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Euclid’s Geometry

One of Euclid’s ideas on geometry was that if you had a straight line extending into infinity, and a point next to it, then there is only one line going through that point that was completely parallel to the first line. Oddly though, he did not consider this a fundamental idea. The origin of geometry in Euclid’s time had very practical roots in land surveying where a flat surface was a good approximation for measuring land. “Geo-metry” was the study of straight lines and angles in two dimensions.

Greeks

Classical Greek philosophers wondered if space and the Universe had an edge. This is understandable if the Universe is flat or like a sphere. Either there is an edge, or it is infinitely big. What happens, they asked, if you go to the ends of the Universe and stuck out your hand, or threw a javelin? Is space a thing in itself with objects in it? Or does it only become something if there are objects in it to create space?

Renaissance Perspective

If you take a walk through any historical collection of paintings from the Renaissance onwards, it is striking that the depiction of space and the landscape seems to expand outwards, as you go forwards in time. In 15th century paintings the landscape is often a highly stylised view through a window in the background. A frame within a frame. Then moving through the next two hundred years the landscape or the environment takes up a larger and larger portion of the picture, as if the idea of our surroundings is opening out.

Vanishing point perspective is very effective at depicting enclosed spaces - the narrow streets and geometric architecture of a medieval Italian town. Perhaps because the northern Italians had the regularity of Classical ruins, they were encouraged to trust a mathematical picturing of space. But in the countryside, a solitary tree or a slowly curving hillside doesn’t easily suggest a mathematical treatment of hidden lines leading to a vanishing point on the horizon.

Add to that the visual experience of wide-open spaces with uninterrupted horizons. Take the rising moon as an example. As it rises just over the horizon, it seems very large, and quickly shrinks in size as it climbs in the sky. It’s not because the moon is somehow closer at ground level. It’s because the Earth’s atmosphere acts like a magnifying lens. Looking along the ground, the atmosphere is effectively thicker (and the magnification greater), than looking straight up through it. Therefore, in the wide-open landscape, objects on the horizon appear larger than they should. So perspective doesn’t work for larger distances.

Imagine that you are far back in time on the frozen tundra, hunting deer. The air is sharp and cold. There are not even trees now, no distinguishing feature that tells you where you are. The ability to make out tiny, dark, moving shapes, against the horizon, is part of your survival skill. Have we been selected by nature to perceive the tiny dot on the horizon? Is it possible that our brains magnify the distant object? The vanishing point perspective works best for an urban environment of narrow streets. For the wide-open circular horizons outside towns, another way of looking is more suitable.

René Descartes (1596-1650)

According to Bertrand Russell, Descartes carried a sword, and he rarely got out of bed before noon. One of the earliest scientists, he was also a mathematician, and a philosopher. Descartes had to watch what he wrote. He thought that the universe was infinite, but he couldn’t publish the idea because it was against the teachings of the Church, and therefore heretical. Descartes invented a way of treating space mathematically, by using “Cartesian” co-ordinates. Though it is not possible to imagine more than 3 perpendicular axes, it is still mathematically possible to define a space with more than 3 dimensions.

Where Descartes’ influence continues to condition our thinking is in the distinction he made between inner thoughts, and the outside world. The subjective, and the objective. By first proving the existence of a Creator, he could then say that our thought processes must therefore also exist. Then follows proof of the existence of an outside world. “Cogito ergo sum”: “I think, therefore I am”. To us today this seems like an odd order.

Philosophy split because of this scheme. Natural philosophy looked at the world of nature. Philosophy in general looked at the world of our perceptions, and language. The split has been very effective for natural philosophy (or Science), allowing the freedoms and methods to study nature, without raising complications in the nature of perception, and language. Though a Creator was implied by Descartes’ scheme, it was customary not to mention Him in the method and results of Science.

The 20th century quantum physicist, Heisenberg, addressed this problem. He pointed out that traditional (or classical) science in the 20th century could not see the proper relation between the observer and the observed in any quantum mechanical experiment, in his view because of the tradition begun by Descartes.

