Draw a rectangle of any width whose length is three times the width.
How many squares has it if the width be taken as the unit?
Make it twice as wide as before.
How many squares has it now?
What two numbers multiplied together will give the number of squares?
Make it three times as wide.
How many squares has it now?
What two numbers multiplied together will give the number of squares?
The cover of a geography is one foot long and one foot wide, how many square feet in it?
How many inches long is the same cover? How many wide?
How many square inches does it contain?
How many square inches are equal to one square foot?
A table is one yard long and one yard wide, how many square yards in it?
How many feet long is the same table?
How many feet wide?
How many square feet does it contain?
One square yard equals how many square feet?
Draw a square whose side is a unit of any length.
Draw another whose side is two units of the same length.
The second square is how many times as large as the first one?
How many squares in half the second square?
Which is greater, two square inches, or two inches square?
Two inches square is how many times two square inches?
Draw a square whose side is three inches.
How many square inches does it contain?
How many times as many squares as the square of one inch?
How many square inches in the bottom row?
How many in all?
Which is greater, three inches square, or three square inches?
Three inches square is how many times three square inches?
PROBLEMS.
An equilateral triangle has each of its sides one inch long, what is its perimeter?
If each side were two inches long, what would be its perimeter?
An isosceles triangle has its two equal sides each three inches long, and its third side five inches long, what is its perimeter?
A right-angled isosceles triangle has its base five inches, and its hypothenuse seven inches long, what is its perimeter?
A square geography-cover is nine inches long on one side, how long all round?
How many square inches in it?
A slate is sixteen inches long and twelve wide, how many inches all round it?
A rectangle is five inches long and three wide, how long all round?
How many square inches in it?
A slate is one foot long and eight inches wide, what is its perimeter?
A room is twenty-four feet long and twenty-one feet wide, how many feet all round it?
How many square feet in the floor?
How many pieces of paper each a foot square would exactly cover it?
A yard of carpet is two feet wide, how many square feet in it?
Charles and Henry start from the same place, and walk in opposite directions; Charles goes twenty yards, and Henry fifteen, how many yards apart are they?
If they start from opposite ends of a straight walk twenty-five feet long, and walk towards each other, how many feet will Charles have to walk to meet Henry who has walked fifteen feet?
A lot is forty rods long and thirty wide, how long must the fence be?
What length of fence will divide it into four equal parts?
Diagram 24.
Diagram 24.
Diagram 24.
If the straight linec awere a string made fast atc, with a sharp pencil-point at the other enda, and the pencil-point were moved towardsd, what line would be drawn?
What kind of a line would it be?
If the pencil-point continued to move in the same direction until it returned to the starting-pointa, what curved line would be drawn, naming it by all the points in it which are marked?
The plane figure bounded by this curve is called a “circle.”
What point is at the centre of this figure?
A circle is a plane figure bounded by a curved line, all points of which are equally distant from the centre.
The curved line is called a “circumference.”
The circumference of a circle is the curve which bounds it.
Name a straight line that joins two points in the circumference.
It is called a “chord.”
A chord is a straight line that joins two points of a circumference.
Read six chords in the diagram.
Which two of these chords pass through the centre?
They are called “diameters.”
A diameter is a chord that passes through the centre.
Name a line that joins the centre with a point of the circumference.
It is called a “radius.”—(Plural,radii.)
A radius is a straight line that joins the centre to a point of the circumference.
Read five radii.
Which is farther from the centre, the pointaor the pointd?
Can the radiusc dbe greater than the radiusc a? Or greater thanc v, orc o?
Then all radii of the same circle are equal to each other.
What do we call the lineso d,c d,c o?
What part of the diametero dis the radiuso c?
Name a chord that is produced without the circle.
It is called a “secant.”
A secant is a chord produced.
Name two secants.
If the chordd iwere made a secant, would it become longer or shorter?
In how many points does the straight linel mtouch the circumference?
It is called a “tangent.”
A tangent is a straight line that touches a circumference in only one point.
Name three tangents.
Read six chords. (Diagram24.)
Why isi da chord?
What is a chord?
Name two diameters.
Why isa ja diameter?
What is a diameter?
Is every chord a diameter?
Is every diameter a chord?
Name five radii.
Why isc aa radius?
What is a radius?
A diameter is equal to how many radii?
Are all radii equal to each other?
Are all chords equal to each other?
Are all diameters equal to each other?
Name two secants.
Why is either one a secant?
What is a secant?
Name three tangents.
Why isa ba tangent?
What is a tangent?
Is a tangent inside of a circle or outside of it?
Is a chord inside or outside of a circle?
Is a secant within or without a circle?
If the radius is three inches, how long is the diameter?
Diagram 25.
