A plain surface is one with which a straight line may eve. ry way coincide: if not, it is curved.

Plain figures, that are bounded by straight lines, take their names from the number and position of their sides, or angles.

A Triangle is a figure having three sides, and three angles, fig. 7.

If C

A right angle triangle has one S a ngles right. fig. 8. Oblique angled triangles have all their angles oblique. they have one obtuse angle they are called obtuse angled triangles, otherwise they are called acute angled triangles, fig. 9.

If all the sides of a triangle be equal, it is called an equilateral triangle-if two sides be equal an isosceles triangle and if the three dides be all unequal a scalene triangle.

11

A

B

12

c

A

65. A figure of four sides is called a quadrangle or quadrilateral. A parallelogram is a quadrilateral which has both A pair of its opposite sides parallel. C A rectangle is a parallelogram, baving all its angles right. A BD C, fig II.

A square is a figure having four equal sides and all is angles right, A B D C, fig. 12.

A rhomboid is an oblique an

gle parallelogram, fig. 13.

A a rhombus is an equal sided rhomboid, fig. 14.

A trapezium is a four sided figure which has neither pair of its opposite sides parallel, fig 15.

A trapezoid has one pair of its opposite sides párallel, fig. 16.

66. Figures of more than four sides are in general called polygons; they however receive particular names according the number of their sides or angles. A figure of five side is called a peniagon; of six sides, a hexagon; of seven. a heptagon; of eight, an octagon; of nine,

a

What is a plain surface?-a curved surface?

From what do plain right lined figures take their names?

What is the meaning of the word coincide?

What is the meanBing of position? What is a triangle? Make one.

What is a right angled triangle ?-an oblique angled tri angle?-an obtuse

angled triangle ?-an Daccute angled triangle? Make the several kinds of triangles.

13

B

14

15

D

16

17

D diagonal.

B

What is an equilateral triangle ?— an isosceles?--a scalene? Make them.

What is the general name of four sided figures? What is a parallelogram? What is a rectangle? a square? a rhomboid? a rhombus? What is a trapezium? a trapezoid? What is a diagonal A diagonal is a right line joining two opposite angles of a figure. Make a square and then a

What is the general name of figures of more than 4 sides? Do they have particular names? From what do they derive their particular names ?

What is a pentagon? Make one. Whatis a hexagon ?—a heptagon? The term polygon is derived from two Greek words, signifying many corners. The particular names are also derived from Greek words denoting the number of corners in the respective figures.

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nonagon; of ten, a decagon; of eleven, au What is an octagon? undecagon, and of twelve a dodecagon. If a nonagon?-an unpolygon have all its sides and all its angles decagon ?-a dodecagon? equal it is a regular polygon; otherwise it is Let the pupil be re

an irregular polygon. The perimeter of a fig-quired to make each figure is the sum of all its sides. ure on a slate or black

67 A circle is a plain figure bounded by aboard. continued curve line called the circumfer What is a regular polence, every part of which is equally distantly gon ?-an irregular polfrom a point within called the centre, fig 17 ygon? What is the peThe circumference itself is sometimes calledrimeter of a figure?a circle. The periphery of a circle is the What is a circle? What same as the circumference. The radius of a is the circumference? eircle is a right line drawn from the centre to What is it sometimes the circumference. CA, or C D, fig. 17. called? What is the perThe diameter of a circle is a right line pass-iphery of a circle? How ing through the centre and terminating iu many degrees in the cirthe circumference. A C B 6g. 17. cumference of a circle?

An arc of a circle is any part of the cir-(28) What is the radius cumference. of a circle? How many A chord is a right line joining the extreme-radii can there be to a ties of an arc. circle? Are they all of A segement is any part of a circle bounded a length? What is the by an arc and its chord. diameter of a circle ?

A semicircle is half a circle, or a segment How much longer than out off by a diameter, A D B fig. 17. a radius? What is an

A sector is any part of a circle bounded by arc of a circle? How an arc and two radii drawn to its extremities. many degrees can an arc A quadrant, or quarter of a circle is a sector have? What is a chord? having a quarter of the circumference for its What is a segment? What arc and its two radii perpendicular to each is the longest chord a other A CD, or B C D, fig. 17. circle can have? Does

The Height or Altitude of a figure is a per-the longest chord alpendicnlar let fall from an angle, or its ver-ways cut off the largest tex, to the opposite side, called the base. segment or longest arc ?

