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

Maths index
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Geometry
Measuring the earth
Euclid's definitions
Squares, Cubes and Roots

Euclid's axiomatic foundations for
geometry
contained:
several
definitions,
some
postulates
and
a few
common notions.
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.
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.
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 rightangled triangle is
that which has a right angle, an obtuseangled triangle that which
has an obtuse angle, and an acuteangled triangle that which has its
three angles acute.
22. Of quadrilateral figures, a square is that which is both
equilateral and rightangled: 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 rightangled.
And let quadrilaterals other than these be called trapezia.
23.
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.
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.
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
Squares
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
Cubes
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 crosssection = 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 crosssection all
along it, you would calculate the area of the crosssection and multiply
that by the length of the ditch to work out its volume. The area of the
crosssection 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
Powers
Roots
Straight Line
A string stretched tight is a straight line.
Euclid's definition
Topology
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 figure8.

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