles of optical theory.

Alberti, in his On painting, described the principles of this method verbally, thus providing a first theoretical formulation of what he termed the best method (modo optimo), now remembered as the legitimate construction (costruzione legittima. Even so, he saw the window, or veil (velo), as the practical equivalent of this method and insisted on its fundamental importance:

“Nor will I hear what some may say, that the painter should not use these things...I do not believe that infinite pains should be demanded of the painter, but paintings which appear in good relief and a good likeness of the subject should be expected. This I do not believe can ever be done without the use of the veil.”

Alberti assumed that optics provided the theory for both his verbal demonstration of the legitimate construction, and for its practical equivalent, which used the window. In the next generation, Francesco di Giorgio Martini and Luca Pacioli also assumed this, although they classed perspective under practical geometry and surveying. Even in the latter sixteenth century perspective continued to be seen in terms of practice as is witnessed by titles such as Barbaro's Practice of perspective, Barozzi's Two rules of practical perspective and Sirigatti's Practice of perspective.

Because authors continued to assume that Euclid's Optics and Elements provided such theory as was necessary for their subject, there was no theory of perspective as such at the time.

Page 48
In the mid-sixteenth century, the collection of mathematical manuscripts was given new impetus by Federico Commandino who launched a programme to edit and publish major Greek and Latin classics. His initiatives attracted a wide circle of scholars and scientists (fig. 38), which included Nicolo Tartaglia, John Dee and Giovanni Battista Benedetti. Of particular interest for our purposes, is Commandino's edition of Ptolemy's Planisphere (1558), in which he related the projections of planispheres with those of linear perspective. This recognition, that perspective involved a special case of projections that occurred also in planispheres and astrolabes, brought to light a need for a more general theory of projections, which Commandino's student, Guidobaldo del Monte, attempted to fulfill in his classic text (1600), in which he demonstrated that a single universal principle underlay practical methods then in use.

In the next generation Galileo, a direct descendent of the Urbino tradition, became the teacher of Bonaventura Cavalieri, Italy's foremost expert on conic sections in the early seventeenth century. These same individuals were involved with projection problems in astrolabes (Commandino, Clavius), sundials (Maurolyco, Clavius) and perspective (Commandino, Guidobaldo del Monte), and became aware that such practical interests were related to the projections involved in conic sections.


Page 62
As in other European centres (cf. 2.1), Paris witnessed a great interest in universal measuring instruments in the latter sixteenth century: Abel Foullon's holomètre, Jacques Besson's cosmolabe, Philippe Danfrie's graphomètre and Henry de Suberville's henrymètre. By about 1609, Jacques Aléaume, who also wrote on perspective, had developed his own version of a sector. Denis Henrion improved upon it. Pierre Hérigone further developed it, and mentioned its application to perspective, a topic which Jean, le Sieur de Vaulezard took further, showing the sector's uses in determining various anamorphic projections in conic mirrors. In his letter to Beeckman about his new science, Descartes had specifically referred to the need for "new kinds of compasses" to demonstrate the new method. Was this what Hérigone and Vaulezard were doing? In any case we find thereafter, especially in the French tradition, an ongoing link between sectors and perspective (e.g. Bosse, Huret, Chales, Lambert).

There was also a religious and magical context to these studies. As early as the 1450's, Giovanni Fontana, a student of optics and author of what may have been the first treatise on perspective, had used his knowledge in these fields to project images of devils onto the walls of buildings presumably to demonstrate that effects claimed to be supernatural by followers of black magic could be accounted for by purely natural causes. In the sixteenth century, Giovanni Battista della Porta, working in the tradition of artist-engineers, pursued this approach to natural magic.

Subsequently Abraham Bosse, first professor of the Royal Academy and a friend of both Poussin and Desargues, continued the debates on Desargues' behalf, and as mentioned earlier (pp. ) brought into focus the dichotomy between projection planes of linear perspective, based on geometry, and the visual angles principles of Euclid's optics. This inspired the further opposition of Jean Le Brun and more writings against him by Jean Le Bicheur and Grégoire Huret, leading eventually to Bosse's expulsion as the academy's professor of perspective. And ironically, Dubreuil's text, with its various faults, went on to become one of the most popular texts with nine issues by the end of the century and another dozen in the eighteenth century, including translations into English and German. Dubreuil was a Jesuit.


Page 64
The combined result of these developments was to convince thinkers that descriptive geometry, and the principles of perspective, must (somehow) correspond to both objective reality and the laws of vision. Vallée played a considerable role in establishing these suppositions, and accordingly, the second half of the century saw a series of works relating perspective to sight (e.g. Malaval), or to observation (e.g., Babinet, Devinat, Watelet), a theme which continued into the twentieth century (e.g. Bocquillon, Legrand), although new evidence concerning discrepancies between vision and representation had arisen in the meantime (see pp. ).

Page 71
Augustin Hirschvogel pursued this approach, explicitly setting out to relate theoretical geometry with practical perspective in a work fittingly entitled: “Geometry. The book geometry is my name. Originally all liberal art from me came. I bring architecture and perspective together”. Hirschvogel also wrote an unpublished treatise on surveying (now Vienna, Stadtmuseum).

Dürer had entitled his book Instruction in measurement...with the compass and ruler (1525), and had emphasized the general problem of practical geometry. Heinrich Lautensack's text (1564) echoed Dürer's title, adding also of perspective. Following a brief introduction on geometry, Lautensack focussed on theoretical and practical aspects of perspective, including anatomical proportion and symmetry, themes which had been explored independently by two of Durer's other students: Hans Beham and Erhard Schön.


Page 100,
3. Optics

Optics and perspective shared a common etymological root in the Latin perspectiva, and fifteenth century authors no doubt tended to see in optics a theoretical basis for linear perspective.

Hence, Ghiberti, active in proto- perspectival demonstrations, also discussed optics at length in his Third commentary. Alberti, in his “On painting”, referred briefly to concepts such as visual angles and central ray, but carefully avoided details of optical debates. So too did Piero della Francesca, Leonardo da Vinci, Pélerin and Dürer. Indeed, none of these authors mentioned any specific sources on optics.

One of the first to do so was Serlio (1540), and then more as a disclaimer: "In this work I will not trouble myself to dispute philosophically what perspective is, or from whence it hath the original, for learned Euclides writeth darkly on the speculation thereof."

Barbaro (1568) referred to Apollonius' Conics and Euclid's Elements with respect to visual angles, but did not mention Euclid's Optics explicitly, although he cited introductory assumptions from that treatise. Egnazio Danti, who produced an Italian translation of Euclid's Optics (1573), did cite the work specifically in his edition of Vignola's The two rules (1583).
No one at the time pointed out basic discrepancies between theories of visual angles and principles of perspectival planes noted above (see p. ). Hence, while most fifteenth and sixteenth century treatises paid lip service to optics, there was no serious consideration of its principles.