The circumference of every circle is sup- What is a semicircle! posed to be divided into 360 equal parts, call- What is a sector? What ed Degrees and each degree into 60 Min-is a quadrant? How utes, each minute into 60 Seconds, and so many quadrants in a seon. Hence a semicircle contains 180 de-micircle?-in a circle } grees, and a quadrant 90 degrees. What is the position of

The Measure of an angle, is an arc of any the radii of a quadrant ? circle contained between the two lines which What is the angle beform that angle, the angular point being thetween them? How macentre; and it is estimated by the number of ny right angles will a degrees contained in that arc. Thus A D is circle measure? How the measure of the angle A C D fig. 17. many degrees in a right A Secant is a line that cuts a circle, lying angle? What is meant partly within, and partly without it. by the altitude of a fig Two triangles, or other right lined figures, ure? How are all cir are said to be mutually equilateral, when all cles supposed to be dithe sides of the one are equal to the corres-vided? How many deponding sides of the other, each to each: andgrees in a semicircle they are said to be mutually equiangular,|—in a quadrant ? What

when the angles of the one are respectively is the measure of an anequal to those of the other. le? How many degrees

1dentical figures, are such as are both m a right angle? What tually equilateral and equiangular; or tha a secant? When are have all the sides and all the angles of theo figures said to be eone, respectively equal to all the sides aneuelateral? What is the all the angles of the other, each to each; seaning of equilateral? that if the one figure were applied to, or lain - When equiangular?upon the other, all the sides of the one woulWhen identical? exactly fall upon and cover all the sides o

the other; the two becoming as it were bu one and the same figure.

Similar figures, are those that have all the angles of the one equal to all the angles of the other, each to each, and the sides about ures? the equal angles proportional.

68. In a right angled triangle (A B C fig.

What are similar fig

What names are some

8, 18,) the side (A C) opposite to the right an- times given to the three gle is called the hypothenuse; and the other sides of a right angled two (A B and B C) the legs, or sides or base triangle ?

and perpendicular.

H

Let A B C, figure 18, be a triangle right angle at B. On the three sides make the squares A D E B, B F G C and A CHI. Continue the side D A to n, IA to k, H C to mand G C till it meets A n. Thro' I draw Is parallel to A B, through k draw kl parallel to A C, set the distance kl from A to o and through o draw oc parrallel to A B. Then will the square A CHI be divided into parts similar and equal to ali the parts which make up the two squares A D E B and B F G C, as may be proved by measuring the parts, or by cutting them out and applying them to each other Hence the square formed on the hypothenuse or longest side of a right angled triangle is equal to the sum of the squares formed on the two legs.

A Proposition, is something which is eith- NOTE There are sever proposed to be done, or to be demonstrat eral other methods of ed, and is either a problem or a theorem demonstrating the pre

A Problem, is something proposed to be ceding proposition, but done. they cannot be well unA Theorem, is something proposed to bederstood without condemoustrated. siderable knowledge of A Lemma, is something which is premised, geometry, or algebra. or demonstrated, in order to render what

follows more easy.

A Corollory, is a consequent truth, gained What is a proposition? immediately from some preceding truth oa problem? a theorem? demonstration. a lemma? a corollory?

A Scholium, is a remark or observation a scholium? made upon something going before it

69. The common Section of 2 Plaues, is the line in which they meet, to cut each other. A Line is Perpendicular to a Plane, when it is perpendicular to every line in that plane which meets it.

What is a plane? What One Plane is Perpendicular to, Another the common section of when every line of the one, which is perpen wo planes? dicular to the line of their common section, When is a line per is perpendicular to the other. pendicular to a plane! Parallel Planes, are such as being produced A plane perpendicular aver so far both ways, will never meet, or o another? which are every where at an equal perpén. When are planes pardicular distance.

A Solid Angle, is that which is made by three or more plane angles, meeting each other in the same point.

lallel?

What is a solid angle?

When are solids sim.

Similar Solids, contained by plane figures, ilar? are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, alike placed.

A Prism, is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms.

When are planes sim ilar?

What is a prism!

What particular names

A Prism takes particular names according to the figure of its base, or ends, whether tri angular, square, rectangular, pentagonal have prisms? hexagonal, &c.

A Right or Upright Prism, is that which has What is the form of the planes of the sides perpendicular to the the sides of a prism? planes of the ends or base.

A Parallelopiped or Parallelopipedon, is a What is a right prism? prism bounded by six parallelograms, every

opposite two of which are equal, alike, and What is a parallelopi. parallel.

A Rectangular Parallelopipedon, is that whose bounding planes are all rectangles. which are perpendicular to each other.

ped?

What is a rectangular parallelopiped?

Jar drawn from the vertex to the opposite of a solid? side or base.