43. A chord is a right line joining the extremities of an arc; as the line AB.

44. A segment is any part of a circle bounded by an arc and its chord.

45. A semicircle is half of a circle, or a segment cut off by the diameter; as ABC.

B

A

C

D

46. A sector is any part of a circle bounded by an arc and two radii.

47. A quadrant is the fourth part of a circle, or a sector bounded by an arc and two radii at right angles to each other; as CDB. Corol. Hence a right angle is said to contain 90°. NOTE.-All Definitions and Rules should be committed to memory.

To draw a Line parallel to a given Line AB
at a given Distance.

From any two points, m and n, in the given line, with the given distance as a radius, describe the arcs r Draw CD to touch

and o.

these arcs, without cutting

C

them, and it will be parallel A

to AB.

m

NOTE. This problem may be more readily performed by a parallel ruler.

PROBLEM IV.

To erect a Perpendicular from a given Point C,

near the middle of a given Line AB.

D

B

PROBLEM V.

To erect a Perpendicular from a given Point C, near the End of a

From a given Point c, to let fall a Perpendicular upon a given

To make a Triangle with three given Lines, any two of which must be greater than the third (Euclid i. 22).

Let the given lines

be AB=10, AC=8, and BC= 6 chains.

From any scale of equal parts (which is to be understood as employed likewise in all the following problems) lay off the base AB. A

With the centre A and radius AC describe an arc.

With the centre

B and radius BC describe another arc cutting the former in c. Draw the lines AC and BC, and the triangle will be completed.

NOTE. Any trapezium may be constructed in the same manner, when the four sides and one of the diagonals are given.

PROBLEM VIII.

Having given the Base, the Perpendicular, and the Place of the Perpendicular upon the Base, to construct a Triangle.

NOTE. A trapezium may be constructed in a similar manner, when one of the diagonals, the two perpendiculars let fall thereon from the opposite angles, and the places of these perpendiculars upon the diagonal are given.

PROBLEM IX.

To describe a Square whose Side shall be equal to a given right Line.

Let the given line AB = 4 chains. Upon one extremity B of the given line, by Problem V., erect the perpendicular BC, which make equal

to AB.

With A and C as centres, and the radius AB, describe arcs cutting each other in D. Draw the lines AD and CD, and the square will be completed.

D

C

B

PROBLEM X.

To describe a Rectangular Parallelogram, whose Length and Breadth shall be equal to Two given Lines.

Let the length AB = 8, and the breadth BC= 4 chains.

At в erect the perpendicular BC, which make equal to 4. With A as a centre, and the radius BC, describe an arc; and with c as a centre, and the radius AB, describe another arc, cutting the former in D. Draw the lines AD and CD, and the rectangle will be completed.

PROBLEM XI.

Upon a given Right Line to construct a Regular Rhombus.

Having any Two Right Lines given, to construct a Rhomboid.

Let the given lines be AB = 7, and BC= 4 chains.

Draw the line AB equal to 7. Take in your compasses the line BC and lay it from A to E. With A and E as centres, and the radius AE, make the intersection D. Then

E

B'

with B as a centre, and

C

B

the same radius, describe an arc; and with D as a centre, and the radius AB, describe another arc, cutting the former in c. Draw the lines AD, DC, and BC, and the rhomboid will be completed.