In the seventeenth century the situation became more complex. For instance, Accolti (1625) cited not only Euclid's Optics, but also Heliodorus of Larissa, Theon of Alexandria, Aristotle, Galen, Hippocrates and Witelo with respect to theory. Nor did he cite blindly. For while acknowledging Witelo's great authority as "the sole and principal head of the school of perspectivists," Accolti noted his mistaken claims with respect to the oblique passage of lights. On the other hand, Accolti continued to claim that "perspective is nothing other in effect than a representative section of the visual pyramid," a view which remained popular (see below 1.4), in spite of efforts by Desargues and Bosse to establish that the laws of perspective had their basis not in Euclidean optics, but Euclidean geometry.

Page 105,
Two Chief Methods
Although we know from Danti (1583) and Guidobaldo del Monte (1600), that there were many competing methods during the sixteenth century, including a number of erroneous ones, there were two methods which gained ascendancy. The chief of these which history has remembered as the legitimate construction, deriving from Benedetti's (1585) phrase: this sole legitimate one (hunc solum legittimam), was linked with the perspectival window, and began simply as a verbal description in Alberti (1434).

Filarete (1464?), added a scale diagram, such that relative sizes and distances could be deduced. Piero della Francesca (c. 1480), added actual measurements, as did Luca Pacioli (1494). But these involved only isolated cases.

Leonardo (1490-1500) described systematic measurements of the diminutions involved. In Piero's treatise the legitimate construction became linked with the ground plan/elevation method. Leonardo evolved new combinations thereof, variations of which were subsequently printed by Barbaro (1568), Danti (1583) and Benedetti (1585).

Alberti, in his “Elements of painting” (c 1435-1440), described an alternative geometrical method involving proportional diminution. Piero della Francesca, developed this in books one and two of “On perspective of painting” (c. 1480). Francesco di Giorgio Martini made this a practical demonstration. With Pélerin (1505), this method emerged as the distance point construction.

Ringelbergius (1531) gave quantitative examples of this method, demonstrating what happened to squares positioned 10, 20 or 40 feet from the eye. Beginning with Serlio (1545), it became customary to acknowledge that there were two methods. Danti (1583), set out to demonstrate that they were actually equivalent.

Meanwhile the earlier geometrical version of this method in terms of proportional diminution continued. Piero della Francesca's example using a cube was adapted by Barbaro (fig. 28.4). His examples of foreshortened columns were adapted in a rough version by Serlio. The architectural applications of the method, explored by Pélerin (fig. 29.1-2), were developed dramatically by Androuet Du Cerceau (fig. 29.3) and later by Bibiena (fig. 29.4). Discussion of both methods remained a standard topic of perspective treatises until well into the eighteenth century.


Page 120,
Spherical Perspective

Sixteenth and seventeenth century treatises on perspective almost always avoided questions of spherical perspective, except in connection with astrolabes (see below p. ), but by the eighteenth century this theme emerged with respect to projection problems in astronomy and geography with contributions by Karsten (1768, 1773), Wright (1772), Kautsch (1784) and Schubert (1784, 1788, 1789, 1790) and later by Germain (1866).

In the nineteenth century, spherical projections were subsumed as a branch of descriptive geometry by Davies (1826, 1832, etc.), Lacroix (1840), Leroy (1850) and Church (1868, etc.).
In connection with optics, there has been some confusion between spherical and cylindrical perspective, the sphere of vision frequently being represented simply as a circle equidistant from the eye.

This confusion led Dürer (1525), for instance, to adopt his method of negative perspective, whereby objects further from the eye were represented as larger in order that their apparent size remain constant. Serlio (1545) took up this principle, as did Barbaro (1568) and thereafter it became a commonplace in both treatises on perspective and architecture (cf. fig. 2.1).

In the nineteenth century, Hauck (1879), believed that the subjective curvatures of Greek architecture corresponded to spherical theories of vision. These possible links between architecture and optics were also touched upon by Maertens (1884) and taken up seriously by Borissavlievich (1921, etc.), who gradually evolved a personal theory of spherical perspective with respect to architecture.

Connections between spherical projections and optics also arose from unexpected quarters such as landscape gardening. Seventeenth century authors, such as Tacquet (1668, etc.) had suggested that trees should be planted in rows of half hyperbolas in order to appear parallel. This problem was taken up by an anonymous author (1719) and Varignon (1720), only to be challenged by Bouguer (1755). Debates concerning curvature of parallel rows continued in the twentieth century with experiments by Hillebrand (1902), Blumenfeld (1913) and Luneberg (1947). In the nineteenth century, Helmholtz (1866, 1896), developed a new demonstration involving a curved checkerboard to illustrate subjective curvatures. Two other important demonstrations evolved: one using the vault of the heavens, e.g. Reimann (1890-1891) and Zoth (1899), the other using the apparent bending of light from lighthouses on the horizon, e.g. Bernstein (1904).

The twentieth century has seen an increasing interest in relating spherical projections of optical theories with painting practice. Deininger, in a lecture to the central organization of Austrian architects on 15 August 1914, outlined what he believed was a new theory of artistic painters' perspective, and its practical results, in which he claimed that: “only (on such) a spherical surface is it possible to represent graphically all those lengths, i.e., all the perspectival dimensions in their correct relations and proper sizes.”

In New Hampshire, an artist and a physicist, Ames and Proctor (1921) did experiments together: “for the purpose of determining the exact nature of the image received by the human eye in the belief that a knowledge of its nature would be of aid in suggesting how the various parts of a picture should be painted to give a technically pleasing and artistic effect.”

Birker (1923), took out two patents for a mechanical means of producing spherical perspective. Stark (1928) and Hegenwald (1932), wished to use the spherical surface of the retina as the basis for their theories of spherical perspective, but were hesitant in their application thereof. An important book, in Canada, by Jobin (1932), argued that with the development of skyscrapers one needed to apply perspective to the vertical as well as the horizontal axis. In part two Jobin set out:

“to show that the curved line, determined by the principles of the optical sphere, today constitutes a theory of vision superior to that of the straight line established by the principles of the optical cone.”

He illustrated his theories with an impressive series of illustrations using a four point, spherical perspective. Similar ideas were explored two years later by Garnier (1934) and Serrano (1934, 1952). These works were virtually ignored, however, and it was over a decade before a next wave of interest was initiated, this time largely by architects, i.e. La Grassa (1947), Giorgi (1947), Mohrle (1949) and Zanetti (1951) and an opthalomologist, Graf (1949). These again had no sustained impact. Another decade passed before the matter was taken up afresh by Barre and Flocon (1962, 1964, 1968). This work excited more attention and was eventually translated into German (1983), Spanish (1985) and English (1988).

During the 1970's problems of spherical perspective inspired the imagination of American artists. Hansen (1973), who has since translated Barre and Flocon into English, developed a five point spherical perspective, which he termed hyperbolic linear perspective. Independently, Termes developed 4, 5 and 6 point spherical perspective methods (fig.**). Turner (1976) and Casas (1983) developed alternative methods to accommodate the complexities of visual perception. An exhibition by Marcia Clark (1988) attested that a number of artists, notably Jacqueline Lima, have been developing their own empirical methods. At the same time there have been developments elsewhere. In Buenos Aires, Reggini (e.g.1973) has written a number of articles on the problem. In London, Shaw (1977) has devoted an important thesis to spherical perspective. In Paris, Blotti (1986, 1987) has created a series of demonstrations including spherical and other alternative projection methods for the Musée des sciences et de l'industrie de la Villette. There have also been books by Elias (1973), Fuentes Alonso (1975), and Bonbon (1983). Indeed, these interests are leading to new links between objective and subjective elements (see below pp. ** ).

Page 167,
Meanwhile a number of perspectival instruments had emerged, which played an important role in making vision and representation into quantitative activities. The simplest of these perspectival aids was the mirror which had interested Witelo, and which Brunelleschi used in his original perspectival construction as suggested by Manetti10 and according to accounts by Filarete:

“look at a pavement of square blocks of wood that is stretched out in front of you.... All the sides are equidistant from one another and [yet] looking at them it will seem that they are greater and less, such that those which are closer to you will appear more equal and to the extent that they are more distant, the more they appear attached together in such a way that they all appear to be one. And if you wish to consider the matter better, take a mirror and look into it. You will see clearly that it is thus. And if they were directly in front of the eyes they would all appear equal. And thus I believe that Pippi di ser Brunellescho, the Florentine, found the way to make this plane, which was truly a subtle and beautiful thing, which finds by demonstration that which shows itself to you in a mirror.”

Window
The perspectival window (velo, rete, pariete), was by far the most popular of perspectival instruments. Alberti (1434) claimed that it was indispensable for perspectival construction. Piero della Francesca (c. 1480) described it, Leonardo drew it (c. 1490), Dürer (1525) published it, and via Barbaro (1568), it subsequently became known as Dürer's window in Italy. Dürer's student, Rodler (1531) used the window for landscapes, an idea which found both military and practical applications (fig. 58.1-2).

The window became much more than a simple mechanical aid. According to Danti (1583), Tommaso Laureti produced a model window specifically to demonstrate the principles of perspective, an idea subsequently taken up by the French Academy of Sciences. As early as the 1490's Leonardo da Vinci devised games which used the window principles to improve one's judgment of distance. Later authors such as Accolti (1625) and Bosse (1648) used the window to reveal reciprocal relationships of size and distance and as a tool for imposing a geometrical framework on nature. This made possible the equations between optics, geometry and perspective, which were taken for granted until the nineteenth century and have continued to the present (cf. above p. ).

the military implications of which were explored by Faulhaber (fig. 54.3). Such versions of the window principle allowed one to take a foreshortened view of a fort and work backwards in determining the layout of its ground-plan, thus illustrating practically the principle of reversibility. The effects of geometrical transformations of images could now be followed.

The window now served to demonstrate the complexities of intersecting planes: not only the interplay between patterns on the floor and window, but also between patterns on the ceiling and window, or what happens when the floor is tilted upwards and downwards. Aleaume (1643) was among the first to study these conditions systematically (figs. 30.1-4). The way had been prepared by Marolois (figs. 31.1).

Desargues (1636, etc.) explored the consequences, for the implication was that all these practical instances were concrete examples of some more universal principle of intersecting planes. This realization did not make the window obsolete. It continued to be a standard item in the introductions to treatises on perspective. Eighteenth century thinkers such as Hamilton (1764) produced ever more elegant illustrations of practical examples of intersections (fig. 31.2). Monge's breakthrough was, in a sense, a new way of relating the practical window to the abstract principles underlying it (fig. 31.3). The window, then, was much more than a simple tool for perspective. It was a means of visualizing mathematical principles of planes and, it could be claimed, made possible a whole school of visual geometry (anschauliche Geometrie) as later developed by Hilbert. As such, the window proved as much a tool for abstraction, as it was an aid in concrete representation.

Page 254,
The advent of manuscripts was a necessary but not sufficient condition for texts on vision and representation.

India developed a complex culture without texts on optics or representation.
China produced texts on representation without reference to theories of vision.
Islam produced texts on optics without reference to representation.
Greece, which produced texts on both vision and representation was an exception. These texts were more ambiguous than Panofsky assumed.
Euclid's theory of vision could be taken to imply spherical, cylindrical or even flat projection planes. Hence Greece did not establish a given projection method for optics and art. Its contribution lay, rather, in using geometry for vision and representation, but geometry without reference to measured size and distance, i.e. without a defined scale.

During the Renaissance, which saw the advent of printing, written theories of vision and representation became more widespread. Geometry was applied increasingly in conjunction with measured size and distance. But it was again the case that no single projection method was used exclusively for vision and art.

Explorations of planar projection methods in linear perspective went hand in hand with study of angular, pyramidal, conic, cylindrical and spherical projection methods.

Hence the important characteristic of the period 1400-1600 was not simply the use of linear perspective but rather a recognition that, depending on the surface used, representation involved a number of projection methods governed by mathematical laws: i.e. that projection depended on geometry rather than optics.

This raised further questions which came into focus in the period 1600-1800: whether there might be different projection methods in everyday vision and representation and whether there could be conflicts between these methods. One reaction was to concentrate on representation. Another was to consider only cases where there was no conflict. Since 1800 there has been increasing interest in what the precise nature of these conflicts might be and has led, more recently, to renewed study of cylindrical, spherical, hyperbolic and other complex planes which frequently reflect personal theories of vision rather than claiming to embody universal optical laws.

Panofsky claimed that each society develops a specific theory of vision and corresponding theory of representation.

Our claim is that articulate theories of vision and representation do not emerge in pre-literate societies; that such theories require manuscripts and only begin to thrive when there are printed texts and that, paradoxically, the advent of print culture, instead of establishing one specific method to the exclusion of others, proliferates the number of methods.

Advances within such cultures cannot therefore be seen as a simple choice of a new method and are better understood in terms of increasing distinctions between vision and representation and attention to personal solutions for bridging these distinctions. So it is not really a question of which kind of perspective is used in a society, but rather a more basic problem of the ways in which mathematics, and particularly geometry are used in explaining vision and representation. If Panofsky were writing today he might have discussed mathematics in optics and representation as a symbolic form.

