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ABC Geometry

Words to help you think about space, shape, measurement and the laws of thought.


The ancient Egyptians and Mesopotamians developed methods of measuring objects and calculating relations that they used to build monuments like the

The Greeks called this geometry, which means earth- measurement.

About 300BC, a Greek called Euclid, who lived in Egypt, developed proofs of the geometric rules that the Egyptians had devised.

Euclid's proofs started from axioms and reasoned logically from them to conclusions.

This has been seen by some philosophers as a model for what science (or part of Science) should be. Hobbes argued for a Social Science based on Euclidian methods. Poincare used Euclidian and other geometries to argue that science is based on imagination.

The 3,4,5 rule

Take three straight lines
Make one 3 units long,
make another 4 units long,
make the other 5 units long.
Join them together to make a triangle.
The angle opposite the longest line will always be a right angle.

The Egyptians, who discovered the 3,4,5 rule, used it to build pyramids.

The 3,4,5 rule is connected to these numbers:

5 x 5 = 25 (the square of 5)

4 x 4 = 16 (the square of 4)
3 x 3 = 9 (the square of 3)

16 + 9 = 25 (4 squared, plus 3 squared, equals 5 squared)

It is a specific example of a general rule that:

In a right angled triangle, the square of the line opposite the right angle is equal to the squares of the other two lines added together.

This formula has been called "Pythagoras's Theorem". I do not know if Pythagoras discovered it, but he and his followers probably saw religious significance in it.

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Measuring the earth

Euclid's definitions

Squares, Cubes and Roots

Euclid's Axioms

Euclid's axiomatic foundations for geometry contained:
several definitions,
some postulates and
a few common notions.

Euclid's definitions

1. A point is that which has no part

2. A line is breadthless length

3. The extremities of a line are points

4. A straight line is a line which lies evenly with the points on itself

5. A surface is that which has length and breadth only

6. The extremities of a surface are lines

7. A plane surface is a surface which lies evenly with the straight lines on itself.

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

9. And when the lines containing the angle are straight, the angle is called rectilinear.

10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle.

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.

See circle formula
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

16. And the point is called the centre of the circle.

17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference
[boundary] of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut of by it.

19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and right-angled: an oblong that which is right angled but not equilateral; a rhombus that which is equilateral but not right angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Euclid's Postulates

Let the following be postulated:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

Euclid's Common Notions

1. Things which are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

The hypotenuse is the longest side of a right angled triangle.

Area is a measurment of the surface of a shape.

A line has no area itself, but it can enclose an area - Like a square or a circle.

Converting lines to areas in graphics representing statistics can give misleading emphasis - See Florence Nightingale and the Crimean War

Squares, Cubes and Roots


As a shape, a square is flat with four straight sides of equal length, at right angles to one another

To find the area of a square, multiply the length of a side by itself. The area of a square with 10 centimetre sides is 10 x 10 = 100 square centimetres.

The square of a number is the number multiplied by itself. So:

    the square of ten is 10 x 10 = 100
The square root of a number is the number that produces that number when squared. So, from the example just given, we can see that:
    the square root of 100 is 10

As an object, a cube is like a child's building brick, a cube of sugar or a dice. It has six sides, all of which are squares of the same size.

To find the volume of a cube, multiply the length of a side by itself twice. The area of a cube with 10 centimetre sides is 10 x 10 x 10 = 1,000 cubic centimetres.

The cube of a number is its square multiplied by the number. So:

    the square of ten is 10 x 10 = 100
    the cube of ten is 100 x 10 = 1,000
The cube root of a number is the number that produces that number when cubed. So, from the example just given we can see that:
    the cube root of 1,000 is 10

Some formulas

area of parallelogram = base x height

area of triangle = half base x height

volume of prism = area of base x length
This is a particular application of a general rule that

volume of any solid of uniform cross-section = the area of the cross section x the length (distance between the two end faces)

So: if you had a (very) neatly dug ditch, with the same cross-section all along it, you would calculate the area of the cross-section and multiply that by the length of the ditch to work out its volume. The area of the cross-section can be calculated by adding parts of it. For example, it may be possible to calculate it by adding the area of two triangles (at the sides) to an oblong (in the centre)

Thank you to Lynn Frizell for the problem



Straight Line
A string stretched tight is a straight line.
Euclid's definition

Topology is the study of what does not change when shapes are twisted or stretched. Size and proportion have no meaning in topology. A small oval is the same as an enormous circle. To topologists, what matters is the number of holes and twists. Thus a teacup is identical to a doughnut, but cannot be twisted into a figure-8.

Bernard Burgoyne draws on topology to illuminate issues of space, boundaries, inside and outside in sociology, politics and psychoanalysis. See 2011. See also Sandler and others 1997


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