10. A plane triangle is a space included by three right lines, and has three angles.

G

11. A right angled triangle has one right angle, as AB C. The side A C, opposite the right angle, is called the hypothenuse; the sides A B and B C are respectively called the base and perpendicular.

12. An obtuse angled triangle has one obtuse angle, as the angle at B.

13. An acute angled triangle has all its three angles acute, as D.

14. An equilateral triangle has three equal sides, and three equal angles, as E.

15. An isosceles triangle has two equal sides, and the third side greater or less than each of the equal sides as F.

16. A quadrilateral figure is a space bounded by four right lines, and has four angles; when its opposite sides are equal, it is called a parallelogram.

17. A square has all its sides equal, and all its angles right angles, as G.

18. A rectangle is a right angled parallelogram, whose length exceeds its breadth, as B, (see figure to definition 2).

19. A rhombus is a parallelogram having all its sides and each pair of its opposite angles equal, as I.

20. A rhomboid is a parallelogram having its opposite sides and angles equal, as K.

21. A trapezium "is bounded by four straight lines, no two of which are parallel to each other, as L A line connecting any two of its angles is called the diagonal, as AB.

3

22. A trapezoid is a quadrilateral, having two of its opposite sides parallel, and the remaining two not, as M.

M

23. Polygons have more than four sides, and receive particular names, according to the number of their sides. Thus, a pentagon has five sides; a hexagon, six; a heptagon, seven; an octagon, eight; &c. They are called regular polygons, when all their sides and angles are equal, otherwise irregular polygons.

24. A circle is a plain figure, bounded by a curve line, called the circumference, which is everywhere equidistant from a point C within, called the centre.

A

B

25. An arc of a circle is a part of the circumference, as A B.

A

E

26. The diameter of a circle is a straight line AB, passing through the centre C, and dividing the circle into two equal parts, each of which is called a semicircle. Half the diameter AC or CB is called the radius. If a radius CD be drawn at right angles to A B, it will divide the semicircle into two equal parts, each of which is called a quadrant, or one fourth of a circle. A chord is a right line joining the extremities of an arc, as FE. It divides the circle into two unequal parts called segments. If the radii CF, CE be drawn, the space, bounded by these radii and the arc FE, will be the sector of a circle.

27. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and each degree into 60 minutes, each minute into 60 seconds, &c. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees. 28. The measure of an angle is an arc of any circle, contained between the two lines which form the angle, the angular point being the centre; and it is estimated by the = number of degrees contained in that arc :—~ thus the arc A B, the centre of which is C, is the measure of the angle AC B. If the

B

A

angle ACB contain 42 degrees, 29 minutes, and 48 seconds

it is thus written 42° 29′ 48′′.

PROBLEMS IN PRACTICAL GEOMETRY.

(In solving the five following problems only a pair of common compasses and a straight edge are required; the problems beyond the fifth require a scale of equal parts; and the two last a line of chords: all of which will be found in a common case of instruments.)

PROBLEM I.

To divide a given straight line AB into two equal parts.

From the centres A and B, with any radius, or opening of the compasses, greater than half A B, describe two arcs, cutting each other in C and D; draw CD, and it will cut A B in the middle point E.

PROBLEM II.

At a given distance E, to draw a straight line CD, parallel to a given straight line A B.

E

D

C

A

B

m

From any two points m and r, in the line A B, with a distance equal to E, describe the arcs n and s::-draw CD to touch these arcs, without cutting them, and it will be the parallel required.

NOTE. This problem, as well as the following one, is usually performed by an instrument called the parallel ruler.

PROBLEM III.

Through a given point r, to draw a straight line CD parallel to a given straight line A B.

Am

From any point n in the line A B, with the distance nr, describe the arc rm:-from centre r, with the same radius, describe the arc ns-take the arc mr in the compasses, and apply it from n to s:-through r and s draw CD, which is the parallel required.

PROBLEM IV.

From a given point P in a straight line A B to erect a perpendicular. 1 When the point is in or near the middle of the line.

On each side of the point P take any two equal distances, Pm, Pn; from the points m and n, as centres, with any radius greater than P m, describe two arcs cutting each other in C; through C, draw CP, and it will be the perpendicular required.

2. When the point P is at the end of the line, With the centre P, and any radius, describe the arc nr s;-then with the same radius, and taking n and r as centres, cut off the equal arcs nr and r s :—again, with centres r and s, describe arcs intersecting in C-draw CP, and it will be the perpendicular required.

P

NOTE. This problem and the following one are usually done with an instrument called the square.

PROBLEM V.

C

From a given point C to let fall a perpendicular to a given line. 1 When the point is nearly opposite the middle of the line. From C, as a centre, describe an arc to cut AB in m and n;-with centres m and n, and the same or any other radius, describe arcs intersecting in o: through C and o draw Co, the perpendicular required.

Am

n

2. When the point is nearly opposite the end of the line. From C draw any line Cm to meet B A, in any point m;-bisect Cm in n, and with the centre n, and radius Cn, or mn, describe an arc cutting BA in P. Draw CP for the perpendicular required.

PROBLEM VI.

m

n

A

To construct a triangle with three given right lines, any two of which must be greater than the third. (Euc. I. 22.) Let the three given lines be 5, 4 and 3 yards. From any scale of equal parts lay

off the base A B

5 yards; with the

centre A and radius AC = 4 yards, de

scribe an arc; with centre B and radius A

=

C

B

CB 3 yards, describe another arc cutting the former arc in C:-draw AC and CB; then ABC is the triangle required.

PROBLEM VII.

Given the base and perpendicular, with the place of the latter on the base, to construct the triangle.

Let the base AB = 7, the perpendicular CD = 3, and the dis

tance AD = 2 chains. Make A B 7 and AD= 2;-at D erect the

C

A

perpendicular DC, which make 3:-draw AC and CB; then ABC is the triangle required.

C

PROBLEM VIII.

To describe a square, whose side shall be of a given length. Let the given line A B be three chains. At the end B of the given line erect the perpendicular B C, (by Prob. IV. 2.) which make AB: with A and C as centres, and radius A B, describe arcs cutting each other in D: draw A D, DC and the square will be comB pleted.

=

PROBLEM IX.

To describe a rectangled parallelogram having a given length and breadth.

Di

C

Let the length AB = 5 chains, and the breadth BC = 2. At B erect the perpendicular BC, and make it = 2:with the centre A and radius B C describe an arc; and with centre C and radius A B, describe another arc, cutting the former in D:— join AD, DC to complete the rectangle.

A

B

PROBLEM X.

The base and two perpendiculars being given to construct a trapezoid

D

A

Ꭰ

Let the base AB = 6, and the perpendiculars AD and B C, 2 and 3 chains respectively. Draw the perpendiculars AD, DC, as given B above, and join DC, thus completing the trapezoid.

PROBLEM XI.

To make a triangle equal to a given trapezium A B C D.

C

B

E

Draw the diagonal DB, and CE parallel to it, meeting A B prolonged in E:-join DE; then shall the triangle ADE be equal to the trapezium ABCD.

PROBLEM XII.

To make a triangle equal to the figure ABCDE A.

Draw the diagonals DA, DB, and the lines EF, CG,