3.Ambiguities of Vision
It is so difficult to discuss these changing relationships largely because vision, which is often assumed to be constant, varies culturally and historically. A paleolithic tribesman, an Athenian Greek, a Renaissance Florentine, an eighteenth century Parisian and a modern New Yorker all have two eyes. Yet what each of them could see varies tremendously. A contemporary native of the Amazon rain forests sees all kinds of dangers which we as casual tourists would never notice: dangerous animals, deadly snakes, poisonous plants, and at the same time is only aware of a small percentage of the 1700 species of birds which fly around him. An ornithologist will see the species, but may not see the trees for the birds. Someone who grows up driving automobiles at 200 kilometers per hour on German highways sees very different things.

Every occupation and profession focuses on certain visual skills at the expense of others: a hunter, a goldsmith, tool and die maker, geologist, detective, botanist and an artist each see different dimensions of what is theoretically one world. Most? cultural history is frequently approached in terms of isolated representations of artists which are then taken as typical of the way in which a culture "saw." Sir Ernst Gombrich has explored other reasons why one might wish to avoid speaking of what persons "saw" altogether. Strictly speaking we can, at best, only attempt a history of what persons recorded in the form of words and pictures as having seen: a second-hand history of sight, as it were. Even in this we have scarcely begun. Histories of graphic methods such as Dubery and Willats are strikingly summary. A comprehensive history of projection methods has yet to be written. A history of how different trades and professions changed the boundaries of the visible has yet to be attempted.
4.The Price of Vision
There are also more subtle problems. We have shown that perspective owed much to a late mediaeval commitment to transform storytelling into painting, and we have emphasized the positive dimensions of this process. It exploded the boundaries of representation. It introduced spatio-temporal dimensions. But all this also came at a price. For it reduced the dynamic act of storytelling to static moments. It took away the performance aspect, the spontaneity, the uniqueness of the process, replacing this with something fixed, motionless, but capable of being reproduced almost exactly. Cultures such as China, India and Africa chose another path. They frequently avoided pictures and even words altogether, preferring to translate their stories into dance. In this form each version of a story was unique, spontaneous, full of motion and life, yet incapable of being repeated exactly. The west wanted repeatability and gradually found it, but again at a price. Each step closer to repeatability meant more mechanical aids, first compasses, then pantographs, gradually cameras, photo copy machines and CAD graphics packages. Every step closer to a perfect copy, threatened to become more impersonal. Fortunately there were also enormous advantages which made this price bearable: these involved systematic control of representation which came through mastery of scale, size and distance.


5.Scale
In pre-literate societies, where there were no fixed rules of vision and representation, there were no concepts of scale, size and distance. A voodoo witch could use a tiny doll to affect a large man whether he was near or far away. The size of the doll, the size of the victim, the distance between the two, were meaningless factors in a magical context. Greek theories of vision and representation included some ideas of lines, angles and proportion. This brought a concept of scale to individual objects rather than contexts, as becomes clear if we return to our earlier example of the planisphere (fig. 64a).

When the tropic of cancer was projected onto the equator its scale was reduced. Whether the drawing involved was small as it is here or as large as the actual size of the earth made no difference. Hence although scale within a drawing was important, the scale of the drawing itself was irrelevant. This applied to all representations (sculptures, paintings and drawings) where proportions were constant (fig. 64.b-c). Hence tiny statuettes and monumental statues had the same effect, and there was no incentive to change scale in order to bring objects into focus and study them more closely. Only changes in angle and proportion were significant (fig. 64. d-e).

Greek projection methods concentrated on scale within an object rather than scale between objects. Whether a diagram was small or large did not matter. Only changes in angle were important.

The advent of perspective in the Renaissance integrated Greek geometrical ideas of line, angle and proportion with measured size and distance. A small picture might represent small objects which were nearby or large objects at a distance. Size now became a relation between apparent and measured size which varied with distance. It was not distance between isolated objects, but distance between planes that counted. If distances were too great and objects too small, this could be solved by changing their scale.

Perspective thus brought incentives for the development of telescopes and later of microscopes. All this shifted attention from scale within an object to scale between objects in a given plane and scale between objects and contexts: persons and buildings to townscapes and landscapes, and these in turn to surveyors' charts, topographical views and maps.

An inverse size/distance law governed these relations and all one needed to do was to decide which scale was appropriate for which purpose. This had enormous consequences for representation: scenes, townscapes, landscapes, views and maps could all be coordinated systematically. A sense of mastery and domination of nature emerged, a sense that everything could be measured, but a key to objective control had been found.

Objectivity was simply a question of changing scale in order to get things into focus.


Page 257,
6. Fractals and Scale

Renaissance perspective was based on the assumption that every object has a fixed measured size, that size and shape within a given plane are constant and that size varies only with distance. The advent of microscopes called these assumptions into question. Within a certain range of magnification objects simply appeared larger, i.e. scale affected only size. Magnification beyond this range affected both the size and shape of objects. This was known in the seventeenth century and it became obvious in the eighteenth century when instrument makers such as Brander and Martin began using telescopes and microscopes in connection with perspective. Electron microscopes introduced more vivid examples. Yet, curiously enough, the philosophical implications of the discrepancy remained unnoticed and it took the development of fractals to recognize that this principle involved something far more basic than special conditions in microscopes and telescopes.

James Gleick used a map of England as an example. If one uses a ruler one kilometer in length then the coast has a certain number of corners and a given length. If one uses a ruler one meter in length than the same coast has a thousand more corners. If one uses a ruler one centimeter in length the same coast may have 100,000 more corners then the same coast measured in terms of kilometers. The coast measured in centimeters will also have a far greater circumference than the coast measured in kilometers. Perspective assumed that scale affects only size. Fractals confirm that scale affects shape as well as size.


7.Passive Recording and Active Intervention
The consequences of this corollary run deep. As long as scale affected only size, it could be claimed that perspective, which recorded the world in different scales, was a passive recording technique which did not interfere with the essential characteristics of that which it examined.

Perspective was therefore a model for the scientific method of passive, objective observation and recording without active subjective intervention. If, however, scale affected the basic shape of what was studied, then perspective, which played with scale, was tampering with the evidence that it claimed to be recording and was subjective in a way that Panofsky and his followers had not suspected.

Heisenberg's indeterminacy principle introduced the idea that one could not study problems in quantum physics without disturbing the evidence. Perhaps we need to see the whole of early modern science in these terms: that universal measurement in the Renaissance needed more than a conviction that the language of nature was written in the alphabet of geometry. It required an active imposition of Euclidean geometry on nature, and this particular kind of active, subjective interference with nature was paradoxically inseparable from the passive objective recording to which it is frequently described as being opposed. And like the size of the mesh in a fish net which determines the size of fish which can be caught, the geometrical grid of Euclidean geometry determined the scale of reality which could be recorded. Accordingly geometrical figures with straight lines and circles replaced the complex curves of botany and biology in early scientific textbooks. Nor did this change dramatically in the nineteenth century with the non-Euclidean geometries of Lobachewsky, Bolyai and Riemann which projected these simple geometrical forms onto spherical and hyperbolic surfaces. Mandelbrot's fractal geometry was the first to offer new possibilities in recording nature's complexity.

