"The great book of Nature lies ever open before our eyes and the true philosophy is written in it... But we cannot read it unless we have first learned the language and the characters in which it is written... It is written in mathematical language and the characters are triangles, circles and other geometric figures..."
Galileo (born 15 Feb 1564 in Pisa, died: 8 Jan 1642 in Arcetri, near Florence)

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.

Rhind papyrus

"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.

Thales

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.

Euclid

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.

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

The Elements

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). The year 888 is nearer to now than it is to Euclid's time!

• 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 p, 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 p was transcendental, that is it is not the root of any polynomial equation with rational coefficients.

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 3rd 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, 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".

Below, the Flemish painting "The Measurers", from the sixteenth century shows a range of geometrical instruments being used.

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 refered to as Mascheroni constructions.

Descartes

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)