Bernhard Marks.

Marks' first lessons in geometry, objectively presented online

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What three names may be given to each?

How do they differ from each other?

What particular name may you give to Fig. F?

What four names has Fig. F?


RECTANGLE AND SQUARE.

In what three things are Figs. G and H alike?

On account of the number of their sides, what may each be called?

Because their opposite sides are parallel, what may each be called?

Because they have right angles, what may they be called?

In what respect is Fig. H different from Fig. G?

On this account, what particular name may be applied to Fig. H?

What three names may be applied to Fig. G?

What _four_ to Fig. H?


RHOMBUS AND SQUARE.

In what three things are Figs. F and H alike?

On account of the number of their sides, what name may be given to
each?

Because their opposite sides are parallel, what name may be given to
each?

Because both are parallelograms, and both have their sides equal, what
name may be given to each?

What particular name has Fig. F?

What particular name has Fig. H?

What four names may be given to Fig. F?

What four to Fig. H?




LESSON TWENTY-NINTH.


REVIEW.

What two names may be given to Fig. A. (DIAGRAM 22.)

To Fig. B?

In what are they alike?

In what do they differ?

By what three names may Fig. C be called?

By what three names may Fig. D be called?

In what two things are they alike?

In what one thing do they differ?

What particular name has C? What one has D?

What three names may be applied to Fig. E?

What four to Fig. F?

What property has F that E has not?

What particular name has it on that account?

What three names has Fig. G?

What four has Fig. H?

What property has Fig. H that G has not?

What particular name has it in consequence?

What four names may you give to Fig. F?

What four to Fig. H?

What three names may be applied to either?

In what three things are they alike?

In what respect do they differ?

What particular name has Fig. F?

What particular name has Fig. H?

[Illustration: Diagram 23.]




LESSON THIRTIETH.


MEASUREMENT OF SURFACES.

In Fig. 1 (DIAGRAM 23.) call the line _a b a_ unit.

Rectangle 1 is how many units long?

How many high?

Because its sides are equal, what is it called?

Rectangle 2 is how many units long?

How many high or wide?

How many squares does it contain?

Rectangle 3 is how many units long?

How many wide?

How many squares does it contain?

If it were four units long and one wide, how many squares would it
contain?

If it were five long and one wide? Six long? &c.

Rectangle 4 is how many units long?

How many wide?

How many squares does it contain?

How many squares in that part which is two units long, _m n_, and one
unit wide, _m l_?

On account of the second unit in width, _l k_, how many times two
squares are there?

If the width were one unit more, how many times two squares would
there be?

Rectangle 5 is how many units long?

How many units wide or high?

How many squares does it contain?

How many squares in that part which is three units long, _o p_, and
one unit wide, _o t_?

The second unit in width, _t q_, gives how many more squares? How many
times three squares?

If another unit were added to the width, how many more squares would
be made?

How many times three squares?

If it were four units wide, how many times three squares would there
be?

Rectangle 6 is how many units long?

How many units high or wide?

How many squares in that part which is four units long and one high?

How many times four squares in that part which is four long and two
high?

How many times four squares when it is four long and three high?

If another unit were added to the height, how many more squares would
be added?

How many times four squares would there be?

If a rectangle were five units long and one unit wide, how many square
units would it contain?

If it were two units wide, how many times five square units would it
contain?

If it were three units wide? Four? &c.

If your ruler is ten inches long and only one inch wide, how many
square inches are there in it?

If it were two inches wide, how many times ten square inches would it
contain?

If your arithmetic-cover is seven inches long and five inches wide,
how many square inches are there in it?

If a wall of this room is twenty feet long, how many square feet are
there in that part which is one foot high? Two high? Three high?
Four high?

If the same wall is sixteen feet high, how many square feet in it?

Fig. 5 has how many times three squares?

Fig. 7 has how many times two squares?

Which has the greater number of squares?

What difference is there between two times three squares and three
times two squares?




LESSON THIRTY-FIRST.


REVIEW.

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?

[Illustration: Diagram 24.]




LESSON THIRTY-SECOND.


THE CIRCLE AND ITS LINES.