Isaac Newton (1642-1727)

Newton was the first to bring mathematics to the problem of objects in motion through space, and in a sense was the first abstract scientist. Newton believed in “absolute space”, that is, space that exists regardless of the objects, and their movements, within it. Though his equations of motion revolutionised ideas about space, they didn’t actually prove its separate existence. There was no way, in Newton’s scheme, that you could tell whether you were moving at a constant speed in a straight line, or were at rest. It all depended on what you were measuring the position and speed relative to.

Newton introduced the idea of gravity, as a universal attractive force that acted on planets just as it acted on us. This idea was as novel in its day as four-dimensional space-time is to us. In the way that non mathematicians of the day grasped these (strictly mathematical) ideas, it was proof of the Divine nature of the Universe, because only a Divine Creator could be responsible for such clockwork perfection. The idea that gravity was an invisible force, capable of action at a distance, seemed at the time to be a very magical idea. The abstract idea of an invisible force between masses has, however, become part of our picture of the world.

Latitude and Longitude:

Descartes’ co-ordinate system, though very useful for mathematical space, was less useful for the surface of the Earth. Even the British Ordinance Survey grid system, which relies on x, and y co-ordinates, has a certain warping of the land surface of the British Isles. The mapmakers define the zero co-ordinates (which are in the extreme south west of England). So the further the position north and east of that point, the more the inaccuracies build up. It is as a result of picturing a curved surface as a flat one.

Latitude and Longitude are a way of avoiding this inaccuracy. Latitude lines are imaginary hoops, equally spaced around the Earth’s surface. Longitudinal lines are only equally spaced at the equator, and slowly converge till they intersect at the north and south poles. In map making these imaginary lines go back to Classical Greece, but in practical navigation, an accurate location in the east west (Longitude) direction was quite recent.

Mercator

There is the problem of how to picture the Earth’s curved surface, as a flat map. One solution is to cut up the earth’s surface, like a peeled orange, and lay it flat. Although it is faithful to the actual surface, you need a lot of imagination to piece it together in the mind’s eye. Mercator, a Flemish mapmaker (died 1594), invented the most common way we think of the world as a flat surface. By stretching the sphere (of the earth) into a cylinder, and then slicing it, and unwrapping the surface, we get the Mercator projection of the world. This picture allowed the Latitude lines to remain parallel, but it distorted the Longitude lines so that they appeared to be parallel from the North Pole to the South Pole. It was a matter of allowing massive distortions in places which were less important to human activity, in return for a world view which was easier on the imagination.

Sailors could pin point their Latitude easily, but Longitude required a comparison of two accurate clocks. One clock to time the local noon, that is, the time of day when the sun was at the maximum height in the sky over the ship’s present position. The other clock to keep the noon time of the port that it had left, perhaps weeks before (whose Longitude was known). Each hour difference between the two clocks was equal to 15 degrees Longitude. This is because it takes the Earth 24 hours to rotate the full 360 degrees. 360/24 = 15 degrees/hour.

The challenge was to invent accurate clocks that could withstand ship movement, and temperature variation. Watch maker John Harrison eventually solved the problem for the Board of Longitude. The Greenwich Observatory in London was the site of an alternative method of calculating Longitude, by studying the cycles in the position of the moon. Though the astronomers largely scorned Harrison’s timepieces, Greenwich became the official seafaring choice for the zero point of Longitude, and also of time.

James Clerk Maxwell (1831-1879)

Newton’s scheme is still the way we think of the world. At normal speeds, masses, and sizes, Newton’s Laws are accurate and beautiful descriptions of motion in the world, but at the speed of light another picture was needed. The Scottish natural philosopher (physicist in today’s language) James Clerk Maxwell, working between 1850 and 1880, was well ahead of his time. Earlier in his short and brilliant career, he worked on colour theory and optics (he was the first to project a colour image, and he also invented the fish-eye lens).