An active choice is required to decide the initial scale of the first object for this determines the shape as well as the size of the object to be recorded. In other words, besides actual measurement of an external physical object, objectivity involves a subjective decision about the scale of measurement which provides parameters and tolerances of what is to be measured.



Page 259,
9.Scales and Samples

Simple as these distinctions may seem, they have basic consequences for concepts of knowledge and are very much bound up with western culture. In cultures where there was no distinction between subjective (inner, mental) and objective (outer, physical) images, knowledge was internal, psychological and lay in a person's conception of an image with no need to study further examples and no incentive to communicate this conception. In this extreme case knowledge could stop with a sample of one.

Already in Greece, the west embarked on a different course. Plato combined this approach with a belief in ideal images which theoretically required only introspection but in practice assumed the external stimulus of dialogue. His student Aristotle argued that knowledge was something to be collected rather than meditated, appropriated from outside rather than sought inside, an approach inherited by Islam and in turn by the Christian West, where it was reinforced by a belief in Creatural realism, particularly through the Franciscans and Dominicans. The development of perspective brought into focus a distinction between outer images which could be seen, recorded and measured and inner images which could not.
Perspective introduced systematic rules for recording external objects. Instruments such as the perspectival window or compass and ruler, and later the proportional compass and pantograph were intended to simplify this process. They also revealed that the process was much more complex in practice than in theory and that the catalogue was potentially much larger than suspected.

An interesting pattern thus emerges. Societies where no distinction was made between inner and outer tended to be closed, and focussed attention on their own culture, rather than looking futher. Societies with some awareness of inner and outer began to appropriate knowledge from other cultures. As the distinction became more clearly defined, this process of appropriation became more visible. The emergence of perspective in early fifteenth century Florence thus went hand in hand with an unprecedented interest in other cultures through the collection of Hebraic, Arabic, Chaldaian, Babylonian and other sources, leading gradually to the disciplines of anthropology, archaeology, ethnography and comparative religion which have developed in the west as nowhere else, although ideologies such as fascism and other-isms have tried to limit this looking outwards and control ideas even more systematically than in earlier closed societies.

Perspective, as described by Panofsky and most subsequent authors, was a phenomenon linked specifically with the Renaissance. Its impact, as they described it, was mainly in the domain of painting, the creation of a particular approach to space.

We have shown that it had fundamental consequences for both science, art and literature; that it changed the way western man constructed his environment, and we are suggesting that it transformed the ways in which he acquired knowledge, that the distinctions between inner and outer which it set in motion, led to an increasing externalization and objectivization of the world, which focussed attention away from verbal universals to visual particulars and thereby challenged him to use ever larger samples in making claims concerning knowledge.
Meanwhile, the concepts of scale which perspective introduced had long term implications for both the recording and organization of knowledge. For these reasons perspective needs to be seen as more than a cultural expression specific to the Renaissance, and recognized as one of the pivotal concepts of European culture, the implications of which are still being explored. For what began as a looking into, a seeing through, has led from a literal fixing of the horizon to a metaphorical looking beyond one's horizons and a study of other cultures.

What began as a symbol for a fixed approach to space has become an emblem of spatial play, of imagination and freedom.

The first paragraph is from Thomas Reid and his Geometry of Visibles. It describes our direct experience of visible figure and extension, as if on the inside of a concave surface, of which the eye is the center. Remember that a Great Circle is the circumference line on the spherical surface, that is made by the intersection of a plane passing through the center of a solid ball at any angle or inclination. Its meaning has been discussed by Yaffe, Van Cleve and Belot in many papers. The rest of the post are the writings of R.B. Angell where I think he highlights a common logical fallacy…

“Every visible right line will appear to coincide with some great circle of the sphere, and the circumference of that great circle, even when it is produced until it returns into itself, will appear to be a continuation of the same visible right line, all the parts of it being visibly in directum.”

But it seems to me that what Reid was saying something was much simpler than that. Leibniz introduced differential calculus in 1684 and Reid wrote his Inquiry 80 year later and was a lecturer in mathematics, so I suppose he could have used concepts of slope and direction drawn from the differential calculus, but it is not clear to me that he did.

Years ago while still in graduate school, I attended a fair in Canada where a motorcyclist performed in a giant spherical cage, doing all sorts of loop-the-loops. I remember thinking at the time, that the bands of steel which made great circles in that cage, would all appear, to anyone in the center of the cage as straight lines. This recognition never left me.

I take Reid’s first line to say simply:

If you are at the center of a sphere, every great circle of that sphere will appear to be a straight line. And conversely every line that we see as appearing to be straight in our visual field, could be a great circle on a sphere in which our eye is at the center.

The rest of his statement I took as simply saying that if you see what appears to be a fragment of a straight line, and then continue that line in your imagination in the same direction in an apparent straight line (which of course involves physically turning your head around while keeping your eye in the same location vis-a-vis other objects) position, then you will come around to your initial fragment, all the while appearing to be following a straight line.

And all this seems to me true.

Other interpretations imply that Reid intended to show that the spherical geometry and visible geometry was one and the same. This is nowhere stated by Reid. In fact in Essays on the Intellectual Powers of Man,(1785)1 Reid wrote, with respect to tangible and visible space,

“I take them to be different conceptions of the same thing; the one very partial, and the other more complete; but both distinct and just, so far as they reach” (p232-3) and “...visible figure and extension are only a partial conception, and the tangible figure and extension a more complete conception of that figure and conextension which is really in the object.”

This is not saying that visible and spherical geometry are one and the same. At various points in the papers, I find a confusion about the relation between spherical geometry and Euclidean geometry.

Prima facie spherical geometry is a part of Euclidean solid geometry. A sphere is a solid Euclidean object. Euclid defines circles and spheres. Actually the great circles on the surface of spheres, are not straight lines but are curved lines.

It is only when people begin to analogize them with straight lines on the surface of a sphere are they indeed equal to the dihedral angles of the planes of those great circles.

If you take the definition of ‘a straight line” as ‘the shortest distance between two points is a straight line” and restrict lines to the surfaces of a sphere, the shortest distance between two points on the surface of sphere is always a segment of a great circle on that surface.

If you wish you may use the term “geodesic” for “shortest distance between two points on a two dimensional surface” and substitute “geodesics” for “straight line segments”, and “spherical triangles” composed of three intersecting geodesics, in place of plane triangles.

