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This is a brief history of straight-edge and compass construction
from Ancient Greece through to the seventeenth century. See the
sources page for further information,
including books and web links.
'Geometry' means 'measure of the earth'. In ancient Egypt, from
which Greece inherited this study, the Nile would flood its banks
each year and when the waters receded the work of re-defining and
re-establishing the boundaries was called geometry. The Rhind papyrus,named
after the Scottish Egyptologist A. Henry Rhind, who purchased it
in 1858, was written around 1650 BC by the scribe Ahmes who is copying
a document which is 200 years older. It shows a number of practical
mathematical problems, several of which are concerned with geometrical
shapes.
The history of straight-edge and compass constructions has its
roots in Greek mathematics. Thales was born about 624 BC in Miletus,
Asia Minor and died about 547 BC in Miletus. Thales seems to be
the first known Greek philosopher, scientist and mathematician although
his occupation was that of an engineer. He is supposed to have visited
Egypt and brought back the study of geometry. In many textbooks
on the history of mathematics he is credited with five theorems
of elementary geometry:
- A circle is bisected by any diameter;
- The base angles of an isosceles triangle are equal;
- The angles between two intersecting straight lines are equal;
- Two triangles are congruent if they have two angles and one
side equal;
- An angle in a semicircle is a right angle. (Try
an activity based on this)
Not a great deal is known about Euclid. It is thought that he was
born about 325 BC and died about 265 BC in Alexandria, Egypt. The
Thirteen Books of Euclid's Elements was one of the first
great works of Mathematics, setting out definitions, postulates
and 'common notions' of the most basic terms in geometry and then
describing in precise order the mathematics that can henceforth
be deduced. Not all the book was original; he used the first great
Greek mathematician Thales as a major source, but nevertheless the
book as served as a model for mathematics writing and research ever
since.
Here are the first four definitions from The Elements:
- A point is that which has no part;
- A line is breadthless length;
- The extremities of a line are points;
- A straight line is a line which lies evenly with the
points on itself.
Is Sir Thomas Heath's translation of The Thirteen Books of Euclid's
Elements, published in 1925, a translation of the words which
Euclid wrote in 300 BC? In fact it is likely to be a translation
of a copy of a copy of a copy of...traced back to a version written
(with alterations and additions) in AD 888 for Arethas, bishop of
Caesarea Cappadociae (now in central Turkey). AD 888 is nearer to
now than it is to Euclid's time!
There are three classical problems in Greek geometry that
have fascinated mathematicans for centuries:
- squaring the circle;
- doubling the cube;
- trisecting an angle.
The first problem was quite popular in 414 BC, and appears in Aristophanes'
Birds 1001-1005. Plutarch writes that Anaxagorus worked on
the problem while in prison. Many ancient Greek philosophers including
Aristotle, Themistius, Philoponus and Simplicius also worked or
commented on the problem. Its history has become linked with that
of PI (π), the ratio of the circumference of a circle
to its diameter. In fact the final solution to the problem of whether
the circle could be squared using ruler and compass methods came
in 1880 when Carl Louis Ferdinand von Lindemann proved that π
was transcendental, that is it is not the root of any polynomial
equation with rational coefficients.
Trisecting an angle: Archimedes' method |
Archimedes of Syracuse (born 287 BC, died 212 BC) is crediting
with one solution of the third problem of trisecting an angle: given
an angle CAB, draw a circle with centre A so that AC and AB are
radii of the circle. From C draw a line to cut BA produced at E.
Have this line cut the circle at F and have the property that EF
is equal to the radius of the circle. Finally draw from A the radius
AX of the circle with AX parallel to EC. Then AX trisects angle
CAB.
The legacy of Greek mathematics, particularly in the fields of
geometry and geometric science, was enormous. From an early period
the Greeks formulated the objectives of mathematics not in terms
of practical procedures but as a theoretical discipline committed
to the development of general propositions and formal demonstrations.
The range and diversity of their findings, especially those of the
masters of the third century BC, supplied geometers with subject
matter for centuries thereafter, even though the tradition that
was transmitted into the Middle Ages and Renaissance was incomplete
and defective. Art, religion, mysticism and architecture have all
been heavily influenced by constructions based on lines and circles
- see sacred geometry.
Geometry flourished in many countries over the centuries. The artist
and mathematician Albrecht Durer (who studied mathematics and architecture
from ancient classics by himself) wrote a Treatise on measurement
with compasses and straight edge in 1525, in which he said...
'And since geometry is the right foundation of all painting, I have
decided to teach its rudiments and principles to all youngsters
eager for art'.
Plato believed that geometry was an incredible mode of immersing
oneself into philosophical contemplation. In fact, the notice above
his porch read, 'Let no one unversed in geometry enter my doors.'
The connection between the practical and philosophical aspects of
geometry are illustrated in Margarita philosophica (Basle,
1583).
In 1672 Georg Mohr, a Danish geometer, published Euclides Danicus,
in which he proved that every compass and straight edge construction
can be done with compasses alone. This amazing fact is usually attributed
to Lorenzo Mascheroni, an Italian mathematician in the eighteen
century, and consequently such constructions are often referred
to as Mascheroni constructions.
The French mathematician René Descartes (1596-1650) signalled
a break from a purely geometric approach with his publication La
Géométrie. His creation of coordinate geometry
(also called analytic geometry, or Cartesian geometry), for which
Pierre de Fermat must also be credited, has been called the first
really great advance in mathematical technique since the Greeks.
It laid the foundations not only for modern mathematics, but for
modern science as well. It led directly to the creation of the calculus
by Newton and Leibniz.
'I have come to know that Geometry is at the very heart of feeling,
and that each expression of feeling is made by a movement governed
by Geometry. Geometry is everywhere in Nature. This is the Concert
of Nature.'
Auguste Rodin (1840-1917)
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