# Definition:Compass and Straightedge Construction

## Definition

A **compass and straightedge construction** is a technique of drawing geometric figures using only a straightedge and a compass.

The operations available are:

- using the straightedge to draw a straight line determined by two given points

- using the compass to draw a circle whose center is at a given point and whose radius is the distance between two given points

- finding the points of intersection between straight lines and circles.

## Also known as

The obvious alternative **straightedge and compass construction** can of course be seen.

Some sources use hyphens: **compass-and-straightedge construction**.

Use whatever you feel comfortable with.

Some sources use the term **Euclidean construction**.

Some sources use the term **Euclidean tools**, while others recognize that term but decry it.

Some sources use the word **ruler** for **straightedge**, but this can be confused with a device which has markings on it to allow measurement; a true **Euclidean straightedge** has no such markings.

## Also see

## Historical Note

Some sources suggest that it was the Greek mathematician Oenopides of Chios who imposed the restriction that compass and straightedge were the only tools that were "philosophically suitable" for geometric constructions.

Others suggest that this restriction originated with Plato's Academy.

The fact that the constructions made in Euclid's *The Elements* implicitly impose such a restriction suggest that Euclid himself may well have been an Academy student.

However, when necessary, the Greeks were quite ready to use other tools to achieve constructions that cannot be made with just a straightedge and a compass.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 40$. Construction with Ruler and Compasses - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Euclidean construction** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Euclidean construction** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euclidean construction**