The results will be not be equivalent to plane geometry. The Pythagorean theorem which holds for right triangles in plane geometry won’t hold for spherical right triangles, and the sum of the internal angles of spherical triangles will always be greater than 180 degrees, unlike plane triangles. But this does not mean that spherical geometry is not Euclidean.

It just means that substitution of geodesics for straight lines won’t always result in sentences with the same truth values.

The words are those of RB Angell, describing the "sphere" of visual geometry.

He's dead some years and I doubt he was a flat earther. He means an "eye" (and not a person) that sees as if at the center of a sphere. He (and Reid) are explaining how spherical geometry best describes the phenomena of visual experience.

Have you read the posts in this thread? If you have, then you know I think this is one of the reasons why the globe deception "works" and can be perpetuated and remain unnoticed by so many. The geometry of vision is not well understood, easily confused with tangible geometry in our perception, yet integral to all optical measurements.

Do I give you the impression of trying to sell something?

Schpankme wrote:
These are your words, as you describe once inside the Sphere, you only see straight lines.

"If you are at the center of a sphere
"every great circle of that sphere will appear to be a straight line"
"every line that we see ... could be a great circle on a sphere"

Nope they are his words, but I happen to agree with him, that the eye sees as if at the center of a sphere.

Schpankme wrote:Yes, I'm aware that you started this TOPIC, what I don't see is how this optical babel, courtesy of RB Angell, has any relation to the flat Earth?

I think it has everything to do with the explanation of perspective and how our incomprehension of it is used in the deception and false explanations of visual phenomena such as sunsets and other orbs in the sky. In my opinion this is an important key to understanding a flat earth.

Schpankme wrote: More specially, it was the STORIES told, and the indoctrination through education that persuaded people to believe that the Sun, Moon and of course the Earth must be Spherical in nature.

Sure, no argument there, but specifically which "schooling" is what I am trying to uncover.

Schpankme wrote: Here's how they sell the spherical Earth; first point to people from the past, lets say 4th century BC, and classify them as the "Ancient Greek philosophers", who proposed the idea that the Earth was sphere shaped.  Now, to be more specific lets classify someone from 330 BC as the "first real thinker", lets call him Aristotle, he will become the first to propose a spherical Earth. Now, it only stands to reason that by the Middle Ages, with it's 98% illiterate population, we had widespread knowledge that the Earth was a sphere.

Sure, no argument there either as regards the big picture. Yes we are being told stories, but for me that doesn't negate the importance of understanding how vision works, or how the math/geometry is manipulated. If you or others think that "optical babble" is useless for your personal further understanding, then that is fine. I am not content with just accepting that they are selling/telling stories. I want to know what and how.

Schpankme wrote: So yeah, I find it odd that you would bring RB Angell here, describing the "sphere" of visual geometry!

You are entitled to your opinion. I thought his description of how substituting geometric definitions produces sentences which skew the truth, quite appropriate to this thread.

Personally, I find it odd that you are dismissing this whole topic as if it had no relevance to the pursuit of truth.

If you and the mods want to move this thread to an "off-topic" forum, I have no issue with that.

We all have been on a roller coaster or an amusement park ride at one stage in our lives. Speaking for myself and others that I have asked about the experience of such rides I have got the same or similar answer you get dizzy, nausea and loss of coordination etc. All these effects are noticeable while on the ride and shortly thereafter the completion of the ride. All these symptoms are obviously to do with our bodies experiencing movement, velocity, inertia etc because our bodies are not used to it. My dilemma is why don’t we experience such effects/symptoms when we travel across earths parallels? If earth was round wouldn't we be completely incapacitated with nausea if we traveled or holiday elsewhere on earth as our bodies are used to the speed earth is rotating at point of origin. For example and I am holidaying in Munich which should be rotating at a lesser speed than where I am originally from(Brisbane) and yet we do not feel any symptoms at all.

I've urged everybody to read "An Inquiry into the Human Mind on the Principles of Common Sense", available on archive.org by Dr. Thomas Reid the founder of Scottish Common Sense Philosophy in the 18th century. A highly skilled polymath nearly forgotten for 300 years, but highly influential in many fields.

Here are some great lectures by Dan Robinson on Thomas Reid's critique of Hume.
https://www.youtube.com/playlist?list=PLBHxLhKiPKxBuGhbDKwahBLVhiUVTf9Ts

Watch especially lectures 2, 4, 6 and 8. You will not be disappointed.

We have been schooled (not educated) to believe in a representational theory of reality. However, Reid's philosophy and our in-born common sense actually gives us all the analytical tools we need to understand how the deception is so easily perpetuated.

vortexpuppy wrote:
We have been schooled (not educated) to believe in a representational theory of reality.
Reid's philosophy and our in-born common sense actually gives us all the analytical tools we need to understand how the deception is so easily perpetuated.

"If there had never been an artificial language every man would be a dancer and a painter." ~ Thomas Reid

The people are educated to read, write, and communicate in a language based on ideologies which have been created for them. Fooling the people is accomplished by controlling what they perceive to be fact and fiction.

Am glad we agree, I just refuse to call it education.

Thomas Reid (Birkwood Collection 3.3.13) <--- No stupid comments about numbers please, I didn't choose them.
Euclid – Simson – Parallelism

VP comments in red:

Below is a Letter from Reid (17??) directed at Prof Simson, the translator and restorer of Euclids Elements from the ancient Greek/Latin into English. He discusses topics such as: corrections to the Elements; deficiencies in Mathematical Definitions; the difference between self-evident truths and strict demonstration (i.e. not using some other assumption or axiom, that is no less self-evident than the thing to be proven); never give up the authority to judge for yourself; find a proof of parallel lines via a just and true definition of Quantity and Right lines.

First, here are the self-evident truths (universal ideas) and axioms / assumptions / necessary concessions that Euclid (or his editors) stated in the Elements, as far as we know.

Universal Ideas

1) Equals to the same are also equal to each other
2) And if to equals, equals are added, the whole are equal;
3) And if from equals, equals are subtracted, the remainders are equal;
4) And things coinciding with each other are equal to each other;
5) And the whole is greater than the part.

Necessary concessions

Let it be conceded that …
(1) from every point to every point a straight line can be drawn;
(2) And a limited straight line can be produced continually in a straight line
(3) And for every centre and distance a circle can be described;
(4) And all right angles are equal to each other;
(5) And if a straight line falling on two straight lines make the angles within and towards the same parts less than two right angles, then the two straight lines being indefinitely produced meet towards the parts where are the angles less than two right angles

Nr 5 is the famous parallel postulate, and the changing of the axioms / assumptions / concessions Nrs. 1,2,4, and 5 are what creates other mathematical Geometries. The 2D Geometry of the Plane (yes, 2D planar) can then be either Elliptic (spherical), Parabolic (Euclidean) or Hyperbolic depending on the axioms & assumptions. This is what makes spherical geometry a working model, but not real.