Diagram 25.
Diagram 25.
What small part of the circumference of circle 1 (Diagram25.) is marked?
It is called an “arc.”
An arc is any part of a circumference.
Read five arcs that are marked.
Which is longer, the arce d, or the arce f?b d, orb d e?a b d, ora b d e?
Name an arc which is half of the circumference.
It is called a “semi-circumference.”
“Semi” means “half.”
A semi-circumference is half of a circumference.
Read three arcs, each of which is one-fourth of the circumference.
If the whole circumference were divided into three hundred and sixty equal arcs, would each arc be large or small?
Each of these arcs would be called a “degree.” [Degrees are marked (°).]
A degree of a circumference is a three hundred and sixtieth part of it.
How many degrees in a semi-circumference?
How many degrees in one-fourth of a circumference?
If a fourth of a circumference were divided into three equal parts, how many degrees would there be in each part?
Into how many parts would each third of a quarter have to be again divided to make single degrees?
Is an arc of ninety-one degrees greater or less than one-fourth of a circumference?
Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference?
Can there be more than three hundred and sixty degrees in a circumference?
If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot.
If a degree were divided into sixty equal parts, each part would be called a minute.
If a minute were divided into sixty equal parts, each part would be called a second.
How many degrees in the large circle of Fig. 2?
How many in the smaller one?
Has a large circle any more degrees than a small circle?
In the large circle how many degrees fromatob?
In the small circle how many fromatob?
Which is greater, an arc of ninety degrees of the large circle, or one of ninety degrees of the small one?
Which is greater, an arc of a degree of the large circle, or one of a degree of the small one?
The anglea o bhas its vertex at what part of the larger circle?
At what part of the smaller circle?
On how many degrees of the larger circle does the angle stand?
On how many degrees of the smaller circle does it stand?
Then it is said to be an angle of 90°.
If the anglea o fis an angle of 30°, how many degrees must there be in the arca f?
If the arcf eis an arc of 60°, what is the size of the anglef o e?
An angle of 10° stands upon an arc of how many degrees? Of 8°? Of 1°?
The anglea o bis what kind of an angle?
Upon how many degrees does it stand?
Then a right angle is an angle of how many degrees?
If an angle stand upon less than 90°, what kind of an angle is it?
If an angle stand upon more than 90°, what kind of an angle is it?
Can an angle have as many degrees as a hundred and eighty?
Read nine arcs whose ends are marked. (Diagram26.)
Read three arcs each of which is one-fourth of a circumference.
Read two arcs each of which is one-half of a circumference.
Why ise gan arc?
What is an arc?
How many degrees in the arcf h? Ine h?
If the arcf hwere divided into three equal parts, how many degrees would there be in each?
How many degrees in a circumference?
In a semi-circumference?
How many more degrees in a large circumference than in a small one?
If the arci fis 40°, what is the size of the anglef o i?
If the anglef o gis an angle of 130°, what is the size of the arcf i h g?
How many degrees in each of the adjacent anglesf o h,h o e?
When two adjacent angles are equal to each other, what is each called?
How many degrees in a right angle?
Diagram 26.
Diagram 26.
Diagram 26.
The part of the circle bounded by the chorda band the arca bis called a segment.
Read three segments, each less than half a circle, thus,—the segment bounded by the chorda dand the arca b d.
A segment is a part of a circle bounded by an arc and a chord.
Read two segments that are each half a circle.
What is the chord called?
What is the arc called?
A segment bounded by a diameter and a semi-circumference is a “semicircle.”
A semicircle is half a circle.
Read four segments each larger than a semicircle.
The part of the circle between the two radiio f,o i, and the arcf i, is called a “sector.”
Read four sectors each less than one-fourth of a circle.[2]
2.Thus, a sector bounded by the two radiio g,o h, and the arcg h.
2.Thus, a sector bounded by the two radiio g,o h, and the arcg h.
A sector is a part of a circle bounded by two radii and an arc.
What part of the whole circle is the sectorf o h?
It is called a “quadrant.”
A quadrant is a sector which is one-fourth of a circle.
Read a sector which is greater than a quadrant.
If the chorde fbe regarded a diameter, what do you call the semicircle below it?
If it be regarded as two radii, what is the semicircle called?
Then a semicircle is both a segment and a sector.
Name ten segments. (Diagram26.)
What is a segment?
Of the segments named, which are less than a semicircle?
Which are greater?
Which two are semicircles?
Which two are on the chorda f?
Name nine sectors.
Why isg o ia sector?
What is a sector?
Which four of the sectors named are each less than a quadrant?
Which three are quadrants?
Which two are greater than a quadrant?
What part of the circle is both a segment and a sector?
How many quadrants in a circle?
How many semicircles?