If the straight line _c a_ were a string made fast at _c_, with a
sharp pencil-point at the other end _a_, and the pencil-point were
moved towards _d_, 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-point _a_, 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 point _a_ or the point _d_?

Can the radius _c d_ be greater than the radius _c a_? Or greater than
_c v_, or _c o_?

_Then all radii of the same circle are equal to each other._

What do we call the lines _o d_, _c d_, _c o_?

What part of the diameter _o d_ is the radius _o 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 chord _d i_ were made a secant, would it become longer or
shorter?

In how many points does the straight line _l m_ touch the
circumference?

It is called a “_tangent_.”

_A tangent is a straight line that touches a circumference in only
one point._

Name three tangents.




LESSON THIRTY-THIRD.


REVIEW.

Read six chords. (DIAGRAM 24.)

Why is _i d_ a chord?

What is a chord?

Name two diameters.

Why is _a j_ a diameter?

What is a diameter?

Is every chord a diameter?

Is every diameter a chord?

Name five radii.

Why is _c a_ a 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 is _a b_ a 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?

[Illustration: Diagram 25.]




LESSON THIRTY-FOURTH.


ARCS AND DEGREES.

What small part of the circumference of circle 1 (DIAGRAM 25.) 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 arc _e d_, or the arc _e f_? _b d_, or _b d e_?
_a b d_, or _a 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 from _a_ to _b_?

In the small circle how many from _a_ to _b_?

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 angle _a o b_ has 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 angle _a o f_ is an angle of 30°, how many degrees must there
be in the arc _a f_?

If the arc _f e_ is an arc of 60°, what is the size of the angle _f o
e_?

An angle of 10° stands upon an arc of how many degrees? Of 8°? Of 1°?

The angle _a o b_ is 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?




LESSON THIRTY-FIFTH.


REVIEW.

Read nine arcs whose ends are marked. (DIAGRAM 26.)

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 is _e g_ an arc?

What is an arc?

How many degrees in the arc _f h_? In _e h_?

If the arc _f h_ were 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 arc _i f_ is 40°, what is the size of the angle _f o i_?

If the angle _f o g_ is an angle of 130°, what is the size of the arc
_f i h g_?

How many degrees in each of the adjacent angles _f 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?

[Illustration: Diagram 26.]




LESSON THIRTY-SIXTH.


PARTS OF THE CIRCLE.

The part of the circle bounded by the chord _a b_ and the arc _a b_ is
called a segment.

Read three segments, each less than half a circle, thus,—the segment
bounded by the chord _a d_ and the arc _a 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 radii _o f_, _o i_, and the arc
_f i_, is called a “_sector_.”

Read four sectors each less than one-fourth of a circle.[2]

Footnote 2:

Thus, a sector bounded by the two radii _o g_, _o h_, and the arc _g
h_.

_A sector is a part of a circle bounded by two radii and an arc._

What part of the whole circle is the sector _f 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 chord _e f_ be 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.




LESSON THIRTY-SEVENTH.


REVIEW.

Name ten segments. (DIAGRAM 26.)

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 chord _a f_?

Name nine sectors.

Why is _g o i_ a 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?




PART SECOND.
AXIOMS AND THEOREMS.




AXIOMS ILLUSTRATED.


AXIOM 1.

The triangle A is equal to the triangle C.

The triangle B is also equal to the triangle C.

What do you think of the two triangles A and B? Why?

[Illustration]

_If two things are separately equal to the same thing, they are equal
to each other._


AXIOM 2.

The square A is equal to the square B.

To the rectangle C add the square A, and we have an L pointing in what
direction?

To the same rectangle C add the square B, and we have an L pointing in
what direction?

[Illustration]

Which is larger, the L pointing to the left, or that pointing to the
right?

To what same thing did you add two equals?

What two equals did you add to it?

What was the first sum?

The second?

What do you think of the two sums?

_If equals be added to the same thing, the sums will be equal._


AXIOM 3.

[Illustration]

The square A is equal to the square B.

From the inverted T take away the square A, and we have an L pointing
in what direction?

From the same Fig. T take away the square B, and we have an L pointing
in what direction?

Which is larger, the L pointing to the right, or that pointing to the
left?

What two equal things did we take away from the same thing?

From what same thing did we take them away?