He took Michael Faraday’s picture of magnetism and electricity, and applied mathematics to it. This picture involved thinking of a “field of force”, with “field lines” (like iron filings around a magnet). Maxwell developed this picture mathematically. He proved that electricity and magnetism were two parts of the same thing, and that together they belonged to a whole family of electromagnetic phenomena which included light. The result was a revolution in ideas of space, to the same degree as Newton in the 17th century, and Einstein in the beginning of the 20th century. To Maxwell, space and electromagnetism were interconnected. Electric charge was not made of particles but a result of displaced stresses in a medium he called “aether”. So space had the potential for transmitting energy. This was the beginning of the field picture versus the particle picture of matter. Maxwell realised that in order to follow his mathematics of space to its logical conclusion, required a conscious effort to stop thinking in terms of pictures of the world around him. In this he was inspired by the philosophy of Immanuel Kant, who believed that space and time were real “in themselves”, but which we could never know about directly.

Maxwell also realised that the speed of light was absolutely fixed with respect to the “aether”. The problem then remained to measure the speed of the earth with respect to this absolute space. The earth travels at 20 miles per second round the sun, and the sun must be moving at speed relative to ....what? The centre of the galaxy? The nearest star? It soon becomes meaningless, unless there is an aether, or fixed space to measure by. The result of an experiment by Michelson and Morley in 1887 to measure the speed of light in the direction of the earth’s movement, and against, showed that, incredibly, the speed was exactly the same, in both directions. The only solution to this was provided by Lorentz and Fitzgerald, who suggested that, if the speed of light doesn’t vary, then measurements of time and distance must vary to compensate.

Special Relativity

It was Albert Einstein who, at the beginning of the 20th century, solved the problem. He showed that, because the speed of light was an absolute constant, then there was no need to think of an “aether” or an “absolute space”. An observer would measure time and space differently at different speeds because these things were less real than the speed of light. The only way for space and time to have any meaning was to treat them as one thing, space-time. The common sense thinking of three space dimensions and one time dimension, now became the mathematics of four space-time dimensions.

The Theory of Relativity has had the same type of influence amongst non scientists as Newton’s clockwork universe. The fixed speed of light in space-time doesn’t necessarily mean that “everything is relative”. If only Einstein had called his theory “the absolute theory of light”, instead of “special relativity”!

General Relativity

But gravity did not yet fit into this theory. Newton’s idea of an invisible force acting at a distance worked very well except for one thing. Nothing can travel faster than the speed of light. So how can a force act immediately across vast distances that light would take years to cross. Einstein looked to Maxwell’s ideas about an extended field of force, something that was a property of “space” itself.

In the Special Theory of Relativity, two observers were travelling at a constant speed relative to each other. Einstein then asked (to get to a general theory), what if one of the observers were undergoing acceleration with respect to the other? By equating gravity with acceleration, Einstein could highlight a very subtle “tidal” change in the property of space-time. This is another huge leap of thinking. Einstein dismissed the major attractive part of gravity as a something unreal but relative to the observer’s frame of reference. He then homed in on the least significant attraction where gravity elongates and squashes an approaching object in a minuscule way. Einstein was inspired by the sceptical philosophy of the 18th century Scottish thinker David Hume. Hume, by stripping away common preconceptions about space and time, cause and effect, allowed Einstein to consider new kinds of geometry as equal in status to Euclid’s centuries-old picture of the world.

The geometry of Space-time

When one observer is undergoing acceleration due to gravity, things very slightly change their shape. They become very slightly warped because the space-time itself is warped. Even light itself is bent near massive objects and depends on the relative movement of the observer. Gravity is not seen as a force, but as a geometric property of space-time.

It turns out that depending on the type of curvature that a region of space-time has, there can be several possible ways to look at parallel lines. One form, elliptical space-time, is like the four-dimensional equivalent of lines of Longitude in that they start off parallel at the “equator”, but converge at the “poles”. Another form is hyperbolic space-time where parallel lines splay apart, again not in three dimensions this time, but in four.

Euclid’s assumption that geometry could be based on flat ground, makes less sense in the curved hills and valleys of this four-dimensional space-time landscape.