Which assumptions, axioms, concessions should be used remained a matter of much discussion for centuries until Riemann’s Habilitation separated the geometry of position and geometry of motion (from mathematics & natural philosophy) into the compartmentalized physical sciences, so that mathematics was free to theorize without being constrained by reality. lol.

Note also that if we have no just definition of quantity (to measure), then we cannot verify things ourselves. And if we allow that, then we effectively give up our authority to judge for ourselves. That is/was stupid of us.

Here is the transcribed letter. I have highlighted in Bold some sentences I think especially relevant.

Thomas Reids Letter:

I am very glad to hear that Dr. Simpson (whom I reverence as the father of the Mathematicians now alive) retains so much vigor of body and mind. I am just now teaching Euclids Elements and I think his corrections and emendations have added much to their perfection and accuracy.

But it mortifies me not a little to find that, in his judgement, the 11 or 12 axiom, upon which so great a part of the system depends, is neither self-evident nor does admit of a demonstration in a strict sense. Is not this acknowledging a defect in the Elements? A defect which is not to be attributed to Theon but to the science itself?

His delicacy in attributing every inaccuracy in the present copies not to Euclid but to Theon or some ignorant scholiast is without doubt just and well founded in the main and I think it an admirable expedient for preserving the entire respect which is due to the Patriarch of Mathematicians without giving up the liberty of judging for ourselves. I should grant the infallibility of the Pope if I am allowed to ascribe everthing amiss in his decrees to the corruption or ignorance of his notaries.

I am ashamed to tell you how much time I consumed long ago upon the axiom, in order to find mathematical evidence for what common sense does not permit any man to doubt.

After having laboured in vain, I quite despaired when I found that Dr. Simpson was of the opinion that it could not be strictly demonstrated. And began to think what the reason could be that a mathematical truth not self-evident upon which a great part of mathematics depends, does not admit of strict demonstration.

And I am now to acquaint you of what occurred to me in as few words as I can & would be very glad to know Dr. Simpsons opinion of it.

In other parts of the Elements the properties of every figure are drawn from the definition of that figure. The definition contains the whole essence of the thing defined and every property of it may be said to be included in its definition, for all of them by mathematical reasoning can be drawn out of it.

Accordingly, we see that in every proposition which relates to the circle, the definition of a circle is quoted or some preceding proposition is given which is deduced from the definition of a circle. And the same thing may be said of other geometrical figures. Their definition contains a Mathematical conception of them from which all their properties necessarily follow. And every property of them is drawn out of the definition. Thus it is with regard to every mathematical curve line. But the case is far other ways with respect to right lines. The definition given of the right line in the Elements gives no mathematical conception of the thing defined. Accordingly, it is never quoted as far as I remember in the whole Elements. Nor do I see how any demonstration can be grounded upon it.

The axioms in Mathematicks seem to me to be only a kind of succedaneum (substitute) to supply the want of proper definition. Where we have a just mathematical definition of anything, we reason about it from its definition and find no need of any axiom related to it.

There are no axioms in Euclid relating to circles, squares or parallelograms. Why? Because these figures are justly and mathematically defined and all their properties are drawn from their definitions. There are no axioms relating to Ratios. Why? Because Euclid has accurately and mathematically defined when one Ratio is said to be the same to, or greater than another, and all the propositions regarding Ratios are drawn out of the equality, majority or minority definitions.

All the axioms in Mathematics relate either to the Equality, Majority or Minority of Quantities in general, or to Right lines.

I do not allow the 11 Ax Lib1 El., That all right angles are equal to be an exception to this rule. For that axiom put in other words is this. That if a right line falls upon a right line making the adjacent angles equal, the sum of these two angles will always be the same, which axiom more properly relates to right lines than to angles. The axioms usually prefixed to the fifth book may all be demonstrated from those of the first.

Now what is the reason that all the axioms that are necessary in mathematics relate either (the Majority, Minority or Equality of) to Quantities in general, or to Right lines?

I take it to be this, that we have no mathematical definition either of:

1) Equality, Majority or Minority of Quantities, or
2) of a Right line.

Since therefore we have no Definition of a Right Line then we cannot reason from it. As to Quantity the Axioms about it I conceive, can never be superseded by definitions. For in order to this there be found to be not only a mathematical definition of Quantity which perhaps might be found; but there be found likewise to be mathematical definitions of addition, subtraction, sum, difference, greater and less.

Now these are too simple objects of thought to admit of definition. Nor is there any occasion for it, since the axioms relating to Quantities in general could receive no addition to their evidence even from demonstration.

But to come to right lines. You will say has not Euclid defined a right line? I say he has not and that definition in the Elements cannot be Euclids, but ought to be attributed to Theon or some interpolator.

1) Because it is no mathematical definition at all. It is no more than if you should say a Right line is a Straight line, that is, giving a synonymous word for it.
2) This definition is never quoted in the whole Elements as far as I remember, nor can any demonstration be drawn from it.
3) We have a better definition of a right line in the Elements Lib II Prop. 1. Latin Text: “Reel lines est quo cum reel lines non convenit in pluribus junetis quam in uno alias sibe ipsis congruent”.

Of the four axioms which we have about right lines, this definition might supersede three for it is easy to demonstrate from it that two right lines cannot have a common section nor include a space and that all right angles are equal between themselves.

If the famous axiom which seems to bring a reproach upon Geometry could be demonstrated from this Definition I would speak beg in its favour and plead that all the four axioms about Right lines as well as the common definition of a Right line, should be turned out of the Elements to make way for it.

But alas! I have spent much labour in vain to deduce that axiom by mathematical reasoning from this definition. And after what Dr. Simpson has said, I almost despair of it.

Shall we say then that this definition of Right Lines is imperfect and does not express the whole nature of them? And therefore all their properties cannot be deduced from it? If it is so, were it not worthwhile to seek for a more perfect definition of Right lines? If that cannot be done and if Right lines are things too simple to admit of a Definition, yet still we ought to lay down only such Axioms concerning them as are self-evident, and to deduce from these what is not so, by mathematical reasoning.

The simplest I can think of, if it deserves the name of an Axiom is this: That if two points of a right line be equally distant from another right line in the same plane, the intermediate points will be at the same distance from that right line. From this I think we might demonstrate the 12 Axiom.

Dr Simpson has very happily deduced the true definition of a plane from that very proposition in which we have the above mentioned definition of a right line. And if he or any other could find out such a Definition of a Right line as might serve for demonstrating the simple properties of Right lines which are assumed in the Elements. This in my judgement would most effectively wipe of that reproach from the Elements of them being in a great measure founded upon an Axiom that is neither self-evident nor can be strictly demonstrated.

I have wearied you and myself upon this subject and therefore must be short in what remains.

