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THE HELPING HAND It may be appropriate here to introduce “The Helping Hand,” a well-known factotum in Greek geometry who takes care that lines are drawn, circles are described, points are taken, perpendiculars are dropped, etc. The perfect imperative passive is its verbal mask: “Let a circle have been described ( ) with center A and radius AB.” No one who has studied the Elements in Greek will have missed it, and never do we find any of the commands or exhortations so familiar from our own classrooms:
Definitions
- Definition 1.
- A point is that which has no part.
- Definition 2.
- A line is breadthless length.
- Definition 3.
- The ends of a line are points.
- Definition 4.
- A straight line is a line which lies evenly with the points on itself.
- Definition 5.
- A surface is that which has length and breadth only.
- Definition 6.
- The edges of a surface are lines.
- Definition 7.
- A plane surface is a surface which lies evenly with the straight lines on itself.
- Definition 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.
- Definition 9.
- And when the lines containing the angle are straight, the angle is called rectilinear.
- Definition 10.
- When a straight line standing 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.
- Definition 11.
- An obtuse angle is an angle greater than a right angle.
- Definition 12.
- An acute angle is an angle less than a right angle.
- Definition 13.
- A boundary is that which is an extremity of anything.
- Definition 14.
- A figure is that which is contained by any boundary or boundaries.
- Definition 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 equal one another.
- Definition 16.
- And the point is called the center of the circle.
- Definition 17.
- 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 circle.
- Definition 18.
- 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.
- Definition 19.
- Rectilinear 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.
- Definition 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.
- Definition 21.
- 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.
- Definition 22.
- 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 and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
- Definition 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.
Postulates
Postulate 3
To describe a circle with any center and radius.- This is the third assumed construction in the Elements.
- It corresponds to drawing a circle with a compass.
Circles were defined in Def.I.15 and Def.I.16 as plane figures with the property that there is a certain point, called the center of the circle, such that all straight lines from the center to the boundary are equal. That is, all the radii are equal.
The given data are (1) a point A to be the center of the circle, (2) another point B to be on the circumference of the circle, and (3) a plane in which the two points lie. In the first few books of the Elements, there is but one plane under consideration and needn’t be mentioned, but in the last three books which develop solid geometry, the plane has to be specified.
Note that this postulate does not allow for the compass to be moved. The usual way that a compass is used is that is is opened to a given width, then the pivot is placed on the drawing surface, then a circle is drawn as the compass is rotated around the pivot. But this postulate does not allow for transferring distances. It is as if the compass collapses as soon as it’s removed from the plane. Proposition I.3, however, gives a construction for transferring distances. Therefore, the same constructions that can be made with a regular compass can also be made with Euclid’s collapsing compass. Definition 1
A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.Definition 2
Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.Definition 3
A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.Definition 4
A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.Definition 5
Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.Definition 6
A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.Definition 7
A straight line is said to be fitted into a circle when its ends are on the circumference of the circle.
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