# Definition:Euclid's Definitions - Book I

## Euclid's Definitions: Book $\text{I}$

These definitions appear at the start of Book $\text{I}$ of Euclid's *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. - A
**surface**is that which has length and breadth only. - The extremities of a surface are lines.
- A
**plane surface**is a surface which lies evenly with the straight lines on itself. - 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. - And when the lines containing the angle are straight, the angle is called
**rectilineal**. - 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. - An
**obtuse angle**is an angle greater than a right angle. - An
**acute angle**is an angle less than a right angle. - A
**boundary**is that which is an extremity of anything. - A
**figure**is that which is contained by any boundary or boundaries. - 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; - And the point is called the
**center of the circle**. - A
**diameter of the circle**is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center. - A
**semicircle**is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. **Rectilineal figures**are those which are contained by straight lines,**trilateral**figures being those contained by three,**quadrilateral**those contained by four, and**multi-lateral**those contained by more than four straight lines.- 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. - 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. - 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 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 either direction, do not meet one another in either direction.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Definitions - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)