I am very much pleased with Dr. Blacks Theory of Latent Heat. I hope you will bring with you a full account of the experiments that confirm it.

I have often wished this winter that the heat that affects our bodies had not been so latent and that the heat that affects the mind in this society had been more latent than it has been. END

VP: Sidenotes on Latent heat: https://en.wikipedia.org/wiki/Latent_heat

Latent heat is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process. An example is latent heat of fusion for a phase change, melting, at a specified temperature and pressure.

The terms ″sensible heat″ and ″latent heat″ are specific forms of energy; they are two properties of a material or in a thermodynamic system. ″Sensible heat″ is a body's internal energy that may be ″sensed″ or felt. ″Latent heat″ is internal energy concerning the phase ( solid / liquid / gas ) of a material and does not affect the temperature.

The English word latent comes from Latin latēns, meaning lying hidden.The term latent heat was introduced into calorimetry around 1750 when Joseph Black, commissioned by producers of Scotch whisky in search of ideal quantities of fuel and water for their distilling process, to studying system changes, such as of volume and pressure, when the thermodynamic system was held at constant temperature in a thermal bath.

Those who have read this thread know that I think that an incorrect and unjust Philosophy that denies Direct Realism is the continuing enemy of all mankind. It goes unchallenged despite its obvious fallacies, absurdities, degeneracy, corruption and overall failure to progress or to improve the life and joy of all people.

It is a self-evident truth that Reality involves two things:

1. A Mind - Body experience (inner mind, inner body and outer surface)
2. Everything else (the external world and all its stuff)

You in your own bubble and then all the rest. This is true for every one of us, past and present. It’s happening right now. It is a continuous uniform law of nature. Ask your parents and grandparents or anybody else. Do you experience it too? Are we good to agree on this?

We share individually and collectively that same experience. We call it life. These two distinct parts of life are always immediately and directly conjoined, together forming the whole from the singular perspective of any one person. The first part is also included as part of the second external world for some other person’s mind-body experience; Somebody else’s knowledge and perceptions; All of this is real. All part of the warp and woof of life’s motley tapestry; The sum of all human knowledge; The common sense of mankind upon which all further reasoning builds.

All knowledge is about the combination of these two things. “Me and the rest”. Two things that separate your life experience, yet cannot be divided when alive. In constant companion until “Me” is no longer distinguishable from the rest; neither by itself nor any other. It is then neither a first or second thing.

Between them they divide the whole and each other in different ratios. Ratios whose measures depend on the physical matter involved, and their rational quantity. This is communicated at a level that has no need for words and often escape exact definition. Original, natural sensory apparatus and a reasoning mind equips us to experience these two things, identifying patterns with Sight, Sound, Taste, Smell and Touch.

If these two things did not conjoin we would not learn to walk, ride a bike, or anything else. We would not survive, nor be here now, nor even be able to reason. The natural language communicates on a level of sensations and perceptions without words. A natural language of motion pictures, sounds, rhythms, and all without words or any artificial language.

Philosophies that deny this direct realism, must postulate or create a third thing. This third thing is some combination of these two original self-evident things. It is given names such as idea, sense data, representation, concept, often disguised as a mental model supposedly helpful in illustrating a meaning. But these “ideas” are not demonstrating a proportion that can be experienced immediately and directly. They are conveying a concept or model, that often defies demonstration. However they are posited to exist and to either “mediate” between the two things or to have a real physical existence. More often than not these models are defective and illusory and send us on intentional Red Herrings and Goose chases, hunting sense data that can never be found. Do not entertain these kind of debates w/o being clear of the first principles and assumptions being made in the Geometry and Maths of your adversary. :-)

* Philosophy: The study of all human knowledge, including all Natural Sciences, Language, Mind-Body, Morals, Psychology, Theology, etc.,

This next post contains selected extracts, from the book that is linked below. This insightful book researches the contributions of famous Mathematicians and Philosophers to the problem of Right lines and Parallelism in Geometry. In doing so the book confirms the Geometry of the Plane can be either Elliptic (Spherical/Riemannian), Hyperbolic or Parabolic (meaning Planar in this case) and also serves as a prime example of what I mean in my first paragraph.

The main question being researched in this mathematical example is: How to define a Right Line and Parallelism? This question of how to define a Straight and Level Line is a necessity before one step in mathematical reasoning can be made. The question persists to this day as evidenced by the flat earth discussion. If it had been justly, conclusively and satisfactorily answered we would not be discussing it again / anymore.

My favourite definitions are from Wallis, Simson, Laplace, Carnot and Taurinus, being most compatible to a Common Sense Philosophy. (Reid doesn’t get a mention. although he made his own attempts).

Wallis:

“VIII. At this stage, presupposing a knowledge of the nature of ratio and the definition of similar figures, I assume as a universal idea: To any given figure whatever, another figure, similar and of any size, is possible. Because continuous quantities are capable both of illimitable division and illimitable increase, this seems to result from the very nature of quantity; namely, that a figure can be continuously diminished or increased illimitably, the form of the figure being retained.”

Laplace:

“The perception of extension contains therefore a peculiar property which is self-evident, without which we could not rigorously establish the doctrine of parallels. The notion of a limited extension (for example, of a circle) does not involve anything that depends on its absolute magnitude; but if we conceive its radius to be diminished, we are forced to diminish in the same proportion its circumference, and the sides of all the inscribed figures.”

Carnot

"The Theory of Parallels rests on a primitive idea which seems to me almost of the same degree of clearness as that of perfect equality or of superposition. This is the idea of Similitude. It seems to me that we may regard as a principle of the first rank that what exists on a large scale, as a ball, a house, or a picture, can be reduced in size, and vice versa and that consequently, for any figure we please to consider, it is possible to imagine others of all sizes similar to it; that is to say, such that all their dimensions continue to be in the same proportions. This idea once admitted, it is easy to establish the Theory of Parallels without resorting to the idea of infinity.

If these (or something similar ;-) had prevailed, we would never have believed it was a Sphere-Pear because similar, scalable figures do not exist in Elliptic or Hyperbolic geometry.

The book justly criticizes modern definitions of straight and level that went on to create the collective delusion most people still suffer from. Unfortunately the author does not question his own beliefs and common sense, but heh nobody is perfect right?

Enjoy the historical overview and remember just because the Maths works it doesn’t make it real or veridical, since a definition can neither be true or false. We should however choose or define one that does not contradict reality. lol. .

Enjoy,
Vortexpuppy

Extracts and highlighted passages from the book …

https://archive.org/details/theoriesofparall00fran

THEORIES OF PARALLELISM AN HISTORICAL CRITIQUE
BY WILLIAM BARRETT FRANKLAND, M.A.
SOMETIME FELLOW OF CLARE COLLEGE, CAMBRIDGE: VICAR OF WRAWBY, 1910