What did we find true of the two remainders?

_If equals be taken from the same thing, the remainders will be
equal._


AXIOM 4.

[Illustration]

The rectangle 1 2 is equal to the rectangle 1 3.

From the rectangle 1 2 take away the square A, and what rectangle
remains?

From the rectangle 1 3 take away the same square A, and what rectangle
remains?

Which is greater, the rectangle B, or the rectangle C?

What same thing did we take away from equals?

From what did we first take it?

What remained?

From what did we next take it?

What remained?

What did we find true of the two remainders?

_If the same thing be taken from equals, the remainders will be
equal._


AXIOM 5.

_If equals be added to equals, the sums will be equal._


AXIOM 6.

_If equals be subtracted from equals, the remainders will be equal._


AXIOM 7.

_If the halves of two things are equal, the wholes will be equal._


AXIOM 8.

_Every Whole is equal to the sum of all its parts._


AXIOM 9.

_From one point to another only one straight line can be drawn._


AXIOM 10.

_A straight line is the shortest distance between two points._


AXIOM 11.

_If two things coincide throughout their whole extent, they are
equal._




THEOREMS ILLUSTRATED.


[Illustration: Diagram 29.]


DEVELOPMENT LESSON.

Do the angles Blue, Red, take up all the space on the line _a b_?

Do the angles Blue, Yellow, Red, take up all the space on the line?

Do the angles Blue, Yellow, Green, Red, take up all the space on the
line?

Is there room between any two of the angles to put in another angle?

Then are not the angles Blue, Yellow, Green, Red, equal to all the
space on the line _a b_?

NOTE.—The word _space_, as here used, means _angular space_; and it
is indispensable that the teacher impress this fact upon the
learner.

By means of former lessons, the pupil has learned positively, that
an angle is the difference between the directions of two lines; and,
impliedly, that the included space has nothing to do with the size
of the angle. There cannot, therefore, be much danger that the pupil
will imbibe any erroneous notion from this style of expression,
which is very much more simple than to say that the difference of
direction of two given lines is equal to the difference of direction
of two other given lines, which style will be used somewhat later in
these lessons.

[Illustration: Diagram 30.]


PROPOSITION I. THEOREM.


DEVELOPMENT LESSON.

Are the adjacent angles Green, Red, equal to all the angular space on
the line _a b_?

Place a paper square corner or right angle on the line _a b_ at the
_left_ of _c d_ with its vertex at _c_.

It will cover all the angle Green and part of the angle Red up to the
line _c d_.

Now place another square corner on the line _a b_ to the _right_ of
the line _c d_, and with its vertex at the point _c_.

It will cover the remaining part of the angle Red, and two edges of
the square corners will meet along the line _c d_.

Are the two right angles equal to all the angular space on the line _a
b_?

Then if the two adjacent angles Green, Red, are equal to all the
angular space on the line _a b_, and the two right angles are also
equal to the same space, what do you infer concerning the _adjacent
angles_ and the _two right angles_?

What axiom do you apply when you say that the _adjacent_ angles are
equal to the _two right angles_?

To what _same thing_ did you find two things separately equal?

What did you first see equal to it?

What did you next see equal to it?

Then what did you _find_ true?

If the angle Red were smaller, and the angle Green larger, would the
adjacent angles still be equal to two right angles?

Then,—

_Any two adjacent angles are equal to two right angles._

If we draw the straight line _c d_ where the edges of the square
corners come together, what kind of angles will _a c d_, _d c b_,
be?

See now if you can understand the following demonstration:—


DEMONSTRATION.

We wish to prove that

_Any two adjacent angles are equal to two right angles._

Let the two straight lines _a b_, _m n_, intersect each other in the
point _c_. (DIAGRAM 30.)

Then will any two adjacent angles, as Green, Red, be equal to two
right angles?

For, from the point _c_, draw the straight line _c d_ so as to make
the angles _a c d_, _d c b_, right angles.

The adjacent angles Green, Red, are equal to all the angular space on
the line _a b_.

The right angles _a c d_, _d c b_, are also equal to all the angular


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Online LibraryBernhard MarksMarks' first lessons in geometry, objectively presented → online text (page 3 of 6)