Translation:Ayil Meshulash/Discourse 1
This chapter contains sections 1 to 28
The point has no length, width or height. The line has length, but no width or height. The plane has length and width, but no height. The cube has length, width and height.
The point, line and plane are not found in reality, though are connected to reality, as a plane is the side of a cube, the line is the side of a plane, and the point is the tip of a line.
The lines on the sides of a plane are called edges, and the point at which they meet is called a vertex (lit. corner).
A plane is surrounded by not less than 3 edges. The type of shape of a plane is determined by its edges; if it has three edges it is called a triangle, if four a square, if five a pentagon and so on.
According to the number of its edges is the number of its corners with the exception of a circle, which is enclosed with one line and has no corners.
The point in the exact middle of a circle is called the center and it is equally distant from the rim on all sides.
A line that crosses the center of a circle and passes from one end the other is called the diameter of the circle, and it is the greatest of all the lines that can pass through a circle. The lines that pass from one end of the circle to the other but do not pass through the center are called chords (lit. 'extension'). A line from the center until the circumference is half the diameter [the radius].
Any circle, whether great or small, is divided into 360 parts, each of which is called a degree. Each degree can be divided into 60 portions, which are second level degrees [also known as minutes], and those, when divided into 60 parts, are third level degrees [also known as seconds]. This process can continue indefinitely.
To determine the angle of a 'corner' [vertex]: Place one leg of a compass [or calipers] on the corner and extend the other leg out a far as is wished. Then draw an arc and complete the circle. The degrees of the arc contained between the two intersecting edges are the same as the angle of the vertex.
For example, if you wish to know the angle of ABC, place on leg of the compass on C and extend the other leg somewhat until point D. Then draw the circle DEH and see how many degrees are in arc DE out of the entire DEH. This will be the number of degrees of the angle ABC.
Similarly, if you draw the circle FGI from the center C, the amount of degrees in arc FG from circle FGI is the same as the degrees in vertex ABC according to arc DE from circle DEH. In the same manner, whatever the distance of the arc that lies between the two lines, it remains in proportion to the circumference, and the angle is the same, regardless whether the length of the lines are in cubits, handbreaths, or finger measurements.
A right angle is one that contains 90 degrees and whose edges are perpendicular, not leaning towards either side. One edge is termed ‘resting’ and the other is termed ‘standing’, like so:
Corner ABC is a right angle and arc AB is a quarter of the circle. Line AC is termed ‘standing’ and line BC is termed ‘resting’.
A corner containing more than 90 degrees is termed a ‘wide angle’ (obtuse), while one contained less than 90 degrees is termed a ‘narrow angle’ [acute], like so:
Corner AB is ‘narrow’ while corner C is ‘wide’.
There are three types of triangles:
1) A ‘right triangle’, where one of the corners is standing [a 90 degree angle]
2) A ‘wide triangle’ [obtuse triangle], where one of the corners is obtuse.
3) A ‘narrow triangle’ [acute triangle], where all the corners are narrow angles.
A corner [vertex] cannot contain 180 degrees.
Two lines that cross each other create four corners, and all the corners together contain 360 degrees.
Even if three or four lines intersect one point [as above in figure B], they will always contain in total 360 degrees.
The corner with the greatest angle has opposite it the longest side, and the reverse is true as well.
The complement of an angle is the amount needed for an acute angle to reach 90 degrees.
The remainder [שארית] of an angle is amount needed for an angle, whether acute or obtuse, to reach 180 degrees.
Parallel lines are two lines that run next to each other separated by a constant distance. These lines will never intersect with one another.
A line that crosses two parallel lines will create equivalent angles: A with B, E with G, D with C, and H with F. If another lines crosses these parallel lines and creates the same angles these two crossing lines are parallel to each other.
Two lines that intersect each other at right angles through their midpoints are both diameters [of the circle they are in], as line BE and GD intersect each other a midpoint I. If a line intersects at the midpoint of another line it is the diameter while the line intersected is a chord, like line BE’s intersection with HC. If neither intersects with the other at a midpoint they are both called chords, as in lines AF and HC. Two diameter lines will always intersect each other at their midpoint but will not always create a right angle, as is seen in lines BE and CF. The intersection of two diameter lines is always at the circle’s center [point I in the figure].
Two vertical parallel lines that cross two other horizontal parallel lines will cross at the same distance from each other on the top and bottom lines, as in lines AC and BD. AB and CD are also crossing both lines at the same distance from each other.
Two triangles, where each of the edges of one is equal to its respective edge on the other, both share the same angles as well.
Similarly, if two sides and one angle of one triangle are equal to the same two sides and one angle in another triangle the remaining sides and angles of these two triangles are equal as well.
Similarly, if two angles and one side of one triangle are equal to the same two angles and side of another triangle the remaining sides and angles of these two triangles are equal as well.
Two triangles, each of whom both has the same angles are not necessarily constructed of line segments that have the same length. The proof of this is found in drawing a line segment across a triangle like so:
Angle BAE is then equal to angle DCE, angle ABE is equal to CDE, and the angle formed at point E is shared by triangles ABE and CDE, even though the line segments of one triangle is not equal to the other.
A triangle that has two sides that measure the same length is called 'equal footed' [Isosceles]. When all the sides have the same length it is called 'equal sided' [Equilateral]
If you wish to create an 'equal sided' triangle place the foot of the compass on a point that will be one end of the third side, widen the compass as much the length of the side that you want to create, and then draw an arc. Afterwards, place the foot of the compass on the other end of line segment that will be the third side and with the same width draw an arc again [in the opposite direction]. From the point at which both arcs intersect two lines should be drawn down to the corners of the third side, like so: